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Applied
Biopharmaceutics &

Pharmacokinetics

 

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Applied
Biopharmaceutics &

Pharmacokinetics
Seventh Edition

EDITORS
Leon Shargel, PhD, RPh
Applied Biopharmaceutics, LLC
Raleigh, North Carolina
Aliate Professor, School of Pharmacy
Virginia Commonwealth University, Richmond, Virginia
Adjunct Associate Professor, School of Pharmacy
University of Maryland, Baltimore, Maryland

Andrew B.C. Yu, PhD, RPh
Registered Pharmacist
Gaithersburg, Maryland
Formerly Associate Professor of Pharmaceutics
Albany College of Pharmacy
Albany, New York
Formerly CDER, FDA
Silver Spring, Maryland

New York Chicago San Francisco Athens London Madrid Mexico City
Milan New Delhi Singapore Sydney Toronto

 

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Contents

Contributors xi
Preface xv
Preface to First Edition xvii

1. Introduction to Biopharmaceutics and Measures of Central Tendency 53

Pharmacokinetics 1 Measures of Variability 54
Hypothesis Testing 56

Drug Product Performance 1 Statistically Versus Clinically Signicant
Biopharmaceutics 1 Differences 58
Pharmacokinetics 4 Statistical Inference Techniques in Hypothesis
Pharmacodynamics 4 Testing for Parametric Data 59
Clinical Pharmacokinetics 5 Goodness of Fit 63
Practical Focus 8 Statistical Inference Techniques for Hypothesis
Pharmacodynamics 10 Testing With Nonparametric Data 63
Drug Exposure and Drug Response 10 Controlled Versus Noncontrolled Studies 66
Toxicokinetics and Clinical Toxicology 10 Blinding 66
Measurement of Drug Concentrations 11 Confounding 66
Basic Pharmacokinetics and Pharmacokinetic Validity 67

Models 15 Bioequivalence Studies 68
Chapter Summary 21 Evaluation of Risk for Clinical Studies 68
Learning Questions 22 Chapter Summary 70
Answers 23 Learning Questions 70
References 25 Answers 72
Bibliography 25 References 73

2. Mathematical Fundamentals in 4. One-Compartment Open Model:
Pharmacokinetics 27 Intravenous Bolus Administration 75
Calculus 27 Elimination Rate Constant 76
Graphs 29 Apparent Volume of Distribution 77
Practice Problem 31 Clearance 80
Mathematical Expressions and Units 33 Clinical Application 85
Units for Expressing Blood Concentrations 34 Calculation of k From Urinary Excretion Data 86
Measurement and Use of Signicant Figures 34 Practice Problem 87
Practice Problem 35 Practice Problem 88
Practice Problem 36 Clinical Application 89
Rates and Orders of Processes 40 Chapter Summary 90
Chapter Summary 42 Learning Questions 90
Learning Questions 43 Answers 92
Answers 46 Reference 96
References 50 Bibliography 96

3. Biostatistics 51 5. Multicompartment Models:
Variables 51 Intravenous Bolus Administration 97
Types of Data (Nonparametric Versus Parametric) 51 Two-Compartment Open Model 100
Distributions 52 Clinical Application 105

v

 

vi CONTENTS

Practice Problem 107 References 175
Practical Focus 107 Bibliography 175
Practice Problem 110
Practical Focus 113 8. Pharmacokinetics of Oral
Three-Compartment Open Model 114
Clinical Application 115 Absorption 177
Clinical Application 116 Introduction 177

Determination of Compartment Models 116 Basic Principles of Physiologically Based

Practical Focus 117 Absorption Kinetics (Bottom-Up Approach) 178

Clinical Application 118 Absoroption Kinetics

Practical Problem 120 (The Top-Down Approach) 182

Clinical Application 121 Pharmacokinetics of Drug Absorption 182

Practical Application 121 Signicance of Absorption Rate Constants 184

Clinical Application 122 Zero-Order Absorption Model 184

Chapter Summary 123 Clinical Application—Transdermal Drug

Learning Questions 124 Delivery 185

Answers 126 First-Order Absorption Model 185

References 128 Practice Problem 191

Bibliography 129 Chapter Summary 199
Answers 200
Application Questions 202

6. Intravenous Infusion 131 References 203
One-Compartment Model Drugs 131 Bibliography 204
Infusion Method for Calculating Patient Elimination

Half-Life 135
Loading Dose Plus IV Infusion—One-Compartment 9. Multiple-Dosage Regimens 205

Model 136 Drug Accumulation 205

Practice Problems 138 Clinical Example 209

Estimation of Drug Clearance and V From Infusion Repetitive Intravenous Injections 210
D

Data 140 Intermittent Intravenous Infusion 214

Intravenous Infusion of Two-Compartment Model Clinical Example 216

Drugs 141 Estimation of k and V of Aminoglycosides in
D

Practical Focus 142 Clinical Situations 217

Chapter Summary 144 Multiple-Oral-Dose Regimen 218

Learning Questions 144 Loading Dose 219

Answers 146 Dosage Regimen Schedules 220

Reference 148 Clinical Example 222

Bibliography 148 Practice Problems 222
Chapter Summary 224
Learning Questions 225

7. Drug Elimination, Clearance, and Answers 226
Renal Clearance 149 References 228
Drug Elimination 149 Bibliography 228
Drug Clearance 150
Clearance Models 152 10. Nonlinear Pharmacokinetics 229
The Kidney 157 Saturable Enzymatic Elimination Processes 231
Clinical Application 162 Practice Problem 232
Practice Problems 163 Practice Problem 233
Renal Clearance 163 Drug Elimination by Capacity-Limited
Determination of Renal Clearance 168 Pharmacokinetics: One-Compartment
Practice Problem 169 Model, IV Bolus Injection 233
Practice Problem 169 Practice Problems 235
Relationship of Clearance to Elimination Half-Life Clinical Focus 242

and Volume of Distribution 170 Clinical Focus 243
Chapter Summary 171 Drugs Distributed as One-Compartment
Learning Questions 171 Model and Eliminated by Nonlinear
Answers 172 Pharmacokinetics 243

 

CONTENTS vii

Clinical Focus 244 Practical Focus 311
Chronopharmacokinetics and Time-Dependent Hepatic Clearance 311

Pharmacokinetics 245 Extrahepatic Metabolism 312
Clinical Focus 247 Enzyme Kinetics—Michaelis–Menten
Bioavailability of Drugs That Follow Nonlinear Equation 313

Pharmacokinetics 247 Clinical Example 317
Nonlinear Pharmacokinetics Due to Drug–Protein Practice Problem 319

Binding 248 Anatomy and Physiology of the Liver 321
Potential Reasons for Unsuspected Hepatic Enzymes Involved in the Biotransformation

Nonlinearity 251 of Drugs 323
Dose-Dependent Pharmacokinetics 252 Drug Biotransformation Reactions 325
Clinical Example 253 Pathways of Drug Biotransformation 326
Chapter Summary 254 Drug Interaction Example 331
Learning Questions 254 Clinical Example 338
Answers 255 First-Pass Effects 338
References 257 Hepatic Clearance of a Protein-Bound Drug:
Bibliography 258 Restrictive and Nonrestrictive Clearance From

Binding 344

11. Physiologic Drug Distribution and Biliary Excretion of Drugs 346
Clinical Example 348

Protein Binding 259 Role of Transporters on Hepatic Clearance
Physiologic Factors of Distribution 259 and Bioavailability 348
Clinical Focus 267 Chapter Summary 350
Apparent Volume Distribution 267 Learning Questions 350
Practice Problem 270 Answers 352
Protein Binding of Drugs 273 References 354
Clinical Examples 275 Bibliography 355
Effect of Protein Binding on the Apparent Volume

of Distribution 276
Practice Problem 279 13. Pharmacogenetics and Drug
Clinical Example 280 Metabolism 357
Relationship of Plasma Drug–Protein Binding to Genetic Polymorphisms 358

Distribution and Elimination 281 Cytochrome P-450 Isozymes 361
Clinical Examples 282 Phase II Enzymes 366
Clinical Example 284 Transporters 367
Determinants of Protein Binding 285 Chapter Summary 368
Clinical Example 285 Glossary 369
Kinetics of Protein Binding 286 Abbreviations 369
Practical Focus 287 References 370
Determination of Binding Constants and Binding

Sites by Graphic Methods 287 14. Physiologic Factors Related to Drug
Clinical Signicance of Drug–Protein

Binding 290 Absorption 373
Clinical Example 299 Drug Absorption and Design

Clinical Example 300 of a Drug Product 373

Modeling Drug Distribution 301 Route of Drug Administration 374

Chapter Summary 302 Nature of Cell Membranes 377

Learning Questions 303 Passage of Drugs Across Cell Membranes 378

Answers 304 Drug Interactions in the Gastrointestinal

References 306 Tract 389

Bibliography 307 Oral Drug Absorption 390
Oral Drug Absorption During Drug Product

Development 401
12. Drug Elimination and Hepatic Methods for Studying Factors That Affect Drug

Clearance 309 Absorption 402
Route of Drug Administration and Extrahepatic Effect of Disease States on Drug Absorption 405

Drug Metabolism 309 Miscellaneous Routes of Drug Administration 407

 

viii CONTENTS

Chapter Summary 408 Other Approaches Deemed Acceptable
Learning Questions 409 (by the FDA) 482
Answers to Questions 410 Bioequivalence Studies Based on Multiple
References 411 Endpoints 482
Bibliography 414 Bioequivalence Studies 482

Design and Evaluation of Bioequivalence

15. Biopharmaceutic Considerations in Studies 484
Study Designs 490

Drug Product Design and In Vitro Drug Crossover Study Designs 491
Product Performance 415 Clinical Example 496
Biopharmaceutic Factors and Rationale for Drug Clinical Example 496

Product Design 416 Pharmacokinetic Evaluation of the Data 497
Rate-Limiting Steps in Drug Absorption 418 The Partial AUC in Bioequivalence
Physicochemical Properties of the Drug 420 Analysis 498
Formulation Factors Affecting Drug Product Examples of Partial AUC Analyses 499

Performance 423 Bioequivalence Examples 500
Drug Product Performance, In Vitro: Dissolution Study Submission and Drug Review Process 502

and Drug Release Testing 425 Waivers of In Vivo Bioequivalence Studies
Compendial Methods of Dissolution 429 (Biowaivers) 503
Alternative Methods of Dissolution Testing 431 The Biopharmaceutics Classication System
Dissolution Prole Comparisons 434 (BCS) 507
Meeting Dissolution Requirements 436 Generic Biologics (Biosimilar Drug
Problems of Variable Control in Dissolution Products) 510

Testing 437 Clinical Signicance of Bioequivalence
Performance of Drug Products: In Vitro–In Vivo Studies 511

Correlation 437 Special Concerns in Bioavailability and
Approaches to Establish Clinically Relevant Drug Bioequivalence Studies 512

Product Specications 441 Generic Substitution 514
Drug Product Stability 445 Glossary 517
Considerations in the Design of a Drug Chapter Summary 520

Product 446 Learning Questions 520
Drug Product Considerations 450 Answers 525
Clinical Example 456 References 526
Chapter Summary 461
Learning Questions 462
Answers 462 1 7. Biopharmaceutical Aspects of the
References 463 Active Pharmaceutical Ingredient and
Bibliography 466 Pharmaceutical Equivalence 529

Introduction 529
16. Drug Product Performance, In Vivo: Pharmaceutical Alternatives 533

Bioavailability and Bioequivalence 469 Practice Problem 534
Drug Product Performance 469 Bioequivalence of Drugs With Multiple
Purpose of Bioavailability and Bioequivalence Indications 536

Studies 471 Formulation and Manufacturing Process
Relative and Absolute Availability 472 Changes 536
Practice Problem 474 Size, Shape, and Other Physical Attributes of
Methods for Assessing Bioavailability and Generic Tablets and Capsules 536

Bioequivalence 475 Changes to an Approved NDA or ANDA 537
In Vivo Measurement of Active Moiety or Moieties The Future of Pharmaceutical Equivalence and

in Biological Fluids 475 Therapeutic Equivalence 538
Bioequivalence Studies Based on Pharmacodynamic Biosimilar Drug Products 539

Endpoints—In Vivo Pharmacodynamic (PD) Historical Perspective 540
Comparison 478 Chapter Summary 541

Bioequivalence Studies Based on Clinical Learning Questions 541
Endpoints—Clinical Endpoint Study 479 Answers 542

In Vitro Studies 481 References 542

 

CONTENTS ix

18. Impact of Biopharmaceutics on Pharmacokinetics of Biopharmaceuticals 630
Bioequivalence of Biotechnology-Derived

Drug Product Quality and Clinical
Drug Products 631

Efficacy 545 Learning Questions 632
Risks From Medicines 545 Answers 632
Risk Assessment 546 References 633
Drug Product Quality and Drug Product Bibliography 633

Performance 547
Pharmaceutical Development 547 21. Relationship Between Pharmacokinetics
Example of Quality Risk 550
Excipient Effect on Drug Product and Pharmacodynamics 635

Performance 553 Pharmacokinetics and Pharmacodynamics 635
Practical Focus 554 Relationship of Dose to Pharmacologic Effect 640
Quality Control and Quality Assurance 554 Relationship Between Dose and Duration of
Practical Focus 555 Activity (t ), Single IV Bolus Injection 643

eff

Risk Management 557 Practice Problem 643
Scale-Up and Postapproval Changes (SUPAC) 558 Effect of Both Dose and Elimination Half-Life on
Practical Focus 561 the Duration of Activity 643
Product Quality Problems 561 Effect of Elimination Half-Life on Duration of
Postmarketing Surveillance 562 Activity 644
Glossary 562 Substance Abuse Potential 644
Chapter Summary 563 Drug Tolerance and Physical Dependency 645
Learning Questions 564 Hypersensitivity and Adverse Response 646
Answers 564 Chapter Summary 673
References 565 Learning Questions 674
Bibliography 565 Answers 677

References 678

1 9. Modified-Release Drug Products and
2 2. Application of Pharmacokinetics to

Drug Devices 567
Modied-Release (MR) Drug Products and Clinical Situations 681

Conventional (Immediate-Release, IR) Medication Therapy Management 681
Drug Products 567 Individualization of Drug Dosage Regimens 682

Biopharmaceutic Factors 572 Therapeutic Drug Monitoring 683
Dosage Form Selection 575 Clinical Example 690
Advantages and Disadvantages of Extended- Clinical Example 692

Release Products 575 Design of Dosage Regimens 692
Kinetics of Extended-Release Dosage Forms 577 Conversion From Intravenous Infusion to
Pharmacokinetic Simulation of Extended-Release Oral Dosing 694

Products 578 Determination of Dose 696
Clinical Examples 580 Practice Problems 696
Types of Extended-Release Products 581 Effect of Changing Dose ond Dosing Interval on

Ç Ç Ç

Considerations in the Evaluation of C , C , and C 697
max min av

Modied-Release Products 601 Determination of Frequency of Drug
Evaluation of Modied-Release Products 604 Administration 698
Evaluation of In Vivo Bioavailability Data 606 Determination of Both Dose and Dosage
Chapter Summary 608 Interval 698
Learning Questions 609 Practice Problem 699
References 609 Determination of Route of Administration 699
Bibliography 613 Dosing Infants and Children 700

Practice Problem 702

2 Dosing the Elderly 702
0. Targeted Drug Delivery Systems and

Practice Problems 703
Biotechnological Products 615 Clinical Example 704
Biotechnology 615 Dosing the Obese Patients 705
Drug Carriers and Targeting 624 Pharmacokinetics of Drug Interactions 706
Targeted Drug Delivery 627 Inhibition of Drug Metabolism 710

 

x CONTENTS

Inhibition of Monoamine Oxidase (MAO) 712 General Approaches for Dose Adjustment in Renal
Induction of Drug Metabolism 712 Disease 777
Inhibition of Drug Absorption 712 Measurement of Glomerular Filtration Rate 779
Inhibition of Biliary Excretion 713 Serum Creatinine Concentration and
Altered Renal Reabsorption Due to Changing Creatinine Clearance 780

Urinary pH 713 Practice Problems 782
Practical Focus 713 Dose Adjustment for Uremic Patients 785
Effect of Food on Drug Disposition 713 Practice Problem 787
Adverse Viral Drug Interactions 714 Practice Problem 792
Population Pharmacokinetics 714 Practice Problems 793
Clinical Example 722 Practice Problem 795
Regional Pharmacokinetics 724 Extracorporeal Removal of Drugs 796
Chapter Summary 725 Practice Problem 799
Learning Questions 725 Clinical Examples 800
Answers 728 Effect of Hepatic Disease
References 731 on Pharmacokinetics 803
Bibliography 732 Practice Problem 805

Chapter Summary 809

23. Application of Pharmaco kinetics to Learning Questions 810
Answers 811

Specific Populations: Geriatric, Obese, References 813
and Pediatric Patients 735 Bibliography 815
Specic and Special Populations 735
Module I: Application of Pharmacokinetics to the 25. Empirical Models, Mechanistic

Geriatric Patients 736
Summary 749 Models, Statistical Moments, and
Learning Questions 749 Noncompartmental Analysis 817
Answers 750 Empirical Models 818
References 751 Mechanistic Models 822
Further Reading 754 Noncompartmental Analysis 835
Module II: Application of Pharmacokinetics to the Comparison of Different Approaches 842

Obese Patients 754 Selection of Pharmacokinetic Models 844
Summary 760 Chapter Summary 845
Learning Questions 760 Learning Questions 845
Answers 761 Answers 846
References 761 References 847
Module III: Application of Pharmacokinetics to the Bibliography 848

Pediatric Patients 763
Summary 769 Appendix A Applications of
Learning Questions 770
Answers 771 Software Packages in
References 773 Pharmacokinetics 851

24. Dose Adjustment in Renal and Hepatic Appendix B Glossary 875
Disease 775
Renal Impairment 775 Index 879
Pharmacokinetic Considerations 775

 

Contributors

S. Thomas Abraham, PhD Philippe Colucci, PhD

Associate Professor Principal Scientist

Department of Pharmaceutical Sciences Learn and Confirm Inc.

College of Pharmacy & Health Sciences Sr. Laurent, QC, Canada

Campbell University
Buies Creek, North Carolina Dale P. Conner, Pharm.D.

Director

Michael L. Adams, PharmD, PhD Office of Bioequivalence

Associate Professor Office of Generic Drugs

Department of Pharmaceutical Sciences CDER, FDA

College of Pharmacy & Health Sciences Silver Spring, Maryland

Campbell University
Buies Creek, North Carolina Barbara M. Davit, PhD, JD

Executive Director

Antoine Al-Achi, PhD Biopharmaceutics

Associate Professor Merck & Co.

Campbell University Kenilworth, New Jersey

College of Pharmacy & Health Sciences
Buies Creek, North Carolina Hong Ding, PhD

Assistant Professor

Lily K. Cheung, PharmD Department of Immunology

Assistant Professor Herbert Wertheim College of Medicine

Department of Pharmacy Practice Florida International University

College of Pharmacy & Health Sciences Miami, Florida

Texas Southern University
Houston, Texas John Z. Duan, PhD

Master Reviewer

Diana Shu-Lian Chow, PhD Office of New Drug Products

Professor of Pharmaceutics Office of Pharmaceutical Quality

Director FDA/CDER

Institute for Drug Education and Research (IDER) Silver Spring, Maryland

College of Pharmacy
University of Houston
Houston, Texas

xi

 

xii CONTRIBUTORS

Murray P. Ducharme, PharmD, FCCP, FCP Minerva A. Hughes, PhD, RAC (US)
President and CEO Senior Pharmacologist
Learn and Confirm Inc. Food and Drug Administration
Sr. Laurent, QC, Canada Center for Drug Evaluation and Research
Professeur Associé Silver Spring, Maryland
Faculté de Pharmacie
University of Montreal, Canada Manish Issar, PhD
Visiting Professor Assistant Professor of Pharmacology
Faculty of Pharmacy College of Osteopathic Medicine of the Pacific
Rhodes University, South Africa Western University of Health Sciences

Pomona, California
Mathangi Gopalakrishnan, MS, PhD
Research Assistant Professor Vipul Kumar, PhD
Center for Translational Medicine Senior Scientist I
School of Pharmacy Nonclinical Development Department
University of Maryland Cubist Pharmaceuticals Inc.
Baltimore, Maryland Lexington, Massachusetts

Phillip M. Gerk, PharmD, PhD S.W. Johnny Lau, RPh, PhD
Associate Professor Senior Clinical Pharmacologist
Department of Pharmaceutics Food and Drug Administration
Virginia Commonwealth University Office of Clinical Pharmacology
MCV Campus Silver Spring, Maryland
School of Pharmacy
Richmond, Virginia David S.H. Lee, PharmD, PhD

Assistant Professor
Charles Herring, BSPharm, PharmD, BCPS, CPP Department of Pharmacy Practice
Associate Professor Oregon State University/Oregon Health and Science
Department of Pharmacy Practice University College of Pharmacy
College of Pharmacy & Health Sciences Portland, Oregon
Campbell University
Clinical Pharmacist Practitioner Patrick J Marroum, PhD
Adult Medicine Team Director
Downtown Health Plaza of Wake Forest Baptist Clinical Pharmacology and Pharmacometrics
Health AbbVie
Winston-Salem, North Carolina North Chicago, Illinois

Christine Yuen-Yi Hon, PharmD, BCOP Shabnam N. Sani, PharmD, PhD
Clinical Pharmacology Reviewer Assistant Professor
Division of Clinical Pharmacology III Department of Pharmaceutical and Administrative
Office of Clinical Pharmacology Sciences
Office of Translational Sciences College of Pharmacy
Center for Drug Evaluation and Research Western New England University
Food and Drug Administration Springfield, Massachusetts
Silver Spring, Maryland

 

CONTRIBUTORS xiii

Leon Shargel, PhD, RPh Vincent H. Tam, PharmD, BCPS (Infectious Diseases)
Manager and Founder Professor Department of Clinical Sciences and
Applied Biopharmaceutics, LLC Administration
Raleigh, North Carolina University of Houston College of Pharmacy
Affiliate Professsor Texas Medical Center Campus
School of Pharmacy Houston, Texas
Virginia Commonwealth University
Richmond, Virginia Dr. Susanna Wu-Pong, PhD

Associate Professor
Sandra Suarez Sharp, PhD Director
Master Biopharmaceutics Reviewer/Biopharmaceutics Pharmaceutical Sciences Graduate Program
Lead VCU School of Pharmacy
Office of New Drug Products/Division of Richmond, Virginia
Biopharmaceutics
Office of Pharmaceutical Quality Andrew B.C. Yu, PhD, RPh
Food and Drug Administration Registered Pharmacist
Silver Spring, Maryland Formerly senior reviewer, CDER, FDA

Associate Pharmaceutics Professor
Rodney Siwale, PhD, MS Albany College of Pharmacy
Assistant Professor Albany, New York
Department of Pharmaceutical and Administrative
Sciences Corinne Seng Yue, BPharm, MSc, PhD
College of Pharmacy Principal Scientist
Western New England University Learn and Confirm Inc.
Springfield, Massachusetts Sr. Laurent, QC, Canada

Changquan Calvin Sun, PhD Hong Zhao, PhD
Associate Professor of Pharmaceutics Clinical Pharmacology Master Reviewer
University of Minnesota Clinical Pharmacology Team Leader
Department of Pharmaceutics Office of Clinical Pharmacology (OCP)
College of Pharmacy Office of Translational Sciences (OTS)
Minneapolis, Minnesota Center for Drug Evaluation and Research (CDER)

U.S. Food and Drug Administration (FDA)
He Sun, PhD Silver Spring, Maryland
President and CEO
Tasly Pharmaceuticals Inc. HaiAn Zheng, PhD
Rockville, Maryland Associate Professor
Professor and Chairman Department of Pharmaceutical Sciences
Department of Pharmaceutical Economics and Policy Albany College of Pharmacy and Health Sciences
School of Pharmaceutical Science and Technology Albany, New York
Tianjin University
Tianjin, P. R. China

 

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Preface

The publication of this seventh edition of Applied in teaching basic concepts that may be applied to
Biopharmaceutics and Pharmacokinetics represents understanding complex issues associated with in vivo
over three decades in print. Since the introduction drug delivery that are essential for safe and efficacious
of classic pharmacokinetics in the first edition, the drug therapy.
discipline has expanded and evolved greatly. The The primary audience is pharmacy students
basic pharmacokinetic principles and biopharma- enrolled in pharmaceutical science courses in phar-
ceutics now include pharmacogenetics, drug recep- macokinetics and biopharmaceutics. This text fulfills
tor theories, advances in membrane transports, and course work offered in separate or combined courses
functional physiology. These advances are applied to in these subjects. Secondary audiences for this text-
the design of new active drug moieties, manufacture book are research, technological and development
of novel drug products, and drug delivery systems. scientists in pharmaceutics, biopharmaceutics, and
Biopharmaceutics and pharmacokinetics play a key pharmacokinetics.
role in the development of safer drug therapy in This edition represents many significant changes
patients, allowing individualizing dosage regimens from previous editions.
and improving therapeutic outcomes.

• The book is an edited textbook with the collabo-
In planning for the seventh edition, we realized

ration of many experts well known in biopharma-
that we needed expertise for these areas. This sev-

ceutics, drug disposition, drug delivery systems,
enth edition is our first edited textbook in which an

manufacturing, clinical pharmacology, clinical
expert with intimate knowledge and experience in

trials, and regulatory science.
the topic was selected as a contributor. We would

• Many chapters have been expanded and updated
like to acknowledge these experts for their precious

to reflect current knowledge and application of
time and effort. We are also grateful to our readers

biopharmaceutics and pharmacokinetics. Many
and colleagues for their helpful feedback and support

new topics and updates are listed in Chapter 1.
throughout the years.

• Practical examples and questions are included
As editors of this edition, we kept the original

to encourage students to apply the principles in
objectives, starting with fundamentals followed by

patient care and drug consultation situations.
a holistic integrated approach that can be applied to

• Learning questions and answers appear at the end
practice (see scope and objectives in Preface to the

of each chapter.
first edition). This textbook provides the reader with

• Three new chapters have been added to this edi-
a basic and practical understanding of the principles

tion including, Biostatistics which provides intro-
of biopharmaceutics and pharmacokinetics that can be

duction for popular topics such as risk concept,
applied to drug product development and drug ther-

non-inferiority, and superiority concept in new
apy. Practice problems, clinical examples, frequently

drug evaluation, and Application of Pharmaco-
asked questions and learning questions are included in

kinetics in Specific Populations which discusses
each chapter to demonstrate how these concepts relate

issues such as drug and patient related pharmacy
to practical situations. This textbook remains unique

xv

 

xvi PREFACE

topics in during therapy in various patient popula- bioavailability of the drug from the drug product
tions, and Biopharmaceutic Aspects of the Active and clinical efficacy.
Pharmaceutical Ingredient and Pharmaceutical
Equivalence which explains the synthesis, Leon Shargel
quality and physical/chemical properties of the Andrew B.C. Yu
active pharmaceutical ingredients affect the

 

Preface to First Edition

The publication of the twelfth edition of this book Online features now supplement the printed
is a testament to the vision and ideals of the original edition. The entire text, updates, reviews of newly
authors, Alfred Gilman and Louis Goodman, who, approved drugs, animations of drug action, and
in 1941set forth the principles that have guided the hyper links to relevant text in the prior edition are
book through eleven editions: to correlate pharma- available on the Goodman & Gilman section of
cology with related medical sciences, to reinterpret McGraw-Hill’s websites, AccessMedicine.com and
the actions and uses of drugs in light of advances AccessPharmacy.com. An Image Bank CD accom-
in medicine and the basic biomedical sciences, to panies the book and makes all tables and figures
emphasize the applications of pharmacodynamics to available for use in presentations.
therapeutics, and to create a book that will be use- The process of editing brings into view many
ful to students of pharmacology and to physicians. remarkable facts, theories, and realizations. Three
These precepts continue to guide the current edition. stand out: the invention of new classes of drugs has

As with editions since the second, expert schol- slowed to a trickle; therapeutics has barely begun
ars have contributed individual chapters. A multiau- to capitalize on the information from the human
thored book of this sort grows by accretion, posing genome project; and, the development of resistance
challenges editors but also offering memorable pearls to antimicrobial agents, mainly through their overuse
to the reader. Thus, portions of prior editions persist in medicine and agriculture, threatens to return us to
in the current edition, and I hasten to acknowledge the the pre-antibiotic era. We have the capacity and inge-
contributions of previous editors and authors, many nuity to correct these shortcomings.
of whom will see text that looks familiar. However, Many, in addition to the contributors, deserve
this edition differs noticeably from its immediate thanks for their work on this edition; they are
predecessors. Fifty new scientists, including a num- acknowledged on an accompanying page. In addition,
ber from out-side. the U.S., have joined as contribu- I am grateful to Professors Bruce Chabner (Harvard
tors, and all chapters have been extensively updated. Medical School/Massachusetts General Hospital)
The focus on basic principles continues, with new and Björn Knollmann (Vanderbilt University Medical
chapters on drug invention, molecular mechanisms School) for agreeing to be associate editors of this
of drug action, drug toxicity and poisoning, princi- edition at a late date, necessitated by the death of my
ples of antimicrobial therapy and pharmacotherapy colleague and friend Keith Parker in late 2008. Keith
of obstetrical and gynecological disorders. Figures and I worked together on the eleventh edition and on
are in full color. The editors have continued to stan- planning this edition. In anticipation of the editorial
dardize the organization of chapters: thus, students work ahead, Keith submitted his chapters before any-
should easily find the basic physiology, biochemis- one else and just a few weeks before his death; thus,
try, and pharmacology set forth in regular type; bullet he is well represented in this volume, which we dedi-
points highlight important lists within the text; the cate to his memory.
clinician and expert will find details in extract type
under clear headings. Laurence L. Brunton

xvii

 

About the Authors

Dr. Leon Shargel is a consultant for the pharmaceuti- Association Pharmaceutical Scientists (AAPS),
cal industry in biopharmaceutics and pharmacokinetics. American Pharmacists Association (APhA), and the
Dr. Shargel has over 35 years experience in both aca- American Society for Pharmacology and Experi-
demia and the pharmaceutical industry. He has been mental Therapeutics (ASPET).
a member or chair of numerous national committees

Dr. Andrew Yu has over 30 years of experience
involved in state formulary issues, biopharmaceutics

in academia, government, and the pharmaceutical
and bioequivalence issues, institutional review boards,

industry. Dr. Yu received a BS in pharmacy from
and a member of the USP Biopharmaceutics Expert

Albany College of Pharmacy and a PhD in pharma-
Committee. Dr. Shargel received a BS in pharmacy

cokinetics from the University of Connecticut. He is
from the University of Maryland and a PhD in phar-

a registered pharmacist and has over 30 publications
macology from the George Washington University

and a patent in novel drug delivery. He had lectured
Medical Center. He is a registered pharmacist and

internationally on pharmaceutics, drug disposition,
has over 150 publications including several leading

and drug delivery.
textbooks in pharmacy. He is a member of vari-
ous professional societies, including the American

 

Introduction to

1 Biopharmaceutics and
Pharmacokinetics
Leon Shargel and Andrew B.C. Yu

Chapter Objectives DRUG PRODUCT PERFORMANCE
»» Define drug product Drugs are substances intended for use in the diagnosis, cure, mitiga-

performance and tion, treatment, or prevention of disease. Drugs are given in a variety
biopharmaceutics. of dosage forms or drug products such as solids (tablets, capsules),

»» Describe how biopharmaceutics semisolids (ointments, creams), liquids, suspensions, emulsions, etc,
affects drug product for systemic or local therapeutic activity. Drug products can be con-
performance. sidered to be drug delivery systems that release and deliver drug to

the site of action such that they produce the desired therapeutic
»» Define pharmacokinetics and

effect. In addition, drug products are designed specifically to meet
describe how pharmacokinetics

the patient’s needs including palatability, convenience, and safety.
is related to pharmacodynamics

Drug product performance is defined as the release of the
and drug toxicity.

drug substance from the drug product either for local drug action
»» Define the term clinical or for drug absorption into the plasma for systemic therapeutic

pharmacokinetics and explain activity. Advances in pharmaceutical technology and manufactur-
how clinical pharmacokinetics ing have focused on developing quality drug products that are
may be used to develop dosage safer, more effective, and more convenient for the patient.
regimens for drugs in patients.

»» Define pharmacokinetic model BIOPHARMACEUTICS
and list the assumptions that
are used in developing a Biopharmaceutics examines the interrelationship of the physical/
pharmacokinetic model. chemical properties of the drug, the dosage form (drug product) in

which the drug is given, and the route of administration on the rate
»» Explain how the prescribing

and extent of systemic drug absorption. The importance of the
information or approved

drug substance and the drug formulation on absorption, and in vivo
labeling for a drug helps the

distribution of the drug to the site of action, is described as a
practitioner to recommend an

sequence of events that precede elicitation of a drug’s therapeutic
appropriate dosage regimen for

effect. A general scheme describing this dynamic relationship is
a patient.

illustrated in Fig. 1-1.
First, the drug in its dosage form is taken by the patient by an

oral, intravenous, subcutaneous, transdermal, etc, route of adminis-
tration. Next, the drug is released from the dosage form in a predict-
able and characterizable manner. Then, some fraction of the drug is
absorbed from the site of administration into either the surrounding
tissue for local action or into the body (as with oral dosage forms),
or both. Finally, the drug reaches the site of action. A pharmacody-
namic response results when the drug concentration at the site of

1

 

2 Chapter 1

Absorption Distribution
Drug release and Drug in systemic Drug in

dissolution circulation tissues

Elimination

Excretion and Pharmacologic or
metabolism clinical effect

FIGURE 11 Scheme demonstrating the dynamic relationship between the drug, the drug product, and the pharmacologic effect.

action reaches or exceeds the minimum effective con- product manufacturers must characterize their drug
centration (MEC). The suggested dosing regimen, and drug product and demonstrate that the drug prod-
including starting dose, maintenance dose, dosage uct performs appropriately before the products can
form, and dosing interval, is determined in clinical become available to consumers in the United States.
trials to provide the drug concentrations that are Biopharmaceutics provides the scientific basis for
therapeutically effective in most patients. This drug product design and drug product development.
sequence of events is profoundly affected—in fact, Each step in the manufacturing process of a finished
sometimes orchestrated—by the design of the dosage dosage form may potentially affect the release of the
form and the physicochemical properties of the drug. drug from the drug product and the availability of the

Historically, pharmaceutical scientists have eval- drug at the site of action. The most important steps in
uated the relative drug availability to the body in vivo the manufacturing process are termed critical manu-
after giving a drug product by different routes to an facturing variables. Examples of biopharmaceutic
animal or human, and then comparing specific phar- considerations in drug product design are listed in
macologic, clinical, or possible toxic responses. For Table 1-1. A detailed discussion of drug product
example, a drug such as isoproterenol causes an design is found in Chapter 15. Knowledge of physio-
increase in heart rate when given intravenously but logic factors necessary for designing oral products is
has no observable effect on the heart when given discussed in Chapter 14. Finally, drug product quality
orally at the same dose level. In addition, the bio- of drug substance (Chapter 17) and drug product testing
availability (a measure of systemic availability of a is discussed in later chapters (18, 19, 20, and 21). It is
drug) may differ from one drug product to another important for a pharmacist to know that drug product
containing the same drug, even for the same route of selection from multisources could be confusing and
administration. This difference in drug bioavailability needs a deep understanding of the testing procedures
may be manifested by observing the difference in the and manufacturing technology which is included in the
therapeutic effectiveness of the drug products. Thus, chemistry, manufacturing, and control (CMC) of the
the nature of the drug molecule, the route of delivery, product involved. The starting material (SM) used to
and the formulation of the dosage form can determine make the API (active pharmaceutical ingredient), the
whether an administered drug is therapeutically processing method used during chemical synthesis,
effective, is toxic, or has no apparent effect at all. extraction, and the purification method can result in

The US Food and Drug Administration (FDA) differences in the API that can then affect drug product
approves all drug products to be marketed in the performance (Chapter 17). Sometimes a by-product of
United States. The pharmaceutical manufacturers the synthetic process, residual solvents, or impurities
must perform extensive research and development that remain may be harmful or may affect the product’s
prior to approval. The manufacturer of a new drug physical or chemical stability. Increasingly, many drug
product must submit a New Drug Application (NDA) sources are imported and the manufacturing of these
to the FDA, whereas a generic drug pharmaceutical products is regulated by codes or pharmacopeia in other
manufacturer must submit an Abbreviated New Drug countries. For example, drugs in Europe may be meet-
Application (ANDA). Both the new and generic drug ing EP (European Pharmacopeia) and since 2006,

 

Introduction to Biopharmaceutics and Pharmacokinetics 3

TABLE 11 Biopharmaceutic Considerations in Drug Product Design
Items Considerations

Therapeutic objective Drug may be intended for rapid relief of symptoms, slow extended action given once per day, or
longer for chronic use; some drug may be intended for local action or systemic action

Drug (active pharmaceutical Physical and chemical properties of API, including solubility, polymorphic form, particle size;
ingredient, API) impurities

Route of administration Oral, topical, parenteral, transdermal, inhalation, etc

Drug dosage and dosage Large or small drug dose, frequency of doses, patient acceptance of drug product, patient compliance
regimen

Type of drug product Orally disintegrating tablets, immediate release tablets, extended release tablets, transdermal, topical,
parenteral, implant, etc

Excipients Although very little pharmacodynamic activity, excipients may affect drug product performance
including release of drug from drug product

Method of manufacture Variables in manufacturing processes, including weighing accuracy, blending uniformity, release tests,
and product sterility for parenterals

agreed uniform standards are harmonized in ICH guid- pharmaceutical equivalence, bioavailability, bioequiv-
ances for Europe, Japan, and the United States. In the alence, and therapeutic equivalence often evolved by
US, the USP-NF is the official compendia for drug necessity. The implications are important regarding
quality standards. availability of quality drug product, avoidance of

Finally, the equipment used during manufactur- shortages, and maintaining an affordable high-quality
ing, processing, and packaging may alter important drug products. The principles and issues with regard
product attribute. Despite compliance with testing and to multisource drug products are discussed in subse-
regulatory guidance involved, the issues involving quent chapters:

Chapter 14 Physiologic Factors How stomach emptying, GI residence time, and gastric window affect drug absorption
Related to Drug
Absorption

Chapter 15 Biopharmaceutic How particle size, crystal form, solubility, dissolution, and ionization affect in vivo dissolution and
Considerations in absorption. Modifications of a product with excipient with regard to immediate or delayed action
Drug Product Design are discussed. Dissolution test methods and relation to in vivo performance

Chapter 16 Drug Product Bioavailability and bioequivalence terms and regulations, test methods, and analysis exam-
Performance, In Vivo: ples. Protocol design and statistical analysis. Reasons for poor bioavailability. Bioavailability
Bioavailability and reference, generic substitution. PE, PA, BA/BE, API, RLD, TE
Bioequivalence SUPAC (Scale-up postapproval changes) regarding drug products. What type of changes will result

in changes in BA, TE, or performances of drug products from a scientific and regulatory viewpoint

Chapter 17 Biopharmaceutic Physicochemical differences of the drug, API due to manufacturing and synthetic pathway.
Aspects of the How to select API from multiple sources while meeting PE (pharmaceutical equivalence) and
Active Pharmaceuti- TE (therapeutic equivalence) requirement as defined in CFR. Examples of some drug failing TE
cal Ingredient and while apparently meeting API requirements. Formulation factors and manufacturing method
Pharmaceutical affecting PE and TE. How particle size and crystal form affect solubility and dissolution. How
Equivalence pharmaceutical equivalence affects therapeutic equivalence. Pharmaceutical alternatives.

How physicochemical characteristics of API lead to pharmaceutical inequivalency

Chapter 18 Impact of Drug Drug product quality and drug product performance
Product Quality and Pharmaceutical development. Excipient effect on drug product performance. Quality control
Biopharmaceutics on and quality assurance. Risk management
Clinical Efficacy

Scale-up and postapproval changes (SUPAC)

Product quality problems. Postmarketing surveillance

 

4 Chapter 1

Thus, biopharmaceutics involves factors that analytical methods for the measurement of drugs
influence (1) the design of the drug product, (2) stabil- and metabolites, and procedures that facilitate data
ity of the drug within the drug product, (3) the manu- collection and manipulation. The theoretical aspect
facture of the drug product, (4) the release of the drug of pharmacokinetics involves the development of
from the drug product, (5) the rate of dissolution/ pharmacokinetic models that predict drug disposi-
release of the drug at the absorption site, and (6) deliv- tion after drug administration. The application of
ery of drug to the site of action, which may involve statistics is an integral part of pharmacokinetic stud-
targeting the drug to a localized area (eg, colon for ies. Statistical methods are used for pharmacokinetic
Crohn disease) for action or for systemic absorption parameter estimation and data interpretation ulti-
of the drug. mately for the purpose of designing and predicting

Both the pharmacist and the pharmaceutical sci- optimal dosing regimens for individuals or groups of
entist must understand these complex relationships to patients. Statistical methods are applied to pharma-
objectively choose the most appropriate drug product cokinetic models to determine data error and struc-
for therapeutic success. tural model deviations. Mathematics and computer

The study of biopharmaceutics is based on fun- techniques form the theoretical basis of many phar-
damental scientific principles and experimental macokinetic methods. Classical pharmacokinetics is
methodology. Studies in biopharmaceutics use both a study of theoretical models focusing mostly on
in vitro and in vivo methods. In vitro methods are model development and parameterization.
procedures employing test apparatus and equipment
without involving laboratory animals or humans.
In vivo methods are more complex studies involving PHARMACODYNAMICS
human subjects or laboratory animals. Some of these
methods will be discussed in Chapter 15. These Pharmacodynamics is the study of the biochemical
methods must be able to assess the impact of the and physiological effects of drugs on the body; this
physical and chemical properties of the drug, drug includes the mechanisms of drug action and the rela-
stability, and large-scale production of the drug and tionship between drug concentration and effect.
drug product on the biologic performance of the drug. A typical example of pharmacodynamics is how a

drug interacts quantitatively with a drug receptor to

PHARMACOKINETICS produce a response (effect). Receptors are the mole-
cules that interact with specific drugs to produce a

After a drug is released from its dosage form, the pharmacological effect in the body.
drug is absorbed into the surrounding tissue, the The pharmacodynamic effect, sometimes referred
body, or both. The distribution through and elimina- to as the pharmacologic effect, can be therapeutic
tion of the drug in the body varies for each patient but and/or cause toxicity. Often drugs have multiple
can be characterized using mathematical models and effects including the desired therapeutic response as
statistics. Pharmacokinetics is the science of the well as unwanted side effects. For many drugs, the
kinetics of drug absorption, distribution, and elimina- pharmacodynamic effect is dose/drug concentration
tion (ie, metabolism and excretion). The description related; the higher the dose, the higher drug concen-
of drug distribution and elimination is often termed trations in the body and the more intense the phar-
drug disposition. Characterization of drug disposition macodynamic effect up to a maximum effect. It is
is an important prerequisite for determination or desirable that side effects and/or toxicity of drugs
modification of dosing regimens for individuals and occurs at higher drug concentrations than the drug
groups of patients. concentrations needed for the therapeutic effect.

The study of pharmacokinetics involves both Unfortunately, unwanted side effects often occur con-
experimental and theoretical approaches. The exper- currently with the therapeutic doses. The relationship
imental aspect of pharmacokinetics involves the between pharmacodynamics and pharmacokinetics is
development of biologic sampling techniques, discussed in Chapter 21.

 

Introduction to Biopharmaceutics and Pharmacokinetics 5

CLINICAL PHARMACOKINETICS TABLE 12 Ratio of Age-Adjusted Death
Rates, by Male/Female Ratio from the

During the drug development process, large numbers 10 Leading Causes of Death* in the US, 2003
of patients are enrolled in clinical trials to determine
efficacy and optimum dosing regimens. Along with Disease Rank Male:Female

safety and efficacy data and other patient information, Disease of heart 1 1.5

the FDA approves a label that becomes the package
Malignant neoplasms 2 1.5

insert discussed in more detail later in this chapter. The
approved labeling recommends the proper starting Cerebrovascular diseases 3 4.0

dosage regimens for the general patient population and Chronic lower respiration 4 1.4

may have additional recommendations for special diseases

populations of patients that need an adjusted dosage Accidents and others* 5 2.2
regimen (see Chapter 23). These recommended dosage

Diabetes mellitus 6 1.2
regimens produce the desired pharmacologic response
in the majority of the anticipated patient population. Pneumonia and influenza 7 1.4

However, intra- and interindividual variations will Alzheimers 8 0.8
frequently result in either a subtherapeutic (drug con-

Nephrotis, nephrotic 9 1.5
centration below the MEC) or a toxic response (drug syndrome, and nephrosis
concentrations above the minimum toxic concentra-
tion, MTC), which may then require adjustment to Septicemia 10 1.2

the dosing regimen. Clinical pharmacokinetics is the *Death due to adverse effects suffered as defined by CDC.

application of pharmacokinetic methods to drug Source: National Vital Statistics Report Vol. 52, No. 3, 2003.

therapy in patient care. Clinical pharmacokinetics
involves a multidisciplinary approach to individually
optimized dosing strategies based on the patient’s

therapeutic drug monitoring (TDM) for very potent
disease state and patient-specific considerations.

drugs, such as those with a narrow therapeutic range,
The study of clinical pharmacokinetics of drugs

in order to optimize efficacy and to prevent any
in disease states requires input from medical and

adverse toxicity. For these drugs, it is necessary to
pharmaceutical research. Table 1-2 is a list of 10 age

monitor the patient, either by monitoring plasma drug
adjusted rates of death from 10 leading causes of

concentrations (eg, theophylline) or by monitoring a
death in the United States in 2003. The influence of

specific pharmacodynamic endpoint such as pro-
many diseases on drug disposition is not adequately

thrombin clotting time (eg, warfarin). Pharmacokinetic
studied. Age, gender, genetic, and ethnic differences

and drug analysis services necessary for safe drug
can also result in pharmacokinetic differences that may

monitoring are generally provided by the clinical
affect the outcome of drug therapy (see Chapter 23).

pharmacokinetic service (CPKS). Some drugs fre-
The study of pharmacokinetic differences of drugs in

quently monitored are the aminoglycosides and anti-
various population groups is termed population

convulsants. Other drugs closely monitored are those
pharmacokinetics (Sheiner and Ludden, 1992; see

used in cancer chemotherapy, in order to minimize
Chapter 22). Application of Pharmacokinetics to

adverse side effects (Rodman and Evans, 1991).
Specific Populations, Chapter 23, will discuss many
of the important pharmacokinetic considerations for
dosing subjects due to age, weight, gender, renal,
and hepatic disease differences. Despite advances in Labeling For Human Prescription Drug and

modeling and genetics, sometimes it is necessary to Biological Products

monitor the plasma drug concentration precisely in a In 2013, the FDA redesigned the format of the
patient for safety and multidrug dosing consider- prescribing information necessary for safe and
ation. Clinical pharmacokinetics is also applied to effective use of the drugs and biological products

 

6 Chapter 1

(FDA Guidance for Industry, 2013). This design was pharmacognosy, pharmacokinetics, pharmacody-
developed to make information in prescription drug namics, pharmacotherapeutics, and toxicology. The
labeling easier for health care practitioners to access application of pharmacology in clinical medicine
and read. The practitioner can use the prescribing including clinical trial is referred to as clinical phar-
information to make prescribing decisions. The macology. For pharmacists and health profession-
labeling includes three sections: als, it is important to know that NDA drug labels

report many important study information under
• Highlights of Prescribing Information (Highlights)—

Clinical Pharmacology in Section 12 of the standard
contains selected information from the Full Pre-

prescription label (Tables 1-3A and 1-3B).
scribing Information (FPI) that health care prac-
titioners most commonly reference and consider
most important. In addition, highlights may contain 12 CLINICAL PHARMACOLOGY
any boxed warnings that give a concise summary 12.1 Mechanism of Action
of all of the risks described in the Boxed Warning 12.2 Pharmacodynamics
section in the FPI. 12.3 Pharmacokinetics

• Table of Contents (Contents)—lists the sections
and subsections of the FPI. Question

• Full Prescribing Information (FPI)—contains the Where is toxicology information found in the pre-
detailed prescribing information necessary for safe scription label for a new drug? Can I find out if a
and effective use of the drug. drug is mutagenic under side-effect sections?

An example of the Highlights of Prescribing Answer
Information and Table of Contents for Nexium Nonclinical toxicology information is usefully in
(esomeprazole magnesium) delayed release capsules Section 13 under Nonclinical Toxicology if avail-
appears in Table 1-3B. The prescribing information able. Mutagenic potential of a drug is usually
sometimes referred to as the approved label or the reported under animal studies. It is unlikely that a
package insert may be found at the FDA website, drug with known humanly mutagenicity will be mar-
Drugs@FDA (http://www.accessdata.fda.gov/scripts keted, if so, it will be labeled with special warning.
/cder/drugsatfda/). Prescribing information is updated Black box warnings are usually used to give warn-
periodically as new information becomes available. ings to prescribers in Section 5 under Warnings and
The prescribing information contained in the label Precautions.
recommends dosage regimens for the average patient
from data obtained from clinical trials. The indica-
tions and usage section are those indications that the Pharmacogenetics
FDA has approved and that have been shown to be Pharmacogenetics is the study of drug effect includ-
effective in clinical trials. On occasion, a practitioner ing distribution and disposition due to genetic differ-
may want to prescribe the drug to a patient drug for a ences, which can affect individual responses to
non-approved use or indication. The pharmacist must drugs, both in terms of therapeutic effect and adverse
decide if there is sufficient evidence for dispensing the effects. A related field is pharmacogenomics, which
drug for a non-approved use (off-label indication). emphasizes different aspects of genetic effect on
The decision to dispense a drug for a non-approved drug response. This important discipline is discussed
indication may be difficult and often made with con- in Chapter 13. Pharmacogenetics is the main reason
sultation of other health professionals. why many new drugs still have to be further studied

after regulatory approval, that is, postapproval phase
Clinical Pharmacology 4 studies. The clinical trials prior to drug approval
Pharmacology is a science that generally deals with are generally limited such that some side effects and
the study of drugs, including its mechanism, effects, special responses due to genetic differences may not
and uses of drugs; broadly speaking, it includes be adequately known and labeled.

 

Introduction to Biopharmaceutics and Pharmacokinetics 7

TABLE 13A Highlights of Prescribing Information for Nexium (Esomeprazole Magnesium)
Delayed Release Capsules

HIGHLIGHTS OF PRESCRIBING INFORMATION

These highlights do not include all the information needed to use NEXIUM safely and effectively. See full prescribing
information for NEXIUM.
NEXIUM (esomeprazole magnesium) delayed-release capsules, for oral use
NEXIUM (esomeprazole magnesium) for delayed-release oral suspension
Initial U.S. Approval: 1989 (omeprazole)

RECENT MAJOR CHANGES

Warnings and Precautions. Interactions with Diagnostic
Investigations for Neuroendocrine Tumors (5.8) 03/2014

INDICATIONS AND USAGE

NEXIUM is a proton pump inhibitor indicated for the following:
• Treatment of gastroesophageal reflux disease (GERD) (1.1)
• Risk reduction of NSAID-associated gastric ulcer (1.2)
• H. pylori eradication to reduce the risk of duodenal ulcer recurrence (1.3)
• Pathological hypersecretory conditions, including Zollinger-Ellison syndrome (1.4)

DOSAGE AND ADMINISTRATION

Indication Dose Frequency

Gastroesophageal Reflux Disease (GERD)
Adults 20 mg or 40 mg Once daily for 4 to 8 weeks

12 to 17 years 20 mg or 40 mg Once daily for up to 8 weeks

1 to 11 years 10 mg or 20 mg Once daily for up to 8 weeks

1 month to less than 1 year 2.5 mg, 5 mg or 10 mg (based on weight). Once daily, up to 6 weeks for erosive esophagitis (EE) due
to acid-mediated GERD only.

Risk Reduction of NSAID-Associated Gastric Ulcer

20 mg or 40 mg Once daily for up to 6 months

H. pylori Eradication (Triple Therapy):

NEXIUM 40 mg Once daily for 10 days

Amoxicillin 1000 mg Twice daily for 10 days

Clarithromycin 500 mg Twice daily for 10 days

Pathological Hypersecretory Conditions

40 mg Twice daily

See full prescribing information for administration options (2)

Patients with severe liver impairment do not exceed dose of 20 mg (2)

DOSAGE FORMS AND STRENGTHS
• NEXIUM Delayed-Release Capsules: 20 mg and 40 mg (3)
• NEXIUM for Delayed-Release Oral Suspension: 2.5 mg, 5 mg, 10 mg, 20 mg, and 40 mg (3)

CONTRAINDICATIONS

Patients with known hypersensitivity to proton pump inhibitors (PPIs) (angioedema and anaphylaxis have occurred) (4)

(Continued)

 

8 Chapter 1

TABLE 13A Highlights of Prescribing Information for Nexium (Esomeprazole Magnesium)
Delayed Release Capsules (Continued)

HIGHLIGHTS OF PRESCRIBING INFORMATION

WARNINGS AND PRECAUTIONS
• Symptomatic response does not preclude the presence of gastric malignancy (5.1)
• Atrophic gastritis has been noted with long-term omeprazole therapy (5.2)
• PPI therapy may be associated with increased risk of Clostriodium difficle-associated diarrhea (5.3)
• Avoid concomitant use of NEXIUM with clopidogrel (5.4)
• Bone Fracture: Long-term and multiple daily dose PPI therapy may be associated with an increased risk for

osteoporosis-related fractures of the hip, wrist, or spine (5.5)
• Hypomagnesemia has been reported rarely with prolonged treatment with PPIs (5.6)
• Avoid concomitant use of NEXIUM with St John’s Wort or rifampin due to the potential reduction in esomeprazole levels

(5.7,7.3)
• Interactions with diagnostic investigations for Neuroendocrine Tumors: Increases in intragastric pH may result in hypergas-

trinemia and enterochromaffin-like cell hyperplasia and increased chromogranin A levels which may interfere with diagnostic
investigations for neuroendocrine tumors (5.8,12.2)

ADVERSE REACTIONS

Most common adverse reactions (6.1):
• Adults (≥18 years) (incidence ≥1%) are headache, diarrhea, nausea, flatulence, abdominal pain, constipation, and dry mouth
• Pediatric (1 to 17 years) (incidence ≥2%) are headache, diarrhea, abdominal pain, nausea, and somnolence
• Pediatric (1 month to less than 1 year) (incidence 1%) are abdominal pain, regurgitation, tachypnea, and increased ALT

To report SUSPECTED ADVERSE REACTIONS, contact AstraZeneca at 1-800-236-9933 or FDA at 1-800-FDA-1088 or
www.fda.gov/medwatch.

DRUG INTERACTIONS
• May affect plasma levels of antiretroviral drugs – use with atazanavir and nelfinavir is not recommended: if saquinavir is used

with NEXIUM, monitor for toxicity and consider saquinavir dose reduction (7.1)
• May interfere with drugs for which gastric pH affects bioavailability (e.g., ketoconazole, iron salts, erlotinib, and digoxin)

Patients treated with NEXIUM and digoxin may need to be monitored for digoxin toxicity. (7.2)
• Combined inhibitor of CYP 2C19 and 3A4 may raise esomeprazole levels (7.3)
• Clopidogrel: NEXIUM decreases exposure to the active metabolite of clopidogrel (7.3)
• May increase systemic exposure of cilostazol and an active metabolite. Consider dose reduction (7.3)
• Tacrolimus: NEXIUM may increase serum levels of tacrolimus (7.5)
• Methotrexate: NEXIUM may increase serum levels of methotrexate (7.7)

USE IN SPECIFIC POPULATIONS
• Pregnancy: Based on animal data, may cause fetal harm (8.1)

See 17 for PATIENT COUNSELING INFORMATION and FDA-approved Medication Guide.
Revised: 03/2014

PRACTICAL FOCUS pharmacogenetics) or pharmacokinetics, the recom-
mended dosage regimen drug may not provide the

Relationship of Drug Concentrations to desired therapeutic outcome. The measurement of
Drug Response plasma drug concentrations can confirm whether the
The initiation of drug therapy starts with the manu- drug dose was subtherapeutic due to the patient’s
facturer’s recommended dosage regimen that individual pharmacokinetic profile (observed by
includes the drug dose and frequency of doses (eg, low plasma drug concentrations) or was not respon-
100 mg every 8 hours). Due to individual differences sive to drug therapy due to genetic difference in
in the patient’s genetic makeup (see Chapter 13 on receptor response. In this case, the drug concentrations

 

Introduction to Biopharmaceutics and Pharmacokinetics 9

TABLE 13B Contents for Full Prescribing Information for Nexium (Esomeprazole Magnesium)
Delayed Release Capsules

FULL PRESCRIBING INFORMATION: CONTENTS*

1. INDICATIONS AND USAGE
1.1 Treatment of Gastroesophageal Reflux Disease (GERD)
1.2 Risk Reduction of NSAID-Associated Gastric Ulcer
1.3 H. pylori Eradication to Reduce the Risk of Duodenal Ulcer Recurrence
1.4 Pathological Hypersecretory Conditions Including Zollinger-Ellison Syndrome

2. DOSAGE AND ADMINISTRATION
3. DOSAGE FORMS AND STRENGTHS
4. CONTRAINDICATIONS
5. WARNINGS AND PRECAUTIONS

5.1 Concurrent Gastric Malignancy
5.2 Atrophic Gastritis
5.3 Clostridium difficile associated diarrhea
5.4 Interaction with Clopidogrel
5.5 Bone Fracture
5.6 Hypomagnesemia
5.7 Concomitant Use of NEXIUM with St John’s Wort or rifampin
5.8 Interactions with Diagnostic Investigations for Neuroendocrine Tumors
5.9 Concomitant Use of NEXIUM with Methotrexate

6. ADVERSE REACTIONS
6.1 Clinical Trials Experience
6.2 Postmarketing Experience

7. DRUG INTERACTIONS
7.1 Interference with Antiretroviral Therapy
7.2 Drugs for Which Gastric pH Can Affect Bioavailability
7.3 Effects on Hepatic Metabolism/Cytochrome P-450 Pathways
7.4 Interactions with Investigations of Neuroendocrine Tumors
7.5 Tacrolimus
7.6 Combination Therapy with Clarithromycin
7.7 Methotrexate

8. USE IN SPECIFIC POPULATIONS
8.1 Pregnancy
8.3 Nursing Mothers
8.4 Pediatric Use
8.5 Geriatric Use

10. OVERDOSAGE
11. DESCRIPTION
12. CLINICAL PHARMACOLOGY

12.1 Mechanism of Action
12.2 Pharmacodynamics
12.3 Pharmacokinetics
12.4 Microbiology

13. NONCLINICAL TOXICOLOGY
13.1 Carcinogenesis, Mutagenesis, Impairment of Fertility
13.2 Animal Toxicology and/or Pharmacology

14. CLINICAL STUDIES
14.1 Healing of Erosive Esophagitis
14.2 Symptomatic Gastroesophageal Reflux Disease (GERD)
14.3 Pediatric Gastroesophageal Reflux Disease (GERD)
14.4 Risk Reduction of NSAID-Associated Gastric Ulcer
14.5 Helicobacter pylori (H. Pylon) Eradication in Patients with Duodenal Ulcer Disease
14.6 Pathological Hypersecretory Conditions Including Zollinger-Ellison Syndrome

16. HOW SUPPLIED/STORAGE AND HANDLING
17. PATIENT COUNSELING INFORMATION

*Sections or subsections omitted from the full prescribing information are not listed.

Source: FDA Guidance for Industry (February 2013).

 

10 Chapter 1

and physiologic effects that influence the interaction of
TOXIC drug with the receptor. The interaction of a drug mole-

cule with a receptor causes the initiation of a sequence
POTENTIALLY TOXIC

of molecular events resulting in a pharmacologic or
toxic response. Pharmacokinetic–pharmacodynamic
models are constructed to relate plasma drug level to
drug concentration at the site of action and establish the

THERAPEUTIC intensity and time course of the drug. Pharmacodynamics
and pharmacokinetic–pharmacodynamic models are
discussed more fully in Chapter 21.

POTENTIALLY SUBTHERAPEUTIC DRUG EXPOSURE AND DRUG
RESPONSE

SUBTHERAPEUTIC

Drug exposure refers to the dose (drug input to the
FIGURE 12 Relationship of drug concentrations to drug body) and various measures of acute or integrated
response. drug concentrations in plasma and other biological

fluid (eg, Cmax, Cmin, Css, AUC) (FDA Guidance for

are in the therapeutic range but the patient does not Industry, 2003). Drug response refers to a direct

respond to drug treatment. Figure 1-2 shows that the measure of the pharmacologic effect of the drug.

concentration of drug in the body can range from Response includes a broad range of endpoints or

subtherapeutic to toxic. In contrast, some patients biomarkers ranging from the clinically remote bio-

respond to drug treatment at lower drug doses that markers (eg, receptor occupancy) to a presumed

result in lower drug concentrations. Other patients mechanistic effect (eg, ACE inhibition), to a poten-

may need higher drug concentrations to obtain a tial or accepted surrogate (eg, effects on blood pres-

therapeutic effect, which requires higher drug doses. sure, lipids, or cardiac output), and to the full range

It is desirable that adverse drug responses occur at of short-term or long-term clinical effects related to

drug concentrations higher relative to the therapeutic either efficacy or safety.

drug concentrations, but for many potent drugs, Toxicologic and efficacy studies provide infor-

adverse effects can also occur close to the same drug mation on the safety and effectiveness of the drug

concentrations as needed for the therapeutic effect. during development and in special patient popula-
tions such as subjects with renal and hepatic insuffi-
ciencies. For many drugs, clinical use is based on
weighing the risks of favorable and unfavorable out-

Frequently Asked Questions
comes at a particular dose. For some potent drugs, the

»»Which is more closely related to drug response, the
doses and dosing rate may need to be titrated in order

total drug dose administered or the concentration
of the drug in the body? to obtain the desired effect and be tolerated.

»»Why do individualized dosing regimens need to be
determined for some patients? TOXICOKINETICS AND CLINICAL

TOXICOLOGY
Toxicokinetics is the application of pharmacoki-

PHARMACODYNAMICS
netic principles to the design, conduct, and inter-

Pharmacodynamics refers to the relationship between pretation of drug safety evaluation studies (Leal et al,
the drug concentration at the site of action (receptor) 1993) and in validating dose-related exposure in
and pharmacologic response, including biochemical animals. Toxicokinetic data aid in the interpretation

DRUG CONCENTRATION

 

Introduction to Biopharmaceutics and Pharmacokinetics 11

of toxicologic findings in animals and extrapolation include sampling blood, spinal fluid, synovial fluid,
of the resulting data to humans. Toxicokinetic stud- tissue biopsy, or any biologic material that requires
ies are performed in animals during preclinical parenteral or surgical intervention in the patient. In
drug development and may continue after the drug contrast, noninvasive methods include sampling of
has been tested in clinical trials. urine, saliva, feces, expired air, or any biologic mate-

Clinical toxicology is the study of adverse effects rial that can be obtained without parenteral or surgi-
of drugs and toxic substances (poisons) in the body. cal intervention.
The pharmacokinetics of a drug in an overmedicated The measurement of drug and metabolite con-
(intoxicated) patient may be very different from the centration in each of these biologic materials yields
pharmacokinetics of the same drug given in lower important information, such as the amount of drug
therapeutic doses. At very high doses, the drug con- retained in, or transported into, that region of the tis-
centration in the body may saturate enzymes involved sue or fluid, the likely pharmacologic or toxicologic
in the absorption, biotransformation, or active renal outcome of drug dosing, and drug metabolite forma-
secretion mechanisms, thereby changing the pharma- tion or transport. Analytical methods should be able
cokinetics from linear to nonlinear pharmacokinetics. to distinguish between protein-bound and unbound
Nonlinear pharmacokinetics is discussed in parent drug and each metabolite, and the pharmaco-
Chapter 10. Drugs frequently involved in toxicity logically active species should be identified. Such
cases include acetaminophen, salicylates, opiates (eg, distinctions between metabolites in each tissue and
morphine), and the tricylic antidepressants (TCAs). fluid are especially important for initial pharmacoki-
Many of these drugs can be assayed conveniently by netic modeling of a drug.
fluorescence immunoassay (FIA) kits.

Drug Concentrations in Blood, Plasma,
MEASUREMENT OF DRUG or Serum

CONCENTRATIONS Measurement of drug and metabolite concentrations
(levels) in the blood, serum, or plasma is the most

Because drug concentrations are an important ele- direct approach to assessing the pharmacokinetics of
ment in determining individual or population phar- the drug in the body. Whole blood contains cellular
macokinetics, drug concentrations are measured in elements including red blood cells, white blood
biologic samples, such as milk, saliva, plasma, and cells, platelets, and various other proteins, such as
urine. Sensitive, accurate, and precise analytical albumin and globulins (Table 1-4). In general, serum
methods are available for the direct measurement of or plasma is most commonly used for drug measure-
drugs in biologic matrices. Such measurements are ment. To obtain serum, whole blood is allowed to
generally validated so that accurate information is clot and the serum is collected from the supernatant
generated for pharmacokinetic and clinical monitor- after centrifugation. Plasma is obtained from the
ing. In general, chromatographic and mass spectro- supernatant of centrifuged whole blood to which an
metric methods are most frequently employed for anticoagulant, such as heparin, has been added.
drug concentration measurement, because chroma- Therefore, the protein content of serum and plasma
tography separates the drug from other related mate- is not the same. Plasma perfuses all the tissues of the
rials that may cause assay interference and mass body, including the cellular elements in the blood.
spectrometry allows detection of molecules or mol- Assuming that a drug in the plasma is in dynamic
ecule fragments based on their mass-to-charge ratio. equilibrium with the tissues, then changes in the

drug concentration in plasma will reflect changes in
Sampling of Biologic Specimens tissue drug concentrations. Drugs in the plasma are
Only a few biologic specimens may be obtained often bound to plasma proteins, and often plasma
safely from the patient to gain information as to the proteins are filtered from the plasma before drug
drug concentration in the body. Invasive methods concentrations are measured. This is the unbound

 

12 Chapter 1

TABLE 14 Blood Components

Blood Component How Obtained Components

Whole blood Whole blood is generally obtained by venous Whole blood contains all the cellular and protein
puncture and contains an anticoagulant such as elements of blood
heparin or EDTA

Serum Serum is the liquid obtained from whole blood Serum does not contain the cellular elements,
after the blood is allowed to clot and the clot is fibrinogen, or the other clotting factors from
removed the blood

Plasma Plasma is the liquid supernatant obtained after Plasma is the noncellular liquid fraction of
centrifugation of non-clotted whole blood that whole blood and contains all the proteins
contains an anticoagulant including albumin

drug concentration. Alternatively, drug concentration involved, such as elimination in the feces, sweat, or
may be measured from unfiltered plasma; this is the exhaled air.
total plasma drug concentration. When interpreting The relationship of the drug level–time curve
plasma concentrations, it is important to understand and various pharmacologic parameters for the drug
what type of plasma concentration the data reflect. is shown in Fig. 1-3. MEC and MTC represent the

minimum effective concentration and minimum toxic
concentration of drug, respectively. For some drugs,

Frequently Asked Questions such as those acting on the autonomic nervous sys-

»»Why are drug concentrations more often measured tem, it is useful to know the concentration of drug

in plasma rather than whole blood or serum? that will just barely produce a pharmacologic effect
(ie, MEC). Assuming the drug concentration in the

»»What are the differences between bound drug,
plasma is in equilibrium with the tissues, the MEC

unbound drug, total drug, parent drug, and metabolite
reflects the minimum concentration of drug needed

drug concentrations in the plasma?

Plasma Drug Concentration–Time Curve
MTC

The plasma drug concentration (level)–time curve is
generated by obtaining the drug concentration in
plasma samples taken at various time intervals after
a drug product is administered. The concentration of
drug in each plasma sample is plotted on rectangular-
coordinate graph paper against the corresponding
time at which the plasma sample was removed. Duration MEC
As the drug reaches the general (systemic) circula-
tion, plasma drug concentrations will rise up to a
maximum if the drug was given by an extravascular
route. Usually, absorption of a drug is more rapid
than elimination. As the drug is being absorbed into
the systemic circulation, the drug is distributed to all
the tissues in the body and is also simultaneously Onset

time
being eliminated. Elimination of a drug can proceed Time

by excretion, biotransformation, or a combination of FIGURE 13 Generalized plasma level–time curve after
both. Other elimination mechanisms may also be oral administration of a drug.

Plasma level

Intensity

 

Introduction to Biopharmaceutics and Pharmacokinetics 13

at the receptors to produce the desired pharmaco- level or maximum drug concentration is related to
logic effect. Similarly, the MTC represents the drug the dose, the rate constant for absorption, and the
concentration needed to just barely produce a toxic elimination constant of the drug. The AUC is related
effect. The onset time corresponds to the time to the amount of drug absorbed systemically. These
required for the drug to reach the MEC. The inten- and other pharmacokinetic parameters are discussed
sity of the pharmacologic effect is proportional to in succeeding chapters.
the number of drug receptors occupied, which is
reflected in the observation that higher plasma drug

Frequently Asked Questions
concentrations produce a greater pharmacologic

»»At what time intervals should plasma drug con-
response, up to a maximum. The duration of drug

centration be taken in order to best predict drug
action is the difference between the onset time and response and side effects?
the time for the drug to decline back to the MEC.

The therapeutic window is the concentrations »»What happens if plasma concentrations fall outside

between the MEC and the MTC. Drugs with a wide of the therapeutic window?

therapeutic window are generally considered safer
than drugs with a narrow therapeutic window.
Sometimes the term therapeutic index is used. This Drug Concentrations in Tissues
term refers to the ratio between the toxic and thera- Tissue biopsies are occasionally removed for diag-
peutic doses. nostic purposes, such as the verification of a malig-

In contrast, the pharmacokineticist can also nancy. Usually, only a small sample of tissue is
describe the plasma level–time curve in terms of removed, making drug concentration measurement
such pharmacokinetic terms as peak plasma level difficult. Drug concentrations in tissue biopsies may
(Cmax), time for peak plasma level (Tmax), and area not reflect drug concentration in other tissues nor the
under the curve, or AUC (Fig. 1-4). The time for drug concentration in all parts of the tissue from
peak plasma level is the time of maximum drug which the biopsy material was removed. For exam-
concentration in the plasma and is a rough marker ple, if the tissue biopsy was for the diagnosis of a
of average rate of drug absorption. The peak plasma tumor within the tissue, the blood flow to the tumor

cells may not be the same as the blood flow to other
cells in this tissue. In fact, for many tissues, blood

MTC
flow to one part of the tissues need not be the same

Peak concentration
as the blood flow to another part of the same tissue.
The measurement of the drug concentration in tissue
biopsy material may be used to ascertain if the drug
reached the tissues and reached the proper concen-
tration within the tissue.

MEC

Drug Concentrations in Urine and Feces

Measurement of drug in urine is an indirect method
AUC

to ascertain the bioavailability of a drug. The rate
and extent of drug excreted in the urine reflects the
rate and extent of systemic drug absorption. The use

Peak of urinary drug excretion measurements to establish
time

Time various pharmacokinetic parameters is discussed in

FIGURE 14 Chapter 4.
Plasma level–time curve showing peak time

and concentration. The shaded portion represents the AUC Measurement of drug in feces may reflect drug
(area under the curve). that has not been absorbed after an oral dose or may

Plasma level

 

14 Chapter 1

reflect drug that has been expelled by biliary secre- methods, such as gas chromatography coupled with
tion after systemic absorption. Fecal drug excretion mass spectrometry, provides information regarding
is often performed in mass balance studies, in which past drug exposure. A study by Cone et al (1993)
the investigator attempts to account for the entire showed that the hair samples from subjects who were
dose given to the patient. For a mass balance study, known drug abusers contained cocaine and 6-acetyl-
both urine and feces are collected and their drug morphine, a metabolite of heroin (diacetylmorphine).
content measured. For certain solid oral dosage
forms that do not dissolve in the gastrointestinal tract Significance of Measuring Plasma Drug
but slowly leach out drug, fecal collection is per- Concentrations
formed to recover the dosage form. The undissolved

The intensity of the pharmacologic or toxic effect of
dosage form is then assayed for residual drug.

a drug is often related to the concentration of the
drug at the receptor site, usually located in the tissue

Drug Concentrations in Saliva
cells. Because most of the tissue cells are richly per-

Saliva drug concentrations have been reviewed for fused with tissue fluids or plasma, measuring the
many drugs for therapeutic drug monitoring plasma drug level is a responsive method of monitor-
(Pippenger and Massoud, 1984). Because only free ing the course of therapy.
drug diffuses into the saliva, saliva drug levels tend Clinically, individual variations in the pharma-
to approximate free drug rather than total plasma cokinetics of drugs are quite common. Monitoring
drug concentration. The saliva/plasma drug concen- the concentration of drugs in the blood or plasma
tration ratio is less than 1 for many drugs. The saliva/ ascertains that the calculated dose actually delivers
plasma drug concentration ratio is mostly influenced the plasma level required for therapeutic effect. With
by the pKa of the drug and the pH of the saliva. some drugs, receptor expression and/or sensitivity in
Weak acid drugs and weak base drugs with pKa sig- individuals varies, so monitoring of plasma levels is
nificantly different than pH 7.4 (plasma pH) gener- needed to distinguish the patient who is receiving
ally have better correlation to plasma drug levels. too much of a drug from the patient who is supersen-
The saliva drug concentrations taken after equilib- sitive to the drug. Moreover, the patient’s physiologic
rium with the plasma drug concentration generally functions may be affected by disease, nutrition, envi-
provide more stable indication of drug levels in the ronment, concurrent drug therapy, and other factors.
body. The use of salivary drug concentrations as a Pharmacokinetic models allow more accurate inter-
therapeutic indicator should be used with caution pretation of the relationship between plasma drug
and preferably as a secondary indicator. levels and pharmacologic response.

In the absence of pharmacokinetic information,
Forensic Drug Measurements plasma drug levels are relatively useless for dosage
Forensic science is the application of science to per- adjustment. For example, suppose a single blood
sonal injury, murder, and other legal proceedings. sample from a patient was assayed and found to con-
Drug measurements in tissues obtained at autopsy or tain 10 mg/mL. According to the literature, the maxi-
in other bodily fluids such as saliva, urine, and blood mum safe concentration of this drug is 15 mg/mL. In
may be useful if a suspect or victim has taken an over- order to apply this information properly, it is important
dose of a legal medication, has been poisoned, or has to know when the blood sample was drawn, what dose
been using drugs of abuse such as opiates (eg, heroin), of the drug was given, and the route of administration.
cocaine, or marijuana. The appearance of social drugs If the proper information is available, the use of phar-
in blood, urine, and saliva drug analysis shows short- macokinetic equations and models may describe the
term drug abuse. These drugs may be eliminated rap- blood level–time curve accurately and be used to
idly, making it more difficult to prove that the subject modify dosing for that specific patient.
has been using drugs of abuse. The analysis for drugs Monitoring of plasma drug concentrations
of abuse in hair samples by very sensitive assay allows for the adjustment of the drug dosage in order

 

Introduction to Biopharmaceutics and Pharmacokinetics 15

to individualize and optimize therapeutic drug regi- The predictive capability of a model lies in the
mens. When alterations in physiologic functions proper selection and development of mathematical
occur, monitoring plasma drug concentrations may function(s) that parameterizes the essential factors
provide a guide to the progress of the disease state governing the kinetic process. The key parameters in
and enable the investigator to modify the drug dos- a process are commonly estimated by fitting the
age accordingly. Clinically, sound medical judgment model to the experimental data, known as variables.
and observation are most important. Therapeutic A pharmacokinetic parameter is a constant for the
decisions should not be based solely on plasma drug drug that is estimated from the experimental data.
concentrations. For example, estimated pharmacokinetic parameters

In many cases, the pharmacodynamic response to such as k depend on the method of tissue sampling,
the drug may be more important to measure than just the timing of the sample, drug analysis, and the pre-
the plasma drug concentration. For example, the elec- dictive model selected.
trophysiology of the heart, including an electrocardio- A pharmacokinetic function relates an indepen-
gram (ECG), is important to assess in patients dent variable to a dependent variable, often through
medicated with cardiotonic drugs such as digoxin. For the use of parameters. For example, a pharmacoki-
an anticoagulant drug, such as dicumarol, prothrom- netic model may predict the drug concentration in the
bin clotting time may indicate whether proper dosage liver 1 hour after an oral administration of a 20-mg
was achieved. Most diabetic patients taking insulin dose. The independent variable is the time and the
will monitor their own blood or urine glucose levels. dependent variable is the drug concentration in the

For drugs that act irreversibly at the receptor liver. Based on a set of time-versus-drug concentra-
site, plasma drug concentrations may not accurately tion data, a model equation is derived to predict the
predict pharmacodynamic response. Drugs used in liver drug concentration with respect to time. In this
cancer chemotherapy often interfere with nucleic case, the drug concentration depends on the time
acid or protein biosynthesis to destroy tumor cells. after the administration of the dose, where the time–
For these drugs, the plasma drug concentration does concentration relationship is defined by a pharmaco-
not relate directly to the pharmacodynamic response. kinetic parameter, k, the elimination rate constant.
In this case, other pathophysiologic parameters and Such mathematical models can be devised to
side effects are monitored in the patient to prevent simulate the rate processes of drug absorption, distri-
adverse toxicity. bution, and elimination to describe and predict drug

concentrations in the body as a function of time.
Pharmacokinetic models are used to:

BASIC PHARMACOKINETICS AND
PHARMACOKINETIC MODELS 1. Predict plasma, tissue, and urine drug levels

with any dosage regimen
Drugs are in a dynamic state within the body as they 2. Calculate the optimum dosage regimen for each
move between tissues and fluids, bind with plasma patient individually
or cellular components, or are metabolized. The 3. Estimate the possible accumulation of drugs
biologic nature of drug distribution and disposition and/or metabolites
is complex, and drug events often happen simulta- 4. Correlate drug concentrations with pharmaco-
neously. Such factors must be considered when logic or toxicologic activity
designing drug therapy regimens. The inherent and 5. Evaluate differences in the rate or extent of
infinite complexity of these events requires the use availability between formulations
of mathematical models and statistics to estimate (bioequivalence)
drug dosing and to predict the time course of drug 6. Describe how changes in physiology or disease
efficacy for a given dose. affect the absorption, distribution, or elimina-

A model is a hypothesis using mathematical tion of the drug
terms to describe quantitative relationships concisely. 7. Explain drug interactions

 

16 Chapter 1

Simplifying assumptions are made in pharmacoki- in separate chapters under the topics of drug absorp-
netic models to describe a complex biologic system tion, drug distribution, drug elimination, and pharma-
concerning the movement of drugs within the body. cokinetic drug interactions involving one or all of the
For example, most pharmacokinetic models assume above processes. Theoretically, an unlimited number
that the plasma drug concentration reflects drug con- of models may be constructed to describe the kinetic
centrations globally within the body. processes of drug absorption, distribution, and elimi-

A model may be empirically, physiologically, or nation in the body, depending on the degree of
compartmentally based. The model that simply detailed information considered. Practical consider-
interpolates the data and allows an empirical formula ations have limited the growth of new pharmacoki-
to estimate drug level over time is justified when netic models.
limited information is available. Empirical models A very simple and useful tool in pharmacokinet-
are practical but not very useful in explaining the ics is compartmentally based models. For example,
mechanism of the actual process by which the drug assume a drug is given by intravenous injection and
is absorbed, distributed, and eliminated in the body. that the drug dissolves (distributes) rapidly in the body
Examples of empirical models used in pharmacoki- fluids. One pharmacokinetic model that can describe
netics are described in Chapter 25. this situation is a tank containing a volume of fluid

Physiologically based models also have limita- that is rapidly equilibrated with the drug. The concen-
tions. Using the example above, and apart from the tration of the drug in the tank after a given dose is
necessity to sample tissue and monitor blood flow to governed by two parameters: (1) the fluid volume of
the liver in vivo, the investigator needs to understand the tank that will dilute the drug, and (2) the elimina-
the following questions. What is the clinical implica- tion rate of drug per unit of time. Though this model
tion of the liver drug concentration value? Should is perhaps an overly simplistic view of drug disposi-
the drug concentration in the blood within the tissue tion in the human body, a drug’s pharmacokinetic
be determined and subtracted from the drug in the properties can frequently be described using a fluid-
liver tissue? What type of cell is representative of the filled tank model called the one-compartment open
liver if a selective biopsy liver tissue sample can be model (see below). In both the tank and the one-
collected without contamination from its surround- compartment body model, a fraction of the drug
ings? Indeed, depending on the spatial location of would be continually eliminated as a function of time
the liver tissue from the hepatic blood vessels, tissue (Fig. 1-5). In pharmacokinetics, these parameters are
drug concentrations can differ depending on distance assumed to be constant for a given drug. If drug con-
to the blood vessel or even on the type of cell in the centrations in the tank are determined at various time
liver. Moreover, changes in the liver blood perfusion intervals following administration of a known dose,
will alter the tissue drug concentration. If heteroge- then the volume of fluid in the tank or compartment
neous liver tissue is homogenized and assayed, the (VD, volume of distribution) and the rate of drug
homogenized tissue represents only a hypothetical elimination can be estimated.
concentration that is an average of all the cells and In practice, pharmacokinetic parameters such as
blood in the liver at the time of collection. Since tis- k and VD are determined experimentally from a set of
sue homogenization is not practical for human sub- drug concentrations collected over various times and
jects, the drug concentration in the liver may be
estimated by knowing the liver extraction ratio for
the drug based on knowledge of the physiologic and

Fluid replenished
biochemical composition of the body organs. Fluid

automatically to keep outlet
A great number of models have been developed volume constant

to estimate regional and global information about
FIGURE 15 Tank with a constant volume of fluid equili-

drug disposition in the body. Some physiologic phar-
brated with drug. The volume of the fluid is 1.0 L. The fluid

macokinetic models are also discussed in Chapter 25. outlet is 10 mL/min. The fraction of drug removed per unit of
Individual pharmacokinetic processes are discussed time is 10/1000, or 0.01 min–1.

 

Introduction to Biopharmaceutics and Pharmacokinetics 17

known as data. The number of parameters needed to compartments, that communicate reversibly with each
describe the model depends on the complexity of the other. A compartment is not a real physiologic or ana-
process and on the route of drug administration. In tomic region but is considered a tissue or group of
general, as the number of parameters required to tissues that have similar blood flow and drug affinity.
model the data increases, accurate estimation of Within each compartment, the drug is considered to
these parameters becomes increasingly more diffi- be uniformly distributed. Mixing of the drug within a
cult. With complex pharmacokinetic models, com- compartment is rapid and homogeneous and is con-
puter programs are used to facilitate parameter sidered to be “well stirred,” so that the drug concentra-
estimation. However, for the parameters to be valid, tion represents an average concentration, and each
the number of data points should always exceed the drug molecule has an equal probability of leaving the
number of parameters in the model. compartment. Rate constants are used to represent

Because a model is based on a hypothesis and the overall rate processes of drug entry into and exit
simplifying assumptions, a certain degree of caution from the compartment. The model is an open system
is necessary when relying totally on the pharmacoki- because drug can be eliminated from the system.
netic model to predict drug action. For some drugs, Compartment models are based on linear assump-
plasma drug concentrations are not useful in predict- tions using linear differential equations.
ing drug activity. For other drugs, an individual’s
genetic differences, disease state, and the compensa-

Mammillary Model
tory response of the body may modify the response
to the drug. If a simple model does not fit all the A compartmental model provides a simple way of

experimental observations accurately, a new, more grouping all the tissues into one or more compart-

elaborate model may be proposed and subsequently ments where drugs move to and from the central or

tested. Since limited data are generally available in plasma compartment. The mammillary model is the

most clinical situations, pharmacokinetic data should most common compartment model used in pharma-

be interpreted along with clinical observations rather cokinetics. The mammillary model is a strongly con-

than replacing sound judgment by the clinician. nected system, because one can estimate the amount

Development of pharmacometric statistical models of drug in any compartment of the system after drug

may help to improve prediction of drug levels among is introduced into a given compartment. In the one-

patients in the population (Sheiner and Beal, 1982; compartment model, drug is both added to and

Mallet et al, 1988). However, it will be some time eliminated from a central compartment. The central

before these methods become generally accepted. compartment is assigned to represent plasma and
highly perfused tissues that rapidly equilibrate with
drug. When an intravenous dose of drug is given, the

Compartment Models drug enters directly into the central compartment.

If the tissue drug concentrations and binding are Elimination of drug occurs from the central compart-

known, physiologic pharmacokinetic models, which ment because the organs involved in drug elimination,

are based on actual tissues and their respective blood primarily kidney and liver, are well-perfused tissues.

flow, describe the data realistically. Physiologic phar- In a two-compartment model, drug can move
macokinetic models are frequently used in describing between the central or plasma compartment to and
drug distribution in animals, because tissue samples from the tissue compartment. Although the tissue
are easily available for assay. On the other hand, tissue compartment does not represent a specific tissue, the
samples are often not available for human subjects, mass balance accounts for the drug present in all
so most physiological models assume an average set the tissues. In this model, the total amount of drug in
of blood flow for individual subjects. the body is simply the sum of drug present in the cen-

In contrast, because of the vast complexity of tral compartment plus the drug present in the tissue
the body, drug kinetics in the body are frequently compartment. Knowing the parameters of either the
simplified to be represented by one or more tanks, or one-compartment or the two-compartment model,

 

18 Chapter 1

one can estimate the amount of drug left in the body k1 k
k 2 23

a
and the amount of drug eliminated from the body at any 1 2 3

time. The compartmental models are particularly useful k21 k32

when little information is known about the tissues.
FIGURE 17 Example of caternary model.

Several types of compartment models are
described in Fig. 1-6. The pharmacokinetic rate con-
stants are represented by the letter k. Compartment 1 more compartments around a central compartment

represents the plasma or central compartment, and like satellites. Because the catenary model does not

compartment 2 represents the tissue compartment. apply to the way most functional organs in the body

The drawing of models has three functions. The are directly connected to the plasma, it is not used as

model (1) enables the pharmacokineticist to write often as the mammillary model.

differential equations to describe drug concentration
changes in each compartment, (2) gives a visual Physiologic Pharmacokinetic Model
representation of the rate processes, and (3) shows (Flow Model)
how many pharmacokinetic constants are necessary Physiologic pharmacokinetic models, also known as
to describe the process adequately. blood flow or perfusion models, are pharmacoki-

netic models based on known anatomic and physi-
Catenary Model ologic data. The models describe the data kinetically,
In pharmacokinetics, the mammillary model must be with the consideration that blood flow is responsible
distinguished from another type of compartmental for distributing drug to various parts of the body.
model called the catenary model. The catenary Uptake of drug into organs is determined by the
model consists of compartments joined to one
another like the compartments of a train (Fig. 1-7).
In contrast, the mammillary model consists of one or EXAMPLE »» »

Two parameters are needed to describe model 1
MODEL 1. One-compartment open model, IV injection. (Fig. 1-6): the volume of the compartment and

k the elimination rate constant, k. In the case of
1

model 4, the pharmacokinetic parameters consist
of the volumes of compartments 1 and 2 and the

MODEL 2. One-compartment open model with rst-order absorption. rate constants—ka, k, k12, and k21—for a total of
six parameters.

ka k
1 In studying these models, it is important to

know whether drug concentration data may be

MODEL 3. Two-compartment open model, IV injection. sampled directly from each compartment. For mod-

k els 3 and 4 (Fig. 1-6), data concerning compartment
12

1 2 2 cannot be obtained easily because tissues are
k21 not easily sampled and may not contain homoge-

k neous concentrations of drug. If the amount of drug
absorbed and eliminated per unit time is obtained

MODEL 4. Two-compartment open model with rst-order absorption. by sampling compartment 1, then the amount of

k drug contained in the tissue compartment 2 can be
k 12

a
1 2 estimated mathematically. The appropriate math-

k21 ematical equations for describing these models and
k evaluating the various pharmacokinetic parameters

are given in subsequent chapters.
FIGURE 16 Various compartment models.

 

Introduction to Biopharmaceutics and Pharmacokinetics 19

binding of drug in these tissues. In contrast to an IV injection

estimated tissue volume of distribution, the actual QH

tissue volume is used. Because there are many tissue Heart

organs in the body, each tissue volume must be
obtained and its drug concentration described. The QM

model would potentially predict realistic tissue drug Muscle

concentrations, which the two-compartment model
fails to do. Unfortunately, much of the information QS

required for adequately describing a physiologic SET

pharmacokinetic model is experimentally difficult
to obtain. In spite of this limitation, the physiologic QR

pharmacokinetic model does provide much better RET

insight into how physiologic factors may change
k

drug distribution from one animal species to another. e
Urine QK

Other major differences are described below. Kidney

First, no data fitting is required in the perfusion
model. Drug concentrations in the various tissues are QL

predicted by organ tissue size, blood flow, and Liver

experimentally determined drug tissue–blood ratios km

(ie, partition of drug between tissue and blood). FIGURE 18 Pharmacokinetic model of drug perfu-
Second, blood flow, tissue size, and the drug sion. The ks represent kinetic constants: ke is the first-order

tissue–blood ratios may vary due to certain patho- rate constant for urinary drug excretion and km is the rate

physiologic conditions. Thus, the effect of these constant for hepatic elimination. Each “box” represents a tissue

variations on drug distribution must be taken into compartment. Organs of major importance in drug absorption
are considered separately, while other tissues are grouped as

account in physiologic pharmacokinetic models.
RET (rapidly equilibrating tissue) and SET (slowly equilibrating

Third, and most important of all, physiologically tissue). The size or mass of each tissue compartment is deter-
based pharmacokinetic models can be applied to sev- mined physiologically rather than by mathematical estimation.

eral species, and, for some drugs, human data may be The concentration of drug in the tissue is determined by the

extrapolated. Extrapolation from animal data is not ability of the tissue to accumulate drug as well as by the rate of
blood perfusion to the tissue, represented by Q.

possible with the compartment models, because the
volume of distribution in such models is a mathemati-
cal concept that does not relate simply to blood volume would make the model very complex and mathemat-
and blood flow. To date, numerous drugs (including ically difficult. A simpler but equally good approach
digoxin, lidocaine, methotrexate, and thiopental) have is to group all the tissues with similar blood perfu-
been described with perfusion models. Tissue levels of sion properties into a single compartment.
some of these drugs cannot be predicted successfully A physiologic based pharmacokinetic model
with compartment models, although they generally (PBPK) using known blood flow was used to describe
describe blood levels well. An example of a perfusion the distribution of lidocaine in blood and various
model is shown in Fig. 1-8. organs (Benowitz et el 1974) and applied in anesthe-

The number of tissue compartments in a perfu- siology in man (Tucker et el 1971). In PBKB models,
sion model varies with the drug. Typically, the tis- organs such as lung, liver, brain, and muscle were
sues or organs that have no drug penetration are individually described by differential equations as
excluded from consideration. Thus, such organs as shown in Fig. 1-8, sometimes tissues were grouped as
the brain, the bones, and other parts of the central RET (rapidly equilibrating tissue) and SET (slowly
nervous system are often excluded, as most drugs equilibrating tissue) for simplicity to account for the
have little penetration into these organs. To describe mass balance of the drug. A general scheme showing
each organ separately with a differential equation blood flow for typical organs is shown in Fig. 1-8.

Venous blood

Arterial blood

 

20 Chapter 1

20 drug metabolism capacity, and blood flow in humans
and other species are often known or can be deter-

10 mined. Thus, physiologic and anatomic parameters

5 can be used to predict the effects of drugs on humans
from the effects on animals in cases where human
experimentation is difficult or restricted.

1

Frequently Asked Questions
0.5

»»What are the reasons to use a multicompartment
Observed

Simulated model instead of a physiologic model?
perfusion model

»»What do the boxes in the mammillary model mean?
0.1

0 60 120 180 240
Time (minutes)

More sophisticated models are introduced as the
FIGURE 19 Observed mean (•) and simulated (—) understanding of human and animal physiology
arterial lidocaine blood concentrations in normal volunteers
receiving 1 mg/kg/min constant infusion for 3 minutes. (From improves. For example, in Chapter 25, special com-
Tucker GT, Boas RA: Pharmacokinetic aspects of intravenous partment models that take into account transporter-
regional anesthesia. Anesthesiology 34(6):538–549, 1971, with mediated drug disposition are introduced for specific
permission.) drugs. This approach is termed Physiologic Pharmaco-

kinetic Model Incorporating Hepatic Transporter-
The data showing blood concentration of lidocaine Mediated Clearance. The differences between the
after an IV dose declining biexponentially (Fig. 1-9) physiologic pharmacokinetic model, the classical
was well predicted by the model. A later PBPK compartmental model, and the noncompartmental
model was applied to model cyclosporine (Fig. 1-10). approach are discussed. It is important to note that
Drug level in various organs were well predicted and mass transfer and balances of drug in and out of the
scaled to human based on this physiologic model body or body organs are fundamentally a kinetic pro-
(Kawai R et al, 1998). The tissue cyclosporine levels cess. Thus, the model may be named as physiologi-
in the lung, muscle, and adipose and other organs are cally based when all drug distributed to body organs
shown in Fig. 1-10. For lidocaine, the tissue such as are identified. For data analysis, parameters are
adipose (fat) tissue accumulates drugs slowly because obtained quantitatively with different assumptions.
of low blood supply. In contrast, vascular tissues, like The model analysis may be compartmental or non-
the lung, equilibrate rapidly with the blood and start compartmental (Chapter 25). One approach is to clas-
to decline as soon as drug level in the blood starts to sify models simply as empirically based models and
fall resulting in curvature of plasma profile. The mechanistic models. Although compartment models
physiologic pharmacokinetic model provides a real- are critically referred to as a “black box” approach
istic means of modeling tissue drug levels. However, and not physiological. The versatility of compartment
drug levels in tissues are not available. A criticism of models and their easy application are based on simple
physiologic pharmacokinetic models in general has mass transfer algorithms or a system of differential
been that there are fewer data points than parameters equations. This approach has allowed many body
that one tries to fit. Consequently, the projected data processes such as binding, transport, and metabolic
are not well constrained. clearance to be monitored. The advantage of a non-

The real significance of the physiologically compartmental analysis is discussed in Chapter 25. In
based model is the potential application of this model Appendix B, softwares used for various type of model
in the prediction of human pharmacokinetics from analysis are discussed, for example, noncompartmen-
animal data (Sawada et al, 1985). The mass of vari- tal analysis is often used for pharmacokinetic and
ous body organs or tissues, extent of protein binding, bioavailability data analysis for regulatory purpose.

Lidocaine hydrochloride (mg/mL blood)

 

Introduction to Biopharmaceutics and Pharmacokinetics 21

1998 Tissue Distribution Kinetics of IV Cyclosporine A (CyA)

Blood Lung Heart Kidney
100 100 100 100

10

10 10 10

1

0.1 1 1 1
0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32

Time (h) Time (h) Time (h) Time (h)

Spleen Liver Gut Skin
100 100 100 10

10 10 10

0.1 1 1 1
0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32

Time (h) Time (h) Time (h) Time (h)

Bone Muscle Fat Thymus
10 10 100 10

1 10

1 0.1 1 1
0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32 0 4 8 12 16 20 24 28 32

Time (h) Time (h) Time (h) Time (h)

FIGURE 110 Measured and best fit predictions of CyA concentration in arterial blood and various organs/tissues in rat. Each
plot and vertical bar represent the mean and standard deviation, respectively. Solid and dotted lines are the physiological-based
pharmacokinetic (PBPK) best fit predictions based on the parameters associated with the linear or nonlinear model, respectively.
(Reproduced with permission from Kawai R, Mathew D, Tanaka C, Rowland M: Physiologically based pharmacokinetics of cyclospo-
rine A: Extension to tissue distribution kinetics in rats and scale-up to human. JPET 287:457–468, 1998.)

CHAPTER SUMMARY
Drug product performance is the release of the drug route of administration on the rate and extent of sys-
substance from the drug product leading to bioavail- temic drug absorption. Pharmacokinetics is the sci-
ability of the drug substance and eventually leading ence of the dynamics (kinetics) of drug absorption,
to one or more pharmacologic effects, both desirable distribution, and elimination (ie, excretion and
and undesirable. Biopharmaceutics provides the sci- metabolism), whereas clinical pharmacokinetics
entific basis for drug product design and drug prod- considers the applications of pharmacokinetics to
uct performance by examining the interrelationship drug therapy.
of the physical/chemical properties of the drug, the The quantitative measurement of drug concen-
drug product in which the drug is given, and the trations in the plasma after dose administration is

Tissue concentration (mg/g) Tissue concentration (mg/g) Blood concentration (mg/mL)

Tissue concentration (mg/g) Tissue concentration (mg/g) Tissue concentration (mg/g)

Tissue concentration (mg/g) Tissue concentration (mg/g) Tissue concentration (mg/g)

Tissue concentration (mg/g) Tissue concentration (mg/g) Tissue concentration (mg/g)

 

22 Chapter 1

important to obtain relevant data of systemic drug drug concentration to the pharmacodynamic effect
exposure. The plasma drug concentration-versus- or adverse response, and enables the development of
time profile provides the basic data from which vari- individualized therapeutic dosage regimens and new
ous pharmacokinetic models can be developed that and novel drug delivery systems.
predict the time course of drug action, relates the

LEARNING QUESTIONS
1. What is the significance of the plasma level– d. Write an expression describing the

time curve? How does the curve relate to the rate of change of drug concentration in
pharmacologic activity of a drug? compartment 1 (dC1/dt).

2. What is the purpose of pharmacokinetic models? 5. Give two reasons for the measurement of
3. Draw a diagram describing a three-compartment the plasma drug concentration, Cp, assuming

model with first-order absorption and drug (a) the Cp relates directly to the pharma-
elimination from compartment 1. codynamic activity of the drug and (b) the

4. The pharmacokinetic model presented in Cp does not relate to the pharmacodynamic
Fig. 1-11 represents a drug that is eliminated activity of the drug.
by renal excretion, biliary excretion, and drug 6. Consider two biologic compartments separated
metabolism. The metabolite distribution is by a biologic membrane. Drug A is found in
described by a one-compartment open model. compartment 1 and in compartment 2 in a
The following questions pertain to Fig. 1-11. concentration of c1 and c2, respectively.
a. How many parameters are needed to a. What possible conditions or situations

describe the model if the drug is injected would result in concentration c1 > c2 at
intravenously (ie, the rate of drug absorp- equilibrium?
tion may be neglected)? b. How would you experimentally demonstrate

b. Which compartment(s) can be sampled? these conditions given above?
c. What would be the overall elimination rate c. Under what conditions would c1 = c2 at

constant for elimination of drug from equilibrium?
compartment 1? d. The total amount of Drug A in each biologic

compartment is A1 and A2, respectively.
Metabolite Describe a condition in which A1 > A2, but

Drug compartment
c1 = c2 at equilibrium.

2 Include in your discussion, how the physico-
chemical properties of Drug A or the biologic

k12 k21 properties of each compartment might influ-

k ence equilibrium conditions.
e km ku

1 3
7. Why is it important for a pharmacist to keep

up with possible label revision in a drug newly
kb

approved? Which part of the label you expect
to be mostly likely revised with more phase 4

FIGURE 111 Pharmacokinetic model for a drug eliminated
information?

by renal and biliary excretion and drug metabolism. km = rate
constant for metabolism of drug; k a. The chemical structure of the drug

u = rate constant for urinary
excretion of metabolites; kb = rate constant for biliary excretion b. The Description section
of drug; and ke = rate constant for urinary drug excretion. c. Adverse side effect in certain individuals

 

Introduction to Biopharmaceutics and Pharmacokinetics 23

8. A pharmacist wishing to find if an excipient 9. A pregnant patient is prescribed pantoprazole
such as aspartame in a product is mostly found sodium (Protonix) delayed release tablets
under which section in the SPL drug label? for erosive gastroesophageal reflux disease
a. How supplied (GERD). Where would you find information
b. Patient guide concerning the safety of this drug in pregnant
c. Description women?

ANSWERS

Frequently Asked Questions What are the reasons to use a multicompartment

Why are drug concentrations more often measured model instead of a physiologic model?

in plasma rather than whole blood or serum? • Physiologic models are complex and require more

• Blood is composed of plasma and red blood cells information for accurate prediction compared to

(RBCs). Serum is the fluid obtained from blood compartment models. Missing information in the

after it is allowed to clot. Serum and plasma do physiologic model will lead to bias or error in the

not contain identical proteins. RBCs may be con- model. Compartment models are more simplistic

sidered a cellular component of the body in which in that they assume that both arterial and venous

the drug concentration in the serum or plasma is drug concentrations are similar. The compartment

in equilibrium, in the same way as with the other model accounts for a rapid distribution phase and

tissues in the body. Whole blood samples are gen- a slower elimination phase. Physiologic clearance

erally harder to process and assay than serum or models postulate that arterial blood drug levels are

plasma samples. Plasma may be considered a liq- higher than venous blood drug levels. In practice,

uid tissue compartment in which the drug in the only venous blood samples are usually sampled.

plasma fluid equilibrates with drug in the tissues Organ drug clearance is useful in the treatment of

and cellular components. cancers and in the diagnosis of certain diseases in-
volving arterial perfusion. Physiologic models are

At what time intervals should plasma drug concen- difficult to use for general application.
tration be taken in order to best predict drug response
and side effects? Learning Questions

• The exact site of drug action is generally un- 1. The plasma drug level–time curve describes the
known for most drugs. The time needed for the pharmacokinetics of the systemically absorbed
drug to reach the site of action, produce a phar- drug. Once a suitable pharmacokinetic model
macodynamic effect, and reach equilibrium are is obtained, plasma drug concentrations may be
deduced from studies on the relationship of the predicted following various dosage regimens
time course for the drug concentration and the such as single oral and IV bolus doses, multiple-
pharmacodynamic effect. Often, the drug concen- dose regimens, IV infusion, etc. If the pharma-
tration is sampled during the elimination phase cokinetics of the drug relates to its pharmaco-
after the drug has been distributed and reached dynamic activity (or any adverse drug response
equilibrium. For multiple-dose studies, both the or toxicity), then a drug regimen based on the
peak and trough drug concentrations are fre- drug’s pharmacokinetics may be designed to
quently taken. provide optimum drug efficacy. In lieu of a direct

 

24 Chapter 1

pharmacokinetic–pharmacodynamic relation- cell (eg, purine drug). Other explanations for
ship, the drug’s pharmacokinetics describes the C1 > C2 may be possible.
bioavailability of the drug including inter- and b. Several different experimental conditions are
intrasubject variability; this information allows needed to prove which of the above hypoth-
for the development of drug products that consis- eses is the most likely cause for C1 > C2.
tently deliver the drug in a predictable manner. These experiments may use in vivo or in vitro

2. The purpose of pharmacokinetic models is to methods, including intracellular electrodes to
relate the time course of the drug in the body to measure pH in vivo, protein-binding studies
its pharmacodynamic and/or toxic effects. The in vitro, and partitioning of drug in chloro-
pharmacokinetic model also provides a basis form/water in vitro, among others.
for drug product design, the design of dosage c. In the case of protein binding, the total
regimens, and a better understanding of the concentration of drug in each compartment
action of the body on the drug. may be different (eg, C1 > C2) and, at the

3. (Figure A-1) same time, the free (nonprotein-bound)
4. a. Nine parameters: V1, V2, V3, k12, k21, ke, kb, drug concentration may be equal in each

km, ku compartment—assuming that the free or
b. Compartment 1 and compartment 3 may be unbound drug is easily diffusible. Similarly,

sampled. if C1 > C2 is due to differences in pH and the
c. k = kb + km + ke nonionized drug is easily diffusible, then the

dC nonionized drug concentration may be the
d. 1

= k21C2 − (k12 + k + k + kb )Cdt m e 1 same in each compartment. The total drug
concentrations will be C1 = C2 when there

6. Compartment 1 Compartment 2 is similar affinity for the drug and similar

C conditions in each compartment.
1 C2

d. The total amount of drug, A, in each com-
a. C1 and C2 are the total drug concentration in partment depends on the volume, V, of the

each compartment, respectively. C1 > C2 may compartment and the concentration, C, of the
occur if the drug concentrates in compart- drug in the compartment. Since the amount
ment 1 due to protein binding (compartment of drug (A) = concentration (C) times volume
1 contains a high amount of protein or special (V), any condition that causes the product,
protein binding), due to partitioning (compart- C1V1 ≠ C2V2, will result in A1 ≠ A2. Thus, if
ment 1 has a high lipid content and the drug is C1 = C2 and V1 ≠ V2, then A1 ≠ A2.
poorly water soluble), if the pH is different in 7. A newly approved NDA generally contains
each compartment and the drug is a weak elec- sufficient information for use labeled. However,
trolyte (the drug may be more ionized in com- as more information becomes available through
partment 1), or if there is an active transport postmarketing commitment studies, more
mechanism for the drug to be taken up into the information is added to the labeling, including

Warnings and Precautions.
8. An excipient such as aspartame in a product

ka
is mostly found under the Description section,

k12 k13 which describes the drug chemical structure
2 1 3 and the ingredients in the drug product.

k21 k31 9. Section 8, Use in Specific Populations, reports
k information for geriatric, pediatric, renal, and

hepatic subjects. This section will report dosing
FIGURE A1 for pediatric subjects as well.

 

Introduction to Biopharmaceutics and Pharmacokinetics 25

REFERENCES
Cone EJ, Darwin WD, Wang W-L: The occurrence of cocaine, pharmacokinetics, with application to cyclosporine. J Pharm

heroin and metabolites in hair of drug abusers. Forensic Sci Biopharm 16:311–327, 1988.
Int 63:55–68, 1993. Pippenger CE, Massoud N: Therapeutic drug monitoring. In Benet

FDA Guidance for Industry: Exposure-Response Relationships— LZ, et al (eds). Pharmacokinetic Basis for Drug Treatment.
Study Design, Data Analysis, and Regulatory Applications, New York, Raven, 1984, chap 21.
FDA, Center for Drug Evaluation and Research, April 2003 Rodman JH, Evans WE: Targeted systemic exposure for pediatric
(http://www.fda.gov/cder/guidance/5341fnl.htm). cancer therapy. In D’Argenio DZ (ed). Advanced Methods of

FDA Guidance for Industry: Labeling for Human Prescription Pharmacokinetic and Pharmacodynamic Systems Analysis.
Drug and Biological Products – Implementing the PLR Con- New York, Plenum Press, 1991, pp 177–183.
tent and Format Requirements, February 2013. Sawada Y, Hanano M, Sugiyama Y, Iga T: Prediction of the disposi-

Kawai R, Mathew D, Tanaka C, Rowland M: Physiologically tion of nine weakly acidic and six weekly basic drugs in humans
based pharmacokinetics of cyclosporine. from pharmacokinetic parameters in rats. J Pharmacokinet

Leal M, Yacobi A, Batra VJ: Use of toxicokinetic principles in Biopharm 13:477–492, 1985.
drug development: Bridging preclinical and clinical studies. In Sheiner LB, Beal SL: Bayesian individualization of pharmaco-
Yacobi A, Skelly JP, Shah VP, Benet LZ (eds). Integration of kinetics. Simple implementation and comparison with non
Pharmacokinetics, Pharmacodynamics and Toxicokinetics in Bayesian methods. J Pharm Sci 71:1344–1348, 1982.
Rational Drug Development. New York, Plenum Press, 1993, Sheiner LB, Ludden TM: Population pharmacokinetics/dynamics.
pp 55–67. Annu Rev Pharmacol Toxicol 32:185–201, 1992.

Mallet A, Mentre F, Steimer JL, Lokiec F: Pharmacometrics: Tucker GT, Boas RA: Pharmacokinetic aspects of intravenous
Nonparametric maximum likelihood estimation for population regional anesthesia.

BIBLIOGRAPHY
Benet LZ: General treatment of linear mammillary models with Gibaldi M: Estimation of the pharmacokinetic parameters of the

elimination from any compartment as used in pharmacokinet- two-compartment open model from post-infusion plasma con-
ics. J Pharm Sci 61:536–541, 1972. centration data. J Pharm Sci 58:1133–1135, 1969.

Benowitz N, Forsyth R, Melmon K, Rowland M: Lidocaine dis- Himmelstein KJ, Lutz RJ: A review of the applications of physio-
position kinetics in monkey and man. Clin Pharmacol Ther logically based pharmacokinetic modeling. J Pharm Biopharm
15:87–98, 1974. 7:127–145, 1979.

Bischoff K, Brown R: Drug distribution in mammals. Chem Eng Lutz R, Dedrick RL: Physiologic pharmacokinetics: Relevance to
Med 62:33–45, 1966. human risk assessment. In Li AP (ed). Toxicity Testing: New

Tucker GT, Boas RA:Pharmacokinetic aspects of intravenous Applications and Applications in Human Risk Assessment.
regional anesthesia. New York, Raven, 1985, pp 129–149.

Bischoff K, Dedrick R, Zaharko D, Longstreth T: Methotrexate Lutz R, Dedrick R, Straw J, et al: The kinetics of methotrexate
pharmacokinetics. J Pharm Sci 60:1128–1133, 1971. distribution in spontaneous canine lymphosarcoma. J Pharm

Chiou W: Quantitation of hepatic and pulmonary first-pass effect and Biopharm 3:77–97, 1975.
its implications in pharmacokinetic study, I: Pharmacokinetics of Metzler CM: Estimation of pharmacokinetic parameters: Statisti-
chloroform in man. J Pharm Biopharm 3:193–201, 1975. cal considerations. Pharmacol Ther 13:543–556, 1981.

Colburn WA: Controversy III: To model or not to model. J Clin Montandon B, Roberts R, Fischer L: Computer simulation of sulfo-
Pharmacol 28:879–888, 1988. bromophthalein kinetics in the rat using flow-limited models

Cowles A, Borgstedt H, Gilles A: Tissue weights and rates of with extrapolation to man. J Pharm Biopharm 3:277–290, 1975.
blood flow in man for the prediction of anesthetic uptake and Rescigno A, Beck JS: The use and abuse of models. J Pharm Bio-
distribution. Anesthesiology 35:523–526, 1971. pharm 15:327–344, 1987.

Dedrick R, Forrester D, Cannon T, et al: Pharmacokinetics of Ritschel WA, Banerjee PS: Physiologic pharmacokinetic models:
1-β-d-arabinofurinosulcytosine (ARA-C) deamination in sev- Applications, limitations and outlook. Meth Exp Clin Pharmacol
eral species. Biochem Pharmacol 22:2405–2417, 1972. 8:603–614, 1986.

Gerlowski LE, Jain RK: Physiologically based pharmacoki- Rowland M, Tozer T: Clinical Pharmacokinetics—Concepts and
netic modeling: Principles and applications. J Pharm Sci 72: Applications, 3rd ed. Philadelphia, Lea & Febiger, 1995.
1103–1127, 1983. Rowland M, Thomson P, Guichard A, Melmon K: Disposition

Gibaldi M: Biopharmaceutics and Clinical Pharmacokinetics, kinetics of lidocaine in normal subjects. Ann NY Acad Sci
3rd ed. Philadelphia, Lea & Febiger, 1984. 179:383–398, 1971.

 

26 Chapter 1

Segre G: Pharmacokinetics: Compartmental representation. Wagner JG: Do you need a pharmacokinetic model, and, if so,
Pharm Ther 17:111–127, 1982. which one? J Pharm Biopharm 3:457–478, 1975.

Tozer TN: Pharmacokinetic principles relevant to bioavailability Welling P, Tse F: Pharmacokinetics. New York, Marcel Dekker,
studies. In Blanchard J, Sawchuk RJ, Brodie BB (eds). Principles 1993.
and Perspectives in Drug Bioavailability. New York, S Karger, Winters ME: Basic Clinical Pharmacokinetics, 3rd ed. Vancouver,
1979, pp 120–155. WA, Applied Therapeutics, 1994.

 

Mathematical

2 Fundamentals in
Pharmacokinetics
Antoine Al-Achi

Chapter Objectives1 CALCULUS
»» Algebraically solve mathematical Pharmacokinetic models consider drugs in the body to be in a

expressions related to dynamic state. Calculus is an important mathematic tool for ana-
pharmacokinetics. lyzing drug movement quantitatively. Differential equations are

»» Express the calculated and used to relate the concentrations of drugs in various body organs
theoretical pharmacokinetic over time. Integrated equations are frequently used to model the
values in proper units. cumulative therapeutic or toxic responses of drugs in the body.

»» Represent pharmacokinetic
data graphically using Cartesian Differential Calculus

coordinates (rectangular Differential calculus is a branch of calculus that involves finding
coordinate system) and the rate at which a variable quantity is changing. For example, a
semilogarithmic graphs. specific amount of drug X is placed in a beaker of water to dis-

solve. The rate at which the drug dissolves is determined by the
»» Use the least squares method

rate of drug diffusing away from the surface of the solid drug and
to find the best fit straight line

is expressed by the Noyes–Whitney equation:
through empirically obtained
data.

dX DA
»» Define various models Dissolution rate = = (C − C

dt l 1 2 )

representing rates and order
of reactions and calculate

where d denotes a very small change; X = drug X; t = time; D =
pharmacokinetic parameters

diffusion coefficient; A = effective surface area of drug; l = length
(eg, zero- and first-order) from

of diffusion layer; C1 = surface concentration of drug in the diffu-
experimental data based on

sion layer; and C2 = concentration of drug in the bulk solution.
these models.

The derivative dX/dt may be interpreted as a change in X (or a
derivative of X) with respect to a change in t.

In pharmacokinetics, the amount or concentration of drug in
the body is a variable quantity (dependent variable), and time is
considered to be an independent variable. Thus, we consider the
amount or concentration of drug to vary with respect to time.

1It is not the objective of this chapter to provide a detailed description of mathematical functions, algebra, or statistics. Readers who are
interested in learning more about these topics are encouraged to consult textbooks specically addressing these subjects.

27

 

28 Chapter 2

y
EXAMPLE »» »

The concentration C of a drug changes as a func-
tion of time t:

dx
C = f (t ) (2.1)

Consider the following data:

Time Plasma Concentration
(hours) of Drug C (μg/mL)

0 12

1 10

2 8 a b x

3 6 FIGURE 2-1 Integration of y = ax or ∫ax·dx.

4 4

5 2 A definite integral of a mathematical function is
the sum of individual areas under the graph of that

The concentration of drug C in the plasma is function. There are several reasonably accurate
declining by 2 mg/mL for each hour of time. The numerical methods for approximating an area. These
rate of change in the concentration of the drug methods can be programmed into a computer for
with respect to time (ie, the derivative of C ) may rapid calculation. The trapezoidal rule is a numerical
be expressed as method frequently used in pharmacokinetics to cal-

culate the area under the plasma drug concentration-
dc

= 2 µg/mL/h versus-time curve, called the area under the curve
dt

(AUC). For example, Fig. 2-2 shows a curve depict-
Here, f(t) is a mathematical equation that describes ing the elimination of a drug from the plasma after a
how C changes, expressed as single intravenous injection. The drug plasma levels

and the corresponding time intervals plotted in
C =12 −2t (2.2) Fig. 2-2 are as follows:

Integral Calculus 40

Integration is the reverse of differentiation and is con-
sidered the summation of f (x) ⋅ dx; the integral sign ∫ 30

implies summation. For example, given the function
y = ax, plotted in Fig. 2-1, the integration is ∫ ax ⋅ dx. 20

Compare Fig. 2-1 to a second graph (Fig. 2-2), where
the function y = Ae–x is commonly observed after an 10
intravenous bolus drug injection. The integration pro-
cess is actually a summing up of the small individual 0
pieces under the graph. When x is specified and is 0 1 2 3 4 5

given boundaries from a to b, then the expression Time (hours)

becomes a definite integral, that is, the summing up FIGURE 2-2 Graph of the elimination of drug from the
of the area from x = a to x = b. plasma after a single IV injection.

Plasma drug level (mg/mL)

 

Mathematical Fundamentals in Pharmacokinetics 29

estimated by back extrapolation of the data points
Plasma Drug Level

Time (hours) (μg/mL) using a log linear plot (ie, log y vs x). The last plasma
level–time curve is extrapolated to t = ∞. In this case

0.5 38.9 the residual area t
[AUC] ∞

t is calculated as follows:
n

1.0 30.3

t Cpn
2.0 18.4 [AUC] ∞ = (2.4)

tn k
3.0 11.1

4.0 6.77 where Cpn = last observed plasma concentration at tn
and k = slope obtained from the terminal portion of

5.0 4.10
the curve.

The trapezoidal rule written in its full form to
The area between time intervals is the area of a calculate the AUC from t = 0 to t = ∞ is as follows:

trapezoid and can be calculated with the following
formula: ∞ t Cpn

[AUC] = Σ[AUC] n +
0 tn−1 k

t C
[ ] n 1 + C

AUC n − n
= (t − t

t 2 n n 1) (2.3)

n−1 This numerical method of obtaining the AUC is
fairly accurate if sufficient data points are available.

where [AUC] = area under the curve, tn = time of As the number of data points increases, the trapezoi-
observation of drug concentration Cn, and tn–1 = time dal method of approximating the area becomes more
of prior observation of drug concentration corre- accurate.
sponding to Cn–1. The trapezoidal rule assumes a linear or straight-

To obtain the AUC from 1 to 4 hours in Fig. 2-2, line function between data points. If the data points
each portion of this area must be summed. The AUC are spaced widely, then the normal curvature of the
between 1 and 2 hours is calculated by proper substi- line will cause a greater error in the area estimate.
tution into Equation 2.3:

t 30.3+18.4 Frequently Asked Questions
[AUC] 2 = (2 −1) = 24.35 µg ⋅h/mL

t1 2 »»What are the units for logarithms?

»»What is the difference between a common log and a
Similarly, the AUC between 2 and 3 hours is calcu- natural log (ln)?
lated as 14.75 mg·h/mL, and the AUC between 3 and
4 hours is calculated as 8.94 mg·h/mL. The total
AUC between 1 and 4 hours is obtained by adding
the three smaller AUC values together. GRAPHS

The construction of a curve or straight line by plot-
t4 t t t

[AUC] = [AUC] 2 + [AUC] 3 + AUC
t t t [ ] 4 ting observed or experimental data on a graph is an
1 1 2 t3

important method of visualizing relationships
= 24.3+14.3+ 8.94

between variables. By general custom, the values of
= 48.04 µg ⋅h/mL the independent variable (x) are placed on the hori-

zontal line in a plane, or on the abscissa (x axis),
The total area under the plasma drug level–time whereas the values of the dependent variable are
curve from time zero to infinity (Fig. 2-2) is obtained placed on the vertical line in the plane, or on the
by summation of each individual area between each ordinate (y axis). The values are usually arranged so
pair of consecutive data points using the trapezoidal that they increase linearly or logarithmically from
rule. The value on the y axis when time equals 0 is left to right and from bottom to top.

 

30 Chapter 2

6 100

5
50

4

3

2 10

1
5

0
0 1 2 3 4 5 6 7 8

FIGURE 2-3 Rectangular coordinates. 1
0 1 2 3 4 5 6 7

FIGURE 2-4 Semilog coordinates.
In pharmacokinetics, time is the independent

variable and is plotted on the abscissa (x axis),
whereas drug concentration is the dependent variable observed data. If the relationship between x and y is
and is plotted on the ordinate (y axis). Two types of linearly related, then the relationship between the
graphs or graph papers are usually used in pharma- two can be expressed as a straight line.
cokinetics. These are Cartesian or rectangular coor- Physiologic variables are not always linearly
dinate (Fig. 2-3) and semilogarithmic graph or graph related. However, the data may be arranged or trans-
paper (Fig. 2-4). Semilogarithmic allows placement formed to express the relationship between the vari-
of the data at logarithmic intervals so that the num- ables as a straight line. Straight lines are very useful
bers need not be converted to their corresponding log for accurately predicting values for which there are
values prior to plotting on the graph. no experimental observations. The general equation

of a straight line is
Curve Fitting

Fitting a curve to the points on a graph implies that y = mx + b (2.5)
there is some sort of relationship between the vari-
ables x and y, such as dose of drug versus pharmaco- where m = slope and b = y intercept. Equation 2.5
logic effect (eg, lowering of blood pressure). could yield any one of the graphs shown in Fig. 2-5,
Moreover, when using curve fitting, the relationship depending on the value of m. The absolute magnitude
is not confined to isolated points but is a continuous of m gives some idea of the steepness of the curve.
function of x and y. In many cases, a hypothesis is For example, as the value of m approaches 0, the line
made concerning the relationship between the vari- becomes more horizontal. As the absolute value of m
ables x and y. Then, an empirical equation is formed becomes larger, the line slopes farther upward or
that best describes the hypothesis. This empirical downward, depending on whether m is positive or
equation must satisfactorily fit the experimental or negative, respectively.

y y y

m = 0 m < 0

b

b

b m > 0

x x x

FIGURE 2-5 Graphic demonstration of variations in slope (m).

 

Mathematical Fundamentals in Pharmacokinetics 31

Linear Regression/Least Squares Method between the variables. If a linear line deviates sub-

This method is often encountered and used in clinical stantially from the data, it may suggest the need for a

pharmacy studies to construct a linear relationship nonlinear regression model, although several vari-

between an independent variable (also known as the ables (multiple linear regression) may be involved.

input factor or the x factor) and a dependent variable Nonlinear regression models are complex mathemati-

(commonly known as an output variable, an outcome, cal procedures that are best performed with a com-

or the y factor). In pharmacokinetics, the relationship puter program.

between the plasma drug concentrations versus time
can be expressed as a linear function. Because of the
availability of computing devices (computer pro-

Frequently Asked Questions
grams, scientific calculators, etc), the development of

»»How is the area under the curve, AUC, calculated?
a linear equation has indeed become a simple task.

What are the units for AUC?
A general format for a linear relationship is often
expressed as: »»How do you know that the line that you fit to pro-

duce a curve on a graph is the line of best fit?
y = mx + b (2.6)

»»What assumptions are made when a line is fitted to
the points on a graph?

where y is the dependent variable, x is the indepen-
dent variable, m is the slope, and b is the y intercept.
The value of the slope and the y intercept may be
positive, negative, or zero. A positive linear relation-
ship has a positive slope, and a negative slope PRACTICE PROBLEM
belongs to a negative linear relationship (Gaddis and
Gaddis, 1990; Munro, 2005). Plot the following data and obtain the equation for

The strength of the linear relationship is the line that best fits the data by (a) using a ruler and
assessed by the correlation coefficient (r). The value (b) using the method of least squares. Data can be
of r is positive when the slope is positive and it is plotted manually or by using a computer spreadsheet
negative when the slope is negative. When r takes program such as Microsoft Excel.
the value of either +1 or −1, a perfect relationship
exists between the variables. A zero value for the

x (mg) y (hours) x (mg) y (hours)
slope (or for r) indicates that there is no linear rela-
tionship existing between y and x. In addition to r, 1 3.1 5 15.3

the coefficient of determination (r2) is often com- 2 6.0 6 17.9
puted to express how much variability in the out-
come is explained by the input factor. For example, 3 8.7 7 22.0

if r is 0.90, then r2 equals to 0.81. This means that 4 12.9 8 23.0

the input variable explains 81% of the variability
observed in y. It should be noted, however, that a
high correlation between the two variables does not Solution

necessarily mean causation. For example, the pas- Many computer programs have a regression analy-
sage of time is not really the cause for the drug sis, which fits data to a straight line by least squares.
concentration in the plasma to decrease. Rather it is In the least squares method, the slope m and the y
the distribution and the elimination functions that intercept b (Equation 2.7) are calculated so that the
cause the level of the drug to decrease over time average sum of the deviations squared is minimized.
(Gaddis and Gaddis, 1990; Munro, 2005). The deviation, d, is defined by

The linear regression/least squares method
assumes, for simplicity, that there is a linear relationship b + mx − y = d (2.7)

 

32 Chapter 2

If there are no deviations from linearity, then d = 0 Problems of Fitting Points to a Graph
and the exact form of Equation 2.7 is as follows: When x and y data points are plotted on a graph, a

relationship between the x and y variables is sought.
b + mx − y = 0

Linear relationships are useful for predicting values
for the dependent variable y, given values for the

To find the slope, m, and the intercept, b, the follow-
independent variable x.

ing equations are used:
The linear regression calculation using the least

squares method is used for calculating a straight line
Σ(x)Σ(y) − nΣ(xy)

m = (2.8) through a given set of points. However, it is impor-
2

[Σ(x)] − nΣ(x2 ) tant to realize that, when using this method, one has
already assumed that the data points are related lin-

where n = number of data points. early. Indeed, for three points, this linear relationship
may not always be true. As shown in Fig. 2-6, Riggs

Σ(x)Σ(xy) (1963) calculated three different curves that fit the
− Σ(x2 )Σy

b = (2.9)
2 data accurately. Generally, one should consider the

[Σ(x)] − nΣ(x2 ) law of parsimony, which broadly means “keep it
simple”; that is, if a choice between two hypotheses

where Σ is the sum of n data points. is available, choose the more simple relationship.
The following graph was obtained by using a If a linear relationship exists between the x and

Microsoft Excel spreadsheet and calculating a y variables, one must be careful as to the estimated
regression line (sometimes referred to as a trendline value for the dependent variable y, assuming a value
in the computer program): for the independent variable x. Interpolation, which

30 means filling the gap between the observed data on
y = 2.9679x + 0.2571

25
R2

= 0.99231 15

20

15

10

10 A
5

0 y
0 2 4 6 8 10

Therefore, the linear equation that best fits the 5

data is

y = 2.97x + 0.257 B
C

0

Although an equation for a straight line is obtained 0 2 4 6 8

by the least squares procedure, the reliability of the x

values should be ascertained. A correlation coeffi- FIGURE 2-6 Three points equally well fitted by different
cient, r, is a useful statistical term that indicates the curves. The parabola, y = 10.5 – 5.25x + 0.75×2 (curve A); the

relationship of the x, y data fit to a straight line. For exponential, y = 12.93e–1.005x + 1.27 (curve B); and the rectangular

a perfect linear relationship between x and y, r = +1. hyperbola, y = 6/x (curve C) all fit the three points (1,6), (2,3), and
(4,1.5) perfectly, as would an infinite number of other curves.

Usually, r ≥ 0.95 demonstrates good evidence or a
(Reprinted with permission from Riggs DS: The Mathematical

strong correlation that there is a linear relationship Approach to Physiological Problems. Baltimore, Williams &
between x and y. Wilkins, 1963.)

 

Mathematical Fundamentals in Pharmacokinetics 33

a graph, is usually safe and assumes that the trend An important rule in using equations with dif-
between the observed data points is consistent and ferent units is that the units may be added or sub-
predictable. In contrast, the process of extrapolation tracted as long as they are alike, but divided or
means predicting new data beyond the observed multiplied if they are different. When in doubt,
data, and assumes that the same trend obtained check the equation by inserting the proper units.
between two data points will extend in either direc- For example,
tion beyond the last observed data points. The use of
extrapolation may be erroneous if the regression

FD
line no longer follows the same trend beyond the AUC 0

= = concentration × time
kV

measured points. D

Graphs should always have the axes (abscissa (2.11)
µg 1mg µg ⋅h

and ordinate) properly labeled with units. For h = =
mL h−1 L mL

example, the amount of drug on the ordinate (y axis)
is given in milligrams and the time on the abscissa
(x axis) is given in hours. The equation that best fits Certain terms have no units. These terms include
the points on this curve is the equation for a straight logarithms and ratios. Percent may have no units
line, or y = mx + b. Because the slope m = ∆y/∆x, the and is expressed mathematically as a decimal
units for the slope should be milligrams per hour between 0 and 1 or as 0% to 100%, respectively.
(mg/h). Similarly, the units for the y intercept b On occasion, percent may indicate mass/volume,
should be the same units as those for y, namely, mil- volume/volume, or mass/mass. Table 2-1 lists com-
ligrams (mg). mon pharmacokinetic parameters with their sym-

bols and units.
A constant is often inserted in an equation to

MATHEMATICAL EXPRESSIONS quantify the relationship of the dependent variable
AND UNITS to the independent variable. For example, Fick’s

Mathematics is a basic science that helps to explain law of diffusion relates the rate of drug diffusion,

relationships among variables. For an equation to be dQ/dt, to the change in drug concentration, C, the

valid, the units or dimensions must be constant on surface area of the membrane, A, and the thick-

both sides of the equation. Many different units are ness of the membrane, h. In order to make this

used in pharmacokinetics, as listed in Table 2-1. For relationship an equation, a diffusion constant D is

an accurate equation, both the integers and the units inserted:

must balance. For example, a common expression
for total body clearance is dQ DA

= × ∆C (2.12)
dt h

Cl = kV 2 1 )
T d ( . 0

To obtain the proper units for D, the units for each of

After insertion of the proper units for each term in the other terms must be inserted:

the above equation from Table 2-1,
mg D(cm2 ) mg

= ×
h cm cm3

mL 1
= mL

h h D = cm2 /h

Thus, the above equation is valid, as shown by the The diffusion constant D must have the units of area/
equality mL/h = mL/h. time or cm2/h if the rate of diffusion is in mg/h.

 

34 Chapter 2

TABLE 2-1 Common Units Used in Pharmacokinetics

Parameter Symbol Unit Example

Rate dD Mass mg/h
dt Time

dC Concentration ug/mL/h
dt Time

Zero-order rate constant K Concentration
0 mg/mL/h

Time

Mass mg/h
Time

First-order rate constant k 1 1/h or h–1

Time

Drug dose D0 Mass mg

Concentration C Mass mg/mL
Volume

Plasma drug concentration Cp Drug mg/mL
Volume

Volume V Volume mL or L

Area under the curve AUC Constration × time mg·h/mL

Fraction of drug absorbed F No units 0 to 1

Clearance Cl Volume mL/h
Time

Half-life t1/2 Time H

UNITS FOR EXPRESSING BLOOD MEASUREMENT AND USE OF
CONCENTRATIONS SIGNIFICANT FIGURES
Various units have been used in pharmacology, toxi- Every measurement is performed within a certain
cology, and the clinical laboratory to express drug degree of accuracy, which is limited by the instru-
concentrations in blood, plasma, or serum. Drug con- ment used for the measurement. For example, the
centrations or drug levels should be expressed as weight of freight on a truck may be measured accu-
mass/volume. The expressions mcg/mL, mg/mL, and rately to the nearest 0.5 kg, whereas the mass of drug
mg/L are equivalent and are commonly reported in the in a tablet may be measured to 0.001 g (1 mg).
literature. Drug concentrations may also be reported Measuring the weight of freight on a truck to the
as mg% or mg/dL, both of which indicate milligrams nearest milligram is not necessary and would require
of drug per 100 mL (1 deciliter). Two older expres- a very costly balance or scale to detect a change in a
sions for drug concentration occasionally used in milligram quantity.
veterinary medicine are the terms ppm and ppb, which Significant figures are the number of accurate
indicate the number of parts of drug per million parts digits in a measurement. If a balance measures the
of blood (ppm) or per billion parts of blood (ppb), mass of a drug to the nearest milligram, measure-
respectively. One ppm is equivalent to 1.0 mg/mL. The ments containing digits representing less than 1 mg
accurate interconversion of units is often necessary to are inaccurate. For example, in reading the weight or
prevent confusion and misinterpretation. mass of a drug of 123.8 mg from this balance, the

 

Mathematical Fundamentals in Pharmacokinetics 35

0.8 mg is only approximate; the number is therefore A pharmacist is interested in learning the time
rounded to 124 mg and reported as the observed mass. needed for 90% of ASA to be released from the

For practical calculation purposes, all figures tablet. To answer her inquiry the following steps are
may be used until the final number (answer) is taken:
obtained. However, the answer should retain only the

1. Calculate the amount of ASA in milligrams
number of significant figures in the least accurate

representing 90% of the drug present in the
initial measurement.

tablet.
2. Replace the value found in step (1) in

PRACTICE PROBLEM Equation 2.14 and solve for time (t):
90% of 325 = (0.9)(325 mg) = 292.5 mg

When a patient swallows a tablet containing 325 mg 292.5 mg = 0.86t − 0.04
of aspirin (ASA), the tablet comes in contact with the 292.5 + 0.04 = 0.86t
contents of the gastrointestinal tract and the ASA is Dividing both sides of the equation by 0.86:
released from the tablet. Assuming a constant amount (292.5 + 0.04)/0.86 = (0.86t)/0.86
of the drug release over time (t), the rate of drug release 340.07 minutes = t
is expressed as: Or it takes 5.7 hours for this amount of ASA

(90%) to be released from the tablet.
d(ASA)

Rate of drug (ASA) release (mg/min) =
dt The above calculations show that this tablet

= k0 releases the drug very slowly over time and it may
not be useful in practice when the need for the drug

where k0 is a rate constant. is more immediate. It should also be emphasized that
Integration of the rate expression above gives only the amount of the drug released and soluble in

Equation 2.13: the GI juices is available for absorption. If the drug
precipitates out in the GI tract, it will not be absorbed

Amount of ASA released (mg) = at + b (2.13) by the GI mucosa. It is also assumed that the unab-
sorbed portion of the drug in the GI tract is consid-
ered to be “outside the body” because its effect

The symbol “a” represents the slope (equivalent to k0), cannot be exerted systematically.
t is time, and b is the y intercept. Assuming that time

To calculate the amount of ASA that was imme-
was measured in minutes, the following mathematical

diately released from the tablet upon contact with
expression is obtained representing Equation 2.13:

gastric juices, the time in Equation 2.14 is set to the
value zero:

Amount of ASA released (mg) = 0.86t − 0.04
(2.14)

Amount of ASA released (mg) = 0.86(0) − 0.04
To calculate the amount of ASA released at

180 seconds, the following algebraic manipulations Amount of ASA released (mg) = −0.04 mg
are needed:

1. Convert 180 seconds to minutes: 3 minutes.
2. Replace t in Equation 2.14 by the value 3. Since an amount released cannot be negative, this
3. Solve the equation for the amount of ASA indicates that no amount of ASA is released from

released. the tablet instantly upon coming in touch with the
juices. Equation 2.14 may be represented graphi-

Amount of ASA released (mg) = 0.86(3) − 0.04 cally using Cartesian or rectangular coordinates
= 2.54 mg (Fig. 2-7).

 

36 Chapter 2

60 the area under the moment curve, whereas MRT is
the mean residence time, which is estimated from the

50
ratio of AUMC(0–infinity)/AUC(0–infinity). These

40 pharmacokinetic terms are discussed in more details
throughout this textbook.

30 The below table (Ravi Shankar et al., 2012)
shows pharmacokinetic data obtained from a study

20
conducted in rabbits following administration of

10 various formulations of rectal suppositories contain-
ing aspirin (600 mg each). Various formulations

0
0 10 20 30 40 50 60 70 were prepared in a suppository base made of a mix-

Time (minutes) ture of gelatin and glycerin. Formulation Fas9 had
the same composition as Fs9 with the exception that

FIGURE 2-7 Amount of ASA released versus time (minutes) Fas9 contained ASA in the form of nanoparticles,
plotted on Cartesian coordinates.

whereas Fs9 had ASA in its free form (so did formu-
lations Fs2, Fs4, and Fs11, but varied in their gelatin/
glycerin composition). The authors concluded that

PRACTICE PROBLEM the incorporation of ASA in the form of nanoparti-
cles increased the Tmax. The other pharmacokinetic

Briefly, Cmax is the maximum drug concentration in parameters taken together indicate that nanoparticles
the plasma and Tmax is the time associated with Cmax. produced a sustained-release profile of ASA when
First-order elimination rate constant signifies the given in this dosage form. In this study, the plasma
fraction of the drug that is eliminated per unit time. concentration was expressed in “micrograms per
The biological half-life of the drug is the time needed milliliter.” If the mg/mL were not specified, it would
for 50% of the drug to be eliminated. The AUC term have been difficult to compare the results from this
or the area under the drug plasma concentration- study with other similar studies. It is imperative,
versus-time curve reflects the extent of absorption therefore, that pharmacokinetic parameters such as
from the site of administration. The term AUMC is Cmax be properly defined by units.

Pharmacokinetic
Parameters Fs2 Fs4 Fs9 Fs11 Fas9

Cmax (mg/mL) 34.93 ± 0.60 31.16 ± 1.04 32.66 ± 1.52 35.33 ± 0.57 31.86 ± 0.41

Tmax (hours) 1 ± 0.01 1 ± 0.03 1 ± 0.06 1 ± 0.09 6 ± 0.03

Elimination rate constant (h–1) 0.14 ± 0.02 0.19 ± 0.06 0.205 ± 0.03 0.17 ± 0.01 0.133 ± 0.004

Half-life (hours) 1.88 ± 0.76 1.9 ± 1.19 1.43 ± 0.56 1.99 ± 0.24 5.11 ± 0.15

AUC(0–t) 127.46 ± 8.9 126.62 ± 2.49 132.11 ± 3.88 127.08 ± 1.95 260.62 ± 4.44

AUC(0–infinity)(ng·h/mL) 138.36 ± 13.87 131.61 ± 0.27 136.89 ± 4.40 133.07 ± 2.97 300.48 ± 24.06

AUMC(0–t)(ng·h2/mL) 524.51 ± 69.64 516.04 ± 28.25 557.84 ± 16.25 501.29 ± 26.65 2006.07 ± 38.00

AUMC(0–infinity)(ng·h2/mL) 382.09 ± 131.45 237.74 ± 64.37 232.93 ± 28.16 257.71 ± 30.04 1494.71 ± 88.21

MRT (hours) 2.45 ± 0.36 2.31 ± 0.80 1.41 ± 0.31 2.95 ± 0.17 8.23 ± 0.06

Ravi Sankar V, Dachinamoorthi D, Chandra Shekar KB: A comparative pharmacokinetic study of aspirin suppositories and aspirin nanoparticles loaded
suppositories. Clinic Pharmacol Biopharm 1:105, 2012.

Amount of ASA released (mg)

 

Mathematical Fundamentals in Pharmacokinetics 37

Expressing the Cmax value by equivalent units units is ([amount][time]/[volume]). Together, the rate
is also possible. For example, converting mg/mL to and extent of absorption refers to the bioavailability
mg/dL follows these steps: of the drug from the site of administration. The term

“absolute bioavailability” is used when the reference
1. Convert micrograms (also written as mcg) to

route of administration is the intravenous injection
milligrams.

(ie, the IV route). If the reference route is different
2. Convert milliliters to deciliters:

from the intravenous route, then the term “relative
Since:

bioavailability” is used. The value for the AUC (0 to
1 mg = 1000 mg, then 31.86 mg/mL = 0.03186

+∞) following the administration of Fs2, Fs4, and
mg/mL

Fas9 was 138.36, 131.61, and 300.48 ng·h/mL,
1 dL = 100 mL, then 31.86 mg/mL = 3186

respectively (Ravi Sankar et al, 2012). The origin of
mg/dL

the AUC units is based on the trapezoidal rule. The
We have to divide the value of Cmax by 1000

trapezoidal rule is a numerical method frequently
and multiply it by 100. The net effect is to divide

used in pharmacokinetics to calculate the area under
the number by 10, or (31.86)(100/1000) =

the plasma drug concentration-versus-time curve,
3.19 mg/dL.

called the area under the curve (AUC). This rule
Expressing the Cmax value 34.93 mg/mL in nano- computes the average concentration value of each
grams per microliter (ng/mL) is done as follows: consecutive concentration and multiplies them by the

difference in their time values. To compute the AUC
1. Convert the number of micrograms to nano-

(0 to time t), the sum of all these products is calcu-
grams.

lated. For example, AUC(0–t) = 127.46 ng·h/mL can
2. Convert milliliters to microliters:

be written as 127.46 (ng/mL)(h).
1 mg = 1000 ng, or 34.93 mg/mL = 34,930 To convert 260.62 ng·h/mL to mg·-h/mL, divide
ng/mL the value by 1000 (recall that 1 mg is 1000 ng).
1 mL = 1000 mL, or 34.93 mg/mL = Therefore, the AUC value becomes 0.26 mg·h/mL.
0.03493 mg/mL Expressing the AUC (0 to +∞) value 300.48
As 34.93 was multiplied and divided by the ng·h/mL in ng·min/mL can be accomplished by
same number (1000), the final answer is dividing 300.48 by 60 (1 hour is 60 min). Thus, the
34.93 ng/mL. AUC value becomes 5.0 ng·min/mL.

Consider the following data:
Express the Cmax value 35.33 mg/mL in %w/v (this is
defined as the number of grams of ASA in 100 mL
plasma). Plasma Time AUC

Concentration (ng/L) (hours) (ng·h/L)

(35.33 mg/mL)(100 mL) = 3533 mg/dL = 0 0 0

3.533 mg/dL = 0.0035 g/dL, or 0.0035% w/v 0.05 1 0.025

(This means that there is 0.0035 g of ASA
0.10 2 0.075

in every 100 mL plasma.)
0.18 3 0.140

The data (Tmax, Cmax) represent a maximum point on 0.36 5 0.540

the plasma drug level-versus-time curve. This point 0.13 7 0.490
reflects the rate of absorption of the drug from its
site of administration. Another pharmacokinetic 0.08 9 0.210

measure obtained from the same curve is the area
under the curve (AUC). It reflects the extent of To compute the AUC value from initial to 9
absorption for a drug from the site of administration hours, sum up the values under the AUC column
into the circulation. The general format for the AUC above (0.025 + 0.075 + … + 0.210 = 1.48 ng·h/L).

 

38 Chapter 2

To convert the AUC value 1.48 ng·h/L to 0.4

mg·min/dL, use the following steps: 0.35

1. Divide the value by 106 to convert the nano- 0.3
grams to milligrams.

2. Divide the value by 60 to convert the hours to 0.25

minutes. 0.2
3. Divide the value by 10 to convert the liters to

0.15
deciliters.

0.1

AUC = (1.48)/[(106 )(60)(10)] 0.05
5 6 7 8 9

–9
= 2.47 ×10 mg·min/dL (2.15) Time (hours)

FIGURE 2-9 The exponential decline in plasma concen-
Figure 2-8 represents the data in a rectangular tration over time portion in Fig. 2-8.

coordinate–type graph. Time is placed on the x axis
(the abscissa) and plasma concentration is placed on bases or weak acids, the pH of the biological fluid
the y axis (the ordinate). The highest point on the determines the degree of ionization of the drug and
graph can simply be determined by spotting it on the this in turns influences the pharmacokinetic profile of
graph. Note that the plasma concentration declines the drug. The pH scale is a logarithmic scale:
exponentially from the apex point on the curve over
time. Figure 2-9 shows the exponential portion of the

pH = −log[H O+ ]= log(1/[H +

3 3O ]) (2.16)
graph on its own.

Exponential and Logarithmic Functions where the symbol “log” is the logarithm to base 10.
The natural logarithm has the symbol “ln,” which is

These two mathematical functions are related to each
the logarithm to base e (the value of e is approxi-

other. For example, the pH of biological fluids (eg,
mately 2.71828). The two functions are linked by the

plasma or urine) can influence all pharmacokinetic
following expression:

aspects including drug dissolution/release in vitro
as well as systemic absorption, distribution, metabo-
lism, and excretion. Since most drugs are either weak ln x = 2.303 log x (2.17)

The concentration of hydronium ions [H3O+]
0.4

can be calculated from Equation 2.16 as follows:
0.35

0.3
[H3O

+ ] 10−pH
= (2.18)

0.25

0.2
For example, the pH of a patient’s plasma is 7.4 at

0.15
room temperature. Therefore, the hydronium ion

0.1
concentration in plasma is:

0.05

0
[H3O

+ ] = 10−7.4
= 3.98 10−8

× M
–0.05

0 2 4 6 8 10
Time (hours) The value (3.98 × 10–8) is the antilogarithm of 7.4.

FIGURE 2-8 Plasma concentration (g/L)-versus-time With the availability of scientific calculators and
(hours) curve plotted on Cartesian coordinates. computers, these functions can be easily calculated.

Plasma concentration (g/L)

Plasma concentration (g/L)

 

Mathematical Fundamentals in Pharmacokinetics 39

–1 The slope of the line is (−0.38). Thus,

Slope = −0.38 = −k
–1.5 1

Multiplying both sides of the equation by (−1)
–2 results in:

k1 = 0.38 h–1
–2.5

5 6 7 8 9 where k1 is the first-order elimination rate constant.
Time (hours) The units for this constant are reciprocal time, such

as h–1 or 1/h. The value 0.38 h–1 means that 38% of
FIGURE 2-10 ln (Plasma concentration)-versus-time the concentration remaining of the drug in plasma is
curve plotted on Cartesian coordinates.

eliminated every hour.
Using Equation 2.17, Equation 2.19 can be con-

Oftentimes, converting plasma concentrations verted to the following expression:
to logarithmic values and plotting the logarithmic
values against time would convert an exponential
relationship to a linear function between the two 2.303 [log (Plasma concentration)] = 0.77 − 0.38

variables. Consider, for example, Fig. 2-9. When the Time (hours)

concentration values are converted to logarithmic
values, the graph now becomes linear (Fig. 2-10).

Dividing both sides of the equation by 2.303:
This same linear function may be obtained by plotting
the actual values of the plasma concentration versus
time using a semilogarithmic graph (Fig. 2-11). The 2.303 [log (Plasma concentration)]/2.303 =
following equation represents the straight line: [0.77 − 0.38 Time (hours)]/2.303

log (Plasma concentration) = 0.334 − 0.17
ln (Plasma concentration) = 0.77 − 0.38 Time (hours) Time (hours)

(2.19) (2.20)

0.4 Equation 2.20 is mathematically equivalent to
Equation 2.19.

0.3 The value 0.77 in Equation 2.19 equals (ln C0),
where C0 is the initial concentration of the drug in

0.2 plasma. Thus,

ln C0 = 0.77

0.1 C0 = e0.77 = 2.16 g/L
0.09
0.08
0.07

5 6 7 8 9 Once k1 is known, the AUC from the last data point
Time (hours) to t–infinity can be calculated as follows:

FIGURE 2-11 Plasma concentration-versus-time curve
using a semilogarithmic graph. AUC = CLast/k1 (2.21)

Plasma concentration (g/L) In (plasma concentration)

 

40 Chapter 2

Applying Equation 2.21 on the data used to obtain the may be defined in terms of specifying its order. In
AUC value in Equation 2.15 results in the following pharmacokinetics, two orders are of importance, the
value: zero order and the first order.

AUC = 0.08/0.38 = 0.21 g·h/L
Zero-Order Process

And the total AUC (t = 0 to t = infinity): The rate of a zero-order process is one that proceeds
over time (t) independent from the concentration of
the drug (c). The negative sign for the rate indicates

AUCTotal = 1.48 + 0.21 = 1.69 g·h/L
that the concentration of the drug decreases over time.

The following rules may be useful in handling
exponential and logarithmic functions. For this, if m −dc/dt = k0 (2.22)
and n are positive, then for the real numbers q and s

dc = −k
(Howard, 1980): 0 dt

c = c0 − k0t (2.23)
Exponent rules:

1. m0 = 1 where c0 is the initial concentration of the drug at
2. m1 = m t = 0 and k0 is the zero-order rate constant. The units
3. m–1 = 1/m1

for k0 are concentration per unit time (eg, [mg/mL]/h)
4. mq/ms = mq–s

or amount per unit time (eg, mg/h).
5. (mq)(ms) = mq+s

For example, calculate the zero-order rate con-
6. (mq)s = mqs

stant ([ng/mL]/min) if the initial concentration of
7. (mq/nq) = (m/n)q

the drug is 200 ng/mL and that at t = 30 minutes is
8. (mq)(nq) = (mn)q

35 ng/mL.
If z is any positive number other than 1 and if

zy = x, then following logarithmic rules apply:
c = c0 − k0 t

Logarithm rules:
35 = 200 − k0 (30)

1. y = logz x (y is the logarithm to the base z of x)
−k0 = (35 − 200)/30 = −5.5

2. For x > 1, then loge x = ln x (where e is approxi-
mately 2.71828) k0 = 5.5 (ng/mL)/min

3. logz x = (ln x/ln z)
4. logz mn = logz m + logz n
5. logq (m/n) = logq m − logq n

When does the concentration of drug equal to 100

6. logz (1/m) ng/mL?
= −logz m

7. ln e = 1
8. For z = 10, then logz 1 = 0 100 = 200 − 5.5 t
9. logz mh = h logz m

10. For z = 10, then (2.303) logz x = ln x (100 − 200)/5.5 = −t

−18.2 = −t

t = 18.2 min
RATES AND ORDERS OF PROCESSES
Oftentimes a process such as drug absorption or drug In pharmacokinetics, the time required for one-
elimination may be described by the rate by which half of the drug concentration to disappear is known
the process proceeds. The rate of a process, in turn, as t½. Thus, for this drug the t½ is 18.2 minutes.

 

Mathematical Fundamentals in Pharmacokinetics 41

In general, t½ may be calculated as follows for a ln 0.5/−k1 = t½
zero-order process:

t½ = −0.693/−k1

c = c0 − k0t t½ = 0.693/k1 (2.27)
(0.5 c0 ) = c0 − k0t12

(0.5 c0 ) − c0 = −k0t1 Unlike a zero-order rate process, the t1/2 for a first-
2

order rate process is always a constant, independent
−0.5 c0 = −k0t12 of the initial drug concentration or amount (Table 2-2,

Fig. 2-12).
t1 = (0.5 c0 )/k0 (2.24)

2 A plot between ln c versus t produces a straight
line. A semilogarithmic graph also produces a

Applying Equation 2.24 to the example above straight line between c and t. The units of the first-
should yield the same result: order rate constant (k1) are in reciprocal time.

t½ = (0.5 c0)/k0 TABLE 2-2 Comparison of Zero- and First-
t½ = (0.5)(200)/5.5 = 18.2 minutes Order Reactions

Zero-Order First-Order

A plot of x versus time on rectangular coordinates Reaction Reaction

produces a straight line with a slope equal to (−k0) Equation –dC/dt = k0 –dC/dt = kC
and a y intercept as c0. In a zero-order process the t1/2 C = –k0t + C0 C = C –

0 e kt
is not constant and depends upon the initial amount
or concentration of drug. Rate constant— (mg/L)/h 1/h

units

First-Order Process Half-life, t1/2 t1/2 = 0.5C/k0 t1/2 = 0.693/k
(units = time) (not constant) (constant)

The rate of a first-order process is dependent upon
the concentration of the drug: Effect of time Zero-order rate First-order rate

on rate is constant with will change with
respect to time respect to time

−dc/dt = k c
1 as concentration

(2.25)
changes

−dc/c = k dt
1

Effect of time on Rate constant Rate con-
rate constant with respect to stant remains

lnc = lnc t (2.26) time changes as constant with
0 − k1

the concentra- respect to time
tion changes

While the rate of the process is a function of the drug
Drug concen- Drug concentra- Drug concentra-

concentration, the t½ is not: trations versus tions decline tions decline
time—plotted linearly for a nonlinearly for
on rectangular zero-order rate a first-order rate

ln c = ln c0 − k1t coordinates process process

ln (0.5 c ug concen- Drug concentra- Drug concentra-
0) = ln c0 − k Dr

1t½
trations versus tions decline tions decline

ln (0.5 c linearly for a
0) − ln c0 = −k time—plotted nonlinearly for a

1t½
on a semiloga- zero-order rate single first-order

ln (0.5 c rithmic graph process rate process
0 /c0) = −k1t½

 

42 Chapter 2

ln 0.3/−k1 = t
100

t = −1.2/−0.04 = 30 hours
50

The value 30 hours may be written as t30 = 30 hours
(it is t30 because 70% of the drug is eliminated).

t1/2

10 Determination of Order

Graphical representation of experimental data pro-
5 vides a visual relationship between the x values

t1/2 (generally time) and the y axis (generally drug
concentrations). Much can be learned by inspecting
the line that connects the data points on a graph.
The relationship between the x and y data will

1
0 4 8 12 14 20 24 determine the order of the process, data quality,

Time (hours) basic kinetics, and number of outliers, and provide
FIGURE 2-12 The t1/2 in a first-order rate process is a the basis for an underlying pharmacokinetic model.
constant. To determine the order of reaction, first plot the

data on a rectangular graph. If the data appear to be a
For a drug with k1 = 0.04 h–1, find t½. curve rather than a straight line, the reaction rate

for the data is non-zero order. In this case, plot the
data on a semilog graph. If the data now appear to

t½ = 0.693/k1 form a straight line with good correlation using
linear regression, then the data likely follow first-

t½ = 0.693/0.04 = 17.3 hours
order kinetics. This simple graph interpretation is
true for one-compartment, IV bolus (Chapter 4).

The value 0.04 h–1 for the first-order rate constant Curves that deviate from this format are discussed
indicates that 4% of the drug disappears every in other chapters in terms of route of administration
hour. and pharmacokinetic model.

Calculate the time needed for 70% of the drug
to disappear.

Frequently Asked Questions

»»How is the rate and order of reaction determined
ln c = ln c0 − k1t graphically?

ln (0.3 c0) = ln c0 − k1t »»What is the difference between a rate and a rate
constant?

ln (0.3 c0) − ln c0 = −k1t

CHAPTER SUMMARY
Pharmacokinetic calculations require basic skills in line by plotting observed or experimental data on a
mathematics. Although the availability of computer graph is an important method of visualizing relation-
programs and scientific calculators facilitate pharma- ships between variables. The linear regression calcu-
cokinetic calculations, the pharmaceutical scientist lation using the least squares method is used for
should be familiar with fundamental rules pertaining calculation of a straight line through a given set of
to calculus. The construction of a curve or straight points. However, it is important to realize that, when

Drug concentration (mg/mL)

 

Mathematical Fundamentals in Pharmacokinetics 43

using this method, one has already assumed that the order. In pharmacokinetics, two orders are of impor-
data points are related linearly. For all equations, both tance, the zero order and the first order. Mathematical
the integers and the units must balance. The rate of a skills are important in pharmacokinetics in particular
process may be defined in terms of specifying its and in pharmacy in general.

LEARNING QUESTIONS
1. Plot the following data on both semilog graph 3. A pharmacist dissolved a few milligrams of

paper and standard rectangular coordinates. a new antibiotic drug into exactly 100 mL of
distilled water and placed the solution in a

Time (minutes) Drug A (mg) refrigerator (5°C). At various time intervals,
the pharmacist removed a 10-mL aliquot from

10 96.0
the solution and measured the amount of drug

20 89.0 contained in each aliquot. The following data
40 73.0 were obtained:

60 57.0
Time (hours) Antibiotic (μg/mL)

90 34.0
0.5 84.5

120 10.0
1.0 81.2

130 2.5
2.0 74.5

a. Does the decrease in the amount of drug A 4.0 61.0
appear to be a zero-order or a first-order

6.0 48.0
process?

b. What is the rate constant k? 8.0 35.0

c. What is the half-life t1/2? 12.0 8.7
d. Does the amount of drug A extrapolate to

zero on the x axis? a. Is the decomposition of this antibiotic a first-
e. What is the equation for the line produced order or a zero-order process?

on the graph? b. What is the rate of decomposition of this
2. Plot the following data on both semilog graph antibiotic?

paper and standard rectangular coordinates. c. How many milligrams of antibiotics were
in the original solution prepared by the

Time (minutes) Drug A (mg) pharmacist?
4 70.0 d. Give the equation for the line that best fits

the experimental data.
10 58.0

4. A solution of a drug was freshly prepared at a
20 42.0 concentration of 300 mg/mL. After 30 days at
30 31.0 25°C, the drug concentration in the solution

was 75 mg/mL.
60 12.0

a. Assuming first-order kinetics, when will
90 4.5 the drug decline to one-half of the original

120 1.7 concentration?
b. Assuming zero-order kinetics, when will

Answer questions a, b, c, d, and e as stated in the drug decline to one-half of the original
Question 1. concentration?

 

44 Chapter 2

5. How many half-lives (t1/2) would it take for 12. The following information was provided by
99.9% of any initial concentration of a drug to Steiner et al (2013):
decompose? Assume first-order kinetics.

“ACT-335827 hydrobromide (Actelion Phar-
6. If the half-life for decomposition of a drug is

maceuticals Ltd., Switzerland) was freshly
12 hours, how long will it take for 125 mg of

prepared in 10% polyethylene glycol 400/0.5%
the drug to decompose by 30%? Assume first-

methylcellulose in water, which served as
order kinetics and constant temperature.

vehicle (Veh). It was administered orally at
7. Exactly 300 mg of a drug is dissolved into an

300 mg/kg based on the weight of the free base,
unknown volume of distilled water. After com-

in a volume of 5 mL/kg, and administered daily
plete dissolution of the drug, 1.0-mL samples

2 h before the onset of the dark phase.”
were removed and assayed for the drug. The
following results were obtained: How many milligrams of ACT-335827 hydro-

bromide would be given orally to a 170-g rat?
13. Refer to Question 12; how many milliliters

Time (hours) Concentration (mg/mL)
of drug solution would be needed for the

0.5 0.45 170-g rat?
2.0 0.3 14. Refer to Question 12; express 0.5% methylcel-

lulose (%w/v) as grams in 1 L solution.
15. The t½ value for aceclofenac tablet following

Assuming zero-order decomposition of the oral administration in Wistar male rats was
drug, what was the original volume of water in reported to be 4.35 hours (Shakeel et al, 2009).
which the drug was dissolved? Assuming a first-order process, what is the

8. For most drugs, the overall rate of drug elimination rate constant value in hours–1?
elimination is proportional to the amount of

16. Refer to Question 15; express the value of t½
drug remaining in the body. What does in minutes.
this imply about the kinetic order of drug

17. Refer to Question 15; the authors reported
elimination? that the relative bioavailability of aceclofenac

9. A single cell is placed into a culture tube from a transdermally applied gel is 2.6 folds
containing nutrient agar. If the number of cells higher compared to that of an oral tablet. The
doubles every 2 minutes and the culture tube following equation was used by the authors to
is completely filled in 8 hours, how long does calculate the relative bioavailability:
it take for the culture tube to be only half full
of cells? F% = {[(AUC sample)(Dose oral)]/

10. Cunha (2013) reported the following: “…CSF
[(AUC oral)(Dose sample)]}*100 (2.28)

levels following 2 g of ceftriaxone are
approximately 257 mcg/mL, which is well
above the minimal inhibitory concentration where AUC/Dose sample is for the gel and
(MIC) of even highly resistant (PRSP) in AUC/Dose oral is for the tablet. F% is the rela-
CSF…” What is the value of 257 mcg/mL tive bioavailability expressed in percent. If
in mg/mL? oral and transdermal doses were the same,

11. Refer to Question 10 above; express the value calculate AUC sample given AUC oral of
257 mcg/mL in mcg/dL. 29.1 mg·h/mL. What are the units for AUC

sample in (mg·day/mL)?

 

Mathematical Fundamentals in Pharmacokinetics 45

18. 300

DMAA_Concentration vs Time
1

200 2
3
4
5
6

100 7
8
Mean

0
0 5 10 15 20 25

Time (hours)

The above figure (from Basu Sarkar et al, The equation in the graph is that for the standard
2013) shows the plasma concentration–time curve generated for progesterone using a high-
profile of DMAA (1,3-dimethylamylamine) in performance liquid chromatography method.
eight men following a single oral dose of the In the equation, y is the area under the curve of
DMAA (25 mg). progesterone peak and x represents the concen-

What type of graph paper is the above graph? tration of the drug in mg/mL. Using this equa-

(Semilogarithmic or rectangular?) tion, predict the AUC for a drug concentration

19. Refer to Question 18; what are the C of 35 mg/mL.
max

and Tmax values for subject #1? (subject #1) 24. Refer to Question 23; predict the concentration

occurred at Tmax of ____ hour. of progesterone (mg/L) for a peak area (AUC)

20. Refer to Question 18; what is the average C of 145.
max

value for all eight subjects? Please use the cor- 25. Consider the following function dc/dt = 0.98

rect units for your answer. with c and t being the concentration of the drug

21. Refer to Question 18; what are the units for and time, respectively. This equation can also

AUC obtained from the graph? be written as ______.

22. Refer to Question 18; for subject #3, the C a. x = x0 − 0.98 t
max

value is approximately 105 ng/mL. Express b. x = 0.98 − t

this concentration in %w/v. c. x = x0 + 0.98 t

23. Consider the following graph (Figure 2a in the d. x = t/0.98

original article) presented in Schilling et al (2013):

180
y = 1.6624x – 0.3

160 R2 = 0.9998
140

120

100

80

60

40

20

0
0 25 50 75 100 125

DMAA_Concentration (ng/mL)

 

46 Chapter 2

ANSWERS

Learning Questions Notice that the answer differs in accordance
with the method used.

1. a. Zero-order process (Fig. A-1). c. t1/2

For zero-order kinetics, the larger the initial
100

amount of drug A0, the longer the t1/2.
Method 1

80 0.5A
t 0
1/2 =

k0

60 0.5(103.5)
t1/2 = = 66 min

0.78

40 Method 2

The zero-order t1/2 may be read directly from
the graph (see Fig. A-1):

20
At t = 0, A0 = 103.5 mg

At t1/2 , A = 51.8 mg
0

0 20 40 60 80 100 120 140
Minutes Therefore, t1/2 = 66 min.

d. The amount of drug, A, does extrapolate to
FIGURE A-1

zero on the x axis.
b. Rate constant, k The equation of the line is

0: e.

Method 1 A = −k0t + A0

Values obtained from the graph (see Fig. A-1): A = −0.78t +103.5

2. a. First-order process (Fig. A-2).
t (minutes) A (mg)

40 70 100

80 41
50

∆Y y − y
lope 1

−k 2
0 = s = =

∆X x
2 − x1

20
41 − 71

−k0 = k0 = 0.75 mg/min
80 − 40

10
Notice that the negative sign shows that the

slope is declining. 5

Method 2

By extrapolation: 2

A0 = 103.5 at t = 0; A = 71 at t = 40 min

A 1
= k0t + A0 0 20 40 60 80 100 120

71= −40k Minutes
0 +103.5

k0 = 0.81 mg/min FIGURE A-2

A (mg)

A (mg)

 

Mathematical Fundamentals in Pharmacokinetics 47

b. Rate constant, k: 3. a. Zero-order process (Fig. A-3).
Method 1

Obtain the first-order t1/2 from the semilog 100

graph (see Fig. A-2):

80
t (minutes) A (mg)

30 30
60

53 15

t1/2 = 23 min 40

0.693 0.693
k = = = 0.03 min−1

t1/2 23 20

Method 2
0

0 2 4 6 8 10 12
−k logY2 − logY

Slope 1 Hours
= =
2.3 X2 − X1 FIGURE A-3
−2.3 (log15− log 30)

k 0.03 min−1
= =

53− 30
∆Y

b. k0 = slope =
∆X

c. t1/2 = 23 min (see Method 1 above). Values obtained from the graph (see
d. The amount of drug, A, does not extrapolate Fig. A-3):

to zero on the x axis.
e. The equation of the line is

t (hours) C (μg/mL)

−kt 1.2 80
log A = − + log A

2.3 0
4.2 60

0.03t
log A = − + log78

2.3 It is always best to plot the data. Obtain a
regression line (ie, the line of best fit), and

A − t
= 78e 0.03

then use points C and t from that line.

On a rectangular plot, the same data show a 60 − 80
−k0 =

curve (not plotted). 4.2 −1.2
k0 = 6.67 µg/mL/h

c. By extrapolation:
At t0, C0 = 87.5 mg/mL.

d. The equation (using a ruler only) is

A = −k0t + A0 = −6.67t + 87.5

A better fit to the data may be obtained by
using a linear regression program. Linear
regression programs are available on spread-
sheet programs such as Excel.

mg/mL

 

48 Chapter 2

4. Given: Method 3

A t1/2 value of 20 days may be obtained
C (mg/mL) t (days) directly from the graph by plotting C against

300 0 t on rectangular coordinates.
5. Assume the original concentration of drug to be

75 30
1000 mg/mL.

−kt Method 1
a. logC = − + logC

2.3 0

No. of Half- No. of Half-
−30k

log75 = − + log300 mg/mL Lives mg/mL Lives
2.3

1000 0 15.6 6
k = 0.046 days−1

500 1 7.81 7

0.693 0.693
t 250 2 3.91 8
1/2 = = = 15 days

k 0.046
125 3 1.95 9

b. Method 1 62.5 4 0.98 10

300 mg/mL 31.3 5
= C0 at t = 0

75 mg/mL = C at t = 30 days 99.9% of 1000 = 999
225 mg/mL = difference between initial and Concentration of drug remaining = 0.1% of

final drug concentration 1000
225 mg/mL 1000 − 999 = 1 mg/mL

k0 = = 7.5 mg/mL/d
30 days It takes approximately 10 half-lives to eliminate

all but 0.1% of the original concentration of
The time, t1/2, for the drug to decompose drug.

to one-half C0 (from 300 to 150 mg/mL) is
Method 2

calculated by (assuming zero order):
Assume any t1/2 value:

150 mg/mL
t1/2 = = 20 days

75 mg/mL/day 0.693
t1/2 =

k

Method 2 Then

C = −k0t + C0 0.693
k =

75 30k 300 t
= − 1/2

0 +

k mg/mL/d −kt
0 = 7.5 logC = + logC

2.3 0

At t1/2 ,C = 150 mg/mL
−kt

log1.0 = + log1000
150 = −7.5t 2.3

1/2 + 300

t = 9.96 t
t 1/2
1/2 = 20 days

 

Mathematical Fundamentals in Pharmacokinetics 49

Substituting 0.693/t1/2 for k: Alternatively, at t = 0.5 hour,

−0.693 t 0.45 = –0.1(0.5) –C0
log1.0 = + log1000

2.3× t1/2 C0 = 0.5 mg/mL

t = 9.96 t1/2 Since the initial mass of drug D0 dissolved is
300 mg and the initial drug concentration C0 is

6. t1/2 = 12 h 0.5 mg/mL, the original volume may be calcu-

0.693 0.693 lated from the following relationship:
k = = = 0.058 h–1

t1/2 12 D
C 0

0 =
V

If 30% of the drug decomposes, 70% is left.
Then 70% of 125 mg = (0.70)(125) = 87.5 mg 300 mg

0.5 mg/mL =
V

A0 = 125 mg V = 600 mL
A = 87.5 mg

8. First order.
k −1

= 0.058 h 9. The volume of the culture tube is not impor-

kt tant. In 8 hours (480 minutes), the culture tube
log A = − + log A

2.3 0 is completely full. Because the doubling time
for the cells is 2 minutes (ie, one t1/2), then in

0.058t
log 87.5 = − + log125 480 minutes less 2 minutes (478 minutes) the

2.3
culture tube is half full of cells.

t = 6.1 hours 10. b. Since 1 mg = 1000 mg, then
(257 mg/mL)/1000 = 0.257 mg/mL.

7. Immediately after the drug dissolves, the drug
11. c. Since 1 dL = 100 mL, then

degrades at a constant, or zero-order rate.
(257 mg/mL) × 100 = 25,700 mg/dL.

Since concentration is equal to mass divided by
12. a. Since 1 kg = 1000 g, then (170 g)/1000 =

volume, it is necessary to calculate the initial
0.17 kg.

drug concentration (at t = 0) to determine the
The oral dose was 300 mg/kg; therefore,

original volume in which the drug was dis-
for 0.17 kg rat, (0.17 kg)(300 mg)/1 kg =

solved. From the data, calculate the zero-order
51 mg.

rate constant, k0: 13. c. The volume given was 5 mL/kg. For

∆Y 0.45− 0.3 0.17 kg rat, (0.17 kg)(5 mL)/1 kg = 0.85 mL.
−k0 = slope = = 14. d. 0.5% of methylcellulose (% w/v) means

∆X 2.0 − 0.5
0.5 g of methylcellulose in 100 mL solution.

k0 = 0.1mg/mL/h Or 5 g of methylcellulose in 1 L solution.
15. b. kel = 0.693/t½ = 0.693/4.35 = 0.16 h–1

Then calculate the initial drug concentration,
16. b. 4.35 hours × 60 min/h = 261 minutes.

C0, using the following equation:
17. c. F%= {[(AUC sample)(Dose oral)]/[(AUC

oral)(Dose sample)]} * 100 (2.28)
C = −k t + C

0 0
F% = [(AUC sample)/AUC oral)] * 100

At t = 2 hours, 2.6 folds higher = 260%

0.3 = −0.1(2) +C 260 = [AUC sample)/29.1] * 100
0

AUC sample = 75.66 mg·h/mL = 0.07566
C0 = 0.5 mg/mL mg·h/mL = 1.8 mg·day/mL

 

50 Chapter 2

18. b. A rectangular coordinate graph. 23. c. y = 1.6624 × −0.3
19. d. According to the figure, the highest plasma y = 1.6624 (35) − 0.3 = 57.9 = AUC

concentration for subject #1 occurred at 24. a. y = 1.6624 × −0.3
24 hours. 145 = 1.6624 × −0.3

20. b. From the graph, the average Cmax was x = 87.4 mg/mL = 87.4 mg/L
between 50 and 100 ng/mL. 25. c. dc/dt = 0.98

21. c. It is (concentration units) × (time) = dc = 0.98 dt
(ng/mL) × (hours) = (ng·h/mL). ∫dc = 0.98 ∫dt

22. c. 105 ng/mL = 10,500 ng/100 mL = c = c0 + 0.98t
10.5 mg/100 mL = 0.0105 mg/100 mL =
0.0000105 g/100 mL.

REFERENCES
Basu Sarkar A, Kandimalla A, Dudley R: Chemical stability of Schilling et al: Physiological and pharmacokinetic effects of oral

progesterone in compounded topical preparations using PLO 1,3-dimethylamylamine administration in men. BMC Phar-
Transdermal Cream™ and HRT Cream™ base over a 90-day macol Toxicol 14:52, 2013. (© 2013 Schilling et al; licensee
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uted under the terms of the Creative Commons Attribution permits unrestricted use, distribution, and reproduction in any
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(© 2012 Ravi Sankar V, et al. This is an open-access article duction in other forums is permitted, provided the original
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reproduction in any medium, provided the original author and demic practice. No use, distribution, or reproduction is permit-
source are credited.) ted which does not comply with these terms.)

 

Biostatistics

3 Charles Herring

Chapter Objectives VARIABLES1

»» Describe basic statistical Several types of variables will be discussed throughout this text.
methodology and concepts A random variable is “a variable whose observed values may be

»» Describe how basic statistical considered as outcomes of an experiment and whose values cannot be
methodology may be used anticipated with certainty before the experiment is conducted”
in pharmacokinetic and (Herring, 2014). An independent variable is defined as the “interven-
pharmacodynamics study tion or what is being manipulated” in a study (eg, the drug or dose of
design the drug being evaluated) (Herring, 2014). “The number of indepen-

dent variables determines the category of statistical methods that are
»» Describe how basic statistical

appropriate to use” (Herring, 2014). A dependent variable is the
methodology may be used in

“outcome of interest within a study.” In bioavailability and bioequiva-
critically evaluating data

lence studies, examples include the maximum concentration of the
»» Describe how basic statistical drug in the circulation, the time to reach that maximum level, and the

methodology may be used to area under the curve (AUC) of drug level-versus-time curve. These
help minimize error, bias, and are “the outcomes that one intends to explain or estimate” (Herring,
confounding, and, therefore, 2014). There may be multiple dependent (aka outcome) variables. For
promote safe and efficacious example, in a study determining the half-life, clearance, and plasma
drug therapy protein binding of a new drug following an oral dose, the independent

»» Provide examples of how basic variable is the oral dose of the new drug. The dependent variables are

statistical methodology may be the half-life, clearance, and plasma protein binding of the drug

used for study design and data because these variables “depend upon” the oral dose given.

evaluation Discrete variables are also known as counting or nonparamet-
ric variables (Glasner, 1995). Continuous variables are also
known as measuring or parametric variables (Glasner, 1995). We
will explore this further in the next section.

TYPES OF DATA NONPARAMETRIC
VERSUS PARAMETRIC
There are two types of nonparametric data, nominal and ordinal. For
nominal data, numbers are purely arbitrary or without regard to any
order or ranking of severity (Gaddis and Gaddis, 1990a; Glasner,

1The 5th edition of Quick Stats: Basics for Medical Literature Evaluation was
utilized for the majority of the following chapter (Herring, 2014). In order to
discuss basic statistics, some background terminology must be defined.

51

 

52 Chapter 3

1995). Nominal data may be either dichotomous or level-versus-time curve, drug clearance, and elim-
categorical. Dichotomous (aka binary) nominal data ination half-life.
evaluate yes/no questions. For example, patients lived
or died, were hospitalized, or were not hospitalized.
Examples of categorical nominal data would be things Frequently Asked Questions

like tablet color or blood type; there is no order or »»Is it appropriate to degrade parametric data to

inherent value for nominal, categorical data. nonparametric data for data analysis?

Ordinal data are also nonparametric and cate- »»What occurs if this is done?
gorical, but unlike nominal data, ordinal data are
scored on a continuum, without a consistent level of
magnitude of difference between ranks (Gaddis and
Gaddis, 1990a; Glasner, 1995). Examples of ordinal Data Scale Summary Example

data include a pain scale, New York Heart Association In pharmacokinetic studies, researchers may be inter-
heart failure classification, cancer staging, bruise ested in testing the difference in the oral absorption of
staging, military rank, or Likert-like scales (poor/ a generic versus a branded form of a drug. In this case,
fair/good/very good/excellent) (Gaddis and Gaddis, “generic or branded” is a nominal scale-type variable,
1990a; DeYoung, 2005). whereas expressing the “rate of absorption” numeri-

Parametric data are utilized in biopharmaceu- cally is a ratio-type scale (Gaddis and Gaddis, 1990a;
tics and pharmacokinetic research more so than Ferrill and Brown 1994; Munro, 2005).
are the aforementioned types of nonparametric
data. Parametric data are also known as continu-
ous or measuring data or variables. There is an DISTRIBUTIONS
order and consistent level of magnitude of differ- Normal distributions are “symmetrical on both sides
ence between data units. There are two types of of the mean” sometimes termed as a bell-shaped
parametric data: interval and ratio. Both interval curve, Gaussian curve, curve of error, or normal
and ratio scale parametric data have a predeter- probability curve (Shargel et al, 2012). An example
mined order to their numbering and a consistent of normally distributed data includes drug elimina-
level of magnitude of difference between the tion half-lives in a specific population, as would be
observed data units (Gaddis and Gaddis, 1990a; the case in a sample of men with normal renal and
Glasner, 1995). However, for interval scale data, hepatic function. As will be discussed later in this
there is no absolute zero, for example, Celsius or chapter, parametric statistical tests like t-test and
Fahrenheit (Gaddis and Gaddis, 1990a; Glasner, various types of analysis of variance (ANOVA) are
1995). For ratio scale data, there is an absolute utilized for normally distributed data.
zero, for example, drug concentrations, plasma
glucose, Kelvin, heart rate, blood pressure, dis-
tance, and time (Gaddis and Gaddis, 1990a;
Glasner, 1995). Although the specific definitions
of these two types of parametric data are listed
above, their definitions are somewhat academic
since all parametric data utilize the same statisti-
cal tests. In other words, regardless of whether the
parametric data are interval or ratio scale, the Sometimes in bioequivalence or pharmacokinetic
same tests are used to detect statistical differ- studies, a bimodal distribution is noted. In this case two
ences. Examples of parametric data include plasma peaks of cluster or areas of high frequency occur. For
protein binding, the maximum concentration of example, a medication that is acetylated at different
the drug in the circulation, the time to reach that rates in humans would be a “bimodal distribution, indi-
maximum level, the area under the curve of drug cating two populations consisting of fast acetylators

 

Biostatistics 53

and slow acetylators” (Gaddis and Gaddis, 1990a;
Glasner, 1995; Shargel et al, 2012).

Mortality

Blood pressure

Skewed distributions occur when data are not MEASURES OF CENTRAL TENDENCY
normally distributed and tail off to either the high or There are several measures of central tendency that
the low end of measurement units. A positive skew are utilized in biopharmaceutical and pharmacoki-
occurs when data cluster on the low end of the x axis netic research. The most common one is the mean,
(Gaddis and Gaddis, 1990a; Glasner, 1995). For or average. It is the “sum of all values divided by the
example, the x axis could be the income of patients total number of values,” is used for parametric data,
seen in inner-city Emergency Department (ED), cost and is affected by outliers or extreme values, which
of generic medications, number of prescribed medica- “deviate far from the majority of the data” (Gaddis
tions in patients younger than 30 years of age. and Gaddis, 1990b; Shargel et al, 2012). Mu (μ) is

the population mean and X-bar (X ) is the sample
mean (Gaddis and Gaddis, 1990b).

Median is also known as the 50th percentile or
y axis mid-most point (Gaddis and Gaddis, 1990b). It is

“the point above which or below which half of the
data points lie” (Gaddis and Gaddis, 1990b). It is not

x axis affected by outliers and may be used for ordinal and
parametric data (Gaddis and Gaddis, 1990b). Median

A negative skew occurs when data cluster on the is used when outliers exist, when a data set spans a
high end of the x axis (Gaddis and Gaddis, 1990a; wide range of values, or “when continuous data are
Glasner, 1995). For example, the x axis could be the not normally distributed” (Gaddis and Gaddis,
income of patients seen in ED of an affluent area, cost 1990b; DeYoung, 2005).
of brand name medications, number of prescribed Mode is the most common value (Gaddis and
medications in patients older than 60 years of age. Gaddis, 1990b). Mode is not affected by outliers

and may be used for nominal, ordinal, or parametric
data (Gaddis and Gaddis, 1990b). As with median,
the mode is not affected by outliers (Gaddis and

y axis Gaddis, 1990b). However, the mode is not helpful
when a data set contains a large range of infre-
quently occurring values (Gaddis and Gaddis,

x axis 1990b).
For normally distributed data, mean, median,

Kurtosis occurs when data cluster on both ends and mode are the same. For positively skewed data,
of the x axis such that the graph tails upward (ie, the mode is less than the median and the median is
clusters on both ends of the graph). For example, the less than the mean. For negatively skewed data, the
J-curve of hypertension treatment; with the J-curve, mode is greater than the median and the median is
mortality increases if blood pressure is either too greater than the mean (Gaddis and Gaddis, 1990b;
high or too low (Glasner, 1995). Glasner, 1995).

 

54 Chapter 3

Normally distributed data (Gaddis and Gaddis, Range is the interval between lowest and highest
1990b; Glasner, 1995) values (Gaddis and Gaddis, 1990b; Glasner, 1995).

Normally distributed data (2, 8) Range only considers extreme values, so it is affected by
outliers (Gaddis and Gaddis, 1990b). Range is descrip-
tive only, so it is not used to infer statistical significance
(Gaddis and Gaddis, 1990b). Interquartile range is the
interval between the 25th and 75th percentiles, so it is
directly related to median, or the 50th percentile (Gaddis
and Gaddis, 1990b). It is not affected by outliers and,

Mode = median = mean along with the median, is used for ordinal scale data
(Gaddis and Gaddis, 1990b).

Positively skewed data (Gaddis and Gaddis, Variance is deviation from the mean, expressed as
1990b; Glasner, 1995) the square of the units used. The data are squared in the

Positively skewed data (2, 8) variance calculations because some deviations are nega-
tive and squaring provides a positive number (Gaddis
and Gaddis, 1990b; Glasner, 1995). “As sample size (n)
increases, variance decreases” (Herring, 2014). Variance
equals the sum of (mean – data point) squared, divided
by n – 1.

∑(X − X)2
Mode < median < mean Variance = (3.1)

n −1

Negatively skewed data (Gaddis and Gaddis,
Standard deviation (SD) is the square root of

1990b; Glasner, 1995)
variance (Gaddis and Gaddis, 1990b; Glasner, 1995).

Negatively skewed data (2, 8) SD estimates the degree of data scatter around the
sample mean. Sixty-eight percent of data lie within ±1
SD of the mean and 95% of data lie within ±2 SD of
the mean (Gaddis and Gaddis, 1990b; Glasner, 1995).
SD is only meaningful when data are normally or
near-normally distributed and, therefore, is only appli-

Mean < median < mode cable to parametric data (Gaddis and Gaddis, 1990b;
Glasner, 1995). Sigma (s) is the population SD and S

Based upon a data set’s mean, median, and mode is the sample SD (Glasner, 1995).
values, one can determine if the data is normally dis-
tributed or skewed when no graphical representation SD = Variance (3.2)
is provided. For biopharmaceutical and pharmacoki-
netic data, this is important to know so that appropri- “Coefficient of variation (or relative standard
ate logarithmic transformation can be performed for deviation) is another measure used when evaluating
skewed data to restore normality. dispersion from one data set to another. The coefficient

A weakness of measures of central tendency is of variation is the SD expressed as a percentage of the
the data does not describe variability or spread of data. mean. This is useful in comparing the relative difference

in variability between two or more samples, or which

MEASURES OF VARIABILITY group has the largest relative variability of values from
the mean” (Herring, 2014). The smaller the coefficient

Measures of variability describe data spread and, in the of variation, the less the variability in the data set.
case of confidence intervals (CIs), can help one infer
statistical significance (Gaddis and Gaddis, 1990b). Coefficientof variation = 100 × SD/X (3.3)

 

Biostatistics 55

Standard error of the mean (SEM) is the SD However, the closer the point estimate lies to the
divided by the square root of n (Gaddis and Gaddis, middle of the CI, the more likely the point estimate
1990b; Glasner, 1995). The larger n is, the smaller represents the population.
SEM is (Gaddis and Gaddis, 1990b; Glasner, 1995). For example, if a point estimate and 95% CI for
SEM is always smaller than SD. drug clearance are 3 L/h (95% CI: 1.5–4.5 L/h), all

“The mean of separate samples from a single values including and between 1.5 and 4.5 L/h are

population will give slightly different parameter statistically possible. However, a point estimate of
estimates. The standard error (SE) is the standard 2.5 L/h is a more accurate representation of the stud-
deviation (SD) of the sampling distribution of a ied population than a point estimate of 1.6 L/h since
statistic and should not be confused with SEM. 2.5 is closer to the sample’s point estimate of 3 than
The distribution of means from random samples is is 1.6. As seen in this example, CI shows the degree
approximately normal. The mean of this ‘distribu- of certainty (or uncertainty) in each comparison in an
tion of means’ is the unknown population mean” easily interpretable way.

(Glasner, 1995)
In addition, CIs make it easier to assess clinical

SD for the distribution of means is estimated by the significance and are less likely to mislead one into
SEM. One “could name the SEM as the standard thinking that nonsignificantly different sample val-
deviation of means of random samples of a fixed size ues imply equal population values
drawn from the original population of interest”
(Herring, 2014). The SEM is the quantification of the 95% CI = X ± 1.96 (SEM) (3.5)
spread of the sample means for a study that is repeated
multiple times. The SEM helps to estimate how well Significance of CIs depends upon the objective
a sample represents the population from which it was of the trial being conducted or evaluated.
drawn (Glasner, 1995). However, the SEM should not In superiority trials, all values within a CI are
be used as a measure of variability when publishing a statistically possible. For differences like differ-
study. Doing so is misleading. The only purpose of ences in half-life, differences in area under the
SEM is to calculate CIs, which contain an estimate of curve (AUC), relative risk reductions/increases
the true population mean from which the sample was (RRRs/RRIs), or absolute risk reductions/increases
drawn (Gaddis and Gaddis, 1990b). (ARRs/ARIs), if the CI includes ZERO (0), then

the results are not statistically significant (NSS). In
SEM = SD/ n (3.4)

the case of a 90% CI, if the CI includes ZERO (0)
Confidence interval (CI) is a method of estimat- for this type of data, it can be interpreted as a p >

ing the range of values likely to include the true value 0.10. In the case of a 95% CI, if the CI includes
of a population parameter (Gaddis and Gaddis, ZERO (0) for this type of data, it can be interpreted
1990b). In medical literature, a 95% CI is most fre- as a p > 0.05. In the case of a 97.5% CI, if the CI
quently used. The 95% CI is a range of values that “if includes ZERO (0) for this type of data, it can be
the entire population could be studied, 95% of the interpreted as a p > 0.025.
time the true population value would fall within the CI For superiority trials, since all values within a
estimated from the sample” (Gaddis and Gaddis, CI are statistically possible, for ratios like relative
1990b). For a 95% CI, 5 times out of 100, the true risk (RR), odds ratio (OR), or hazards ratio (HR), if
population parameter may not lie within the CI. For a the CI includes ONE (1.0), then the results are not
97.5% CI, 2.5 times out of 100, the true population statistically significant (NSS). In the case of a 90%
parameter may not lie within the CI. Therefore, a CI, if the CI includes ONE (1.0) for this type of data,
97.5% CI is more likely to include the true population it can be interpreted as a p > 0.10. In the case of a
value than a 95% CI (Gaddis and Gaddis, 1990b). 95% CI, if the CI includes ONE (1.0) for this type of

The true strength of a CI is that it is both data, it can be interpreted as a p > 0.05. In the case
descriptive and inferential. “All values contained in of a 97.5% CI, if the CI includes ONE (1.0) for this
the CI are statistically possible” (Herring, 2014). type of data, it can be interpreted as a p > 0.025.

 

56 Chapter 3

HYPOTHESIS TESTING are willing to accept and is denoted in trials as a p ≤
0.05 (Gaddis and Gaddis, 1990c). So the p-value is

For superiority trials, the null hypothesis (H0) is that the calculated chance that a type 1 error has occurred
no difference exists between studied populations (Gaddis and Gaddis, 1990c). In other words, it tells us
(Gaddis and Gaddis, 1990c). For superiority trials, the likelihood of obtaining a statistically significant
the alternative hypothesis (H1) is that a difference result if H0 were true. “At p = 0.05, the likelihood is 5%.
does exist between studied populations (Gaddis and At p = 0.10, the likelihood is 10%” (Herring, 2014).
Gaddis, 1990c). A p ≤ a means the observed treatment difference is

H statistically significant, it does not indicate the size or
0: There is no difference in the AUC for drug

formulation A relative to formulation B. direction of the difference. The size of the p-value is

H not related to the importance of the result (Gaddis and
1 (aka Ha): There is a difference in AUC for

drug formulation A relative to formulation B. Gaddis, 1990f; Berensen, 2000). Smaller p-values
simply mean that “chance” is less likely to explain

H1 is sometimes directional. For example, observed differences (Gaddis and Gaddis, 1990f;
H1: We expect AUC for drug formulation A to Berensen, 2000). Also, “a small p-value does not cor-

be 25% higher than that of formulation B. rect for systematic error (bias)” from a poorly designed
H0 is tested instead of H1 because there are an study (DeYoung, 2005).

infinite number of alternative hypotheses. It would be A type 2 error occurs if one accepts the H0 when,
impossible to calculate the required statistics for each in fact, the H0 is false (Gaddis and Gaddis, 1990c).
of the infinite number of possible magnitudes of dif- For superiority trials this is when one concludes there
ference between population samples H1 hypothesizes is no difference between treatment groups, when in
(Gaddis and Gaddis, 1990c). H0 is used to determine fact, a difference does exist. Beta (b) is the probability
“if any observed differences between groups are due of making a type 2 error (Gaddis and Gaddis, 1990c).
to chance alone” or sampling variation. By convention, an acceptable b is 0.2 (20%) or less

Statistical significance is tested (hypothesis test- (Gaddis and Gaddis, 1990c).
ing) to indicate if H0 should be accepted or rejected Regardless of the trial design (superiority,
(Gaddis and Gaddis, 1990c). For superiority trials, if equivalence, or non-inferiority), a and b are interre-
H0 is “rejected,” this means a statistically significant lated (Gaddis and Gaddis, 1990c). All else held
difference between groups exists (results unlikely due constant, a and b are inversely related (Gaddis and
to chance) (Gaddis and Gaddis, 1990c). For superior- Gaddis, 1990c). In other words, as a is decreased, b
ity trials, if H0 is “accepted,” this means no statisti- is increased, and as a is increased, b is decreased (ie,
cally significant difference exists (Gaddis and Gaddis, as risk for a type 1 error is increased, risk for a type
1990c). However, “failing to reject H0 is not sufficient 2 error is decreased and vice versa) (Gaddis and
to conclude that groups are equal” (DeYoung, 2005). Gaddis, 1990c). The most common use of b is in

A type 1 error occurs if one rejects the H0 when, calculating the approximate sample size required for
in fact, the H0 is true (Gaddis and Gaddis, 1990c). a study to keep a and b acceptably small (Gaddis
For superiority trials this is when one concludes and Gaddis, 1990c).
there is a difference between treatment groups, when
in fact, no difference exists (Gaddis and Gaddis,
1990c). Frequently Asked Questions

Alpha (a) is defined as the probability of making
»»For a superiority trial, if a statistically significant

a type 1 error (Gaddis and Gaddis, 1990c). When a difference were detected, is there any way that the
level is set a priori (or before the trial), the H0 is study was underpowered?
rejected when p ≤ a (Gaddis and Gaddis, 1990c). By

»»For a superiority trial, if a statistically significant dif-
convention, an acceptable a is usually 0.05 (5%),

ference were detected, is there any way a type 2 error
which means that 1 time out of 20, a type 1 error will

could have occurred?
be committed. This is a consequence that investigators

 

Biostatistics 57

TABLE 31 Type 1 and 2 Error for Superiority Trials

Reality

Difference Exists (H0 False) No Difference Exists (H0 True)

Decision from Statistical Test

Difference found (Reject H0) Correct No error Incorrect Type 1 error (false positive)

No difference found (Accept H0) Incorrect Type 2 error (false negative) Correct No error

Delta (∆) is sometimes referred to as the “effect For superiority trials, inadequate power may
size” and is a measure of the degree of difference cause one to conclude that no difference exists when,
between tested population samples (Gaddis and in fact, a difference does exist. As described above,
Gaddis, 1990c). For parametric data, the value of ∆ this would be a type 2 error (Gaddis and Gaddis,
is the ratio of the clinical difference expected to be 1990c). Note that in most cases, power is an issue
observed in the study to the standard deviation (SD) only if one accepts the H0. If one rejects the H0, there
of the variable: is no way that one could have made a type 2 error

(see Table 3-1). Therefore, power to detect a differ-
∆ = (ma − m0)/SD (3.6)

ence would not be an issue in most of these cases. An
where μa is the alternative hypothesis value expected exception to this general rule would be if one wanted
for the mean and μ0 is the null hypothesis value for to decrease data variability or spread. For example,
the mean. if one wanted to narrow the 95% CI, increasing

One-tailed versus two-tailed tests: It is easier to power by increasing sample size could help.
show a statistically significant difference with a For research purposes, power calculations are
one-tailed test than with a two-tailed test, because generally used to determine the required sample size
with a one-tailed test a statistical test result must not when designing a study (ie, prior to the study).
vary as much from the mean to achieve significance Power calculations are generally based upon the
at any level of a chosen (Gaddis and Gaddis, 1990c). primary endpoint of the study and, as is depicted in
However, most reputable journals require that inves- the examples below, the a priori (prespecified) a, b,
tigators perform statistics based upon a two-tailed ∆, SD, and whether a one-tailed or two-tailed design
test even if it innately makes sense that a differ- is used.
ence would only occur unidirectionally (Al-Achi A,
discussions). Parametric Data Sample Size/Power

Power is the ability of an experiment to detect a
Examples

statistically significant difference between samples,
when in fact, a significant difference truly exists The way a study is set up will determine the required

(Gaddis and Gaddis, 1990c). Said another way, sample size. In other words, the preset a, b, ∆, SD,

power is the probability of making a correct decision and tailing (one-tailed vs two-tailed) affect sample

when H0 is false. size required for a study (Drew R, discussions and
provisions).

Power = 1 − b (3.7) Utilizing a larger standard deviation (SD) will
As stated in the section on type 2 error risk, by con- require a larger sample size. Also, a one-tailed test
vention, an acceptable b is 0.2 (20%) or less; there- requires a smaller sample size than a two-tailed test
fore, most investigators set up their studies, and their to detect differences between groups (Drew R, dis-
sample sizes, based upon an estimated power of at cussions and provisions). This is due to the fact that
least 80%. given everything else is the same, a one-tailed test

 

58 Chapter 3

has more power to reject the null hypothesis than a size required to detect that difference (Drew R, dis-
two-tailed test. cussions and provisions).

Sample Size Sample Size
Statistical Statistical

Differences Limits Differences Limits
One- Two- One- Two-

SD ∆ (%) ` a tailed tailed SD ∆ (%) ` a tailed tailed

1 (68% 10 0.05 0.20 1237 1570 2 (95% 10 0.05 0.20 4947 6280
of data) of data)

2 (95% 10 0.05 0.20 4947 6280 2 (95% 20 0.05 0.20 1237 1570
of data) of data)

An example for estimating the sample size for a
Increasing the accepted type 1 (a) and type 2 (b) study would be as follows:
statistical error risks will decrease the sample size a = 0.05
required. b = 0.20

Decreasing the acceptable type 1 (a) and type ∆ = 0.25
2 (b) statistical error risks will increase the required SD = 2.0
sample size (Drew R, discussions and provisions). Statistical test = two-sided t-test

Single sample

Statistical Sample Size From a statistics table, the total sample size
Differences Limits

One- Two- needed for this study is 128, or 64 in each group.
SD ∆ (%) α β tailed tailed This also indicates that the investigators are inter-

ested in detecting a clinically meaningful difference
2 (95% 10 0.05 0.20 4947 6280
of data) of 0.50 unit:

2 (95% 10 0.10 0.20 3607 4947 ∆ = (ma –m0)/SD
of data)

0.25 = (ma – m0)/2.0
(ma – m0) = (2.0) × (0.25) = 0.50 unit

Power = 1 – b, so a larger sample size is required for
smaller b and higher power (Drew R, discussions and In other words, in order for the researchers to

provisions). significantly detect the difference of 0.50 units, they
would need a sample size of 128 patients. This test
would have an estimated power of 80% (since b =

Sample Size 0.20) and a confidence level of 95% (since a = 0.05).
Statistical

It is important to reemphasize here that the smaller
Differences Limits

One- Two- the value for ∆, the greater would be the sample size
SD ∆ (%) a b tailed tailed needed for the study.

2 (95% 10 0.05 0.10 6852 8406
of data)

STATISTICALLY VERSUS CLINICALLY
2 (95% 10 0.05 0.20 4947 6280
of data) SIGNIFICANT DIFFERENCES

Statistically significant differences do not necessar-
A smaller difference (∆) between groups ily translate into clinically significant differences

increases the sample size required to detect that dif- (Gaddis and Gaddis, 1990c). If the sample size of a
ference. A larger difference (∆) decreases the sample trial is large enough, nonclinically meaningful,

 

Biostatistics 59

statistically significant differences may be detected. differences between these is solely academic.
For example, grapefruit juice induces enzymatic Parametric tests are more powerful than nonparametric
activity with some drugs such that their elimination tests (Gaddis and Gaddis, 1990d). Also, more infor-
t½ becomes shorter. Current data support that consis- mation about data is generated from parametric tests
tent grapefruit consumption statistically and clini- (Gaddis and Gaddis, 1990d).
cally significantly decreases the elimination t½ of The t-test (aka Student’s t-test) is the method of
these drugs. However, a one-time, single glass of choice when making a single comparison between
grapefruit juice may statistically significantly two groups. A non-paired t-test is used when obser-
decrease the value of t½ by only 1%, which would vations between groups are independent as in the
not be considered clinically meaningful. case of a parallel study as seen in the example below.

Also, lack of statistical significance does not Exp represents the experimental group and Ctrl repre-
necessarily mean the results are not clinically signifi- sents the control group.
cant; consider power, trial design, and populations

Population
studied (Gaddis and Gaddis, 1990f). A nonstatistically

Exp Endpoint
significant difference is more likely to be accepted as
being clinically significant in the instance of safety Sample

issues (like adverse effects), than for endpoint improve-
ments. For example, if a trial were to find a nonsta-
tistically significant increase in the risk for invasive

Ctrl Endpoint
breast cancer with a particular medication, many
clinicians would deem this as being clinically mean-
ingful such that they would avoid using the agent

Randomization Analysis
until further data were obtained. Also, suppose that a
study were conducted to examine the response rate
for a drug in two different populations. The response A paired t-test is used when observations

rates were 55% and 72% for groups 1 and 2, respec- between groups are dependent, as would be the case

tively. This difference in response rate is 17% (72 – 55 in a pretest/posttest study or a crossover study

= 17%) with a 95% CI of –3% to 40%. Since the 95% (Gaddis and Gaddis, 1990d). Initially in a crossover

CI includes zero, the difference is not statistically design, group A receives the experimental drug (Exp)

significant. Let’s also further assume that the mini- while group B receives the control (Ctrl: placebo or

mum clinically acceptable difference in response rate gold standard treatment). After a washout period,

for the particular disease is 15%. Since the response group A receives the control (Ctrl) and group B

rate is 17% (which is greater than 15%), it may very receives the experimental drug (Exp). It is very

well be clinically meaningful (significant) such that important to ensure adequate time for washout to

another, more adequately powered study may be prevent carry-over effects.

worth conducting. Population Washout period

Exp Endpoint Exp Endpoint
STATISTICAL INFERENCE
TECHNIQUES IN HYPOTHESIS Ctrl Endpoint Ctrl Endpoint

TESTING FOR PARAMETRIC DATA
Randomization Analysis

Parametric statistical methods (t-test and ANOVA)
are used for analyzing normally distributed, para- However, when making either multiple com-
metric data (Gaddis and Gaddis, 1990d). Parametric parisons between two groups or a single comparison
data include interval and ratio data, but since the between multiple groups, type 1 error risk increases
same parametric tests are used for both, knowing the if utilizing a t-test. For example, when rolling dice,

 

60 Chapter 3

think of rolling ones on both dice (snake eyes) as Investigators should make their best effort to
being a type 1 error. For each roll of the dice, there keep the type 1 error risk ≤ 5% (ie, ≤0.05). The best
is a 1 in 36 chances (2.78%) of rolling snake eyes. way of doing so for multiple comparisons is by
For each statistical analysis, we generally accept a avoiding unnecessary comparisons or analyses,
1 in 20 chances (5%) of a type 1 error. Although the using the appropriate statistical test(s) for multiple
chance for snake eyes is the same for each roll and comparisons, and using an alpha spending function
the chance for type 1 error is the same for each for interim analyses. However, if investigators fail to
analysis, increasing the number of rolls and analyses do so, there is a crude method for adjusting the pre-
increases the opportunity for snake eyes and type 1 set a level based upon the number of comparisons
errors, respectively. Said another way, the more being made: the Bonferroni correction. This simply
times one rolls the dice, the more opportunity one divides the preset a level by the number of compari-
has to roll snake eyes. It’s the same with statistical sons being made (Gaddis and Gaddis, 1990d). This
testing. The more times one performs a statistical estimates the a level that is required to reach statisti-
test on a particular data set, whether it be multiple cal significance (Gaddis and Gaddis, 1990d). However,
comparisons of two groups, a single comparison of Bonferroni is very conservative as the number of
multiple groups, or multiple comparisons of multiple comparisons increases. A less conservative and more
groups, the more likely one is to commit a type 1 error. accepted way of minimizing type 1 error risk for

As an example of multiple comparisons of two multiple comparisons with parametric data is through
groups for which the authors and/or statisticians did utilization of one of several types of analysis of vari-
not make type 1 error risk corrections, a trial evalu- ance (ANOVA).
ated chlorthalidone versus placebo for the primary ANOVA holds a level (type 1 error risk) constant
endpoint of blood pressure. In addition to this, there when comparing more than two groups (Gaddis and
were other evaluated endpoints (including potassium Gaddis, 1990d). It tests for statistically significant
concentration, serum creatinine, BUN:SCr ratio, difference(s) among a group’s collective values
calcium concentration, and others), and the authors (Gaddis and Gaddis, 1990d). In other words, intra-
did not control for these additional comparisons. and intergroup variability is what is being analyzed
Let’s say there were a total of 20 comparisons instead of the means of the groups (Gaddis and
including the primary endpoint of blood pressure. If Gaddis, 1990d). It involves calculation of an F-ratio,
the original a level were p = 0.05, the corrected a which answers the question, “is the variability
would be 1 – (1 – 0.05)20 = 0.64. This means that if between the groups large enough in comparison to
the original p-value threshold of 0.05 were used, the variability of data within each group to justify the
there would be a 64% chance of inappropriately conclusion that two or more of the groups differ”
rejecting the null hypothesis (ie, committing a type 1 (Gaddis and Gaddis, 1990d)?
error) for at least one of the 20 comparisons (Gaddis The most commonly used ANOVAs are for inde-
and Gaddis, 1990d). pendent (aka non-paired) samples as is the case for

As an example of a single comparison of multi- a parallel design.
ple groups for which the authors and/or statisticians The first is 1-way ANOVA, which is used if there
did not make type 1 error risk corrections, a trial are no confounders and at least three independent
evaluated the difference in cholesterol among four (aka non-paired) samples. For example, if investiga-
lipid-lowering medications. With four groups, there tors wanted to evaluate the excretion rate (percent of
were six paired comparisons. If the original a level dose excreted unchanged in the urine) of different
were p = 0.05, the corrected a would be 1 – (1 – blood pressure medications, they could use a 1-way
0.05)6 = 0.26. Therefore, if the original p-value ANOVA if (1) each sample were independent (ie, a
threshold of 0.05 were used, there would be a 26% parallel design), (2) there were at least three samples
chance of inappropriately rejecting the null hypothe- (ie, at least three different blood pressure medica-
sis (type 1 error) for at least one of the six compari- tions), and (3) the experimental groups differed in
sons (Gaddis and Gaddis, 1990d). only one factor, which for this case would be the

 

Biostatistics 61

type of blood pressure drug being used (ie, there blood pressure medications, they could use a 4-way
were no differences between the groups with regard ANOVA if (1) each sample were independent (ie, a
to confounding factors like age, gender, kidney function, parallel design), (2) there were at least two samples
plasma protein binding, etc). (ie, at least two different blood pressure medica-

Multifactorial ANOVAs include any type of tions), and (3) the experimental groups differed in
ANOVA that controls for at least one confounder for four factors, which for this case would be the type of
at least two independent (non-paired) samples as is blood pressure drug being used and three confound-
the case for a parallel design. ing variables (eg, differences between the groups’

A 2-way ANOVA is used if there is one identifiable renal function, plasma protein binding, and average
confounder and at least two independent (aka non- patient age).
paired) samples. For example, if investigators wanted There are also ANOVAs for related (aka paired,
to evaluate the excretion rate (percent of dose matched, or repeated) samples as is the case for a
excreted unchanged in the urine) of different blood crossover design. These include the repeated mea-
pressure medications, they could use a 2-way sures ANOVA, which is used if there are no con-
ANOVA if (1) each sample were independent (ie, a founders and at least three related (aka paired)
parallel design), (2) there were at least two samples samples. For example, if investigators wanted to
(ie, at least two different blood pressure medications), evaluate the bioavailability of different cholesterol-
and (3) the experimental groups differed in only two lowering medications to determine Cmax, they could
factors, which for this case would be the type of use a repeated measures ANOVA if (1) each subject
blood pressure drug being used and one confounding served as his/her own control (ie, a crossover
variable (eg, differences between the groups’ renal design), (2) there were at least three samples (ie, at
function). least three different cholesterol medications), and (3)

Other types of multifactorial ANOVAs include the experimental groups differed in only one factor,
analyses of covariance (ANACOVA or ANCOVA). which for this case would be the type of cholesterol
These are used if there are at least two confounders drug being used (ie, there were no identified con-
for at least two independent (non-paired) samples as founders like fluctuations in renal function, adminis-
is the case for a parallel design. These include the tration times, etc).
3-way ANOVA, 4-way ANOVA, etc. A second type of ANOVA for related (aka

A 3-way ANOVA is used if there are two identifi- paired, matched, or repeated) samples is the 2-way
able confounders and at least two independent (aka repeated measures ANOVA, which is used if there is
non-paired) samples. For example, if investigators one identifiable confounder and at least two related
wanted to evaluate the excretion rate (percent of (aka paired) samples. For example, if investigators
dose excreted unchanged in the urine) of different wanted to evaluate the bioavailability of different
blood pressure medications, they could use a 3-way cholesterol-lowering medications to determine Cmax,
ANOVA if (1) each sample were independent (ie, a they could use a 2-way repeated measures ANOVA if
parallel design), (2) there were at least two samples (1) each subject served as his/her own control (ie, a
(ie, at least two different blood pressure medica- crossover design), (2) there were at least two sam-
tions), and (3) the experimental groups differed in ples (ie, at least two different cholesterol medica-
three factors, which for this case would be the type tions), and (3) the experimental groups differed in
of blood pressure drug being used and two con- only two factors, which for this case would be the
founding variables (eg, differences between the type of cholesterol drug being used and one con-
groups’ renal function and plasma protein binding). founding variable (eg, fluctuations in renal

A 4-way ANOVA is used if there are three iden- function).
tifiable confounders and at least two independent Beyond that, repeated measures regression
(aka non-paired) samples. For example, if investiga- analysis is used if there are two or more related (aka
tors wanted to evaluate the excretion rate (percent of paired) samples and two or more confounders. For
dose excreted unchanged in the urine) of different example, if investigators wanted to evaluate the

 

62 Chapter 3

bioavailability of different cholesterol-lowering Sometimes, otherwise parametric data are not
medications to determine Cmax, they could use a normally distributed (ie, are skewed) such that afore-
repeated measures regression analysis if (1) each mentioned parametric testing methods, t-test and the
subject served as his/her own control (ie, a crossover various types of ANOVA, would be inaccurate for
design), (2) there were at least two samples (ie, at data analysis. In these cases, investigators can loga-
least two different cholesterol medications), and (3) rithmically transform the data to normalize data
the experimental groups differed in at least three factors, distribution such that t-test or ANOVA can be used
which for this case would be the type of cholesterol for data analysis (Shargel et al, 2012).
drug being used and at least two confounding vari- When performing statistical analyses of subgroup
ables (eg, fluctuations in renal function and adminis- data sets, the term interaction or p for interaction is
tration times). often heard (Shargel et al, 2012). P for interaction (aka

ANOVA will indicate if differences exist p-value for interaction) simply detects heterogeneity or
between groups, but will not indicate where these differences among subgroups. A significant p for inter-
differences exist. For example, if an investigator is action generally ranges from 0.05 to 0.1 depending on
interested in comparing the volume of distribution of the analysis. In other words if a subgroup analysis finds
a drug among various species, both clearance and the a p for interaction <0.05 (or <0.1 for some studies) for
elimination rate constant must be considered. half-life by male versus female patients, then there is
Clearance and the elimination rate constant may be possibly a significant difference in half-life based upon
species dependent (ie, rats vs dogs vs humans) and gender. This difference may be worth investigating in
thus, they are expected to produce different out- future analyses. Just as with other types of subgroup
comes (ie, volumes of distribution). However, a sta- analyses, p for interaction solely detects hypothesis-
tistically significant ANOVA does not point to where generating differences. However, if multiple similar
these differences exist. To find where the differences studies are available, a properly performed meta-
lie, post hoc multiple comparison methods must be analysis may help answer the question of gender and
performed. half-life differences.

Multiple comparison methods are types of post
hoc tests that help determine which groups in a statis-
tically significant ANOVA analysis differ (Gaddis Pharmacokinetic Study Example

and Gaddis, 1990d). These methods are based upon Incorporating Parametric Statistical

the t-test but have built-in corrections to keep a level Testing Principles
constant when >1 comparison is being made. In other The t½ of phenobarbital in a population is 5 days with
words, these help control for type 1 error rate for a standard deviation of 0.5 days. A clinician observed
multiple comparisons (Gaddis and Gaddis, 1990d). that patients who consumed orange juice 2 hours prior

Examples include (1) least significant difference, to dosing with phenobarbital had a reduction in their
which controls individual type 1 error rate for each t½ by 10%. To test this hypothesis, the clinician
comparison, (2) layer (aka stepwise) methods, which selected a group of 9 patients who were already taking
gradually adjust the type 1 error rate and include phenobarbital and asked them to drink a glass of
Newman-Keuls and Duncan, and (3) experiment-wise orange juice 2 hours prior to taking the medication.
methods, which hold type 1 error rate constant for a The average calculated t½ value from this sample of
set of comparisons and include Dunnett, which tests 12 patients was 4.25 days. The clinician has to decide
for contrasts with a control only; Dunn, which tests from the results obtained from the study whether
for small number of contrasts; Tukey, which tests for orange juice consumption decreases the value of t½.
a large number of contrasts when no more than two Assuming that alpha was 0.05 (5%), there are several
means are involved; and Scheffe, which tests for a ways to reach the conclusion. Based on the statement
large number of contrasts when more than two means of the null hypothesis, “drinking orange juice 2 hours
are involved (Gaddis and Gaddis, 1990d). prior to taking phenobarbital does not affect t½ of the

 

Biostatistics 63

drug” (remember that H0 is a statement of no differ- It is a way to describe the “agreement between model
ence, meaning that whether orange juice was or was and data” (Anonymous, 2003). This is done by plot-
not consumed the t½ of phenobarbital is the same), the ting the residuals (RES; the difference between
conclusion of the test is written with respect to H0. observed and predicted values) versus predicted
The alternative hypothesis is that “orange juice lowers (PRED) data points. In addition to this plot, GOF
the t½ value of phenobarbital.” The alternative hypoth- analysis includes other plots such as PRED versus
esis has the symbol of H1 or Ha. One way to analyze observed (OBS) or PRED versus time (Brendel et al,
the result is to calculate a p-value for the test (Ferrill 2007). GOF methodology is often used in population
and Brown, 1994). The p-value is the exact probability pharmacokinetic studies. For example, the pharma-
of obtaining a test value of 4.25 days or less, given cokinetic profile of the antiretroviral drug nelfinavir
that H0: μ0 = 5 days: and its active metabolite M8 was investigated with

the aim of optimizing treatment in pediatric popula-
Pr. [y-bar ≤ 4.25 µ0 = 5] (3.8) tion (Hirt et al, 2006). The authors used GOF in their

assessment of the proposed pharmacokinetic models
Equation 3.8 can be evaluated by standardizing

to compare the population predicted versus the
the data using a standard normal curve (this curve

observed nelfinavir and M8 concentrations.
has an average of μ = 0 and a standard deviation of
s = 1):

STATISTICAL INFERENCE
Pr. [z ≤ (y-bar − m)/s/(n)0.5] (3.9)

TECHNIQUES FOR HYPOTHESIS
Pr. [z ≤ (4.25 − 5)/0.5/(9)0.5] TESTING WITH NONPARAMETRIC

= Pr. [z ≤ −1.28] = 10.03%
Or DATA

Nonparametric statistical methods are used for analyz-
p = 0.1003

ing data that are not normally distributed and cannot be

Since the p-value for the test is greater than a of 5% defined as parametric data (Gaddis and Gaddis, 1990e).

(p > 0.05), then we conclude that drinking orange For nominal data, the most common tests for propor-

juice 2 hours prior to taking phenobarbital dose does tions and frequencies include chi-square (c 2) and

not decrease the value of t½. It should be noted that Fisher’s exact. These tests are “used to answer ques-

the value calculated from Equation 3.9 is for a one- tions about rates, proportions, or frequencies” (Gaddis

tailed test. In order to calculate the p-value for a and Gaddis, 1990e). Fisher’s exact test is only used for

two-tailed test, the value computed from Equation 3.9 very small data sets (N ≤ 20). Chi-square (c 2) is used

is multiplied by 2 (p = 2 × 0.1003 = 0.2006). for all others. For matrices that are larger than 2 × 2, c 2
While z-test and t-test are used for one-sample tests will detect difference(s) between groups, but will

and two-sample comparisons, they cannot be used if not indicate where the difference(s) lie(s) (Gaddis and

the researcher is interested in comparing more than Gaddis, 1990e). To find this, post hoc tests are needed.

two samples at one time. As was explained earlier in These post hoc tests should only be performed if the c 2
this chapter, the parametric analysis of variance test was statistically significant. Doing otherwise will

(ANOVA) test is used to compare two or more groups increase type 1 error risk.

with respect to their means. For ordinal data, the most appropriate test
depends upon the number of groups being compared,

GOODNESS OF FIT the number of comparisons being made, and whether
the study is of parallel or crossover design. The most

The idea of “goodness of fit” (GOF) in pharmacoki- commonly used ordinal tests are Mann–Whitney U,
netic data analysis is an important concept to assure Wilcoxon Rank Sum, Kolmogorov–Smirnov, Wilcoxon
the reliability of proposed pharmacokinetic models. Signed Rank, Kruskal–Wallis, and Friedman.

 

64 Chapter 3

The procedure for utilizing all of these tests is the t-distribution are “less pinched.” The mean of the
very similar to the example provided in the paramet- t-distribution is zero, and its standard deviation is a
ric data testing section: function of the sample size (or the degrees of freedom).

The larger the sample size, the closer the value of the
1. State the null and alternate hypotheses at a

standard deviation is to 1 (recall that the standard devia-
given alpha value.

tion for the z distribution, the standard normal curve, is
2. Calculate test statistics (a computed value for

always 1). With the advances in computer technology
Chi-square or z, depending on the test being

and the availability of software programs that readily
used).

calculate these statistics, the function of the researcher
3. Compare the calculated value with a tabulated

is to enter the data in a computer database, calculate the
value.

slope, and find the p-value associated with the slope. If
4. Build a confidence interval on the true propor-

the p-value is less than a, then the slope is different
tion that is expected in the population.

from zero. Otherwise, do not reject the null hypothesis
5. Make a decision whether or not to reject the

and declare the slope is zero. Similar analysis can be
null hypothesis.

done on the y intercept using a t-test. For the signifi-
Many statistical software programs perform the cance of the regression coefficient (r), a critical value is

above tests or other similar tests found in the litera- obtained from statistics tables at a given degrees of
ture. Computer programs calculate a p-value for the freedom (n – 2), a two- or one-tailed test, and a selected
test to determine whether or not the results are sig- a value. If the observed r value equals or exceeds the
nificant. This is, of course, accomplished by compar- critical value, then r is significant (ie, reject H0 of r =
ing the computed p-value with a predetermined a 0); otherwise, r is statistically insignificant. For exam-
value. In the practice of pharmacokinetics, it is rec- ple, a calculated r value of 0.75 was computed based on
ommended to have computer software for calculating 30 pairs of x and y values. The following calculations
pharmacokinetic parameters and another software are taken in the analysis:
program for statistical analysis of experimental data.

1. State the null hypothesis and alternate hypothesis:
H0: r = 0
H1: r is not equal to zero

Frequently Asked Question
Two-tailed test

»»How do nonparametric statistical tests differ from
2. State the alpha value:

parametric statistical test regarding power?
a = 0.05

3. Find the critical value of r (tables for this may

Least Squares method be found in statistical textbooks):
Degrees of freedom = n – 2 = 30 – 2 = 28

Statistical testing is also applicable to the linear least
Critical value = 0.361

squares method (Gaddis and Gaddis, 1990f; Ferrill and
4. Since the calculated value (r = 0.75) is greater

Brown 1994). In this instance, the analysis focuses on
than 0.361, then the null hypothesis is rejected

whether the slope of the line is different from zero as a
5. A linear relationship exists between variables x

slope of zero means that no linear relationship exists
and y

between the variables x and y. To that end, testing for
the significance of the slope (a statistically significant Another way to test the significance of r is to build a
test is that when the H0 is rejected; an insignificant confidence interval on the true value of r in the popu-
result means that the null hypothesis is not rejected) lation. The procedure for this test includes the fol-
requires the use of a Student’s t-test. This test replaces lowing steps:
the z distribution whenever the standard deviation of
the variable in the population is unknown (ie, s is 1. Convert the observed r value to zr value, also
unknown). The t-test uses a bell-shaped distribution known as Fisher’s z:
similar to that of the z distribution; however, the tails of r = 0.75, then zr = 0.973

 

Biostatistics 65

2. Fisher’s z distribution has a bell-shaped distri- bias occurs when investigators select included and/or
bution with a mean equal to zero and a standard excluded samples or data. Diagnostic or detection
error of the mean (SE) equal to [1/(n – 3)0.5]: bias can occur when outcomes are detected more or

SE = [1/(30 – 3)0.5] = 0.192 less frequently. For example, this can be from changes
in the sensitivity of instruments used to detect drug

3. Construct a confidence interval on the true concentrations. Observer or investigator bias may
value of Fisher’s z in the population: occur when an investigator favors one sample over

95% CIZr = Zr ± 1.96 (SE) another. This is most problematic with “open” or
unblinded study designs. Misclassification bias may

95% CIZr = 0.973 ± 1.96 (0.192)
occur when samples are inappropriately classified and

95% CIZr = [0.60, 1.35]
may bias in favor of one group over another or in

4. Convert the interval found in (3) above to a favor of finding no difference between the groups.
confidence interval on the true value of r in the Bias can also occur when there is a significant dropout
population: rate or loss to follow-up such that data collection is

incomplete. Channeling bias is sometimes called con-
95% CIr = [0.54, 0.88]

founding by indication and can occur when one group
5. If the interval in step (4) contains the value or sample is “channeled” into receiving one treatment

of zero, then do not reject the null hypothesis over another.
(H0: the true value of r in the population is Bias is minimized through a combination of
zero); otherwise, reject H0 and declare that r proper study design, methods, and analysis. Proper
is statistically significantly different from zero analysis cannot “de-flaw” a study with poor design
(this indicates that a linear relationship exists or methodology (DeYoung, 2000). There are several
between the variables x and y): means of minimizing bias. Randomization is some-

times referred to as allocation. In this process, sam-
Since the 95% CI does not contain the value

ples are divided into groups by chance alone such
zero, reject the null hypothesis and conclude

that potential confounders are divided equally among
that r is statistically significant.

the groups and bias is minimized. Doing so helps
ensure that all within a studied sample have an equal

Accuracy Versus Precision and independent opportunity of being selected as

“Accuracy refers to the closeness of the observation to part of the sample. This can be carried a step further

the actual or true value. Precision (or reproducibility) in that once the subject has been selected for a sample,

refers to the closeness of repeated measurements” he/she has an equal opportunity of being selected for

(Shargel et al, 2012). any of the study arms. An example of simple ran-
domization would be drawing numbers from a hat.
Its advantage is that it is simple. Its disadvantage is

Error Versus Bias that if a study were stopped early, there is no assur-
Error occurs when mistakes that neither systemati- ance of similar numbers of subjects in each group at
cally under- nor overestimate effect size are made any given point in time. Block randomization
(Drew, 2003). This is sometimes referred to as ran- involves randomizing subjects into small groups
dom error. An example would be if a coin were tossed called blocks. These blocks generally range from
10 times, yielding 8 “heads,” leading one to conclude 4 to 20 subjects. Block randomization is advanta-
that the probability of heads is 80% (Drew, 2003). geous in that there are nearly equal numbers of subjects
Bias refers to systematic errors or flaws in study in each group at any point during a study. Therefore,
design that lead to incorrect results (Drew, 2003). In if a study is stopped early, equal comparisons and
other words, bias is “error with direction” leading to more valid conclusions can be made.
systematic under- or overestimation of effect size Other means of minimizing bias include utiliz-
(Drew, 2003). There are many types of bias. Selection ing objective study endpoints, proper and accurate

 

66 Chapter 3

means of defining exposures and endpoints, accurate is/are equally efficacious, non-inferior, or superior to
and complete sources of information, proper controls “standard” treatment.
to allow investigators to minimize outside influences
when evaluating treatments or exposures, proper
selection of study subjects, which would require BLINDING
proper inclusion and exclusion criteria, minimizing

Blinding limits investigators’ treating or assessing one
loss of data, appropriate statistical tests for data analy-

group differently from another. It is especially impor-
sis, blinding as described later in this chapter, and

tant if there is any degree of subjectivity associated
matching, which involves identifying characteristics

with the outcome(s) being assessed. However, it is
that are a potential source of bias and matching con-

expensive and time consuming. There are several types
trols based upon those characteristics (DeYoung,

of blinding but we will only discuss the three most
2000, 2005; Drew, 2003).

common forms. In a single-blind study, someone, usu-
ally the subject, but in rare cases it may be the investi-
gator, is unaware of what treatment or intervention the

CONTROLLED VERSUS subject is receiving. In a double-blind study, neither the
NONCONTROLLED STUDIES investigator nor the subject is aware of what treatment

or intervention the subject is receiving. In a double-
Uncontrolled studies do not utilize a control group

dummy study, if one is comparing two different dosage
such that outside influences may affect study results.

forms (eg, intranasal sumatriptan vs injectable sumat-
Using controls helps minimize bias through keeping

riptan), and doesn’t want the patient or investigator to
study groups as similar as possible and minimizing

know in which arm a patient is participating, then one
outside influences. Ideally, groups will differ only in

group would receive intranasal sumatriptan and a placebo
the factor being studied. There are many types of

injection and the other group would receive intranasal
controls. “Utilizing a placebo control is not always

placebo and a sumatriptan injection. Another example
practical or ethical, but one or more groups receive(s)

would be for a trial evaluating a tablet versus an inhaler.
active treatment(s) while the control group receives

Some trials that claim to be blinded are not. For example,
a placebo” (Drew, 2003). Historical control studies

a medication may have a distinctive taste, physiologic
are generally less expensive to perform but this

effect, or adverse effect that un-blinds patients and/or
design introduces problems with diagnostic, detec-

investigators.
tion, and procedure biases. “Data from a group of
subjects receiving the experimental drug or interven-
tion are compared to data from a group of subjects CONFOUNDING
previously treated during a different time period,
perhaps in a different place” (Herring, 2014). Confounding occurs when variables, other than the
Crossover control is very efficient at minimizing one(s) being studied, influence study results.
bias while maximizing power when used appropri- Confounding variables are difficult to detect some-
ately. Each subject serves as his/her own control. times and are linked to study outcome(s) and may be
Initially, group A receives the experimental drug linked to hypothesized cause(s). As discussed in
while group B receives the control (placebo or gold more detail later in this chapter, validity of a study
standard treatment). After a washout period, group A depends upon how well investigators minimize the
receives the control and group B receives the experi- influence of confounders (DeYoung, 2000).
mental drug. Standard treatment (aka active treat- For example, atherosclerosis and myocardial
ment) control is very practical and ethical. The infarction (MI): There is an association between
control group receives “standard” treatment while atherosclerosis and smoking, smoking and risk for
the other group(s) receives experimental treatment(s). an MI, and atherosclerosis and risk for an MI. The
This type of control is used when the investigator proposed cause is atherosclerosis and the potential
wishes to demonstrate that the experimental treatment(s) confounder is smoking.

 

Biostatistics 67

Proposed cause Confounder the national cholesterol guidelines utilize multiple
(atherosclerosis) (smoking)

regression to help establish atherosclerotic cardio-
vascular disease (ASCVD) risk for patients based
upon population data. A patient’s ASCVD risk is the

Outcome studied dependent variable because its estimate “depends
(heart attack) upon” several independent variables. The indepen-

dent variables include gender, race, age, total choles-
Another example of confounding is the relation-

terol, HDL-cholesterol, smoking status, systolic
ship between fasting blood glucose (FBG) in patients

blood pressure, and whether or not a patient is being
being treated for diabetes with medication. One

treated for hypertension, or has diabetes. All of these
confounding factor on their FBG is their diet. For

independent variables are used to help predict a
example, dietary cinnamon consumption can lower

patient’s ASCVD risk. Similar factors to those listed
blood glucose. If patients regularly consume cinna-

above can influence a multitude of pharmacokinetic
mon, FBG could be lowered beyond the diabetic

parameters as well.
medication’s capabilities. In this case, although cin-

As previously discussed, various types of
namon may not affect the proposed cause (type of

ANOVAs help account for confounding: multivariate
diabetes medication that is being used), it very well

ANOVAs for non-paired data, and two-way repeated
may affect FBG concentrations, possibly resulting in

measures ANOVA for paired data.
biased results by augmenting the diabetes drug’s
FBG lowering effect, and therefore affecting its
pharmacodynamic profile. VALIDITY

Proposed cause Confounder Internal validity addresses how well a study was
(diabetes drug) (cinnamon intake) conducted: if appropriate methods were used to

minimize bias and confounding and ensure that
exposures, interventions, and outcomes were mea-
sured correctly (DeYoung, 2000). This includes

Outcome studied
(fasting blood glucose) ensuring the study accurately tested and measured

what it claims to have tested and measured (DeYoung,
As with bias, confounding is minimized through 2000; Anonymous, 2003). Internal validity directly

the combination of proper study design and method- affects external validity; without internal validity, a
ology, including randomization, proper inclusion study has no external validity. Presuming internal
and exclusion criteria, and matching if appropriate. validity, external validity addresses the application
However, unlike bias, confounding may also be of study findings to other groups, patients, systems,
minimized through proper statistical analysis. or the general population (DeYoung, 2000; Drew,
Stratification separates subjects into nonoverlapping 2003). “A high degree of internal validity is often
groups called strata, where specific factors (eg, gen- achieved at the expense of external validity” (Drew,
der, ethnicity, race, smoking status, weight, diet) are 2003). For example, excluding diabetic hypertensive
evaluated for any influence on study results (DeYoung, patients from a study may provide very clean statisti-
2000). “Stratification has limits” (Herring, 2014). As cal endpoints. However, clinicians who treat mainly
one stratifies, subgroup sample sizes decrease, so one’s diabetic hypertensive patients may be unable to uti-
ability to detect meaningful influences in each sub- lize the results from such a trial (Drew, 2003).
group will also decrease.

Multivariate (or multiple) regression analysis
(MRA) is a possible solution (DeYoung, 2000). With Frequently Asked Question
MRA, “multiple predictor variables (aka indepen- »»Are there any types of statistical tests that can be
dent variables) can be used to predict outcomes (aka used to correct for a lack of internal validity?
dependent variables)” (Herring, 2014). For example,

 

68 Chapter 3

BIOEQUIVALENCE STUDIES nominal outcomes, but in rare cases may be applied
to ordinal outcomes. The following calculations for

“Statistics have wide application in bioequiva- cohort and randomized controlled trial (RCT) are the
lence studies for the comparison of drug bio-

same, but nomenclature is different. For a cohort
availability for two or more drug products. The

study, the exposed group is referred to as such. For an
FDA has published Guidance for Industry for the
statistical determination of bioequivalence (1992, RCT, the exposed group may be referred to as the

2001) that describes the comparison between a interventional, experimental, or treatment group. For

test (T) and reference (R) drug product. These a cohort study, the unexposed group is referred to as
trials are needed for approval of new or generic such. For an RCT, the unexposed group is referred to
drugs. If the drug formulation changes, bio- as the control group. For the following examples, the
equivalence studies may be needed to compare subscript “E” will refer to the exposed or experimen-
the new drug formulation to the previous drug tal (treatment, interventional) group and the subscript
formulation. For new drugs, several investiga- “C” will refer to the unexposed or control group.
tional formulations may be used at various Absolute risk (AR) is simply another term for
stages, or one formulation with several strengths

incidence. It is the number of new cases that occur
must show equivalency by extent and rate

during a specified time period divided by the number
(eg, 2 × 250-mg tablet vs 1 × 500-mg tablet,
suspension vs capsule, immediate-release vs of subjects initially followed to detect the outcome(s)

extended-release product). The blood levels of of interest (Gaddis and Gaddis, 1990c).

the drug are measured for both the new and the
reference formulation. The derived pharmacoki- Number who develop
netic parameters, such as maximum concentra- the outcome of interest

during a specified time period
tion (Cmax) and area under the curve (AUC), must AR =
meet accepted statistical criteria for the two Number initially followed to detect

the outcome of interest
drugs to be considered bioequivalent. In bio-
equivalence trials, a 90% confidence interval of (3.10)
the ratio of the mean of the new formulation to Absolute risk reduction (ARR) is a measure of the
the mean of the old formulation (Test/Reference) absolute incidence differences in the event rate
is calculated. That confidence interval needs to between the studied groups. Absolute differences are
be completely within 0.80–1.25 for the drugs to more meaningful than relative differences in out-
be considered bioequivalent. Adequate power comes when evaluating clinical trials (DeYoung,
should be built into the design and validated

2005). When outcomes are worse for the experimental
methods used for analysis of the samples.

group, the absolute risk difference is termed absolute
Typically, both the rate (reflected by Cmax) and
the extent (AUC) are tested. The ANOVA may risk increase (ARI).

also reveal any sequence effects, period effects, ARR (or ARI) = ARC – ARE (3.11)
treatment effects, or inter- and intrasubject vari-
ability. Because of the small subject population Numbers needed to treat (NNT) is the “reciprocal of
usually employed in bioequivalence studies, the the ARR” (DeYoung, 2000).
ANOVA uses log-transformed data to make an

1
inference about the difference of the two groups” NNT = (3.12)

ARR
(Shargel et al, 2012).

When outcomes are worse for the experimental
group, there is an ARI and this calculation is referred

EVALUATION OF RISK FOR to as numbers needed to harm (NNH).

CLINICAL STUDIES 1
NNH =

ARI (3.13)
Risk calculations estimate the magnitude of associa-
tion between exposure and outcome (DeYoung, These calculations help in understanding the magnitude
2000). These effect measurers are mainly used for of an intervention’s effectiveness (DeYoung, 2000).

 

Biostatistics 69

A weakness of these is that they “assume baseline utilizing logistic or multivariate regression analysis
risk is the same for all patients or that it is unrelated simply because these analyses automatically calculate
to relative risk” (DeYoung, 2000). Although rarely OR. They do so because regression analysis is utilized
seen, “confidence intervals (CIs) may be calculated to adjust for confounding and adjustments are easier
for NNT and NNH” (DeYoung, 2005). to perform with OR than with RR (De Muth, 2006).

Relative risk (RR) compares the AR (incidence) OR is presented differently for case-control studies
of the experimental group to that of the control than for RCTs. For RCTs, OR is presented in the
group (DeYoung, 2000). It is simply a ratio of the same way as RR. For example, in an RCT evaluating
AR for the experimental or exposed group to the AR an association of an intervention and death rate, an
of the control or unexposed group. RR is sometimes OR of 0.75 would be reported as patients receiving
called risk ratio, rate ratio, or incidence rate ratio. the intervention were 25% less likely, or 75% as

AR likely, to have died than controls. Since case-control
E

RR = (31.4)
AR studies identify patients based upon disease rather

C

Relative risk differences are sometimes presented in than intervention, OR is presented differently than for
studies and these estimate the percentage of baseline an RCT; it compares the odds that a case was exposed
risk that is changed between the exposed or experi- to a risk factor to the odds that a control was exposed to
mental group and the unexposed or control group. a risk factor. For example, in a case-control study evalu-
The relative risk difference is termed relative risk ating an association of a rare type of cancer and expo-
reduction (RRR) when risk is decreased. The relative sure to pesticides, an OR of 1.5 would be reported as
risk difference is termed relative risk increase (RRI) cases (those with the rare cancer) were 50% more likely,
when risk is increased. RRR and RRI can be calcu- or 1½ times as likely, to have been exposed to pesticides
lated in two different ways: than controls. CIs should always be provided for RR,

OR, and HR.
RRR (or RRI) = 1 – RR (3.15)

These above calculations and principles are com-
or monly utilized for interpreting data in FDA-approved

ARR (or ARI)
RRR (or RRI) = (3.16) package inserts. For example, in the Coreg®

ARC (carvedilol) package insert, there are several major
Hazard ratio (HR) is used with Cox proportional studies that are presented. The Copernicus trial evalu-

hazards regression analysis. It is used when a study is ated carvedilol’s efficacy against that of placebo for
evaluating the length of time required for an outcome patients with severe systolic dysfunction heart failure
of interest to occur (Katz, 2003). HR is often used over a median of 10 months (GlaxoSmithKline,
similarly to RR, and is a reasonable estimate of RR 2008). The primary endpoint of mortality occurred in
as long as adequate data are collected and outcome 190 out of 1133 patients taking placebo and 130 out
incidence is <15% (Katz, 2003; Shargel et al, 2012). of 1156 patients taking carvedilol. This means that the
However, whereas RR only represents the probability AR for patients taking placebo was 190/1133 = 0.17
of having an event between the beginning and the end or 17% and the AR for patients taking carvedilol was
of a study, HR can represent the probability of having 130/1156 = 0.11 or 11%. The RR would be 0.11/0.17
an event during a certain time interval between the or 11%/17% = 0.65 or 65%. RRR would be 1 – 0.65
beginning and the end of the study (DeYoung, 2005). = 0.35 or 35%. Therefore, patients treated with

Odds ratio (OR) is mainly used in case-control carvedilol were 35% less likely to die than were
studies as an estimate of RR since incidence cannot be patients treated with placebo. However, sometimes
calculated. Estimation accuracy decreases as outcome RR and RRR can be deceptive, so one should always
or disease incidence increases. However, OR is fairly calculate the ARR or ARI and NNT or NNH. In this
accurate as long as disease incidence is <15%, which case, carvedilol improved the death rate, so one would
is usually the case since case-control studies evaluate calculate ARR and NNT. The AAR is simply the dif-
potential risk factors for rare diseases (Katz, 2003). In ference between the AR of each agent: 17% – 11% =
addition, OR is sometimes reported for RCTs 6% or 0.17 – 0.11 = 0.06. NNT is the reciprocal of

 

70 Chapter 3

ARR, so 1/0.06 = 17. Therefore, since the median Frequently Asked Question

follow-up of this trial was 10 months, one would need »»Which are more important: relative or absolute
to treat 17 patients for 10 months with carvedilol differences?

rather than placebo to prevent 1 death.

CHAPTER SUMMARY

Statistical applications are vital in conducting and In this chapter, we have presented very basic,
evaluating biopharmaceutical and pharmacokinetic practical principles in hopes of guiding the reader
research. Utilization includes, but is not limited to, throughout the research process. For readers who
studies involving hypothesis testing, finding ways to are interested in learning about this topic in more
improve a product, its safety, or performance. Proper depth, we recommend statistics textbooks or
statistics are required for experimental planning, online resources and/or taking a research-based
data collection, analysis, and interpretation of results, statistics course at the college or university of their
allowing for rational decision making throughout choosing.
these processes (Durham, 2008; Shargel et al, 2012).

LEARNING QUESTIONS

The column for concentration (ng/mL) refers to
1. The following data represent the concentration

the concentration of vitamin C in infant urine.
of vitamin C in infant urine:

Calculate the arithmetic mean for vitamin C in
the urine.

Age (months) Gender Conc. (ng/mL) 2. Refer to Question 1; find the standard deviation
for the concentration of vitamin C in urine for

1 F 2.7
the male infants.

3 F 2.8 3. Refer to Question 1; find the coefficient of
4 M 2.9 variation (%) value for the variable age.

4. Refer to Question 1; consider the following
6 M 2.9

graph representing the data:
7 M 2.3

3
9 M 2.3

2.5
12 F 1.5

2
15 F 1.1

16 M 1.3 1.5

17 F 1.3 1

18 F 1.1 0.5

24 F 1.5
0

0 5 10 15 20 25 30 35
25 F 1.0

Age (months)
29 M 0.4

Based on the above graph, the value for the
30 F 0.2

correlation coefcient is most likely_______.

Vitamin C urine
concentration (ng/mL)

 

Biostatistics 71

5. Refer to Questions 1 and 4. The older the 9. Which statistics did you use in answering
infant, the _______ is the concentration of Question 8?
vitamin C in the urine. 10. Investigators want to perform a study comparing

6. The p-value associated with the slope of the two doses of an investigational anticoagulant
line in Question 4 is less than 0.0001 for prevention of thromboembolism. They
(p < 0.0001). For a of 5%, the slope value is calculate that a sample size of 400 subjects
statistically _______. (200 in each arm) will be needed to show a dif-

7. Find the slop value for the graph in Question 4. ference (based upon an alpha of 0.05 and beta
8. The following results were presented by Chin of 0.20). They predict that given the patient

KH, Sathyyasurya DR, Abu Saad H, Jan population, approximately 50% of subjects will
Mohamed HJB: Effect of ethnicity, dietary drop out of the study. Based upon the dropout
intake and physical activity on plasma rate, how many subjects will be needed in each
adiponectin concentrations among Malaysian treatment arm?
patients with type 2 diabetes mellitus. Int 11. A superiority trial evaluating the doses of a
J Endocrinol Metab 11(3):16–174, 2013. new cholesterol medication was performed
DOI:10.5812/ijem.8298.) (Copyright © 2013, comparing AUC. There were 200 patients in
Research Institute for Endocrine Sciences this trial and differences were statistically
and Iran Endocrine Society; Licensee Kowsar significant. Was this study underpowered?
Ltd. This is an Open Access article distributed 12. A study is planned to evaluate differences
under the terms of the Creative Commons in half-life (t½) of three different metoprolol
Attribution License [http://creativecommons. formulations. The investigators plan to include
org/licenses/by/3.0], which permits unre- 150 subjects (50 in each arm) to reach statisti-
stricted use, distribution, and reproduction in cal significance based upon a beta of 0.20 and
any medium, provided the original work is alpha of 0.05. Which statistical test would
properly cited): be the most appropriate? (Hint: Assume no

confounders.)

Malay Chinese Indian 13. If you conduct a pharmacokinetic study that
utilizes appropriate methodology and a broad

Adiponectin 6.85 (4.66) 6.21 (3.62) 4.98 (2.22) population base for inclusion, how will this
(μg/mL)

affect the strength of internal and external
(Chin KH, Sathyyasurya DR, Abu Saad H, Jan Mohamed HJB: Effect of validity?
ethnicity, dietary intake and physical activity on plasma adiponectin

14. Investigators wish to study the differences in
concentrations among Malaysian patients with type 2 diabetes mellitus.
Int J Endocrinol Metab 11(3):16–174, 2013.) patients with subtherapeutic concentrations

of vancomycin via two difference delivery
The concentration of adiponectin (a protein systems. The results from this 2-week study
produced by adipocytes) in plasma is reported are listed below:
in Malaysian patients with three different ethnici-
ties. The values in the table above are given as
arithmetic mean (standard deviation). The signi- Formulation A Formulation B
cance of adiponectin plasma concentration is that (FA) (n = 55) (FB) (n = 62)

its plasma levels correlate well with the clinical
Subtherapeutic 35 17

response to administered insulin in patients with vancomycin
type 2 diabetes. Referring to the results above, concentrations
which group of patients is more variable with
respect to its mean than the other two groups? How should these results be reported?

 

72 Chapter 3

ANSWERS
Learning Questions 11. Power is associated with beta: power = 1 – beta.

Beta is the risk of committing a type 2 error. If
1. Using a scientific calculator, the arithmetic mean a statistically significant difference is detected,

for vitamin C in infant urine was 1.69 ng/mL. a type 2 error could not occur. Therefore, the
2. Using a scientific calculator, the standard devia- trial was not underpowered. With this scenario,

tion of vitamin C in urine for male infants was there are only two possibilities: either (1) the
0.98 ng/mL. findings were correct or (2) a type 1 error

3. The coefficient of variation (%) for age was occurred.
(SD/mean) × 100 = (9.49/14.4) × 100 = 66% 12. Differences in half-life (t½) are parametric data

4. The slope of the line depicted in the graph was since they are scored on a continuum and there
negative. Therefore, the correlation coefficient is a consistent level of magnitude of differ-
must be a negative value. ence between data units. Since there are three

5. A negative linear relationship was observed metoprolol formulations being evaluated, and
between age of infants and the concentration of no identified confounders, a 1-way ANOVA is
vitamin C in urine. Thus, the vitamin C concen- appropriate. If there were only two groups and
tration in urine in older infants would be lower no identified confounders, a t-test would be
than that found in younger infants. appropriate.

6. Since p-value is less than 0.05, the results were 13. Utilizing appropriate methodology helps
statistically significant. increase internal validity. Including a broad

7. The slope of the line is negative. The value of the population helps increase external validity.
slope may be obtained by a scientific calculator. 14. There are several ways the results could be

8. The coefficient of variation (%) for Malay, Chi- reported. ARFA = 35/55 = 0.64 or 64%, ARFB =
nese, and Indian patients was 68.03%, 58.29%, 17/62 = 0.27 or 27%, ARI = ARC – ARE = 0.27
and 44.58%, respectively. Recall that CV (%) = – 0.64 = 0.37 or 37%, NNH= 1/0.37 = 2.7, so
(SD/mean) × 100. Since Malay patients had the 3 patients over 2 weeks. In other words, one
highest CV (%) value, then adiponectin plasma would need to treat 3 patients over 2 weeks
concentration was more variable with respect to with formulation A rather than formulation B to
its mean than the other two values. cause one episode of a subtherapeutic vanco-

9. The coefficient of variation (%). mycin concentration. RR = 0.64/0.27 = 2.3.

10. Sample size (corrected for drop-outs) The results could be reported as those utilizing
formulation A were 2.3 times as likely to be

Number of patients subtherapeutic as those being given formula-
=
1–% of expected drop-outs tion B. Since RRI = 1 – RR = 1 – 2.3 = –1.3,

another way of explaining the results would be
200 in each arm/(1 – 0.5) = 200/0.5 = 400 in

that those utilizing formulation A were 130%
each treatment arm. If the question had asked

more likely to have subtherapeutic vancomycin
how many total subjects would be needed

concentrations than those being given
(ie, both arms), the answer would have been

formulation B.
400/(1 – 0.5) = 400/0.5 = 800.

 

Biostatistics 73

REFERENCES
Anonymous: SOP 13 : Pharmacokinetic data analysis. Onkologie Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 3, Sen-

26(suppl 6):56–59, 2003. sitivity, specificity, predictive value, and hypothesis testing.
Al-Achi A, PhD. Discussions. Ann Emerg Med 19(5):591–597, 1990c.
Berensen NM: Biostatistics review with lecture for the MUSC/ Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 4, Statis-

VAMC BCPS study group, Charleston, SC, July 17, 2000. tical inference techniques in hypothesis testing. Ann Emerg Med
Brendel K, Dartois C, Comets E, et al: Are population 19(7):820–825, 1990d.

pharmacokinetic-pharmacodynamic models adequately evalu- Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 5, Statisti-
ated? A survey of the literature from 2002 to 2004. Clin Phar- cal inference techniques for hypothesis testing with nonparamet-
macokinet 46(3):221–234, 2007. ric data. Ann Emerg Med 19(9):1054–1059, 1990e.

De Muth JE: In Chow SC (ed). Basic Statistics and Pharmaceutical Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 6,
Statistical Applications, 2nd ed. Boca Raton, London, New York, Correlation and regression. Ann Emerg Med 19(12):1462–1468,
Chapman & Hall/CRC, Taylor & Francis Group, 2006. 1990f.

DeYoung GR: Clinical Trial Design (handout). 2000 Updates in Glasner AN: High Yield Biostatistics. PA, Williams & Willkins,
Therapeutics: The Pharmacotherapy Preparatory Course, 2000. 1995.

DeYoung GR: Understanding Statistics: An Approach for the Clini- GlaxoSmithKline. Coreg CR [package insert], https://www.gsk-
cian. Science and Practice of Pharmacotherapy PSAP Book 5, source.com/gskprm/htdocs/documents/COREG-CR-PI-PIL.
5th ed. American College of Clinical Pharmacy, Kansas City, PDF. Research Triangle Park, NC, 2008.
MO, 2005. Herring C: Quick Stats: Basics for Medical Literature Evaluation,

Drew R: Clinical Research Introduction (handout). Drug Literature 5th ed. Massachusetts, USA, Xanedu Publishing Inc., 2014.
Evaluation/Applied Statistics Course. Campbell University Hirt D, Urien S, Jullien V, et al: Age-related effects on nelfina-
School of Pharmacy, 2003. vir and M8 pharmacokinetics: A population study with 182

Durham TA, Turner JR: Introduction to Statistics in Pharmaceutical children. Antimicrob Agents Chemother 50(3):910–916,
Clinical Trials. London, UK, Pharmaceutical Press, RPS Pub- 2006.
lishing, 2008. Katz MH: Multivariable analysis: A primer for readers of medical

Ferrill JM, Brown LD: Statistics for the Nonstatistician: A System- research. Annal Internal Med 138:644–650, 2003.
atic Approach to Evaluating Research Reports. US Pharmacist Munro HB: Statistical Methods for Health Care Research.
July:H-3-H-16, 1994. Lippincott Williams & Wilkins, Philadelphia, PA, 2005.

Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 1, Basic Drew R, PharmD, MS, BCPS. Discussions and provisions.
concepts. Ann Emerg Med 19(1):86–89, 1990a. Shargel L, Wu-Pong S, Yu A: Statistics. Applied Biopharma-

Gaddis ML, Gaddis GM: Introduction to biostatistics: Part 2, ceutics and Pharmacokinetics, 6th ed. New York, NY, USA,
Descriptive statistics. Ann Emerg Med 19(3):309–315, 1990b. McGraw-Hill, 2012, Appendix.

 

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One-Compartment Open

4 Model: Intravenous Bolus
Administration
David S.H. Lee

Chapter Objectives While the oral route of drug administration is the most convenient,
intravenous (IV) administration is the most desirable for critical

»» Describe a one-compartment
care when reaching desirable drug concentrations quickly is

model, IV bolus injection.
needed. Examples of when IV administration is desirable include

»» Provide the pharmacokinetic antibiotic administration during septic infections or administration
terms that describe a one- of antiarrhythmic drugs during a myocardial infarction. Because
compartment model, IV bolus pharmacokinetics is the science of the kinetics of drug absorption,
injection, and the underlying distribution, and elimination, IV administration is desirable in
assumptions. understanding these processes since it simplifies drug absorption,

»» Explain how drugs follow one- essentially making it complete and instantaneous. This leaves only

compartment kinetics using the processes of drug distribution and elimination left to study. This

drug examples that follow one- chapter will introduce the concepts of drug distribution and elimi-

compartment kinetics. nation in the simplest model, the one-compartment open model.
The one-compartment open model assumes that the body can

»» Calculate pharmacokinetic be described as a single, uniform compartment (ie, one compart-
parameters from drug ment), and that drugs can enter and leave the body (ie, open
concentration–time data using a model). The simplest drug administration is when the entire drug is
one-compartment model. given in a rapid IV injection, also known as an IV bolus. Thus, the

»» Simulate one-compartment one-compartment open model with IV bolus administration is the
plasma drug level graphically simplest pharmacokinetic model. It assumes that the drug is admin-
using the one-compartment istered instantly into the body, it is instantaneously and rapidly
model equation. distributed throughout the body, and drug elimination occurs

immediately upon entering the body. This model is a simplistic
»» Calculate the IV bolus dose

representation of the processes in the body that determine drug
of a drug using the one-

disposition, but nonetheless, it can be useful to describe and predict
compartment model equation.

drug disposition.
»» Relate the relevance of the In reality, when a drug is administered intravenously, the drug

magnitude of the volume of travels through the bloodstream and distributes throughout the
distribution and clearance of bloodstream in the body. While this process is not truly instanta-
various drugs to underlying neous, it is relatively rapid enough that we can make this assump-
processes in the body. tion for most drugs. Through the bloodstream, the drug is

»» Derive model parameters from distributed to the various tissue organs in the body. The rate and

slope and intercept of the extent of distribution to the tissue organs depends on several pro-

appropriate graphs. cesses and properties. Tissues in the body are presented the drug at
various rates, depending on the blood flow to that organ, and the
drug may have different abilities to cross from the vasculature to

75

 

76 Chapter 4

the organ depending on the molecular weight of the k

drug. Tissues also have different affinity for the drug, IV DB, VD

depending on lipophilicity and drug binding. Finally,
large organs may have a large capacity for drugs to FIGURE 4-1 Pharmacokinetic model for a drug admin-
distribute to. istered by rapid intravenous injection. DB = drug in body; VD =

While drug distribution is complex, if these pro- apparent volume of distribution; k = elimination rate constant.

cesses are rapid enough, we can simplify our con-
ceptualization as if the drug uniformly distributes
into a single (one) compartment of fluid. The volume
of this single compartment is termed the apparent the plasma, but does not predict the concentrations in

volume of distribution, VD. The apparent volume of tissues. However, using this model, which assumes

distribution is not an actual volume in the body, but distribution to tissues is rapid, we can assume the

is a theoretical volume that the drug uniformly dis- declines in drug concentration in the plasma and tis-

tributes to immediately after being injected into the sues will be proportional. For these reasons, the one-

body. This uniform and instantaneous distribution is compartment open model is useful for predicting

termed a well-stirred one-compartment model. The concentrations in the plasma, and declines in plasma

apparent volume of distribution is a proportion concentrations will be proportional to declines in

between the dose and the concentration of the drug tissue concentrations.

in plasma, C0
p , at that time immediately after being

injected.
Most drugs are eliminated from the body by ELIMINATION RATE CONSTANT

liver metabolism and/or renal excretion. All of the The rate of elimination for most drugs from a tissue
processes of drug elimination can be described by or from the body is a first-order process, in which the
the elimination rate constant, k. The elimination rate rate of elimination at any point in time is dependent
constant is the proportion between the rate of drug on the amount or concentration of drug present at
elimination and the amount of drug in the body. that instance. The elimination rate constant, k, is a
Because the amount of drug in the body changes first-order elimination rate constant with units of
over time, the rate of drug elimination changes, but time–1 (eg, h–1 or 1/h). Generally, the injected drug is
the elimination rate constant remains constant for measured in the blood or plasma, sometimes termed
first-order elimination. This makes it convenient to the vascular compartment. Total removal or elimina-
summarize drug elimination from the body indepen- tion of the injected drug from this compartment is
dent of time or the amount of drug in the body. affected by metabolism (biotransformation) and
However, because it’s difficult to measure the amount excretion. The elimination rate constant represents
of drug in the body, DB, pharmacokineticists and the sum of each of these processes:
pharmacists also prefer to convert drug amounts to
drug concentrations in the body using the apparent

k = km + ke (4.1)
volume of distribution. Thus, the elimination rate
constant also describes the proportion between the
rate of change of drug concentration and drug con- where km = first-order rate process of metabolism
centration in the compartment. and ke = first-order rate process of excretion. There

The one-compartment open model with IV may be several routes of elimination of drug by
bolus dosing describes the distribution and elimina- metabolism or excretion. In such a case, each of
tion after an IV bolus administration and is shown in these processes has its own first-order rate constant.
Fig. 4-1. The fluid that the drug is directly injected A rate expression for Fig. 4-1 is
into is the blood, and generally, drug concentrations
are measured in plasma since it is accessible. dD

B
= −kD (4.2)

Therefore, this model predicts the concentrations in dt B

 

One-Compartment Open Model: Intravenous Bolus Administration 77

This expression shows that the rate of elimination of APPARENT VOLUME
drug in the body is a first-order process, depending

OF DISTRIBUTION
on the overall elimination rate constant, k, and the
amount of drug in the body, DB, remaining at any In general, drug equilibrates rapidly in the body.
given time, t. Integration of Equation 4.2 gives the When plasma or any other biologic compartment is
following expression: sampled and analyzed for drug content, the results are

usually reported in units of concentration instead of
−kt amount. Each individual tissue in the body may con-

log DB = + log D0 (4.
2.3 B 3)

tain a different concentration of drug due to differ-
ences in blood flow and drug affinity for that tissue.

where D 0 The amount of drug in a given location can be related
B = the drug in the body at time t and DB is

the amount of drug in the body at t = 0. When log D to its concentration by a proportionality constant that
B

is plotted against t for this equation, a straight line is reflects the apparent volume of fluid in which the

obtained (Fig. 4-2). In practice, instead of transform- drug is dissolved. The volume of distribution repre-

ing values of DB to their corresponding logarithms, sents a volume that must be considered in estimating

each value of DB is placed at logarithmic intervals on the amount of drug in the body from the concentra-

semilog paper. tion of drug found in the sampling compartment. The

Equation 4.3 can also be expressed as volume of distribution is the apparent volume (VD) in
which the drug is dissolved (Equation 4.5).
Because the value of the volume of distribution does

D D0e−kt
B = B (4.4) not have a true physiologic meaning in terms of an

anatomic space, the term apparent volume of distri-
bution is used.

The amount of drug in the body is not deter-
Frequently Asked Questions mined directly. Instead, blood samples are collected
»»What is the difference between a rate and a rate at periodic intervals and the plasma portion of blood

constant? is analyzed for their drug concentrations. The VD
»»Why does k always have the unit 1/time (eg, h–1), relates the concentration of drug in plasma (Cp) and

regardless of what concentration unit is plotted? the amount of drug in the body (DB), as in the fol-
lowing equation:

DB = VDCp (4.5)
100

D 0
B

Substituting Equation 4.5 into Equation 4.3, a
similar expression based on drug concentration in
plasma is obtained for the first-order decline of drug

0 Slope = –k
1 plasma levels:

2.3

−kt
logCp= + logC0 (4.6)

2.3 p

1 where Cp = concentration of drug in plasma at time t
0 1 2 3 4 5 and C0

p = concentration of drug in plasma at t = 0.
Time Equation 4.6 can also be expressed as

FIGURE 4-2 Semilog graph of the rate of drug elimina-
tion in a one-compartment model. C C0e−kt

p = p (4.7)

Drug in body (DB)

 

78 Chapter 4

The relationship between apparent volume, drug
concentration, and total amount of drug may be bet- where D = total amount of drug, V = total volume, and

ter understood by the following example. C = drug concentration. From Equation 4.8, which is
similar to Equation 4.5, if any two parameters are
known, then the third term may be calculated.

EXAMPLE »» »

Exactly 1 g of a drug is dissolved in an unknown vol-
ume of water. Upon assay, the concentration of this The one-compartment open model considers the

solution is 1 mg/mL. What is the original volume of body a constant-volume system or compartment.

this solution? Therefore, the apparent volume of distribution for

The original volume of the solution may be any given drug is generally a constant. If both the

obtained by the following proportion, remember- concentration of drug in the plasma and the apparent

ing that 1 g = 1000 mg: volume of distribution for the drug are known, then
the total amount of drug in the body (at the time in
which the plasma sample was obtained) may be cal-

1000mg 1mg
= culated from Equation 4.5.

xmL mL

x =1000mL Calculation of Volume of Distribution

Therefore, the original volume was 1000 mL or 1 L. In a one-compartment model (IV administration),

This is analogous to how the apparent volume of the VD is calculated with the following equation:

distribution is calculated.
Dose D0

If, in the above example, the volume of the V B
D= = (4.9)

C0 0
p C

solution is known to be 1 L, and the amount of drug p

dissolved in the solution is 1 g, what is the concen-
When C0

p is determined by extrapolation, C0
p repre-

tration of drug in the solution?
sents the instantaneous drug concentration after drug
equilibration in the body at t = 0 (Fig. 4-3). The dose

1000 mg
=1mg/mL of drug given by IV bolus (rapid IV injection) repre-

1000 mL
sents the amount of drug in the body, D0

B, at t = 0.
Because both D0

B and C0
p are known at t = 0, then the

Therefore, the concentration of the drug in the
solution is 1 mg/mL. This is analogous to calculat-
ing the initial concentration in the plasma if the 100

0
apparent volume of distribution is known. Cp

From the preceding example, if the volume
of solution in which the drug is dissolved and the
drug concentration of the solution are known, then
the total amount of drug present in the solution 10

may be calculated. This relationship between drug
concentration, volume in which the drug is dis-
solved, and total amount of drug present is given in
the following equation:

1
0 1 2 3 4 5

Dose D0
V B

= 4 8
D=

( . ) Time
C 0 0
p Cp FIGURE 4-3 Semilog graph giving the value of C 0

p by
extrapolation.

Plasma level (Cp )

 

One-Compartment Open Model: Intravenous Bolus Administration 79

apparent volume of distribution, VD, may be calcu- The calculation of the apparent VD by means of
lated from Equation 4.9. Equation 4.13 is a model-independent or noncom-

From Equation 4.2 (repeated here), the rate of partmental model method, because no pharmacoki-
drug elimination is netic model is considered and the AUC is determined

directly by the trapezoidal rule.
dDB

= −kD
dt B Significance of the Apparent Volume of

Distribution
Substituting Equation 4.5, DB = VDCp, into The apparent volume of distribution is not a true

Equation 4.2, the following expression is obtained: physiologic volume, but rather reflects the space the
drug seems to occupy in the body. Equation 4.9

dD shows that the apparent VD is dependent on C0
p , and

B
= −kV C (4.10)

dt D p thus is the proportionality constant between C0
p and

dose. Most drugs have an apparent volume of distri-

Rearrangement of Equation 4.10 gives bution smaller than, or equal to, the body mass. If a
drug is highly bound to plasma proteins or the mol-
ecule is too large to leave the vascular compartment,

dDB = −kVDCpdt (4.11) then C0
p will be higher, resulting in a smaller appar-

ent VD. For example, the apparent volume of distri-
bution of warfarin is small, approximately 0.14 L/kg,

As both k and V are constants, Equation 4.10
D much less than the total body mass. This is because

may be integrated as follows:
warfarin is highly bound to plasma proteins, making
it hard to leave the vascular compartment.

D0 ∞
∫ dD kV C d

0 B = − D ∫ t (4 12) For some drugs, the volume of distribution may
0 p .

be several times the body mass. In this case, a very
small C0

p may occur in the body due to concentration
Equation 4.12 shows that a small change in time of the drug in peripheral tissues and organs, resulting

(dt) results in a small change in the amount of drug in a large VD. Drugs with a large apparent VD are
in the body, DB. more concentrated in extravascular tissues and less

The integral ∫ C e r s n s t e A C∞

0 pdt r p e e t h U 0 , concentrated intravascularly. For example, the appar-
which is the summation of the area under the curve ent volume of distribution of digoxin is very high,
from t = 0 to t = ∞. Thus, the apparent V may also 7.0 L/kg, much greater than the body mass. This is

D

be calculated from knowledge of the dose, elimina- because digoxin binds extensively to tissues, espe-
tion rate constant, and the area under the curve cially muscle tissues. Consequently, binding of a
(AUC) from t = 0 to t = ∞. This is usually estimated drug to peripheral tissues or to plasma proteins will
by the trapezoidal rule (see Chapter 2). After integra- significantly affect the VD.
tion, Equation 4.12 becomes The apparent VD is a volume term that can be

expressed as a simple volume or in terms of percent
of body weight. In expressing the apparent VD in

D0 = kVD [AUC]0 terms of percent of body weight, a 1-L volume is
assumed to be equal to the weight of 1 kg. For

which upon rearrangement yields the following example, if the VD is 3500 mL for a subject weighing

equation: 70 kg, the VD expressed as percent of body weight is

D
V 0 3.5 kg

D = (4.13)
∞ ×100 = 5%of body weight

k [AUC]0 70 kg

 

80 Chapter 4

In the example of warfarin above, 0.14 L/kg is For each drug, the apparent VD is a constant. In
estimated to be 14% of body weight. certain pathologic cases, the apparent VD for the drug

If VD is a very large number—that is, >100% of may be altered if the distribution of the drug is
body weight—then it may be assumed that the drug changed. For example, in edematous conditions, the
is concentrated in certain tissue compartments. In total body water and total extracellular water
the digoxin example above, 7.0 L/kg is estimated to increases; this is reflected in a larger apparent VD
be 700% of body weight. Thus, the apparent VD is a value for a drug that is highly water soluble. Similarly,
useful parameter in considering the relative amounts changes in total body weight and lean body mass
of drug in the vascular and in the extravascular (which normally occur with age, less lean mass, and
tissues. more fat) may also affect the apparent VD.

Pharmacologists often attempt to conceptualize
the apparent VD as a true physiologic or anatomic
fluid compartment. By expressing the VD in terms of Frequently Asked Question
percent of body weight, values for the VD may be »»If a drug is distributed in the one-compartment model,
found that appear to correspond to true anatomic does it mean that there is no drug in the tissue?
volumes (Table 4-1). In the example above where the
VD is 5% of body weight, this is approximately the
volume of plasma, and it would be assumed that this
drug occupies the vascular compartment with very CLEARANCE
little distributing to tissues outside the vascular com-
partment. However, it may be only fortuitous that the Clearance is a measure of drug elimination from the

value for the apparent VD of a drug has the same body without identifying the mechanism or process.

value as a real anatomic volume. If a drug is to be Clearance is also discussed in subsequent chapters.

considered to be distributed in a true physiologic Clearance (drug clearance, systemic clearance, total

volume, then an investigation is needed to test this body clearance, ClT) considers the entire body or

hypothesis. compartment (in the case of a one-compartment

Given the apparent VD for a particular drug, the model) as a drug-eliminating system from which

total amount of drug in the body at any time after many elimination processes may occur.

administration of the drug may be determined by the
measurement of the drug concentration in the plasma
(Equation 4.5). Because the magnitude of the appar- Drug Clearance in the One-Compartment

ent VD is a useful indicator for the amount of drug Model

outside the sampling compartment (usually the The body may be considered a system of organs
blood), the larger the apparent VD, the greater the perfused by plasma and body fluids. Drug elimina-
amount of drug in the extravascular tissues. tion from the body is an ongoing process due to both

metabolism (biotransformation) and drug excretion
through the kidney and other routes. The mecha-

TABLE 4-1 Fluid in the Body nisms of drug elimination are complex, but collec-
tively drug elimination from the body may be

Percent of
Water Percent of Total Body quantitated using the concept of drug clearance.

Compartment Body Weight Water Drug clearance refers to the volume of plasma fluid
that is cleared of drug per unit time. Clearance may

Plasma 4.5 7.5
also be considered the fraction of drug removed per

Total extracellular water 27.0 45.0 unit time. The rate of drug elimination may be

Total intracellular water 33.0 55.0 expressed in several ways, each of which essentially
describes the same process, but with different levels

Total body water 60.0 100.0
of insight and application in pharmacokinetics.

 

One-Compartment Open Model: Intravenous Bolus Administration 81

Drug Elimination Expressed as Clearance is a concept that expresses “the rate of
Amount per Unit Time drug removal” in terms of the volume of drug in

The expression of drug elimination from the body in solution removed per unit time (at whatever drug

terms of mass per unit time (eg, mg/min, or mg/h) is concentration in the body prevailing at that time)

simple, absolute, and unambiguous. For a zero-order (Fig. 4-4B). In contrast to a solution in a bottle, the

elimination process, expressing the rate of drug drug concentration in the body will gradually decline

elimination as mass per unit time is convenient by a first-order process such that the mass of drug

because the elimination rate is constant (Fig. 4-4A). removed over time is not constant. The plasma vol-

However, drug clearance is not constant for a drug ume in the healthy state is relatively constant because

that has zero-order elimination (see Chapter 6). For water lost through the kidney is rapidly replaced

most drugs, the rate of drug elimination is a first- with fluid absorbed from the gastrointestinal tract.

order elimination process, that is, the elimination Since a constant volume of plasma (about

rate is not constant and changes with respect to the 120 mL/min in humans) is filtered through the glom-

drug concentration in the body. For first-order elimi- eruli of the kidneys, the rate of drug removal is

nation, drug clearance expressed as volume per unit dependent on the plasma drug concentration at all

time (eg, L/h or mL/min) is convenient because it is times. This observation is based on a first-order pro-

a constant. cess governing drug elimination. For many drugs,
the rate of drug elimination is dependent on the
plasma drug concentration, multiplied by a constant

Drug Elimination Expressed as
factor (dC/dt = kC). When the plasma drug concen-

Volume per Unit Time tration is high, the rate of drug removal is high, and
The concept of expressing a rate in terms of volume vice versa.
per unit time is common in pharmacy. For example, a Clearance (volume of fluid removed of drug) for
patient may be dosed at the rate of 2 teaspoonfuls a first-order process is constant regardless of the
(10 mL) of a liquid medicine (10 mg/mL) daily, or drug concentration because clearance is expressed in
alternatively, a dose (weight) of 100 mg of the drug volume per unit time rather than drug amount per
daily. Many intravenous medications are adminis- unit time. Mathematically, the rate of drug elimina-
tered as a slow infusion with a flow rate (30 mL/h) tion is similar to Equation 4.10:
of a sterile solution (1 mg/mL).

dD
A. Mass approach B 4 1 )

= −kC
d pV

( . 0
t D

Dose = 100 mg Amount eliminated/minute
Fluid volume = 10 mL = 10 mg/min
Conc. = 10 mg/mL Dividing this expression on both sides by Cp yields

Equation 4.14:

B. Clearance (volume) approach

Dose = 100 mg dD /dt −kC V
B p D

Volume eliminated/minute = (4.14)
Fluid volume = 10 mL = 1 mL/min Cp Cp
Conc. = 10 mg/mL

dD d
C. Fractional approach B / t

= −kV = −Cl (4.15)
C D

p
Dose = 100 mg Fraction eliminated/minute

Fluid volume = 10 mL = 1 mL/10 mL/min
Conc. = 10 mg/mL = 1/10/min

where dDB/dt is the rate of drug elimination from the
body (mg/h), Cp is the plasma drug concentration

FIGURE 4-4 Diagram illustrating three different ways of
(mg/L), k is a first-order rate constant (h–1 or 1/h),

describing drug elimination after a dose of 100 mg injected IV
into a volume of 10 mL (a mouse, for example). and VD is the apparent volume of distribution (L).

 

82 Chapter 4

Cl is clearance and has the units L/h in this example. Clearance and Volume of Distribution
In the example in Fig. 4-4B, Cl is in mL/min. Ratio, Cl/VD

Clearance, Cl, is expressed as volume/time.
Equation 4.15 shows that clearance is a constant

EXAMPLE »» »
because VD and k are both constants. DB is the amount
of drug in the body, and dDB/dt is the rate of change

Consider that 100 mg of drug is dissolved in 10 mL
(of amount) of drug in the body with respect to time.

of fluid and 10 mg of drug is removed in the first
The negative sign refers to the drug exiting from the

minute. The drug elimination process could be
body. In many ways, Cl expressed as a flow rate

described as:
makes sense since drugs are presented to the elimi-
nating organs at the flow rate of blood to that organ: a. Number of mg of drug eliminated per minute
1000 mL/min to the kidneys and 1500 mL/min to the (mg/min)
liver. Clearance is a reflection of what percentage of b. Number of mL of fluid cleared of drug per minute
drug is eliminated when passing through these organs. c. Fraction of drug eliminated per minute

The relationship of the three drug elimination

Drug Elimination Expressed as Fraction processes is illustrated in Fig. 4-4A–C. Note that in

Eliminated per Unit Time Fig. 4-4C, the fraction Cl/VD is dependent on both
the volume of distribution and the rate of drug

Consider a compartment volume, containing VD
clearance from the body. This clearance concept

liters. If Cl is expressed in liters per minute (L/min),
forms the basis of classical pharmacokinetics and is

then the fraction of drug cleared per minute in the
later extended to flow models in pharmacokinetic

body is equal to Cl/VD.
modeling. If the drug concentration is Cp, the rate

Expressing drug elimination as the fraction of
of drug elimination (in terms of rate of change in

total drug eliminated is applicable regardless of
concentration, dCp/dt) is:

whether one is dealing with an amount or a volume
(Fig. 4-4C). This approach is most flexible and con- dC

p
= −(Cl/V ) ( . 6

dt D ×C 4 1 )
venient because of its dimensionless nature in terms p

of concentration, volume, or amounts. Thus, it is
valid to express drug elimination as a fraction (eg, For a first-order process,

one-tenth of the amount of drug in the body is elimi- dCp
nated or one-tenth of the drug volume is eliminated = −kC = rate of drug elimination (4.17)

dt p

per unit time). Pharmacokineticists have incorpo-
rated this concept into the first-order equation (ie, k) Equating the two expressions yields:

that describes drug elimination from the one-com-
kCp = Cl/V 4 1 )

D × C ( . 8
partment model. Indeed, the universal nature of many p

processes forms the basis of the first-order equation
Cl

of drug elimination (eg, a fraction of the total drug k = (4.19)
V

molecules in the body will perfuse the glomeruli, a D

fraction of the filtered drug molecules will be reab- Thus, a first-order rate constant is the fractional
sorbed at the renal tubules, and a fraction of the fil- constant Cl/VD. Some pharmacokineticists regard
tered drug molecules will be excreted from the body, drug clearance and the volume of distribution as
giving an overall first-order drug elimination rate independent parameters that are necessary to
constant, k). The rate of drug elimination is the prod- describe the time course of drug elimination. They
uct of k and the drug concentration (Equation 4.2a). also consider k to be a secondary parameter that
The first-order equation of drug elimination can also comes about as a result of Cl and VD. Equation 4.19
be based on probability and a consideration of the is a rearrangement of Equation 4.15 given earlier.
statistical moment theory (see Chapter 25).

 

One-Compartment Open Model: Intravenous Bolus Administration 83

One-Compartment Model Equation in Terms first-order elimination processes are involved, clear-
of Cl and V ance represents the sum of the clearances for both

D

Equation 4.20 may be rewritten in terms of clearance renal and nonrenal clearance, each drug-eliminating

and volume of distribution by substituting Cl/VD for k. organ as shown in Equation 4.22:

The clearance concept may also be applied to a bio-
logic system in physiologic modeling without the need ClT = ClR +ClNR (4.22)
of a theoretical compartment.

C C0e−kt
p = p (4.20) where ClR is renal clearance or drug clearance

through the kidney, and ClNR is nonrenal clearance
C −( D )

p=D0 /V e Cl /V t
D (4.21) through other organs. ClNR is assumed to be due

primarily to hepatic clearance (ClH) in the absence of

Equation 4.21 is applied directly in clinical phar- other significant drug clearances, such as elimina-

macy to determine clearance and volume of distribu- tion through the lung or the bile, as shown in

tion in patients. When only one sample is available, Equation 4.23:

that is, Cp is known at one sample time point, t, after a
given dose, the equation cannot be determined unam- ClT = ClR +ClH (4.23)

biguously because two unknown parameters must be
solved, that is, Cl and VD. In practice, the mean values Drug clearance considers that the drug in the
for Cl and VD of a drug are obtained from the popula- body is uniformly dissolved in a volume of fluid
tion values (derived from a large population of sub- (apparent volume of distribution, VD) from which
jects or patients) reported in the literature. The values drug concentrations can be measured easily.
of Cl and VD for the patient are adjusted using a com- Typically, plasma drug concentration is measured
puter program. Ultimately, a new pair of Cl and VD and drug clearance is then calculated as the fixed
values that better fit the observed plasma drug concen- volume of plasma fluid (containing the drug) cleared
tration is found. The process is repeated through itera- of drug per unit of time. The units for clearance are
tions until the “best” parameters are obtained. Since volume/time (eg, mL/min, L/h).
many mathematical techniques (algorithms) are avail- Alternatively, ClT may be defined as the rate of
able for iteration, different results may be obtained drug elimination divided by the plasma drug concen-
using different iterative programs. An objective test to tration. Thus, clearance is expressed in terms of the
determine the accuracy of the estimated clearance and volume of plasma containing drug that is eliminated
VD values is to monitor how accurately those parame- per unit time. This clearance definition is equivalent
ters will predict the plasma level of the drug after a to the previous definition and provides a practical
new dose is given to the patient. In subsequent chap- way to calculate clearance based on plasma drug
ters, mean predictive error will be discussed and cal- concentration data.
culated in order to determine the performance of
various drug monitoring methods in practice. Elimination rate

The ratio of Cl/VD may be calculated regardless ClT = (4.24)
Plasma concentration (Cp )

of compartment model type using minimal plasma
samples. Clinical pharmacists have applied many
variations of this approach to therapeutic drug moni- (dDE /dt)

ClT = = (µg/min)/(µg/mL) = mL/min
toring and drug dosage adjustments in patients. Cp

(4.25)
Clearance from Drug-Eliminating Tissues

Clearance may be applied to any organ that is involved where DE is the amount of drug eliminated and
in drug elimination from the body. As long as dDE/dt is the rate of drug elimination.

 

84 Chapter 4

Rearrangement of Equation 4.25 gives ∞

Because [AUC] is calculated from the plasma
0

Equation 4.26: drug concentration–time curve from 0 to infinity (∞)
using the trapezoidal rule, no compartmental model

dD is assumed. However, to extrapolate the data to infin-
Rate of Drug elimination E

= C
dt pClT (4.26)

ity to obtain the residual [AUC] or (Cpt/k), first-
0

order elimination is usually assumed. In this case, if

Therefore, ClT is a constant for a specific drug the drug follows the kinetics of a one-compartment

and represents the slope of the line obtained by plot- model, the ClT is numerically similar to the product

ting dDE/dt versus Cp, as shown in Equation 4.26. of VD and k obtained by fitting the data to a one-

For drugs that follow first-order elimination, the compartment model.

rate of drug elimination is dependent on the amount The approach (Equation 4.29) of using [AUC]0

of drug remaining in the body. to calculate body clearance is preferred by some
statisticians/pharmacokineticists who desire an
alternative way to calculate drug clearance without

dD
E

= kDB = kC 2 )
p V (4. 7 a compartmental model. The alternative approach

dt D
is often referred to as a noncompartmental method
of analyzing the data. The noncompartmental

Substituting the elimination rate in Equation 4.26 approach may be modified in different ways in order
for kC to avoid subjective interpolation or extrapolation

pVD in Equation 4.27 and solving for ClT gives
Equation 4.28: (see Chapters 7 and 25 for more discussion). While

the advantage of this approach is not having to
make assumptions about the compartmental model,

kC
Cl P VD the disadvantage of the noncompartmental approach

T = = kV 4.2 )
C D ( 8

p is that it does not allow for predicting the concentra-
tion at any specific time.

Equation 4.28 shows that clearance, ClT, is the In the noncompartmental approach, the two

product of VD and k, both of which are constant. This model parameters, (1) clearance and (2) volume of

Equation 4.28 is similar to Equation 4.19 shown ear- distribution, govern drug elimination from the physi-

lier. As the plasma drug concentration decreases dur- ologic (plasma) fluid directly and no compartment

ing elimination, the rate of drug elimination, dDE/dt, model is assumed. The preference to replace k with

will decrease accordingly, but clearance will remain Cl/VD was prompted by Equation 4.19 as rearranged

constant. Clearance will be constant as long as the in the above section:

rate of drug elimination is a first-order process. Cl
For some drugs, the elimination rate processes k = (4.19)

V
D

are not well known and few or no model assumptions
are desirable; in this situation, a noncompartment For a drug to be eliminated from the body fluid,

method may be used to calculate certain pharmaco- the volume cleared of drug over the size of the pool

kinetic parameters such as clearance, which can be indicates that k is really computed from Cl and VD.

determined directly from the plasma drug concentra- In contrast, the classical one-compartment model

tion–time curve by is described by two model parameters: (1) elimination
constant, k, and (2) volume of distribution, VD.
Clearance is derived from Cl = kVD. The classical

D approach considers VD the volume in which the drug
Cl 0

T = (4.29)

[AUC] appears to dissolve, and k reflects how the drug
0

declines due to excretion or metabolism over time. In
chemical kinetics, the rate constant, k, is related to

∞ ∞
where D0 is the dose and [AUC] = ∫ C dt.

0 0 p “encounters” or “collisions” of the molecules involved

 

One-Compartment Open Model: Intravenous Bolus Administration 85

when a chemical reaction takes place. An ordinary and the intercept of the plasma drug concentration–
hydrolysis or oxidation reaction occurring in the test time curve obtained after IV bolus injection. This
tube can also occur in the body. Classical pharmaco- approach is particularly useful for a new or investi-
kineticists similarly realized that regardless of whether gational drug when little pharmacokinetic informa-
the reaction occurs in a beaker or in the body fluid, the tion is known. In practice, rapid bolus injection is
drug molecules must encounter the enzyme molecule often not desirable for many drugs and a slow IV
for biotransformation or the exit site (renal glomeruli) drip or IV drug infusion is preferred. Rapid injection
to be eliminated. The probability of getting to the of a large drug dose may trigger adverse drug reac-
glomeruli or metabolic site during systemic circula- tions (ADR) that would have been avoided if the
tion must be first order because both events are prob- body had sufficient time to slowly equilibrate with
ability or chance related (ie, a fraction of drug the drug. This is particularly true for certain classes
concentration will be eliminated). Therefore, the rate of antiarrhythmics, anticonvulsants, antitumor, anti-
of elimination (dC/dt) is related to drug concentration coagulants, oligonucleotide drugs, and some sys-
and is aptly described by temic anesthetics. Immediately after an intravenous

injection, the concentrated drug solution/vehicle is
dC

= k × Cp (4.30) directly exposed to the heart, lung, and other vital
dt

organs before full dilution in the entire body. During
The compartment model provides a useful way to the drug’s first pass through the body, some tissues

track mass balance of the drug in the body. It is virtu- may react adversely to a transient high drug concen-
ally impossible to account for all the drug in the body tration because of the high plasma/tissue drug con-
with a detailed quantitative model. However, keeping centration difference (gradients) that exists prior to
track of systemic concentrations and the mass balance full dilution and equilibration. Most intravenous
of the dose in the body is still important to understand drugs are formulated as aqueous solutions, lightly
a drug’s pharmacokinetic properties. For example, the buffered with a suitable pH for this reason. A poorly
kinetic parameters for drugs such as aspirin and acet- soluble drug may precipitate from solution if injected
aminophen were determined using mass balance, too fast. Suspensions or drugs designed for IM injec-
which indicates that both drugs are over 90% metabo- tion only could cause serious injury or fatality if
lized (acetaminophen urinary excretion = 3%; aspirin injected intravenously. For example, the antibiotic
urinary excretion = 1.4%). It is important for a phar- Bicillin intended for IM injection has a precaution
macist to apply such scientific principles during drug that accompanies the packaging to ensure that the
modeling in order to optimize dosing, such as if a drug will not be injected accidentally into a vein.
patient has liver failure and metabolism is decreased. Pharmacists should be especially alert to verify
Drug metabolism may be equally well described by extravascular injection when drugs are designed for
applying clearance and first-order/saturation kinetics IM injection.
concepts to kinetic models. With many drugs, the initial phase or transient

plasma concentrations are not considered as impor-
tant as the steady-state “trough” level during long-

Frequently Asked Question term drug dosing. However, drugs with the

»»How is clearance related to the volume of distribu- therapeutic endpoint (eg, target plasma drug concen-

tion and k? tration) that lie within the steep initial distributive
phase are much harder to dose accurately and not
overshoot the target endpoint. This scenario is par-

CLINICAL APPLICATION ticularly true for some drugs used in critical care
where rapid responses are needed and IV bolus

IV bolus injection provides a simple way to study the routes are used more often. Many new biotechno-
pharmacokinetics of a drug. The pharmacokinetic logical drugs are administered intravenously because
parameters of the drug are determined from the slope of instability or poor systemic absorption by the oral

 

86 Chapter 4

route. The choice of a proper drug dose and rate of
Log keD

0
infusion relative to the elimination half-life of the B

drug is an important consideration for safe drug
administration. Individual patients may behave very

Slope = –k
differently with regard to drug metabolism, drug 2.3
transport, and drug efflux in target cell sites. Drug
receptors and enzymes may have genetic variability
making some people more prone to allergic reac-
tions, drug interactions, and side effects. Simple
kinetic half-life determination coupled with a careful

Time
review of the patient’s chart by a pharmacist can
greatly improve drug safety and efficacy. FIGURE 4-5 Graph of Equation 4.33: log rate of drug

excretion versus t on regular paper.

Frequently Asked Question

»»If we use a physiologic model, are we dealing with A straight line is obtained from this equation by
actual volumes of blood and tissues? Why do vol- plotting log dDu/dt versus time on regular paper or
umes of distribution emerge for drugs that often are on semilog paper dDu/dt against time (Figs. 4-5 and
greater than the real physical volume?

4-6). The slope of this curve is equal to –k/2.3 and
the y intercept is equal to k D0

e B. For rapid intravenous
administration, D0

B is equal to the dose D0. Therefore,
if D0

B is known, the renal excretion rate constant (ke)
CALCULATION OF k FROM URINARY can be obtained. Because both ke and k can be deter-

EXCRETION DATA mined by this method, the nonrenal rate constant
(knr) for any route of elimination other than renal

The elimination rate constant k may be calculated excretion can be found as follows:
from urinary excretion data. In this calculation the
excretion rate of the drug is assumed to be first order. k − ke = knr (4.34)
The term ke is the renal excretion rate constant, and
Du is the amount of drug excreted in the urine.

dD
u 1000

= k D 4 3 )
dt e B ( . 1

keD 0
B

From Equation 4.4, DB can be substituted for
D0 e–kt

B : Slope = –k
2.3

100

dD
u k D0e−kt

=
dt e B (4.32)

Taking the natural logarithm of both sides and
10

then transforming to common logarithms, the fol- 0 1 2 3 4 5

lowing expression is obtained: Time (hours)

FIGURE 4-6 Semilog graph of rate of drug excretion
dD kt

log u −
0

= + log k D (4.33) versus time according to Equation 4.33 on semilog paper
dt 2.3 e B (intercept = k D0).

e B

Rate of drug excretion (dDu/dt)

Log dDu/dt

 

One-Compartment Open Model: Intravenous Bolus Administration 87

Substitution of km for knr in Equation 4.34 gives Solution
Equation 4.1. Because the major routes of elimina- Set up the following table:
tion for most drugs are renal excretion and metabo-
lism (biotransformation), knr is approximately equal

Time
to km. (hours) Du (mg) Du/t mg/h t* (hours)

0.25 160 160/0.25 640 0.125
k = k (4.35)

nr m
0.50 140 140/0.25 560 0.375

There are practical considerations of collecting 1.0 200 200/0.5 400 0.750

urine for drug analysis since urine is produced at an 2.0 250 250/1 250 1.50

approximate rate of 1 mL/min and collected in the
4.0 188 188/2 94 3.0

bladder until voided for collection. Thus, the drug
urinary excretion rate (dDu/dt) cannot be determined 6.0 46 46/2 23 5.0

experimentally for any given instant. In practice, Here t* = midpoint of collection period and t = time interval for collection

urine is collected over a specified time interval, and of urine sample.

the urine specimen is analyzed for drug. An average
Construct a graph on a semilogarithmic scale of

urinary excretion rate is then calculated for that col-
D

lection period. Therefore, the average rate of uri- u/t versus t*. The slope of this line should equal
–k/2.3. It is usually easier to determine the elimination

nary drug excretion, Du/t, is plotted against the time
t

corresponding to the midpoint of the collection ½ directly from the curve and then calculate k from

interval, t*, for the collection of the urine sample. 0.693
k =

The average value of dDu/dt is plotted on a semiloga- t1/2
rithmic scale against the time that corresponds to the

In this problem, t1/2 = 1.0 hour and k = 0.693 h–1. Note
midpoint (average time) of the collection period.

that the slope of the log excretion rate constant is a
function of the elimination rate constant k and not of
the urinary excretion rate constant ke (Fig. 4-6).

PRACTICE PROBLEM
A similar graph of the Cp values versus t should

A single IV dose of an antibiotic was given to a yield a curve with a slope having the same value as
50-kg woman at a dose level of 20 mg/kg. Urine and that derived from the previous curve. Note that this
blood samples were removed periodically and method uses the time of plasma sample collection,
assayed for parent drug. The following data were not the midpoint of collection.
obtained: An alternative method for the calculation of the

elimination rate constant k from urinary excretion data
is the sigma-minus method, or the amount of drug

Time (hours) Cp (µg/mL) Du (mg)
remaining to be excreted method. The sigma-minus

0.25 4.2 160 method is sometimes preferred over the previous
method because fluctuations in the rate of elimination

0.50 3.5 140
are minimized.

1.0 2.5 200 The amount of unchanged drug in the urine can
2.0 1.25 250 be expressed as a function of time through the fol-

lowing equation:
4.0 0.31 188

6.0 0.08 46 k
eDD 0

u = (1 e−kt
− ) (4.36)

k

What is the elimination rate constant, k, for this where Du is the cumulative amount of unchanged
antibiotic? drug excreted in the urine.

 

88 Chapter 4

The amount of unchanged drug that is ulti- PRACTICE PROBLEM
mately excreted in the urine, D∞

u , can be determined
by making time t equal to ∞. Thus, the term e–kt Using the data in the preceding problem, determine

becomes negligible and the following expression is the elimination rate constant.

obtained:
Solution

k
∞ eD

D 0
u = (4.37) Construct the following table:

k

Time Cumulative
Substitution of D∞

u for keD0/k in Equation 4.36 (hours) Du (mg) D ∞

u D −D
u u

and rearrangement yields
0.25 160 160 824

0.50 140 300 684
D∞ D D∞ e−kt

u − u = u (4.38)
1.0 200 500 484

2.0 250 750 234
Equation 4.38 can be written in logarithmic

form to obtain a linear equation: 4.0 188 938 46

6.0 46 984 0
−kt

log(D∞

u − Du )

= + log D
2.3 u (4.39)

Plot log (D∞

u − Du ) versus time. Use a semiloga-
rithmic scale for ( ∞

Du − Du ). Evaluate k and t1/2 from
Equation 4.39 describes the relationship for the the slope.

amount of drug remaining to be excreted (D∞

u − Du )
versus time.

Comparison of the Rate and the
A linear curve is obtained by graphing the loga-

Sigma-Minus Methods
rithm scale of the amount of unchanged drug yet to
be eliminated, log ( ∞

Du − Du ), versus time. On semi- The rate method is highly dependent on the accurate

log paper, the slope of this curve is –k/2.3 and the y measurement of drug in the urine at each time point.

intercept is (D∞

u ) (Fig. 4-7). Fluctuations in the rate of drug elimination and
experimental errors including incomplete bladder
emptying for a collection period cause appreciable
departure from linearity using the rate method,

1000 whereas the accuracy of the sigma-minus method is
less affected. The rate method is applicable to zero-
order drug elimination process, while the sigma-
minus method is not. Lastly, the renal drug excretion
rate constant may be obtained from the rate method

100
but not from the sigma-minus method.

The sigma-minus method requires knowing the
D∞

u and even a single missed urine collection will
invalidate the entire urinary drug excretion study.
This method also requires the collection of urine until

10
0 1 2 3 4 urinary drug excretion is complete; prematurely end-

Time ing the study early will invalidate the study. Finally,

FIGURE 4-7 a small error in the assessment of D introduces an
Sigma-minus method, or the amount of drug u

remaining to be excreted method, for the calculation of the error in terms of curvature of the plot, because each
elimination rate constant according to Equation 4.39. point is based on log ( ∞

Du − Du ) versus time.

D

u − Du

 

One-Compartment Open Model: Intravenous Bolus Administration 89

CLINICAL APPLICATION 30 hours. However, the urinary drug excretion rate
method data were more scattered (variable) and the

The sigma-minus method and the excretion rate correlation coefficient r was equal to 0.744 (Fig. 4-9),
method were applied to the urinary drug excretion in compared to the correlation coefficient r of 0.992
subjects following the smoking of a single marijuana using the sigma-minus method (Fig. 4-8).
cigarette (Huestis et al, 1996). The urinary excretion
curves of 11-nor-carboxy 9-tetrahydrocannabinol

Problems in Obtaining Valid Urinary
(THCCOOH), a metabolite of marijuana, in one

Excretion Data
subject from 24 to 144 hours after smoking one
marijuana cigarette are shown in Figs. 4-8 and 4-9. Certain factors can make it difficult to obtain valid

A total of 199.7 mg of THCCOOH was excreted in urinary excretion data. Some of these factors are as

the urine over 7 days, which represents 0.54% of the follows:

total 9-tetrahydrocannabinol available in the ciga- 1. A significant fraction of the unchanged drug
rette. Using either urinary drug excretion method, must be excreted in the urine.
the elimination half-life was determined to be about 2. The assay technique must be specific for the

unchanged drug and must not include interfer-
ence due to drug metabolites that have similar
chemical structures.

5.5 3. Frequent sampling is necessary for a good
5.0 Subject B curve description.
4.5 4. Urine samples should be collected periodically
4.0 until almost all of the drug is excreted. A graph
3.5 t1/2 = 29.9 h of the cumulative drug excreted versus time
3.0 r = 0.992 will yield a curve that approaches an asymp-
2.5 tote at “infinite” time (Fig. 4-10). In practice,

0 24 48 72 96 120 144 168
approximately seven elimination half-lives are

Time (hours)
needed for 99% of the drug to be eliminated.

FIGURE 4-8 Amount remaining to be excreted method. 5. Variations in urinary pH and volume may cause
The half-life of THCCOOH was calculated to be 29.9 hours from significant variation in urinary excretion rates.
the slope of this curve; the correlation coefficient r was equal

6. Subjects should be carefully instructed as to the
to 0.992. (Data from Huestis et al, 1996.)

necessity of giving a complete urine specimen
(ie, completely emptying the bladder).

4.0

3.5 Subject B

3.0

2.5
t1/2 = 30.7 h

2.0 r = 0.744

1.5
0 24 48 72 96 120 144 168

Time (hours)

FIGURE 4-9 Excretion rate method. The half-life of Time
THCCOOH was calculated to be 30.7 hours from the slope of
this curve; the correlation coefficient r was equal to 0.744. FIGURE 4-10 Graph showing the cumulative urinary
(Data from Huestis et al, 1996.) excretion of drug as a function of time.

Log excretion rate Log THCCOOH
remaining to be excreted

Cumulative amount of drug in urine

 

90 Chapter 4

CHAPTER SUMMARY
The one-compartment model assumes that the drug This approach is most flexible and convenient
is uniformly distributed within a single hypothetical because of its dimensionless nature in terms of
compartment volume from which the drug concen- amount or volume (k is expressed as h–1 or min–1).
tration can be sampled and assayed easily. The one- Clearance may be computed by Cl = kVD. This
compartment model, IV bolus drug injection, method is preferred by many pharmacists since it can
provides the simplest approach for estimating the be calculated from two concentration measurements,
apparent volume of distribution, VD, and the elimina- making it more clinically feasible than a full pharma-
tion rate constant, k. If VD, k, and the drug dose are cokinetic study. Many pharmacokineticists do not
known, the model equation allows drug concentra- prefer this method since k is considered a secondary
tion in the compartment (body) at any time to be model parameter, while VD and Cl are considered to
calculated. The volumes of plasma fluid and extra- be independent model parameters. That is, VD and Cl
cellular fluid may be relatively constant under nor- give k its properties. Instead, many prefer the non-
mal conditions. However, these volumes added compartmental approach using area under the con-
together do not usually numerically equal to the centration-time curve to calculate Cl; this method
(apparent) volume of distribution of the drug, which avoids the basic assumptions inherent in the one-
may be larger or smaller depending on how widely compartmental model but requires a full pharmacoki-
the drug distributes into tissues. netic study to determine the area under the curve.

The one-compartment model may be described Drug clearance is constant for a first-order process
with the two model parameters, clearance and vol- regardless of the drug concentration. Clearance is
ume of distribution. Alternatively, the one-compart- expressed as the apparent volume of fluid of the dis-
ment model can also be described by two model solved drug that is removed per unit time. The one-
parameters, the elimination constant, k, and volume compartment model may assume either a first-order
of distribution. The latter model explains that drugs or a zero-order elimination rate depending on whether
are fractionally removed at any time, whatever the the drug follows linear kinetics or not. The disadvan-
initial drug concentration is, and k as a ratio of Cl/VD. tage of the noncompartmental approach is that pre-
Expressing drug elimination as the fraction of total dicting concentrations at specific times may not hold
drug eliminated per time is applicable regardless of true, while using a one-compartmental model allows
whether one is dealing with an amount or a volume. for predicting the concentration at any time point.

LEARNING QUESTIONS
1. A 70-kg volunteer is given an intravenous dose

of an antibiotic, and serum drug concentrations t (hours) Cp (µg/mL)
were determined at 2 hours and 5 hours after

0.25 8.21
administration. The drug concentrations were
1.2 and 0.3 mg/mL, respectively. What is the 0.50 7.87

biologic half-life for this drug, assuming first- 1.00 7.23
order elimination kinetics?

3.00 5.15
2. A 50-kg woman was given a single IV dose of

an antibacterial drug at a dose level of 6 mg/kg. 6.00 3.09

Blood samples were taken at various time inter- 12.0 1.11
vals. The concentration of the drug (Cp) was

18.0 0.40
determined in the plasma fraction of each blood
sample and the following data were obtained:

 

One-Compartment Open Model: Intravenous Bolus Administration 91

a. What are the values for VD, k, and t1/2 for this 6. A drug has an elimination t1/2 of 6 hours and
drug? follows first-order kinetics. If a single 200-mg

b. This antibacterial agent is not effective dose is given to an adult male patient (68 kg)
at a plasma concentration of less than by IV bolus injection, what percent of the dose
2 mg/mL. What is the duration of activity is lost in 24 hours?
for this drug? 7. A rather intoxicated young man (75 kg, age

c. How long would it take for 99.9% of this 21 years) was admitted to a rehabilitation cen-
drug to be eliminated? ter. His blood alcohol content was found to be

d. If the dose of the antibiotic was doubled 210 mg%. Assuming the average elimination
exactly, what would be the increase in dura- rate of alcohol is 10 mL of ethanol per hour,
tion of activity? how long would it take for his blood alcohol

3. A new drug was given in a single intravenous concentration to decline to less than the legal
dose of 200 mg to an 80-kg adult male patient. blood alcohol concentration of 100 mg%?
After 6 hours, the plasma drug concentration of (Hint: Alcohol is eliminated by zero-order
drug was 1.5 mg/100 mL of plasma. Assuming kinetics.) The specific gravity of alcohol is 0.8.
that the apparent VD is 10% of body weight, The apparent volume of distribution for alcohol
compute the total amount of drug in the body is 60% of body weight.
fluids after 6 hours. What is the half-life of this 8. A single IV bolus injection containing 500 mg
drug? of cefamandole nafate (Mandol, Lilly) is given

4. A new antibiotic drug was given in a single to an adult female patient (63 years, 55 kg) for
intravenous bolus of 4 mg/kg to 5 healthy male a septicemic infection. The apparent volume
adults ranging in age from 23 to 38 years of distribution is 0.1 L/kg and the elimination
(average weight 75 kg). The pharmacokinetics half-life is 0.75 hour. Assuming the drug is
of the plasma drug concentration–time curve eliminated by first-order kinetics and may be
for this drug fits a one-compartment model. The described by a one-compartment model, calcu-
equation of the curve that best fits the data is late the following:

a. The C0
p

Cp 78e−0.46t
= b. The amount of drug in the body 4 hours after

the dose is given
Determine the following (assume units of mg/mL c. The time for the drug to decline to 0.5 mg/mL,
for Cp and hours for t): the minimum inhibitory concentration for
a. What is the t1/2? streptococci
b. What is the VD? 9. If the amount of drug in the body declines from
c. What is the plasma level of the drug after 100% of the dose (IV bolus injection) to 25%

4 hours? of the dose in 8 hours, what is the elimination
d. How much drug is left in the body after half-life for this drug? (Assume first-order

4 hours? kinetics.)
e. Predict what body water compartment this 10. A drug has an elimination half-life of 8 hours

drug might occupy and explain why you and follows first-order elimination kinetics. If a
made this prediction. single 600-mg dose is given to an adult female

f. Assuming the drug is no longer effective patient (62 kg) by rapid IV injection, what per-
when levels decline to less than 2 mg/mL, cent of the dose is eliminated (lost) in 24 hours
when should you administer the next dose? assuming the apparent VD is 400 mL/kg? What

5. Define the term apparent volume of distribution. is the expected plasma drug concentration (Cp)
What criteria are necessary for the measure- at 24 hours postdose?
ment of the apparent volume of distribution to 11. For drugs that follow the kinetics of a one-
be useful in pharmacokinetic calculations? compartment open model, must the tissues

 

92 Chapter 4

and plasma have the same drug concentration?
t (hours) Amount of Drug in Urine (mg)

Why?
12. An adult male patient (age 35 years, weight 0 0

72 kg) with a urinary tract infection was 4 100
given a single intravenous bolus of an

8 26
antibiotic (dose = 300 mg). The patient was
instructed to empty his bladder prior to being
medicated. After dose administration, the a. Assuming first-order elimination, calculate
patient saved his urine specimens for drug the elimination half-life for the antibiotic in
analysis. The urine specimens were analyzed this patient.
for both drug content and sterility (lack of b. What are the practical problems in obtaining
bacteriuria). The drug assays gave the follow- valid urinary drug excretion data for the deter-
ing results: mination of the drug elimination half-life?

ANSWERS

Frequently Asked Questions • The first-order rate constant k has no concentration

What is the difference between a rate and a rate or mass units. In the calculation of the slope, k, the

constant? unit for mass or concentration is canceled when
taking the log of the number:

• A rate represents the change in amount or concen-
tration of drug in the body per time unit. For exam- ln y2 − ln y1 ln (y2 /y )Slope 1

= =
ple, a rate equal to –5 mg/h means the amount of x2 − x1 x2 − x1
drug is decreasing at 5 mg/h. A positive or negative
sign indicates that the rate is increasing or decreas- If a drug is distributed in the one-compartment model,
ing, respectively. Rates may be zero order, first does it mean that there is no drug in the tissue?
order, or higher orders. For a first-order rate, the • The one-compartment model uses a single homo-
rate of change of drug in the body is determined by

geneous compartment to represent the fluid and
the product of the elimination rate constant, k, and

the vascular tissues. This model ignores the het-
the amount of drug remaining in the body, that is,

erogeneity of the tissues in the body, so there is
rate = –kDB, where k represents “the fraction” of

no merit in predicting precise tissue drug levels.
the amount of drug in the body that is eliminated

However, the model provides useful insight into
per hour. If k = 0.1 h–1 and DB = 10 mg, then the

the mass balance of drug distribution in and out
rate = 0.1 h–1 × 10 mg = 1 mg/h. The rate constant

of the plasma fluid in the body. If VD is larger than
in this example shows that one-tenth of the drug

the physiologic vascular volume, the conclusion is
is eliminated per hour, whatever amount of drug is

that there is some drug outside the vascular pool,
present in the body. For a first-order rate, the rate

that is, in the tissues. If VD is small, then there is
states the absolute amount eliminated per unit

little extravascular tissue drug storage, except
time (which changes with the amount of drug in

perhaps in the lung, liver, kidney, and heart. With
the body), whereas the first-order rate constant, k,

some knowledge about the lipophilicity of the drug
gives a constant fraction of drug that is eliminated

and an understanding of blood flow and perfusion
per unit time (which does not change with the

within the body, a postulation may be made as to
amount of drug in the body).

which organs are involved in storing the extravas-
Why does k always have the unit 1/time (eg, h–1), cular drug. The concentration of a biopsy sample
regardless of what concentration unit is plotted? may support or refute the postulation.

 

One-Compartment Open Model: Intravenous Bolus Administration 93

How is clearance related to the volume of distribution These data may also be plotted on a semilog
and k? graph and t1/2 obtained from the graph.

2. Dose (IV bolus) = 6 mg/kg × 50 kg = 300 mg
• Clearance is the volume of plasma fluid that is

cleared of drug per unit time. Clearance may also dose 300 mg 300 mg
a. VD = = =

be derived for the physiologic model as the frac- C0 8.4 µg/mL 8.4 mg/L
P

tion of drug that is eliminated by an organ as blood
= 35.7 L

flows through it. The former definition is equiva-
lent to Cl = kVD and is readily adapted to dosing (1) Plot the data on semilog graph paper
since VD is the volume of distribution. If the drug is and use two points from the line of
eliminated solely by metabolism in the liver, then best fit.
ClH = Cl. ClH is usually estimated by the differ-
ence between Cl and ClR. ClH is directly estimated t (hours) Cp (µg/mL)

by the product of the hepatic blood flow and the
2 6

extraction ratio.
6 3

If we use a physiologic model, are we dealing with
actual volumes of blood and tissues? Why do vol- (2) t1/2 (from graph) = 4 hours
umes of distribution emerge for drugs that often are
greater than the real physical volume? 0.693

k = = 0.173 h−1
4

• Since mass balance (ie, relating dose to plasma
drug concentration) is based on volume of distri- b. C0

p = 8.4 µg/mL Cp = 2 µg/mL k = 0.173 h−1
bution rather than blood volume, the compartment
model is used in determining dose. Generally, the kt

logC logC0
p = − +

total blood concentrations of most drugs are not 2.3 P

known, since only the plasma or serum concentra-
0.173t

tion is assayed. Some drugs have an RBC/plasma log 2 = − + log 8.4
2.3

drug ratio much greater than 1, making the appli-
cation of the physiologic model difficult without t = 8.29 h

knowing the apparent volume of distribution.
Alternatively, time t may be found from a

Learning Questions graph of Cp versus t.
1. The Cp decreased from 1.2 to 0.3 mg/mL in c. Time required for 99.9% of the drug to be

3 hours. eliminated:
(1) Approximately 10 t1/2

t (hours) Cp (µg/mL)
t = 10(4) = 40 h

2 1.2

5 0.3 (2) C0
p = 8.4 µg/mL

kt
logCp logC0

= − + With 0.1% of drug remaining,
2.3 P

k(3) Cp = 0.001 (8.4 µg/mL) = 0.0084 µg/mL
log 0.3 = − + log 1.2

2.3
k = 0.173 h−1

k = 0.462 h−1
−0.173t

0.693 0.693 log 0.0084 = + log 8.4
t1/2 = = 2.3

k 0.462

t1/2 = 1.5 h t = 39.9 h

 

94 Chapter 4

d. If the dose is doubled, then C0 will also
p 0.693 0.693

double. However, the elimination half-life a. t1/2 = = = 1.5 h
k 0.46

or first-order rate constant will remain the
same. Therefore, dose 300,000 µg

b. VD = = = 3846 mL
C0 78 µg/mL

p

C0 = µ C k 1
p 16.8 g/mL p = 2µg/mL = 0.173 h−

Dose = 4 mg/kg × 75 kg = 300 mg

0.173t
log 2 = + log16.8 c.

2.3 0.46(4)
(1) logCp = + log78 = 1.092

t = 12.3 h−1 2.3

Cp = 12.4 µg/mL
Notice that doubling the dose does not
double the duration of activity. (2) C = 78e−0.46(4) = 78e−18.4

p = 78 (0.165)

3. D0 = 200 mg Cp = 12.9 µg/mL

VD = 10% of body weight = 0.1 (80 kg) d. At 4 hours:

DB = CpVD = 12.4 µg/mL × 3846 mL
= 8000 mL = 8 L

= 47.69 mg

At 6 hours:

Cp = 1.5 mg/100 mL e. VD = 3846 mL

Average weight = 75 kg
drug in body (D )

V B
D = Percent body wt = (3.846 kg/75 kg) ×100

Cp
= 5.1%

1.5
DB = CpVD = (8000 mL) = 120 mg The apparent VD approximates the plasma

100 mL
volume.

kt f. Cp = 2 mg/mL

log D D0
B = − + log

2.3 B
Find t.

k(6) 0.46t
log120 = − + log 200 log 2 = − + log 78

2.3 2.3

2.3 (log 2− log 78)
k −1

= 0.085 h t = −
0.46

0.693 0.693 t = 7.96 h ≈ 8 h
t1/2 = = = 8.1 h

k 0.085
Alternate Method

4. Cp = 78e–0.46t (the equation is in the form
2 78e−0.46t

=
C C0e−kt

= )
p p

2
0.0256 e−0.46t

ln Cp = ln 78 − 0.46t = =
78

0.46t −37 = −0.46t

logCp = − + log 78
2.3

37
Thus, k = 0.46 h−1,C0 t = = 8 h

p = 78 µg/mL. 0.46

 

One-Compartment Open Model: Intravenous Bolus Administration 95

6. For first-order elimination kinetics, one-half of where
the initial quantity is lost each t1/2. The follow- DB = amount of drug remaining in the body
ing table may be developed: D0 = dose = 200 mg

k = elimination rate constant
Amount 0.693

= = 0.1155 h−1
of Drug Percent Percent

t1/2
Time Number in Body of Drug of Drug
(hours) of t1/2 (mg) in Body Lost t = 24 h

0 0 200 100 0 −0.1155(24)
logDB = + log200

2.3
6 1 100 50 50

DB = 12.47 mg ≈12.5 mg
12 2 50 25 75

200 −12.512 2 50 25 75 % of drug lost = ×100 = 93.75%
200

18 3 25 12.5 87.5
7. The zero-order rate constant for alcohol is

24 4 12.5 6.25 93.75 10 mL/h. Since the specific gravity for alcohol
is 0.8,

Method 1 x(g)
From the above table the percent of drug remaining 0.8 g/mL =

10 mL
in the body after each t1/2 is equal to 100% times

x = 8 g
(1/2)n, where n is the number of half-lives, as shown
below: Therefore, the zero-order rate constant, k0,

is 8 g/h.
Percent of Drug

Drug in body at t = 0:
Number Percent of Remaining in Body
of t1/2 Drug in Body after n t1/2 210 mg

D0
B = CpVD = × (0.60)(75 L) = 94.5 g

0 100 0.100 L

1 50 100 × 1/2
Drug in body at time t:

2 25 100 × 1/2 × 1/2
100 mg

3 12.5 100 × 1/2 × 1/2 × 1/2 DB = CpVD = × (0.60)(75 L) = 45.0 g
0.100 L

N 100 × (1/2)n

For a zero-order reaction:
100

Percent of drug remaining n , where n = number D k t D0
B = − 0 + B

2
of t1/2 45 = −8t + 94.5

100
Percent of drug lost = 100 −

2n t = 6.19 h
At 24 hours, n = 4, since t1/2 = 6 hours.

100 dose 500 mg
Percent of drug lost = 100 − = 93.75% 8. a. C0

p = = = 90.9 mg/L
16 VD (0.1L/kg)(55 kg)

Method 2
−kt

The equation for a first-order elimination after IV b. logD D0
B = + log

2.3 B

bolus injection is
(0.693/0.75)(4)

logDB = + log500
−kt 2.3

logDB = + logD
2.3 0 DB = 12.3 mg

 

96 Chapter 4

−(0.693/0.75)t the body (in plasma and tissues). At equilib-
c. log 0.5 = + log 90.0

2.3 rium, the drug concentration in the tissues may
differ from the drug concentration in the body

t = 5.62 h
because of drug protein binding, partitioning

−kt of drug into fat, differences in pH in different
9. log D 0

B = + log D
2.3 B regions of the body causing a different degree

−k(8) of ionization for a weakly dissociated electro-
log 25 = + log100

2.3 lyte drug, an active tissue uptake process, etc.
12. Set up the following table:

k = 0.173 h−1

0.693 Time (hours) Du (mg) dDu/t mg/h t*
t1/2 = = 4 h

0.173
0 0

−kt
10. logD logD0 4 100 100/4 25 2

B = +
2.3 B

8 26 26/4 6.5 6
(−0.693/8)(24)

= + log 600
2.3

The elimination half-life may be obtained
DB = 74.9 mg graphically after plotting mg/h versus t*.

600 t aphically is approximately
− 74.9 The 1/2 obtained gr

Percent drug lost = ×100
600 2 hours.

= 87.5%

dDu −kt
C log k 0

= + log
p at t = 24 hours: dt 2.3 eDB

74.9 mg
−k logY logY log 6.5 log 2.5

Cp = = 3.02 mg/L Slope 2 − 1 −
= = =

(0.4 L/kg)(62 kg) 2.3 X2 − X1 6− 2

11. The total drug concentration in the plasma is k 0.336 h−1
=

not usually equal to the total drug concentra-
tion in the tissues. A one-compartment model 0.693 0.693

t
implies that the drug is rapidly equilibrated in 1/2 = = = 2.06 h

k 0.336

REFERENCE
Huestis MA, Mitchell J, Cone EJ: Prolonged urinary excretion of

marijuana metabolite (abstract). Committee on Problems of
Drug Dependence, San Juan, PR, June 25, 1996.

BIBLIOGRAPHY
Gibaldi M, Nagashima R, Levy G: Relationship between drug Riegelman S, Loo J, Rowland M: Concepts of volume of distribu-

concentration in plasma or serum and amount of drug in the tion and possible errors in evaluation of this parameter. Science
body. J Pharm Sci 58:193–197, 1969. 57:128–133, 1968.

Riegelman S, Loo JCK, Rowland M: Shortcomings in pharmaco- Wagner JG, Northam JI: Estimation of volume of distribution
kinetic analysis by conceiving the body to exhibit properties of and half-life of a compound after rapid intravenous injection.
a single compartment. J Pharm Sci 57:117–123, 1968. J Pharm Sci 58:529–531, 1975.

 

Multicompartment Models:

5 Intravenous Bolus
Administration
Shabnam N. Sani and Rodney C. Siwale

Chapter Objectives Pharmacokinetic models are used to simplify all the complex pro-
cesses that occur during drug administration that include drug

»» Define the pharmacokinetic
distribution and elimination in the body. The model simplification

terms used in a two- and three-
is necessary because of the inability to measure quantitatively all

compartment model.
the rate processes in the body, including the lack of access to bio-

»» Explain using examples why logical samples from the interior of the body. As described in
drugs follow one-compartment, Chapter 1, pharmacokinetic models are used to simulate drug
two-compartment, or three- disposition under different conditions/time points so that dosing
compartment kinetics. regimens for individuals or groups of patients can be designed.

»» Use equations and graph Compartmental models are classic pharmacokinetic models

to simulate plasma drug that simulate the kinetic processes of drug absorption, distribution,

concentration at various and elimination with little physiologic detail. In contrast, the more

time periods after an IV bolus sophisticated physiologic model is discussed in Chapter 25. In

injection of a drug that follows compartmental models, drug tissue concentration, Ct, is assumed to

the pharmacokinetics of a two- be uniform within a given hypothetical compartment. Hence, all

and three-compartment model muscle mass and connective tissues may be lumped into one hypo-

drug. thetical tissue compartment that equilibrates with drug from the
central (composed of blood, extracellular fluid, and highly per-

»» Relate the relevance of the fused organs/tissues such as heart, liver, and kidneys) compart-
magnitude of the volume of ment. Since no data are collected on the tissue mass, the theoretical
distribution and clearance of tissue concentration cannot be confirmed and used to forecast
various drugs to underlying actual tissue drug levels. Only a theoretical, Ct, concentration of
processes in the body. drug in the tissue compartment can be calculated. Moreover, drug

»» Estimate two-compartment concentrations in a particular tissue mass may not be homoge-
model parameters by using the neously distributed. However, plasma concentrations, Cp, are
method of residuals. kinetically simulated by considering the presence of a tissue or a

group of tissue compartments. In reality, the body is more complex
»» Calculate clearance and alpha

than depicted in the simple one-compartment model and the elimi-
and beta half-lives of a two-

nating organs, such as the liver and kidneys, are much more com-
compartment model drug.

plex than a simple extractor. Thus, to gain a better appreciation
»» Explain how drug metabolic regarding how drugs are handled in the body, multicompartment

enzymes, transportors, and models are found helpful. Contrary to the monoexponential decay
binding proteins in the body in the simple one-compartment model, most drugs given by IV
may modify the distribution bolus dose decline in a biphasic fashion, that is, plasma drug con-
and/or elimination phase of a centrations rapidly decline soon after IV bolus injection, and then
drug after IV bolus. decline moderately as some of the drug that initially distributes

(equilibrates) into the tissue moves back into the plasma. The early

97

 

98 Chapter 5

decline phase is commonly called the distribution 50

phase (because distribution into tissues primarily
determines the early rapid decline in plasma concen-
tration) and the latter phase is called the terminal or

10
elimination phase. During the distribution phase, se

E
changes in the concentration of drug in plasma pri- l

5 imi
a na

marily reflect the movement of drug within the body, tion ph
rather than elimination. However, with time, distribu- ase
tion equilibrium is established in more and more tis- b

sues between the tissue and plasma, and eventually 1
0 3 6 9 12

changes in plasma concentration reflect proportional Time
changes in the concentrations of drug in all other tis-

FIGURE 5-1 Plasma level–time curve for the two-
sues. During this proportionality phase, the body

compartment open model (single IV dose) described in Fig. 5-2
kinetically acts as a single compartment and because (model A).
decline of the plasma concentration is now associated
solely with elimination of drug from the body, this
phase is often called the elimination phase. is completed, the plasma drug concentrations decline

Concentration of the drug in the tissue compart- more gradually when eventually plasma drug equilib-
ment (Ct), is not a useful parameter due to the non- rium with peripheral tissues occurs. Drug kinetics
homogenous tissue distribution of drugs. However, after distribution is characterized by the composite
amount of the drug in the tissue compartment (Dt) is rate constant, b (or b), which can be obtained from
useful because it is an indication of how much drug the terminal slope of the plasma level–time curve in
accumulates extravascularly in the body at any given a semilogarithmic plot (Fig. 5-1).
time. The two-compartment model provides a simple Nonlinear plasma drug level–time decline occurs
way to keep track of the mass balance of the drug in because some drugs distribute at various rates into
the body. different tissue groups. Multicompartment models

Multicompartment models provide answers to were developed to explain and predict plasma and
such questions as: (1) How much of a dose is elimi- tissue concentrations for those types of drugs. In con-
nated? (2) How much drug remains in the plasma trast, a one-compartment model is used when a drug
compartment at any given time? and (3) How much appears to distribute into tissues instantaneously and
drug accumulates in the tissue compartment? The uniformly or when the drug does not extensively dis-
latter information is particularly useful for drug tribute into extravascular tissues such as aminoglyco-
safety since the amount of drug in a deep tissue com- sides. Extent of distribution is partially determined by
partment may be harder to eliminate by renal excre- the physical-chemical properties of the drug. For
tion or by dialysis after drug overdose. instance, aminoglycosides are polar molecules; there-

Multicompartment models explain the observa- fore, their distribution is primarily limited to extracel-
tion that, after a rapid IV bolus drug injection, the lular water. Lipophilic drugs with more extensive
plasma level–time curve does not decline linearly, distribution into tissues such as the benzodiazepines
implying that the drug does not equilibrate rapidly in or those with extensive intracellular uptake may be
the body, as observed for a single first-order rate better described by more complex models. For both
process in a one-compartment model. Instead, a one- and multicompartment models, the drug in those
biphasic or triphasic drug concentration decline is tissues that have the highest blood perfusion equili-
often observed. The initial decline phase represents brates rapidly with the drug in the plasma. These
the drug leaving the plasma compartment and enter- highly perfused tissues and blood make up the central
ing one or more tissue compartments as well as being compartment (often called the plasma compartment).
eliminated. Later, after drug distribution to the tissues While this initial drug distribution is taking place,

Plasma level

Distrib
utio

n p
ha

 

Multicompartment Models: Intravenous Bolus Administration 99

multicompartment drugs are delivered concurrently MDR1, a common transport protein of the ABC
to one or more peripheral compartments (often con- [ATP-binding cassette] transporter subfamily found
sidered as the tissue compartment that includes fat, in the body). Drug transporters are now known to
muscle, and cerebrospinal fluid) composed of groups influence the curvature in the log plasma drug con-
of tissues with lower blood perfusion and different centration–time graph of drugs. The drug isotretinoin
affinity for the drug. A drug will concentrate in a tis- has a long half-life because of substantial distribution
sue in accordance with the affinity of the drug for that into lipid tissues.
particular tissue. For example, lipid-soluble drugs Kinetic analysis of a multicompartment model
tend to accumulate in fat tissues. Drugs that bind assumes that all transfer rate processes for the pas-
plasma proteins may be more concentrated in the sage of drug into or out of individual compartments
plasma, because protein-bound drugs do not diffuse are first-order processes. On the basis of this assump-
easily into the tissues. Drugs may also bind with tis- tion, the plasma level–time curve for a drug that
sue proteins and other macromolecules, such as DNA follows a multicompartment model is best described
and melanin. by the summation of a series of exponential terms,

Tissue sampling often is invasive, and the drug each corresponding to first-order rate processes
concentration in the tissue sample may not represent associated with a given compartment. Most multi-
the drug concentration in the entire organ due to the compartment models used in pharmacokinetics are
nonhomogenous tissue distribution of drugs. In mamillary models. Mamillary models are well con-
recent years, the development of novel experimental nected and dynamically exchange drug concentra-
methods such as magnetic resonance spectroscopy tion between compartments making them very
(MRS), single photon emission computed tomogra- suitable for modeling drug distribution.
phy (SPECT), and tissue microdialysis has enabled Because of all these distribution factors, drugs
us to study the drug distribution in the target tissues will generally concentrate unevenly in the tissues,
of animals and humans (Eichler and Müller, 1998, and different groups of tissues will accumulate the
and Müller, 2009). These innovative technologies drug at very different rates. A summary of the
have enabled us to follow the path of the drug from approximate blood flow to major human tissues is
the plasma compartment into anatomically defined presented in Table 5-1. Many different tissues and
regions or tissues. More importantly, for some classes rate processes are involved in the distribution of any
of drugs the concentration in the interstitial fluid drug. However, limited physiologic significance has
space of the target tissue can be measured. This also been assigned to a few groups of tissues (Table 5-2).
affords a means to quantify, for the first time, the The nonlinear profile of plasma drug concentra-
inter- or intraindividual variability associated with tion–time is the result of many factors interacting
the in vivo distribution process. Although these novel together, including blood flow to the tissues, the per-
techniques are promising, measurement of drug or meability of the drug into the tissues (fat solubility),
active metabolite concentrations in target tissues and partitioning, the capacity of the tissues to accumulate
the subsequent development of associated pharmaco- drug, and the effect of disease factors on these pro-
kinetic models is not a routine practice in standard cesses (see Chapter 11). Impaired cardiac function
drug development and certainly is not mandated by may produce a change in blood flow and these affect
regulatory requirements. Occasionally, tissue sam- the drug distributive phase, whereas impairment of the
ples may be collected after a drug overdose episode. kidney or the liver may decrease drug elimination as
For example, the two-compartment model has been shown by a prolonged elimination half-life and cor-
used to describe the distribution of colchicine, even responding reduction in the slope of the terminal
though the drug’s toxic tissue levels after fatal over- elimination phase of the curve. Frequently, multiple
doses have only been recently described (Rochdi factors can complicate the distribution profile in such
et al, 1992). Colchicine distribution is now known to a way that the profile can only be described clearly
be affected by P-gp (also known as ABCB1 or with the assistance of a simulation model.

 

100 Chapter 5

TABLE 5-1 Blood Flow to Human Tissues

Percent Percent Blood Flow
Tissue Body Weight Cardiac Output (mL/100 g tissue per min)

Adrenals 0.02 1 550

Kidneys 0.4 24 450

Thyroid 0.04 2 400

Liver
Hepatic 2.0 5 20
Portal 20 75

Portal-drained viscera 2.0 20 75

Heart (basal) 0.4 4 70

Brain 2.0 15 55

Skin 7.0 5 5

Muscle (basal) 40.0 15 3

Connective tissue 7.0 1 1

Fat 15.0 2 1

Data from Spector WS: Handbook of Biological Data, Saunders, Philadelphia, 1956; Glaser O: Medical Physics, Vol II, Year Book Publishers, Chicago,
1950; Butler TC: Proc First International Pharmacological Meeting, vol 6, Pergamon Press, 1962.

TABLE 5-2 General Grouping of Tissues According to Blood Supplya

Blood Supply Tissue Group Percent Body Weight

Highly perfused Heart, brain, hepatic-portal system, kidney, and endocrine glands 9
Skin and muscle 50
Adipose (fat) tissue and marrow 19

Slowly perfused Bone, ligaments, tendons, cartilage, teeth, and hair 22

aTissue uptake will also depend on such factors as fat solubility, degree of ionization, partitioning, and protein binding of the drug.

Adapted with permission from Eger (1963).

TWO-COMPARTMENT OPEN MODEL distributes into two compartments, the central com-
partment and the tissue, or peripheral, compartment.

Many drugs given in a single intravenous bolus dose The drug distributes rapidly and uniformly in the
demonstrate a plasma level–time curve that does not central compartment. A second compartment, known
decline as a single exponential (first-order) process. as the tissue or peripheral compartment, contains tis-
The plasma level–time curve for a drug that follows a sues in which the drug equilibrates more slowly.
two-compartment model (Fig. 5-1) shows that the Drug transfer between the two compartments is
plasma drug concentration declines biexponentially assumed to take place by first-order processes.
as the sum of two first-order processes—distribution There are several possible two-compartment
and elimination. A drug that follows the pharmacoki- models (Fig. 5-2). Model A is used most often and
netics of a two-compartment model does not equili- describes the plasma level–time curve observed in
brate rapidly throughout the body, as is assumed for Fig. 5-1. By convention, compartment 1 is the cen-
a one-compartment model. In this model, the drug tral compartment and compartment 2 is the tissue

 

Multicompartment Models: Intravenous Bolus Administration 101

Model A increases up to a maximum in a given tissue, whose
k

Central compartment 12 Tissue compartment value may be greater or less than the plasma drug
Dp Vp Cp Dt Vt Ct

k21 concentration. At maximum tissue concentrations,
k10 the rate of drug entry into the tissue equals the rate

of drug exit from the tissue. The fraction of drug in
Model B

k the tissue compartment is now in equilibrium (distri-
Central compartment 12 Tissue compartment

Dp Vp Cp Dt Vt Ct bution equilibrium) with the fraction of drug in the
k21

k20 central compartment (Fig. 5-3), and the drug concen-
trations in both the central and tissue compartments

Model C decline in parallel and more slowly compared to the
k

Central compartment 12 Tissue compartment distribution phase. This decline is a first-order pro-
Dp Vp Cp Dt Vt Ct

k21 cess and is called the elimination phase or the beta
k10 k20

(b) phase (Fig. 5-1, line b). Since plasma and tissue

FIGURE 5-2 concentrations decline in parallel, plasma drug con-
Two-compartment open models, intrave-

nous injection. centrations provide some indication of the concen-
tration of drug in the tissue. At this point, drug

compartment. The rate constants k12 and k21 repre- kinetics appears to follow a one-compartment model
sent the first-order rate transfer constants for the in which drug elimination is a first-order process
movement of drug from compartment 1 to com- described by b (also known as b). A typical tissue
partment 2 (k12) and from compartment 2 to com- drug level curve after a single intravenous dose is
partment 1 (k21). The transfer constants are sometimes shown in Fig. 5-3.
termed microconstants, and their values cannot be Tissue drug concentrations in the pharmacoki-
estimated directly. Most two-compartment models netic model are theoretical only. The drug level in the
assume that elimination occurs from the central theoretical tissue compartment can be calculated
compartment model, as shown in Fig. 5-2 (model A), once the parameters for the model are estimated.
unless other information about the drug is known. However, the drug concentration in the tissue com-
Drug elimination is presumed to occur from the cen- partment represents the average drug concentration
tral compartment, because the major sites of drug in a group of tissues rather than any real anatomic
elimination (renal excretion and hepatic drug metab- tissue drug concentration. In reality, drug concentra-
olism) occur in organs such as the kidney and liver, tions may vary among different tissues and possibly
which are highly perfused with blood. within an individual tissue. These varying tissue

The plasma level–time curve for a drug that fol-
lows a two-compartment model may be divided into
two parts, (a) a distribution phase and (b) an elimina-

300
tion phase. The two-compartment model assumes 200
that, at t = 0, no drug is in the tissue compartment. 100
After an IV bolus injection, drug equilibrates rapidly 50 Plasma

in the central compartment. The distribution phase
of the curve represents the initial, more rapid decline

10
of drug from the central compartment into the tissue 5
compartment (Fig. 5-1, line a). Although drug elimi- Tissue

nation and distribution occur concurrently during the
distribution phase, there is a net transfer of drug 1

Time
from the central compartment to the tissue compart-

FIGURE 5-3 Relationship between tissue and plasma
ment because the rate of distribution is faster than

drug concentrations for a two-compartment open model. The
the rate of elimination. The fraction of drug in the maximum tissue drug concentration may be greater or less
tissue compartment during the distribution phase than the plasma drug concentration.

Drug concentration

 

102 Chapter 5

drug concentrations are due to differences in the The relationship between the amount of drug in each
partitioning of drug into the tissues, as discussed in compartment and the concentration of drug in that
Chapter 11. In terms of the pharmacokinetic model, compartment is shown by Equations 5.3 and 5.4:
the differences in tissue drug concentration are
reflected in the k12/k21 ratio. Thus, tissue drug con- Dp

Cp = (5.3)
centration may be higher or lower than the plasma Vp

drug concentrations, depending on the properties of
D

the individual tissue. Moreover, the elimination rates t
Ct = (5.4)

V
of drug from the tissue compartment may not be the t

same as the elimination rates from the central com-
partment. For example, if k12·Cp is greater than k21·Ct where Dp = amount of drug in the central compart-
(rate into tissue > rate out of tissue), tissue drug ment, Dt = amount of drug in the tissue compartment,
concentrations will increase and plasma drug con- Vp = volume of drug in the central compartment, and
centrations will decrease. Real tissue drug concen- Vt = volume of drug in the tissue compartment.
tration can sometimes be calculated by the addition
of compartments to the model until a compartment

dC
that mimics the experimental tissue concentrations is p D Dp D

= t p
k2 − k k

dt 1 V 12 − (5.5)
t V 10

p V
found. d

In spite of the hypothetical nature of the tissue
dC D

t p D
compartment, the theoretical tissue level is still valu- = t

k12 − k21 (5.6)
dt Vp Vt

able information for clinicians. The theoretical tissue
concentration, together with the blood concentra-
tion, gives an accurate method of calculating the Solving Equations 5.5 and 5.6 using Laplace trans-

total amount of drug remaining in the body at any forms and matrix algebra will give Equations 5.7 and

given time (see digoxin example in Table 5-5). This 5.8, which describe the change in drug concentration

information would not be available without pharma- in the blood and in the tissue with respect to time:

cokinetic models.
D0

In practice, a blood sample is removed periodi- = p  k21 − α k2 β
C

− α e−α t 1 −
 + −β 

e t
p Vp β α β ( .7)

 − 
5

cally from the central compartment and the plasma is
analyzed for the presence of drug. The drug plasma
level–time curve represents a phase of initial rapid k 0

21DP
equilibration with the central compartment (the dis- C = α β (e−βt

t − e−α t) (5.8)
Vt ( − )

tribution phase), followed by an elimination phase
after the tissue compartment has also equilibrated
with drug. The distribution phase may take minutes k2 − α −α k − β

DP = D0  1 2 β 
P e t

β − α + 1
e− t

 α − β  (5.9)
or hours and may be missed entirely if the blood is
sampled too late or at wide intervals after drug
administration.

k 0
21DIn the model depicted above, k12 and k21 are P

D β −α
t =

( β (e− t − e t

α ) (5.10)
− )

first-order rate constants that govern the rate of drug
distribution into and out of the tissues and plasma:

where D0
P = dose given intravenously, t = time after

administration of dose, and a and b are constants
dC

t = k12Cp − k21C (5.1)
dt t that depend solely on k12, k21, and k10. The amount of

drug remaining in the plasma and tissue compart-
dC ments at any time may be described realistically by

p
= k21Ct − k12Cp − k 0C (5 2

dt 1 p . ) Equations 5.9 and 5.10.

 

Multicompartment Models: Intravenous Bolus Administration 103

The rate constants for the transfer of drug Method of Residuals
between compartments are referred to as microcon- The method of residuals (also known as feathering,
stants or transfer constants. They relate the amount peeling, or curve stripping) is a commonly employed
of drug being transferred per unit time from one technique for resolving a curve into various expo-
compartment to the other. The values for these micro- nential terms. This method allows the separation of
constants cannot be determined by direct measure- the monoexponential constituents of a biexponential
ment, but they can be estimated by a graphic method. plot of plasma concentration against time and there-

fore, it is a useful procedure for fitting a curve to the
α + β = k + k + k10 (5.11) experimental data of a drug when the drug does not

12 21

clearly follow a one-compartment model. For exam-
αβ = k k (5.12) ple, 100 mg of a drug was administered by rapid IV

21 10

injection to a healthy 70-kg adult male. Blood sam-
ples were taken periodically after the administration

The constants a and b are hybrid first-order rate of drug, and the plasma fraction of each sample was
constants for the distribution phase and elimination assayed for drug. The following data were obtained:
phase, respectively. The mathematical relationships
of a and b to the rate constants are given by
Equations 5.11 and 5.12, which are derived after Plasma Concentration

Time (hour) (μg/mL)
integration of Equations 5.5 and 5.6. Equation 5.7
can be transformed into the following expression: 0.25 43.00

0.5 32.00

C = Ae−α t Be−βt
p + (5.13) 1.0 20.00

1.5 14.00

The constants a and b are rate constants for the
2.0 11.00

distribution phase and elimination phase, respec-
tively. The constants A and B are intercepts on the 4.0 6.50

y axis for each exponential segment of the curve in 8.0 2.80

Equation 5.13. These values may be obtained graph-
12.0 1.20

ically by the method of residuals or by computer.
Intercepts A and B are actually hybrid constants, as 16.0 0.52

shown in Equations 5.14 and 5.15, and do not have
actual physiologic significance. When these data are plotted on semilogarithmic

graph paper, a curved line is observed (Fig. 5-4). The
curved-line relationship between the logarithm of the

D0 (α − k
21)

A = (5.14)
VP (α − β plasma concentration and time indicates that the drug

)
is distributed in more than one compartment. From
these data a biexponential equation, Equation 5.13,

D (k − )
0 21 β

B = (5.15) may be derived, either by computer or by the method
VP (α − β)

of residuals.
As shown in the biexponential curve in Fig. 5-4,

Please note that the values of A and B are empirical the decline in the initial distribution phase is
constants directly proportional to the dose admin- more rapid than the elimination phase. The rapid
istered. All the rate constants involved in two- distribution phase is confirmed with the constant a
compartment model will have units consistent with being larger than the rate constant b. Therefore, at
the first-order process (Jambhekar SS and Breen JP. some later time (generally at a time following the
2009). attainment of distribution equilibrium), the term

 

104 Chapter 5

50 From Equation 5.17, the rate constant can be
Cp = 45e–1.8t + 15e–0.21t obtained from the slope (−b/2.3) of a straight line

representing the terminal exponential phase (Fig. 5-4).
10 The t1/2 for the elimination phase (beta half-life) can

be derived from the following relationship:
5

0.693
a b t1/2β = (5.18)

β
1

0.5 In the sample case considered here, b was found
to be 0.21 h−1. From this information the regression
line for the terminal exponential or b phase is extrap-

0.1 olated to the y axis; the y intercept is equal to B, or
0 4 8 12 16 15 mg/mL. Values from the extrapolated line are then

Time (hours) subtracted from the original experimental data points
FIGURE 5-4 Plasma level–time curve for a two- (Table 5-3) and a straight line is obtained. This line
compartment open model. The rate constants and intercepts represents the rapidly distributed a phase (Fig. 5-4).
were calculated by the method of residuals. The new line obtained by graphing the loga-

rithm of the residual plasma concentration (Cp −Cp′ )

Ae−a t will approach 0, while Be−b t will still have a against time represents the a phase. The value for a

finite value. At this later time Equation 5.13 will is 1.8 h−1, and the y intercept is 45 mg/mL. The elimi-

reduce to: nation t1/2b is computed from b by the use of
Equation 5.18 and has the value of 3.3 hours.

A number of pharmacokinetic parameters may
Cp = Be−βt (5.16)

be derived by proper substitution of rate constants
a and b and y intercepts A and B into the following

which, in common logarithms, is: equations:

βt αβ(A + B)
logCp = logB − (5.17) k10 = (5.19)

2.3 Aβ + Bα

TABLE 5-3 Application of the Method of Residuals

Time Cp Observed Plasma Cp Extrapolated Cp – Cp Residual
(hour) Level Plasma Concentration Plasma Concentration

0.25 43.0 14.5 28.5

0.5 32.0 13.5 18.5

1.0 20.0 12.3 7.7

1.5 14.0 11.0 3.0

2.0 11.0 10.0 1.0

4.0 6.5

8.0 2.8

12.0 1.2

16.0 0.52

Blood level (mg/mL)

 

Multicompartment Models: Intravenous Bolus Administration 105

AB(β − α )2 TABLE 5-4 Two-Compartment Model
k12 = (5.20)

(A + B)(Aβ + Bα ) Pharmacokinetic Parameters of Digoxin

Parameters Unit Normal Renal Impaired
Aβ + Bα

k21 = (5.21)
A + B k12 h–1 1.02 0.45

k21 h–1 0.15 0.11
When an administered drug exhibits the characteris-

k h–1 0.18 0.04
tics of a two-compartment model, the difference
between the distribution rate constant a and the slow Vp L/kg 0.78 0.73

post-distribution/elimination rate constant b plays a D mg/kg 3.6 3.6
critical role. The greater the difference between a and

a 1/h 1.331 0.593
b, the greater is the need to apply two-compartment
model. Failure to do so will result in false clinical b 1/h 0.019 0.007

predictions (Jambhekar SS and Breen JP. 2009). On
the other hand, if this difference is small, it will not myocardium digoxin level. In the simulation below,
cause any significant difference in the clinical predic- the amount of the drug in the plasma compartment at
tions, regardless of the model chosen to describe the any time divided by Vp (54.6 L for the normal subject)
pharmacokinetics of a drug. Then, it may be prudent will yield the plasma digoxin level. At 4 hours after
to follow the principle of PARSIMONY when select-
ing the compartment model by choosing the simpler

1000.00
of the two available models (eg, one-compartment Two-Compartment Model Parameters of Digoxin

versus two) (Jambhekar SS and Breen JP. 2009). Parameter Unit NORM RF
k12 t/h 1.02 0.45
k21 t/h 0.15 0.11

CLINICAL APPLICATION k t/h 0.18 0.04
Vp L/kg 0.76 0.73
D mcg/kg 3.6 3.6

Digoxin in a Normal Patient and in a
a t/h 1.331 0.593

Renal-Failure Patient—Simulation of Plasma b t/h 0.019 0.007

and Tissue Level of a Two-Compartment RF tissue

Model Drug

Once the pharmacokinetic parameters are determined NORM tissue
for an individual, the amount of drug remaining in the

100.00
plasma and tissue compartments may be calculated
using Equations 5.9 and 5.10. The pharmacokinetic
data for digoxin were calculated in a normal and in a
renal-impaired, 70-kg subject using the parameters in RF
Table 5-4 as reported in the literature. The amount of
digoxin remaining in the plasma and tissue compart-
ments is tabulated in Table 5-5 and plotted in Fig. 5-5.
It can be seen that digoxin stored in the plasma NORM

declines rapidly during the initial distributive phase,
while drug amount in the tissue compartment takes
3–4 hours to accumulate for a normal subject. It is
interesting that clinicians have recommended that 10.00

0 5 10 15 20 25
digoxin plasma samples be taken at least several hours Hour

after IV bolus dosing (3–4+ hours, Winters, 1994, and FIGURE 5-5 Amount of digoxin (simulated) in the plasma
4–8 hours, Schumacher, 1995) for a normal subject, and tissue compartment after an IV dose to a normal and a
since the equilibrated level is more representative of renal-failure (RF) patient.

Digoxin amount in plasma (mcg)

 

106 Chapter 5

an IV dose of 0.25 mg, Cp = Dp/Vp = 24.43 µg/54.6 L = since the amount of drug is calculated using mass
0.45 ng/mL, corresponding to 3 × 0.45 ng/mL = balance. The rate of drug entry into the tissue in
1.35 ng/mL if a full loading dose of 0.75 mg is given micrograms per hour at any time is k12Dp, while the
in a single dose. Although the initial plasma drug levels rate of drug leaving the tissue is k21Dt in the same
were much higher than after equilibration, the digoxin units. Both of these rates may be calculated from
plasma concentrations are generally regarded as not Table 5-5 using k12 and k21 values listed in Table 5-4.
toxic, since drug distribution is occurring rapidly. Although some clinicians assume that tissue and

The tissue drug levels were not calculated. The plasma concentrations are equal when at full equili-
tissue drug concentration represents the hypothetical bration, tissue and plasma drug ratios are determined
tissue pool, which may not represent actual drug by the partition coefficient (a drug-specific physical
concentrations in the myocardium. In contrast, the ratio that measures the lipid/water affinity of a
amount of drug remaining in the tissue pool is real, drug) and the extent of protein binding of the drug.

TABLE 5-5 Amount of Digoxin in Plasma and Tissue Compartment after an IV Dose of
0.252 mg in a Normal and a Renal-Failure Patient Weighing 70 kga

Digoxin Amount

Normal Renal Function Renal Failure (RF)

Time (hour) Dp (µg) Dt (µg) Dp (μg) Dt (μg)

0.00 252.00 0.00 252.00 0.00

0.10 223.68 24.04 240.01 11.01

0.60 126.94 105.54 189.63 57.12

1.00 84.62 140.46 158.78 85.22

2.00 40.06 174.93 107.12 131.72

3.00 27.95 181.45 78.44 156.83

4.00 24.43 180.62 62.45 170.12

5.00 23.17 177.91 53.48 176.88

6.00 22.53 174.74 48.39 180.04

7.00 22.05 171.50 45.45 181.21

8.00 21.62 168.28 43.69 181.29

9.00 21.21 165.12 42.59 180.77

10.00 20.81 162.01 41.85 179.92

11.00 20.42 158.96 41.32 178.89

12.00 20.03 155.97 40.89 177.77

13.00 19.65 153.04 40.53 176.60

16.00 18.57 144.56 39.62 173.00

24.00 15.95 124.17 37.44 163.59

aDp drug in plasma compartment; D
 t′ drug in tissue compartment.

Source: Data generated from parameters published by Harron (1989).

 

Multicompartment Models: Intravenous Bolus Administration 107

Figure 5-5 shows that the time for the RF (renal- reproducible because they are affected by short-term
failure or renal-impaired) patient to reach stable tis- physiologic changes. For example, stress may result
sue drug levels is longer than the time for the normal in short-term change of the hematocrit or plasma
subject due to changes in the elimination and trans- volume and possibly other hemodynamic factors.
fer rate constants. As expected, a significantly higher
amount of digoxin remains in both the plasma and
tissue compartments in the renally impaired subject Frequently Asked Questions

compared to the normal subject. »»Are “hypothetical” or “mathematical” compartment
models useful in designing dosage regimens in the
clinical setting? Does “hypothetical” mean “not real”?

PRACTICE PROBLEM
»»If physiologic models are better than compartment

From Figure 5-5 or Table 5-4, how many hours does models, why not just use physiologic models?

it take for maximum tissue concentration to be »»Since clearance is the term most often used in clinical
reached in the normal and the renal-impaired patient? pharmacy, why is it necessary to know the other

pharmacokinetic parameters?
Solution

At maximum tissue concentration, the rate of drug
entering the tissue compartment is equal to the rate Apparent Volumes of Distribution
of leaving (ie, at the peak of the tissue curve, where As discussed in Chapter 4, the apparent VD is a use-
the slope = 0 or not changing). This occurs at about ful parameter that relates plasma concentration to the
3–4 hours for the normal patient and at 7–8 hours amount of drug in the body. For drugs with large
for the renal-impaired patient. This may be verified extravascular distribution, the apparent volume of
by examining at what time Dpk12 = Dtk21 using the distribution is generally large. Conversely, for polar
data from Tables 5-4 and 5-5. Before maximum Ct drugs with low lipid solubility, the apparent VD is
is reached, there is a net flux of drug into the tissue, generally small. Drugs with high peripheral tissue
that is, Dpk12 > Dtk21, and beyond this point, there is binding also contribute to a large apparent VD. In
a net flux of drug out of the tissue compartment, multiple-compartment kinetics, such as the two-
that is, Dtk12 > Dpk12. compartment model, several types of volumes of

distribution, each based on different assumptions,
can be calculated. Volumes of distribution generally

PRACTICAL FOCUS
reflect the extent of drug distribution in the body on

The distribution half-life of digoxin is about 31 minutes a relative basis, and the calculations depend on the
(t availability of data. In general, it is important to refer

½a = 0.694/a = 0.694/1.331 = 31 min) based on
Table 5-4. Both clinical experience and simulated tis- to the same volume parameter when comparing
sue amount in Table 5-4 recommend “several hours” kinetic changes in disease states. Unfortunately, val-
for equilibration, longer than 5t½a or 5 × 32 minutes. ues of apparent volumes of distribution of drugs
(1) Is digoxin elimination in tissue adequately mod- from tables in the clinical literature are often listed
eled in this example? (2) Digoxin was not known to without specifying the underlying kinetic processes,
be a P-gp substrate when the data were analyzed; can model parameters, or methods of calculation.
the presence of a transporter at the target site change
tissue drug concentration, necessitating a longer Volume of the Central Compartment
equilibration time? This is a proportionality constant that relates the

Generally, the ability to obtain a blood sample amount or mass of drug and the plasma concentration
and get accurate data in the alpha (distribution) immediately (ie, at time zero) following the adminis-
phase is difficult for most drugs because of its short tration of a drug. The volume of the central compart-
duration. Moreover, the alpha phase may not be very ment is useful for determining the drug concentration

 

108 Chapter 5

directly after an IV injection into the body. In clinical volume of distribution will be 3 L; if it is not, then
pharmacy, this volume is also referred to as Vi or the distribution of drug may also occur outside the vas-
initial volume of distribution as the drug distributes cular pool into extra- and intracellular fluid.
within the plasma and other accessible body fluids.
This volume is generally smaller than the terminal D

0
Vp = (5.22)

C0
volume of distribution after drug distribution to tissue p

is completed. The volume of the central compartment
is generally greater than 3 L, which is the volume of At zero time (t = 0), the entire drug in the body is in

the plasma fluid for an average adult. For many polar the central compartment. C0
p can be shown to be equal

drugs, an initial volume of 7–10 L may be interpreted to A + B by the following equation:

as rapid drug distribution within the plasma and some
extracellular fluids. For example, the Vp of moxalac- C = Ae−α t + Be−βt

p (5.23)
tam ranges from 0.12 to 0.15 L/kg, corresponding to
about 8.4–10.5 L for a typical 70-kg patient At t = 0, e0 = 1. Therefore,
(Table 5-6). In contrast, the Vp of hydromorphone is
about 24 L, possibly because of its rapid exit from the C0

p = A + B (5.24)
plasma into tissues even during the initial phase.

As in the case of the one-compartment model, Vp Vp is determined from Equation 5.25 by measuring
may be determined from the dose and the instanta-

0 A and B after feathering the curve, as discussed
neous plasma drug concentration, Cp . Vp is also use-

previously:
ful in the determination of drug clearance if k (or t½)
is known, as in Chapter 4. D

0
V

In the two-compartment model, Vp may also be p = (5.25)
A + B

considered a mass balance factor governed by the
mass balance between dose and concentration, that Alternatively, the volume of the central compart-

is, drug concentration multiplied by the volume of ment may be calculated from the [AUC] in a manner
0

the fluid must equal the dose at time = 0. At time = 0, similar to the calculation for the apparent VD in the one-
no drug is eliminated, D0 = VpCp. The basic model compartment model. For a one-compartment model
assumption is that plasma drug concentration is rep-
resentative of drug concentration within the distribu- D

[AUC] 0
=

0 (5.26)
tion fluid of plasma. If this statement is true, then the kVD

TABLE 5-6 Pharmacokinetic Parameters (mean ± SD) of Moxalactam in Three Groups of Patients

A B ` a k
Group μg/mL μg/mL h–1 h–1 h–1

1 138.9 ± 114.9 157.8 ± 87.1 6.8 ± 4.5 0.20 ± 0.12 0.38 ± 0.26

2 115.4 ± 65.9 115.0 ± 40.8 5.3 ± 3.5 0.27 ± 0.08 0.50 ± 0.17

3 102.9 ± 39.4 89.0 ± 36.7 5.6 ± 3.8 0.37 ± 0.09 0.71 ± 0.16

Cl Vp Vt (VD)ss (VD)β
Group mL/min L/kg L/kg L/kg L/kg

1 40.5 ± 14.5 0.12 ± 0.05 0.08 ± 0.04 0.20 ± 0.09 0.21 ± 0.09

2 73.7 ± 13.1 0.14 ± 0.06 0.09 ± 0.04 0.23 ± 0.10 0.24 ± 0.12

3 125.9 ± 28.0 0.15 ± 0.05 0.10 ± 0.05 0.25 ± 0.08 0.29 ± 0.09

 

Multicompartment Models: Intravenous Bolus Administration 109

In contrast, [AUC] for the two-compartment Substituting Equation 5.31 into Equation 5.32, and
0

model is: expressing Dp as VpCp, a more useful equation for
the calculation of (VD)ss is obtained:

D

[AUC] 0
=

0 (5.27)
kVp CpVp + k12VpCp /k21

(VD )ss = (5.33)
Cp

Rearrangement of this equation yields:

D which reduces to
0

Vp = (5.28)

k [AUC]0 k
(V 2

D )
1

ss = Vp + V (5 34)
k p .
21

Apparent Volume of Distribution at
Steady State In practice, Equation 5.34 is used to calculate

This is a proportionality constant that relates the plasma (VD)ss. The (VD)ss is a function of the transfer con-

concentration and the amount of drug remaining in the stants, k12 and k21, which represent the rate constants

body at a time, following the attainment of practical of drug going into and out of the tissue compartment,

steady state (which is reached at a time greater by at respectively. The magnitude of (VD)ss is dependent on

least four elimination half-lives of the drug). At steady- the hemodynamic factors responsible for drug distri-

state conditions, the rate of drug entry into the tissue bution and on the physical properties of the drug,

compartment from the central compartment is equal to properties which, in turn, determine the relative

the rate of drug exit from the tissue compartment into amount of intra- and extravascular drug remaining in

the central compartment. These rates of drug transfer the body.

are described by the following expressions:
Extrapolated Volume of Distribution

The extrapolated volume of distribution (V
= D)exp is

Dtk21 Dpk12 (5.29)
calculated by the following equation:

k12Dp
D = (5.30) D

t k (VD ) = 0
21 exp (5.35)

B

Because the amount of drug in the central compart-
where B is the y intercept obtained by extrapolation

ment, Dp, is equal to VpCp, by substitution in the above
of the b phase of the plasma level curve to the y axis

equation,
(Fig. 5-4). Because the y intercept is a hybrid con-

k stant, as shown by Equation 5.15, (VD)exp may also
12CpVp

D = (5.31)
t k be calculated by the following expression:

21

The total amount of drug in the body at steady α − β
(V =  

D )state is equal to the sum of the amount of drug in the exp Vp  k2 − β
(5.36)

1

tissue compartment, Dt, and the amount of drug in
the central compartment, Dp. Therefore, the apparent This equation shows that a change in the distribution
volume of drug at steady state (VD)ss may be calcu- of a drug, which is observed by a change in the value
lated by dividing the total amount of drug in the for Vp, will be reflected in a change in (VD)exp.
body by the concentration of drug in the central
compartment at steady state: Volume of Distribution by Area

The volume of distribution by area (VD)area, also
Dp + Dt

( known as (VD)b, is obtained through calculations
VD )ss = (5.32)

Cp similar to those used to find Vp, except that the rate

 

110 Chapter 5

constant b is used instead of the overall elimination Substituting kVp for clearance in Equation 5.38, one
rate constant k. This volume represents a proportion- obtains:
ality factor between plasma concentrations and
amount of drug in body during the terminal or b kVp

(VD )β = (5.39)
phase of disposition. (VD)b is often calculated from β
total body clearance divided by b and is influenced
by drug elimination in the beta, or b, phase. This

Theoretically, the value for b may remain
volume will be considered a time-dependent and

unchanged in patients showing various degrees of
clearance-dependent volume of distribution parameter.

moderate renal impairment. In this case, a reduction
The value of (VD)b is affected by elimination, and it

in (VD)b may account for all the decrease in Cl, while
changes as clearance is altered. Reduced drug clear-

b is unchanged in Equation 5.39. Within the body, a
ance from the body may increase AUC (area under

redistribution of drug between the plasma and the
the curve), such that (VD)b is either reduced or

tissue will mask the expected decline in b. The fol-
unchanged depending on the value of b, as shown by

lowing example in two patients shows that the b
Equation 5.36.

elimination rate constant remains the same, while
the distributional rate constants change. Interestingly,

D
(V 7

D )β = (V 4 3
D )

area = 0 ( . ) Vp is unchanged, while (VD)b would be greatly

β [AUC]0 changed in the simulated example. An example of a
drug showing a constant b slope while the renal

A slower clearance allows more time for drug equili- function as measured by Clcr decreases from 107 to
bration between plasma and tissues yielding a 56, 34, and 6 mL/min (see Chapter 7) has been
smaller (VD)b. The lower limit of (VD)b is Vss: observed with the aminoglycoside drug gentamicin

in various patients after IV bolus dose (Schentag
Lim(VD )β = Vss et al, 1977). Gentamicin follows polyexponential

decline with a significant distributive phase. The
Cl → 0

following simulation problem may help clarify the
situation by changing k and clearance while keeping

Thus, (VD)b has value in representing Vss for low-
b constant.

clearance drugs as well as estimating terminal or b
phase. Smaller (VD)b values than normal are often
observed in patients with renal failure because of the PRACTICE PROBLEM
reduced Cl. This is a consequence of the Cl-dependent
time of equilibration between plasma and tissue. Thus, Simulated plasma drug concentrations after an IV
Vss is preferred in separating alterations in elimina- bolus dose (100 mg) of an antibiotic in two patients,
tion from those in distribution. patient 1 with a normal k, and patient 2 with a

Generally, reduced drug clearance is also reduced k, are shown in Fig. 5-6. The data in the two
accompanied by a decrease in the constant b (ie, an patients were simulated with parameters using the
increase in the b elimination half-life). For example, two-compartment model equation. The parameters
in patients with renal dysfunction, the elimination used are as follows:
half-life of the antibiotic amoxacillin is longer

Normal subject, k = 0.3 h−1, Vp = 10 L, Cl = 3 L/h
because renal clearance is reduced.

Because total body clearance is equal to k12 = 5 h−1, k21 = 0.2 h−1

D0 / [AUC] , (V ) y b x r s e
0 D β ma e e p e s d in terms of

clearance and the rate constant b: Subject with moderate renal impairment,
k = 0.1 h−1, Vp = 10 L, Cl = 1 L/h

Cl
(VD )β = (5.38)

β k12 = 2 h−1, k21 = 0.25 h−1

 

Multicompartment Models: Intravenous Bolus Administration 111

10 reflects the data. A decrease in the (VD)b with b
unchanged is possible, although this is not the
common case. When this happens, the termi-

Patient 2 nal data (see Fig. 5-6) conclude that the beta
elimination half-lives of patients 1 and 2 are

1
the same due to a similar b. Actually, the real

Patient 1 elimination half-life of the drug derived from
k is a much better parameter, since k reflects
the changes in renal function, but not b, which

0.1 remains unchanged since it is masked by the
1 2

Time (hours) changes in (VD)b.
3. Both patients have the same b value (b =

FIGURE 5-6 Simulation of plasma drug concentration 0.011 h−1); the terminal slopes are identical.
after an IV bolus dose (100 mg) of an antibiotic in two patients,

Ignoring early points by only taking terminal
one with a normal k (patient 1) and the other with reduced k
(patient 2). data would lead to an erroneous conclusion

that the renal elimination process is unchanged,
while the volume of distribution of the renally

Questions impaired patient is smaller. In this case, the
1. Is a reduction in drug clearance generally renally impaired patient has a clearance of

accompanied by an increase in plasma drug 1 L/h compared with 3 L/h for the normal
concentration, regardless of which compart- subject, and yet the terminal slopes are the
ment model the drug follows? same. The rapid distribution of drug into the

2. Is a reduction in drug clearance generally tissue in the normal subject causes a longer and
accompanied by an increase in the b elimina- steeper distribution phase. Later, redistribution
tion half-life of a drug? [Find (VD)b using of drug out of tissues masks the effect of rapid
Equation 5.38, and then b using Equation 5.39.] drug elimination through the kidney. In the

3. Many antibiotics follow multiexponential renally impaired patient, distribution to tissue is
plasma drug concentration profiles indicating reduced; as a result, little drug is redistributed
drug distribution into tissue compartments. In out from the tissue in the b phase. Hence, it
clinical pharmacokinetics, the terminal half- appears that the beta phases are identical in the
life is often determined with limited early data. two patients.
Which patient has a greater terminal half-life
based on the simulated data?

Significance of the Volumes of Distribution

From Equations 5.38 and 5.39 we can observe that
Solutions (VD)b is affected by changes in the overall elimina-

1. A reduction in drug clearance results in less tion rate (ie, change in k) and by change in total body

drug being removed from the body per unit clearance of the drug. After the drug is distributed,

time. Drug clearance is model independent. the total amount of drug in the body during the

Therefore, the plasma drug concentration elimination of b phase is calculated by using (VD)b.

should be higher in subjects with decreased Vp is sometimes called the initial volume of

drug clearance compared to subjects with distribution and is useful in the calculation of drug

normal drug clearance, regardless of which clearance. The magnitudes of the various apparent

compartment model is used (see Fig. 5-6). volumes of distribution have the following relation-

2. Clearance in the two-compartment model is ships to each other:

affected by the elimination rate constant, b, and
the volume of distribution in the b phase, which (VD )exp > (VD )β >Vp

Plasma drug
concentration (mg/mL)

 

112 Chapter 5

Calculation of another VD, (VD)ss, is possible in mul- changes in pharmacokinetic parameters should not
tiple dosing or infusion (see Chapters 6 and 9). (VD)ss be attributed to physiologic changes without careful
is much larger than Vp; it approximates (VD)b but consideration of method of curve fitting and inter-
differs somewhat in value, depending on the transfer subject differences. Equation 5.39 shows that, unlike
constants. a simple one-compartment open model, (VD)b may

In a study involving a cardiotonic drug given be estimated from k, b, and Vp. Errors in fitting are
intravenously to a group of normal and congestive easily carried over to the other parameter estimates
heart failure (CHF) patients, the average AUC for even if the calculations are performed by computer.
CHF was 40% higher than in the normal subjects. The terms k12 and k21 often fluctuate due to minor
The b elimination constant was 40% less in CHF fitting and experimental difference and may affect
patients, whereas the average (VD)b remained essen- calculation of other parameters.
tially the same. In spite of the edematous conditions
of these patients, the volume of distribution appar-
ently remained constant. No change was found in the Frequently Asked Questions

Vp or (VD)b. In this study, a 40% increase in AUC in »»What is the significance of the apparent volume of
the CHF subjects was offset by a 40% smaller b distribution?

elimination constant estimated by using computer
»»Why are there different volumes of distribution in the

methods. Because the dose was the same, the (VD)b multiple-compartment models?
would not change unless the increase in AUC is not
accompanied by a change in b elimination constant.

From Equation 5.38, the clearance of the drug in Drug in the Tissue Compartment
CHF patients was reduced by 40% and accompanied

The apparent volume of the tissue compartment (Vt) by a corresponding decrease in the b elimination
is a conceptual volume only and does not represent

constant, possibly due to a reduction in renal blood
true anatomic volumes. The Vt may be calculated

flow as a result of reduced cardiac output in CHF
from knowledge of the transfer rate constants and Vp:patients. In physiologic pharmacokinetics, clearance

(Cl) and volume of distribution (VD) are assumed to
be independent parameters that explain the impact of Vpk12

Vt = (5.40)
disease factors on drug disposition. Thus, an increase k21
in AUC of a cardiotonic in a CHF patient was
assumed to be due to a reduction in drug clearance, The calculation of the amount of drug in the tis-
since the volume of distribution was unchanged. The sue compartment does not entail the use of Vt.
elimination half-life was reduced due to reduction in Calculation of the amount of drug in the tissue com-
drug clearance. In reality, pharmacokinetic changes partment provides an estimate for drug accumulation
in a complex system are dependent on many factors in the tissues of the body. This information is vital in
that interact within the system. Clearance is affected estimating chronic toxicity and relating the duration
by drug uptake, metabolism, binding, and more; all of pharmacologic activity to dose. Tissue compart-
of these factors can also influence the drug distribu- ment drug concentration is an average estimate of the
tion volume. Many parameters are assumed to be tissue pool and does not mean that all tissues have
constant and independent for simplification of the this concentration. The drug concentration in a tissue
model. Blood flow is an independent parameter that biopsy will provide an estimate for drug in that tissue
will affect both clearance and distribution. However, sample. Due to differences in blood flow and drug
blood flow is, in turn, affected and regulated by partitioning into the tissue, and heterogenicity, even
many physiologic compensatory factors. a biopsy from the same tissue may have different

For drugs that follow two-compartment model drug concentrations. Together with Vp and Cp, used to
kinetics, changes in disease states may not result in calculate the amount of drug in the plasma, the com-
different pharmacokinetic parameters. Conversely, partment model provides mass balance information.

 

Multicompartment Models: Intravenous Bolus Administration 113

Moreover, the pharmacodynamic activity may cor- 6–8 hours apart to minimize potential side effects
relate better with the tissue drug concentration–time from overdigitization. If the entire loading dose were
curve. To calculate the amount of drug in the tissue administered intravenously, the plasma level would
compartment Dt, the following expression is used: be about 4–5 ng/mL after 1 hour, while the level

would drop to about 1.5 ng/mL at about 4 hours. The
k 0
12Dp exact level after a given IV dose may be calculated

D ( −β
= e t − e−α t

t ) (5.41)
α − β using Equation 5.7 at any time desired. The pharma-

cokinetic parameters for digoxin are available in
Table 5-4.

PRACTICAL FOCUS In addition to metabolism, digoxin distribution is
affected by a number of processes besides blood

The therapeutic plasma concentration of digoxin is flow. Digoxin and many other drugs are P-gp
between 1 and 2 ng/mL; because digoxin has a long (P-glycoprotein) substrates, a transporter that is often
elimination half-life, it takes a long time to reach a located in cell membranes that efflux drug in and out
stable, constant (steady-state) level in the body. A of cells, and can theoretically affect k12 (cell uptake)
loading dose is usually given with the initiation of and k21 (cell efflux). Some transporters such as P-gp
digoxin therapy. Consider the implications of the or ABC transporters exhibit genetic variability and
loading dose of 1 mg suggested for a 70-kg subject. therefore can contribute to pharmacokinetic varabil-
The clinical source cited an apparent volume of dis- ity between patients. For example, if drug transport-
tribution of 7.3 L/kg for digoxin in determining the ers avidly carry drug to metabolic sites, then
loading dose. Use the pharmacokinetic parameters metabolism would increase, and plasma levels AUC
for digoxin in Table 5-4. would decrease. The converse is also true; examples

of drugs that are known to increase digoxin level
Solution include amidiodarone, quinidine, and verapamil.

The loading dose was calculated by considering the Verapamil is a potent P-gp inhibitor and a common

body as one compartment during steady state, at agent used to test if an unknown substrate can be

which time the drug well penetrates the tissue com- blocked by a P-gp inhibitor.

partment. The volume of distribution (VD)b of digoxin Many anticancer drugs such as taxol, vincris-

is much larger than Vp, or the volume of the plasma tine, and vinblastine are P-gp substrates. P-gp can

compartment. be located in GI, kidney, liver, and entry to BBB

Using Equation (5.39), (see Chapter 11 for distribution and Chapter 13 for
genetically expressed transporters). There are other
organic anion and cation transporters in the body that

kVp
(VD )β = contribute to efflux of drug into and out of cells.

β
Efflux and translocation of a drug can cause a drug to
lose efficacy (MDR resistance) in many anticancer

0.18/h × 0.78 L/kg
= = 7.39 L/kg

0.019/h drugs. It may not always be possible to distinquish a
specific drug transporter in a specific organ or tissue

mL ng
DL = 7390 × 70 kg ×1.5 in vivo due to ongoing perfusion and the potential for

kg mL multiple transporter/carriers involved. These factors;
drug binding to proteins in blood, cell, and cell mem-

where DL = (VD)b ⋅ (Cp)ss. The desired steady plasma branes; and diffusion limiting processes contribute to
concentration, (Cp)ss, was selected by choosing a “multiexponential” drug distribution kinetically for
value in the middle of the therapeutic range. The many drugs. Much of in vivo kinetics information
loading dose is generally divided into two or three can be learned by examining the kinetics of the IV
doses or is administered as 50% in the first dose bolus time-concentration profile when a suitable sub-
with the remaining drug given in two divided doses strate probe is administered.

 

114 Chapter 5

Drug Clearance consideration in understanding drug permeation and

The definition of clearance of a drug that follows a toxicity. For example, the plasma–time profiles of

two-compartment model is similar to that of the one- aminoglycosides, such as gentamicin, are more use-

compartment model. Clearance is the volume of ful in explaining toxicity than average plasma or

plasma that is cleared of drug per unit time. Clearance drug concentration taken at peak or trough time.

may be calculated without consideration of the com-
partment model. Thus, clearance may be viewed as a Elimination Rate Constant
physiologic concept for drug removal, even though In the two-compartment model (IV administration),
the development of clearance is rooted in classical the elimination rate constant, k, represents the elimi-
pharmacokinetics. nation of drug from the central compartment, whereas

Clearance is often calculated by a noncompart- b represents drug elimination during the beta or
mental approach, as in Equation 5.37, in which the elimination phase, when distribution is mostly com-
bolus IV dose is divided by the area under the plete. Because of redistribution of drug out of the
plasma concentration–time curve from zero to infin- tissue compartment, the plasma drug level curve

∞ ∞

ity, [AUC] . In evaluating the AUC
0 [ ] , early time

0 declines more slowly in the b phase. Hence b is
points must be collected frequently to observe the smaller than k; thus k is a true elimination constant,
rapid decline in drug concentrations (distribution whereas b is a hybrid elimination rate constant that is
phase) for drugs with multicompartment pharmaco- influenced by the rate of transfer of drug into and out
kinetics. In the calculation of clearance using the of the tissue compartment. When it is impractical to
noncompartmental approach, underestimating the determine k, b is calculated from the b slope. The t1/2b
area can inflate the calculated value of clearance. is often used to calculate the drug dose.

D
0

Cl = (5.42)

[AUC] THREE-COMPARTMENT
0

OPEN MODEL
Equation 5.42 may be rearranged to Equation 5.43

to show that Cl in the two-compartment model is the The three-compartment model is an extension of the

product of (VD)b and b. two-compartment model, with an additional deep
tissue compartment. A drug that demonstrates the
necessity of a three-compartment open model is

Cl = (VD )β β (5.43) distributed most rapidly to a highly perfused central
compartment, less rapidly to the second or tissue

If both parameters are known, then calculation of compartment, and very slowly to the third or deep
clearance is simple and more accurate than using the tissue compartment, containing such poorly per-
trapezoidal rule to obtain area. Clearance calculations fused tissue as bone and fat. The deep tissue com-
that use the two-compartment model are viewed as partment may also represent tightly bound drug in
model dependent because more assumptions are the tissues. The three-compartment open model is
required, and such calculations cannot be regarded as shown in Fig. 5-7.
noncompartmental. However, the assumptions pro- A solution of the differential equation describ-
vide additional information and, in some sense, spe- ing the rates of flow of drug into and out of the
cifically describe the drug concentration–time profile central compartment gives the following equation:
as biphasic.

Clearance is a term that is useful in calculating C = Ae−α t + Be−βt
p +Ce−δ t (5.44)

average drug concentrations. With many drugs, a
biphasic profile suggests a rapid tissue distribution where A, B, and C are the y intercepts of extrapolated
phase followed by a slower elimination phase. lines for the central, tissue, and deep tissue compart-
Multicompartment pharmacokinetics is an important ments, respectively, and a, b, and g are first-order

 

Multicompartment Models: Intravenous Bolus Administration 115

k21 k13
Tissue compartment Central compartment Deep tissue compartment

Vt C t Dt Vp Cp Dp Vdt Cdt Ddt
k12 k31

k

FIGURE 5-7 Three-compartment open model. This model, as with the previous two-compartment
models, assumes that all drug elimination occurs via the central compartment.

rate constants for the central, tissue, and deep tissue rate constant k, volume of the central compartment,
compartments, respectively. and area are shown in the following equations:

A three-compartment equation may be written
by statisticians in the literature as (A + B +C)αβδ

k = (5.45)
Aβδ + Bαδ +Cαβ

= −λ1t + −λ2t −λ t
Cp Ae Be +Ce 3 (5.44a)

D
0

Vp = (5.46)
Instead of a, b, g, etc, l1, l2, l3 are substituted to A + B +C

express the triexponential feature of the equation.
A B C

Similarly, the n-compartment model may be expressed [AUC] = + + (5.47)
α β δ

with l1, l2, …, ln. The preexponential terms are some-
times expressed as C1, C2, and C3.

The parameters in Equation 5.44 may be solved
CLINICAL APPLICATION

graphically by the method of residuals (Fig. 5-8) or
by computer. The calculations for the elimination Hydromorphone (Dilaudid)

Three independent studies on the pharmacokinetics

100 of hydromorphone after a bolus intravenous injection
reported that hydromorphone followed the pharma-

Cp = 70e–1.5t + 20e–0.2t + 24e–0.03t
cokinetics of a one-compartment model (Vallner et al,

50
1981), a two-compartment model (Parab et al, 1988),
or a three-compartment model (Hill et al, 1991),
respectively. A comparison of these studies is listed in

20 Table 5-7.

A C
10 Comments

The adequacy of the pharmacokinetic model will
5 B depend on the sampling intervals and the drug assay.

The first two studies showed a similar elimination
half-life. However, both Vallner et al (1981) and Parab

2 et al (1988) did not observe a three-compartment
pharmacokinetic model due to lack of appropriate
description of the early distribution phases for

1
0 10 20 30 40 50 hydromorphone. After an IV bolus injection, hydro-

Time (hours) morphone is very rapidly distributed into the tissues.

FIGURE 5-8 Hill et al (1991) obtained a triexponential function
Plasma level–time curve for a three-

compartment open model. The rate constants and intercepts by closely sampling early time periods after the
were calculated by the method of residuals. dose. Average distribution half-lives were 1.27 and

Blood level (mg/mL)

 

116 Chapter 5

TABLE 5-7 Comparison of Hydromorphone Pharmacokinetics

Study Timing of Blood Samples Pharmacokinetic Parameters

6 Males, 25–29 years; mean weight, 76.8 kg 0, 15, 30, 45 minutes One-compartment model

Dose, 2-mg IV bolus (Vallner et al, 1981) 1, 1.5, 2, 3, 4, 6, 8, 10, 12 hours Terminal t1/2 = 2.64 (± 0.88) hours

8 Males, 20–30 years; weight, 50–86 kg 0, 3, 7, 15, 30, 45 minutes Two-compartment model

Dose, 2-mg IV bolus (Parab et al, 1988) 1, 1.5, 2, 3, 4, 6, 8, 10, 12 hours Terminal t1/2 = 2.36 (± 0.58) hours

10 Males, 21–38 years; mean weight, 72.7 kg 1, 2, 3, 4, 5, 7, 10, 15, 20, 30, 45 minutes Three-compartment model

Dose, 10, 20, and 40 mg/kg IV bolus 1, 1.5, 2, 2.5, 3, 4, 5 hours Terminal t1/2 = 3.07 (± 0.25) hours

(Hill et al, 1991)

14.7 minutes, and the average terminal elimination concentration often lies close to the distributive phase,
was 184 minutes (t1/2b). The average value for sys- since its beta elimination half-life is very short, and
temic clearance (Cl) was 1.66 L/min; the initial dilu- ignoring the alpha phase will result in a large error in
tion volume was 24.4 L. If distribution is rapid, the dosing projection.
drug becomes distributed during the absorption
phase. Thus, hydromorphone pharmacokinetics fol-
lows a one-compartment model after a single oral CLINICAL APPLICATION
dose. Loperamide (Imodium®) is an opioid anti-diarrhea

Hydromorphone is administered to relieve acute agent that is useful for illustrating the importance of
pain in cancer or postoperative patients. Rapid pain understanding drug distribution. Loperamide has lit-
relief is obtained by IV injection. Although the drug is tle central opiate effect. Loperamide is a P-gp (an
effective orally, about 50%–60% of the drug is cleared efflux transporter) substrate. The presence of P-gp
by the liver through first-pass effects. The pharmaco- transporter at the blood–brain barrier allows the drug
kinetics of hydromorphone after IV injection suggests to be pumped out of the cell at the cell membrane
a multicompartment model. The site of action is prob- surface without the substrate (loperamide) entering
ably within the central nervous system, as part of the into the interior of the cell. Mice that have had the
tissue compartment. The initial volume or initial dilu- gene for P-gp removed experimentally show pro-
tion volume, Vp, is the volume into which IV injec- found central opioid effects when administered loper-
tions are injected and diluted. Hydromorphone follows amide. Hypothesizing the presence of a tissue
linear kinetics, that is, drug concentration is propor- compartment coupled with a suitable molecular
tional to dose. Hydromorphone systemic clearance is probe can provide a powerful approach toward eluci-
much larger than the glomerular filtration rate (GFR) dating the mechanism of drug distribution and
of 120 mL/min (see Chapter 7), hence the drug is improving drug safety.
probably metabolized significantly by the hepatic
route. A clearance of 1.66 L/min is faster than the
blood flow of 1.2–1.5 L/min to the liver. The drug DETERMINATION OF
must be rapidly extracted or, in addition, must have COMPARTMENT MODELS
extrahepatic elimination. When the distribution phase
is short, the distribution phase may be disregarded Models based on compartmental analysis should
provided that the targeted plasma concentration is suf- always use the fewest number of compartments neces-
ficiently low and the terminal elimination phase is sary to describe the experimental data adequately.
relatively long. If the drug has a sufficiently high tar- Once an empirical equation is derived from the experi-
get plasma drug concentration and the elimination mental observations, it becomes necessary to examine
half-life is short, the distributive phase must not be how well the theoretical values that are calculated
ignored. For example, lidocaine’s effective target from the derived equation fit the experimental data.

 

Multicompartment Models: Intravenous Bolus Administration 117

The observed number of compartments or expo- dose or the assay for the drug cannot measure very
nential phases will depend on (1) the route of drug low plasma drug concentrations.
administration, (2) the rate of drug absorption, (3) the The total time for collection of blood samples is
total time for blood sampling, (4) the number of usually estimated from the terminal elimination half-
samples taken within the collection period, and (5) the life of the drug. However, lower drug concentrations
assay sensitivity. If drug distribution is rapid, then, may not be measured if the sensitivity of the assay is
after oral administration, the drug will become distrib- not adequate. As the assay for the drug becomes
uted during the absorption phase and the distribution more sensitive in its ability to measure lower drug
phase will not be observed. For example, theophylline concentrations, then another compartment with a
follows the kinetics of a one-compartment model after smaller first-order rate constant may be observed.
oral absorption, but after intravenous bolus (given as In describing compartments, each new compart-
aminophylline), theophylline follows the kinetics of a ment requires an additional first-order plot.
two-compartment model. Furthermore, if theophyl- Compartment models having more than three com-
line is given by a slow intravenous infusion rather than partments are rarely of pharmacologic significance.
by intravenous bolus, the distribution phase will not In certain cases, it is possible to “lump” a few com-
be observed. Hydromorphone (Dilaudid), which fol- partments together to get a smaller number of com-
lows a three-compartment model, also follows a one- partments, which, together, will describe the data
compartment model after oral administration, since adequately.
the first two distribution phases are rapid. An adequate description of several tissue com-

Depending on the sampling intervals, a com- partments can be difficult. When the addition of a
partment may be missed because samples may be compartment to the model seems necessary, it is
taken too late after administration of the dose to important to realize that the drug may be retained or
observe a possible distributive phase. For example, slowly concentrated in a deep tissue compartment.
the data plotted in Fig. 5-9 could easily be mistaken
for those of a one-compartment model, because the
distributive phase has been missed and extrapola- PRACTICAL FOCUS
tion of the data to C0

p will give a lower value than
was actually the case. Slower drug elimination Two-Compartment Model: Relation Between

compartments may also be missed if sampling is Distribution and Apparent (Beta) Half-Life

not performed at later sampling times, when the The distribution half-life of a drug is dependent on the
type of tissues the drug penetrates as well as blood
supply to those tissues. In addition, the capacity of the

300 tissue to store drug is also a factor. Distribution half-
200

life is generally short for many drugs because of the
100 ample blood supply to and rapid drug equilibration in

the tissue compartment. However, there is some sup-
porting evidence that a drug with a long elimination
half-life is often associated with a longer distribution

10
phase. It is conceivable that a tissue with little blood
supply and affinity for the drug may not attain a suf-
ficiently high drug concentration to exert its impact on
the overall plasma drug concentration profile during

1
Time (hours) rapid elimination. In contrast, drugs such as digoxin

have a long elimination half-life, and drug is elimi-
FIGURE 5-9 The samples from which data were obtained

nated slowly to allow more time for distribution to
for this graph were taken too late to show the distributive
phase; therefore, the value of C 0

p obtained by extrapolation tissues. Human follicle-stimulating hormone (hFSH)
(straight broken line) is deceptively low. injected intravenously has a very long elimination

Plasma level (mcg/mL)

 

118 Chapter 5

half-life, and its distribution half-life is also quite CLINICAL APPLICATION
long. Drugs such as lidocaine, theophylline, and mil-
rinone have short elimination half-lives and generally Moxalactam Disodium—Effect of Changing
relatively short distributional half-lives. Renal Function in Patients with Sepsis

In order to examine the effect of changing k The pharmacokinetics of moxalactam disodium, a
(from 0.6 to 0.2 h−1) on the distributional (alpha phase) recently discontinued antibiotic (see Table 5-6), was
and elimination (beta phase) half-lives of various examined in 40 patients with abdominal sepsis
drugs, four simulations based on a two-compartment (Swanson et al, 1983). The patients were grouped
model were generated (Table 5-8). The simulations according to creatinine clearances into three groups:
show that a drug with a smaller k has a longer beta

Group 1: Average creatinine clearance = 35.5 mL/
elimination half-life. Keeping all other parameters

min/1.73 m2
(k12, k21, Vp) constant, a smaller k will result in a

Group 2: Average creatinine clearance = 67.1 ± 6.7 mL/
smaller a, or a slower distributional phase. Examples

min/1.73 m2
of drugs with various distribution and elimination

Group 3: Average creatinine clearance = 117.2 ±
half-lives are shown in Table 5-8.

29.9 mL/min/1.73 m2

TABLE 5-8 Comparison of Beta Half-Life and After intravenous bolus administration, the

Distributional Half-Life of Selected Drugs serum drug concentrations followed a biexponential
decline (Fig. 5-10). The pharmacokinetics at steady

Beta Distributional state (2 g every 8 hours) was also examined in these
Drug Half-Life Half-Life

300
Lidocaine 1.8 hours 8 minutes

Cocaine 1 hours 18 minutes 200

Theophylline 4.33 hours 7.2 minutes

Ergometrine 2 hours 11 minutes
100

Hydromorphone 3 hours 14.7 minutes Group 1

Milrinone 3.6 hours 4.6 minutes

Procainamide 2.5–4.7 hours 6 minutes 50

Quinidine 6–8 hours 7 minutes
Group 2

Lithium 21.39 hours 5 hours

Digoxin 1.6 days 35 minutes

Human FSH 1 day 60 minutes

IgG1 kappa MAB 9.6 days 6.7 hours
(monkey) 10

Simulation 1 13.26 hours 36.24 minutes
Group 3

Simulation 2 16.60 hours 43.38 minutes
0 2 4 6 8

Simulation 3 26.83 hours 53.70 minutes
Time (hours)

Simulation 4 213.7 hours 1.12 hours
FIGURE 5-10 Moxalactam serum concentration in three

Simulation was performed using Vp of 10 L; dose = 100 mg; k12 = 0.5 h–1; groups of patients: group 1, average creatinine concentration =
k21 = 0.1 h–1; k = 0.6, 0.4, 0.2, and 0.02 hour for simulations 1–4, respec- 35.5 mL/min/1.73 m2; group 2, average creatinine concentra-
tively (using Equations 5.11 and 5.12). tion = 67.1 ± 6.7 mL/min/1.73 m2; group 3, average creatinine
Source: Data from manufacturer and Schumacher (1995). concentration = 117.2 ± 29.9 mL/min/1.73 m2.

Moxalactam Cp (mcg/mL)

 

Multicompartment Models: Intravenous Bolus Administration 119

patients. Mean steady-state serum concentrations Clinical Example—Azithromycin
ranged from 27.0 to 211.0 mg/mL and correlated Pharmacokinetics
inversely with creatinine clearance (r = 0.91, Following oral administration, azithromycin
p < 0.0001). The terminal half-life ranged from 1.27 (Zithromax®) is an antibiotic that is rapidly absorbed
to 8.27 hours and reflected the varying renal func- and widely distributed throughout the body.
tion of the patients. Moxalactam total body clear- Azithromycin is rapidly distributed into tissues, with
ance (Cl) had excellent correlation with creatinine high drug concentrations within cells, resulting in
clearance (r2 = 0.92). Cl determined by noncom- significantly higher azithromycin concentrations in
partmental data analysis was in agreement with tissue than in plasma. The high values for plasma
Cl determined by nonlinear least squares regression clearance (630 mL/min) suggest that the prolonged
(r = 0.99, p < 0.0001). Moxalactam total body clear- half-life is due to extensive uptake and subsequent
ance was best predicted from creatinine clearance release of drug from tissues.
corrected for body surface area. Plasma concentrations of azithromycin decline

in a polyphasic pattern, resulting in an average termi-
Questions (Refer to Table 5-6) nal half-life of 68 hours. With this regimen, Cmin and

1. Calculate the beta half-life of moxalactam in Cmax remained essentially unchanged from day 2
the most renally impaired group. through day 5 of therapy. However, without a loading

2. What indicator is used to predict moxalactam dose, azithromycin Cmin levels required 5–7 days to
clearance in the body? reach desirable plasma levels.

3. What is the beta volume of distribution of The pharmacokinetic parameters of azithromycin
patients in group 3 with normal renal function? in healthy elderly male subjects (65–85 years) were

4. What is the initial volume (V similar to those in young adults. Although higher
i) of moxalactam?

peak drug concentrations (increased by 30%–50%)

Solutions were observed in elderly women, no significant accu-
mulation occurred.

1. Mean beta half-life is 0.693/0.20 = 3.47 hours
in the most renally impaired group.

2. Creatinine is mainly filtered through the kidney, Questions
and creatinine clearance is used as an indicator 1. Do you agree with the following statements for
of renal glomerular filtration rate. Group 3 has a drug that is described by a two-compartment
normal renal function (average creatinine clear- pharmacokinetic model? At peak Ct, the drug
ance = 117.2 mL/min/1.73 m2) (see Chapter 7). is well equilibrated between the plasma and the

3. Beta volume of distribution: Moxalactam tissue compartment, Cp = Ct, and the rates of
clearance in group 3 subjects is 125.9 mL/min. drug diffusion into and from the plasma com-
From Equation 5.38, partment are equal.

2. What happens after peak Ct?Cl
(VD )β = 3. Why is a loading dose used?

β
4. What is Vi? How is this volume related to Vp?

125.9 mL/min × 60min /h 5. What population factors could affect the con-
=

0.37 h−1 centration of azithromycin?

= 20,416 mL or 20.4 L

4. The volume of the plasma compartment, Vp, is Solutions

sometimes referred to as the initial volume. V 1. For a drug that follows a multicompartment
p

ranges from 0.12 to 0.15 L/kg among the three model, the rates of drug diffusion into the tissues
groups and is considerably smaller than the from the plasma and from the tissues into the
steady-state volume of distribution. plasma are equal at peak tissue concentrations.

 

120 Chapter 5

However, the tissue drug concentration is gener- plasma compartment (also referred to as the ini-
ally not equal to the plasma drug concentration. tial volume by some clinicians), which includes

2. After peak Ct, the rate out of the tissue exceeds some extracellular fluid.
the rate into the tissue, and Ct falls. The decline 3. Etoposide is a drug that follows a two-
of Ct parallels that of Cp, and occurs because compartment model with a beta elimination
distribution equilibrium has occurred. phase. Within the first few minutes after an intra-

3. When drugs are given in a multiple-dose regi- venous bolus dose, most of the drug is distributed
men, a loading dose may be given to achieve in the plasma fluid. Subsequently, the drug will
desired therapeutic drug concentrations more diffuse into tissues and drug uptake may occur.
rapidly (see Chapter 9). Eventually, plasma drug levels will decline due

4. The volume of the plasma compartment, Vp, is to elimination, and some redistribution as etopo-
sometimes referred to as the initial volume. side in tissue diffuses back into the plasma fluid.

5. Age and gender may affect the Cmax level of the The real tissue drug level will differ from
drug. the plasma drug concentration, depending on

the partitioning of drug in tissues and plasma.

PRACTICAL PROBLEM This allows the AUC, the volume distribution
(VD)b, to be calculated, an area that has been

Clinical Example—Etoposide related to toxicities associated with many cancer
Pharmacokinetics chemotherapy agents.

Etoposide is a drug used for the treatment of lung The two-compartment model allows contin-

cancer. Understanding the distribution of etoposide uous monitoring of the amount of the drug pres-

in normal and metastatic tissues is important to avoid ent in and out of the vascular system, including

drug toxicity. Etoposide follows a two-compartment the amount of drug eliminated. This information

model. The (VD)b is 0.28 L/kg, and the beta elimina- is important in pharmacotherapy.

tion half-life is 12.9 hours. Total body clearance is 4. (VD)b may be determined from the total drug

0.25 mL/min/kg. clearance and beta:

Questions Cl = β × (VD )β

1. What is the (VD)b in a 70-kg subject?
(VD)b is also calculated from Equation 5.37 where

2. How is the (VD)b different than the volume of
the plasma fluid, Vp? D

3. Why is the (V (VD )β = (VD )area = 0

D)b useful if it does not represent β [AUC]
a real tissue volume? 0

4. How is (VD)b calculated from plasma time–
This method for (VD)b determination using

concentration profile data for etoposide? Is ∞ ∞

[AUC] is popular because AUC
0 [ ] is easily cal-

(V 0
D)b related to total body clearance?

5. Etopside was recently shown to be a P-gp culated using the trapezoidal rule. Many values

substrate. How may this affect drug tolerance in for apparent volumes of distribution reported in

different patients? the clinical literature are obtained using the area
equation. In general, both volume terms reflect

Solutions extravascular drug distribution. (VD)b appears
1. (VD)b of etoposide in a 70-kg subject is 0.28 L/kg × to be affected by the dynamics of drug disposi-

70 kg = 19.6 L. tion in the beta phase. In clinical practice, many
2. The plasma fluid volume is about 3 L in a potent drugs are not injected by bolus dose.

70-kg subject and is much smaller than (VD)b. Instead, these drugs are infused over a short
The apparent volume of distribution, (VD)b, is interval, making it difficult to obtain accurate
also considerably larger than the volume of the information on the distributive phase. As a result,

 

Multicompartment Models: Intravenous Bolus Administration 121

many drugs that follow a two-compartment The distributive phase is not a major issue if the distri-
model are approximated using a single compart- bution phase has a short duration (Fig. 5-11) relative to
ment. It should be cautioned that there are sub- the beta phase for chronic dosing. However, from the
stantial deviations in some cases. When in doubt, adverse reaction perspective, injury may occur even
the full equation with all parameters should be with short exposure to sensitive organs or enzyme
applied for comparison. A small bolus (test) dose sites. The observation of where the therapeutically
may be injected to obtain the necessary data if effective levels are relative to the time-concentration
a therapeutic dose injected rapidly causes side profile presents an interesting case below.
effects or discomfort to the subject.

PRACTICAL APPLICATION
Frequently Asked Questions

Drugs A, B, and C are investigated for the treatment of
»»What is the error assumed in a one-compartment

arrhythmia (Fig. 5-12). Drug A has a very short dis-
model compared to a two-compartment or multi-

tributive phase. The short distributive phase does not
compartment model?

distort the overall kinetics when drug A is modeled by
»»What kind of improvement in terms of patient care or the one-compartment model. Simple one-compart-

drug therapy is made using the compartment model? ment model assumptions are often made in practice
and published in the literature for simplicity.

Drugs B and C have different distributive pro-
CLINICAL APPLICATION files. Drug B has a gradual distributive phase fol-

lowed by a slower elimination (beta phase). The
Dosing of Drugs with Different pharmacokinetic profile for drug C shows a longer
Biexponential Profiles and steeper distributive phase. Both drugs are well
Drugs are usually dosed according to clearance described by the two-compartment model.
principles with an objective of achieving a steady- Assuming drugs A and B both have the same
state therapeutic level after multiple dosing (see effective level of 0.1 mg/mL, which drug would you
Chapter 9). The method uses a simple well-stirred prefer for dosing your patient based on the above
one-compartment or noncompartmental approach. plasma profiles provided and assuming that both

100

10

1

0.1

0.01

0.001
0 1 2 3 4 5 6 7 8 9

Time (hours)

FIGURE 5-11 A two-compartment model drug showing a short distributive phase.
The graph shows the log of the drug concentrations (mg/mL) versus time (hours). Drug
mass rapidly distributes within the general circulation and highly vascular organs (central
compartment) and is gradually distributed into other tissues or bound to cellular trans-
porters or proteins.

Drug concentration (mg/mL)

 

122 Chapter 5

100

10

1
Drug A

0.1

Drug B
0.01

Drug C

0.001
0 1 2 3 4 5 6 7 8 9

Time (hours)

FIGURE 5-12 Plasma drug concentration profile of three drugs after IV bolus injec-
tion. Plasma drug concentration (Cp)–time profiles of three drugs (A, B, C) with different
distributive (α) phase after single IV bolus injection are plotted on a semilogarithmic
scale. Plasma concentrations are in mg/mL (x axis) and time in hours (y axis). Drugs A, B,
and C are each given at a dose of 10 mg/kg to subjects by IV bolus injection, and each
drug has minimum effective concentration of 0.1 mg/mL.

drugs have the same toxic endpoint (as measured by However, for a drug with the therapeutic endpoint (eg,
plasma drug level)? target plasma drug concentration) that lies within the

At what time would you recommend giving a steep initial distributive phase, it is much harder to
second dose for each drug? Please state your support- dose accurately and not overshoot the target endpoint.
ive reasons. Hints: Draw a line at 0.1 mg/mL and see This scenario is particularly true for some drugs used
how it intersects the plasma curve for drugs B and C. in critical care where rapid responses are needed and

If you ignore the distributive phase and dose a IV bolus routes are used more often. Many new bio-
drug based only on clearance or the terminal half- technological drugs are administered intravenously
life, how would this dose affect the duration above because of instability by oral route. The choice of a
minimum effective drug concentration of 0.1 mg/mL proper dose and rate of infusion relative to the half-life
for each drug after an IV bolus dose? of the drug is an important consideration for safe drug

Drug A represents a drug that has limited tissue administration. Individual patients may behave very
distribution with mostly a linear profile and is dosed differently with regard to drug metabolism, drug
by the one-compartment model. Can you recognize transport, and drug efflux in target cell sites. Drug
when the terminal phase starts for drugs B and C? receptors can be genetically expressed differently

Drug A—short distribution, drug B—intermediate making some people more prone to allergic reactions
distribution, drug C—long distribution phase due to and side effects. Simple kinetic half-life determination
transporter or efflux. coupled with a careful review of the patient’s chart by

a pharmacist can greatly improve drug safety.
• Which drug is acceptable to be modeled by a simple

one compartment model?
• When re-dosed (ie, at 0.1 mg/mL), which drug was CLINICAL APPLICATION

equilibrated with the tissue compartment?
Lidocaine is a drug with low toxicity and a long his-
tory of use for anesthetization and for treating ven-

Significance of Distribution Phase tricular arrhythmias. The drug has a steep distributive
With many drugs, the initial phase or transient concen- phase and is biphasic. The risk of adverse effects is
tration is not considered as important as the steady- dose related and increases at intravenous infusion
state “trough” level during long-term drug dosing. rates of above 3 mg/min. Dosage and dose rate are

Drug concentration (mg/mL)

 

Multicompartment Models: Intravenous Bolus Administration 123

important for proper use (Greenspon et al, 1989). the site of application even if the route is not directly
A case of inappropriate drug use was reported intravenous. It is important to note that for a drug
(Avery, 1998). with a steeply declining elimination plasma profile, it

An overdose of lidocaine was given to a patient is harder to maintain a stable target level with dosing
to anesthetize the airway due to bronchoscopy by an because a small change on the time scale (x axis) can
inexperienced hospital personnel. The patient was greatly alter the drug concentration (y axis). Some
then left unobserved and subsequently developed drugs that have a steep distributive phase may easily
convulsions and cardiopulmonary arrest. He survived cause a side effect in a susceptible subject.
with severe cerebral damage. His lidocaine concen-
tration was 24 mg/mL about 1 hour after initial
administration (a blood concentration over 6 mg/mL
is considered to be toxic). What is the therapeutic Frequently Asked Questions
plasma concentration range? Is the drug highly pro- »»A new experimental drug can be modeled by a two-
tein bound? Is VD sufficiently large to show extra- compartment model. What potential adverse event
vascular distribution? could occur for this drug if given by single IV bolus

A second case of adverse drug reaction (ADR) injection?
based on inappropriate use of this drug due to rapid

»»A new experimental drug can be modeled by a three-
absorption was reported by Pantuck et al (1997). A compartment model. What potential adverse event
40-year-old woman developed seizures after lido- could occur for this drug if given by multiple IV bolus
caine gel 40 mL was injected into the ureter. Vascular injections?
absorption can apparently be very rapid depending on

CHAPTER SUMMARY
Compartment is a term used in pharmacokinetic time. Pharmacokinetic parameters are numerical
models to describe a theoreticized region within the values of model descriptors derived from data that
body in which the drug concentrations are presumed are fitted to a model. These parameters are initially
to be uniformly distributed. estimated and later refined using computing curve-

fitting techniques such as least squares.
• A two-compartment model typically shows a biex-

ponential plasma drug concentration–time curve • Mamillary models are pharmacokinetic models
with an initial distributive phase and a later termi- that are well connected or dynamically exchange
nal phase. drug concentration between compartments. The

• One or more tissue compartments may be present two- and three-compartment models are examples.
in the model depending on the shape of the poly- • Compartment models are useful for estimating
exponential curve representing log plasma drug the mass balance of the drug in the body. As more
concentration versus time. physiological and genetic information is known,

• The central compartment refers to the volume of the model may be rened. Efux and special trans-
the plasma and body regions that are in rapid equi- porters are now known to inuence drug distri-
librium with the plasma. bution and plasma prole. The well-known ABC

• The amount of drug within each compartment transporters (eg, P-gp) are genetically expressed
after a given dose at a given time can be calculated and vary among individuals. These drug trans-
once the model is developed and model parameters porters can be kinetically simulated using trans-
are obtained by data tting. fer constants in a compartment model designed to

mimic drug efux in and out of a cell or compart-
A pharmacokinetic model is a quantitative

ment model.
description of how drug concentrations change over

 

124 Chapter 5

During curve fitting, simplifying the two- • An important consideration is whether the effec-
compartment model after an IV bolus dose and tive concentration lies near the distributive phase
ignoring the presence of the distributive phase may after the IV bolus dose is given.
cause serious errors unless the beta phase is very
long relative to the distributive phase.

LEARNING QUESTIONS
1. A drug was administered by rapid IV injection 3. Mitenko and Ogilvie (1973) demonstrated

into a 70-kg adult male. Blood samples were that theophylline followed a two-compartment
withdrawn over a 7-hour period and assayed pharmacokinetic model in human subjects.
for intact drug. The results are tabulated below. After administering a single intravenous dose
Using the method of residuals, calculate the (5.6 mg/kg) in nine normal volunteers, these
values for intercepts A and B and slopes a, b, investigators demonstrated that the equation
k, k12, and k21. best describing theophylline kinetics in humans

was as follows:
Time Cp Time Cp

C = 12e−.58t 6
p 18e−0.1 t

+
(hours) (µg/mL) (hours) (µg/mL)

What is the plasma level of the drug 3 hours
0.00 70.0 2.5 14.3

after the IV dose?
0.25 53.8 3.0 12.6 4. A drug has a distribution that can be described
0.50 43.3 4.0 10.5 by a two-compartment open model. If the drug

is given by IV bolus, what is the cause of the
0.75 35.0 5.0 9.0

initial or rapid decline in blood levels (a phase)?
1.00 29.1 6.0 8.0 What is the cause of the slower decline in blood
1.50 21.2 7.0 7.0 levels (b phase)?

5. What does it mean when a drug demonstrates a
2.00 17.0

plasma level–time curve that indicates a three-
2. A 70-kg male subject was given 150 mg of compartment open model? Can this curve be

a drug by IV injection. Blood samples were described by a two-compartment model?
removed and assayed for intact drug. Calculate 6. A drug that follows a multicompartment
the slopes and intercepts of the three phases of pharmacokinetic model is given to a patient
the plasma level–time plot from the results tab- by rapid intravenous injection. Would the drug
ulated below. Give the equation for the curve. concentration in each tissue be the same after

the drug equilibrates with the plasma and all
Time Cp Time Cp the tissues in the body? Explain.

(hours) (μg/mL) (hours) (μg/mL) 7. Park and associates (1983) studied the pharma-
cokinetics of amrinone after a single IV bolus

0.17 36.2 3.0 13.9
injection (75 mg) in 14 healthy adult male

0.33 34.0 4.0 12.0 volunteers. The pharmacokinetics of this drug
0.50 27.0 6.0 8.7 followed a two-compartment open model and

fit the following equation:
0.67 23.0 7.0 7.7

C = Ae−α t
+ Be−βt

1.00 20.8 18.0 3.2 p

where
1.50 17.8 23.0 2.4

A = 4.62 ± 12.0 mg/mL
2.00 16.5 B = 0.64 ± 0.17 mg/mL

 

Multicompartment Models: Intravenous Bolus Administration 125

a = 8.94 ± 13 h−1 10. The toxicokinetics of colchicine in seven
b = 0.19 ± 0.06 h−1 cases of acute human poisoning was studied

by Rochdi et al (1992). In three further cases,
From these data, calculate:

postmortem tissue concentrations of colchi-
a. The volume of the central compartment

cine were measured. Colchicine follows the
b. The volume of the tissue compartment

two-compartment model with wide distribution
c. The transfer constants k12 and k21 in various tissues. Depending on the time of
d. The elimination rate constant from the cen-

patient admission, two disposition processes
tral compartment

were observed. The first, in three patients,
e. The elimination half-life of amrinone after

admitted early, showed a biexponential plasma
the drug has equilibrated with the tissue

colchicine decrease, with distribution half-
compartment

lives of 30, 45, and 90 minutes. The second, in
8. A drug may be described by a three-compartment

four patients, admitted late, showed a mono-
model involving a central compartment and two

exponential decrease. Plasma terminal half-
peripheral tissue compartments. If you could

lives ranged from 10.6 to 31.7 hours for both
sample the tissue compartments (organs), in

groups.
which organs would you expect to find a drug

11. Postmortem tissue analysis of colchicine
level corresponding to the two theoretical periph-

showed that colchicine accumulated at high
eral tissue compartments?

concentrations in the bone marrow (more than
9. A drug was administered to a patient at 20 mg

600 ng/g), testicle (400 ng/g), spleen (250 ng/g),
by IV bolus dose and the time–plasma drug

kidney (200 ng/g), lung (200 ng/g), heart
concentration is listed below. Use a suitable

(95 ng/g), and brain (125 ng/g). The pharmaco-
compartment model to describe the data and list

kinetic parameters of colchicine are:
the fitted equation and parameters. What are the

Fraction of unchanged colchicine in
statistical criteria used to describe your fit?

urine = 30%
Renal clearance = 13 L/h

Hour mg/L
Total body clearance = 39 L/h

0.20 3.42 Apparent volume of distribution = 21 L/kg

0.40 2.25 a. Why is colchicine described by a mono-

0.60 1.92 exponential profile in some subjects and a
biexponential in others?

0.80 1.80
b. What is the range of distribution of half-life

1.00 1.73 of colchicine in the subjects?

2.00 1.48 c. Which parameter is useful in estimating
tissue drug level at any time?

3.00 1.28
d. Some clinical pharmacists assumed that, at

4.00 1.10 steady state when equilibration is reached

6.00 0.81 between the plasma and the tissue, the tissue
drug concentration would be the same as the

8.00 0.60
plasma. Do you agree?

10.00 0.45 e. Which tissues may be predicted by the tissue

12.00 0.33 compartment?

14.00 0.24

18.00 0.13

20.00 0.10

 

126 Chapter 5

ANSWERS

Frequently Asked Questions storage (DB = Dt + Dp). Assuming steady state, the

Are “hypothetical” or “mathematical” compart- tissue drug concentration is equal to the plasma

ment models useful in designing dosage regimens in drug concentration, (Cp)ss, and one may determine

the clinical setting? Does “hypothetical” mean “not size of the tissue volume using Dt /(Cp)ss. This vol-

real”? ume is really a “numerical factor” that is used to
describe the relationship of the tissue storage drug

• Mathematical and hypothetical are indeed vague relative to the drug in the blood pool. The sum of the
and uninformative terms. Mathematical equations two volumes is the steady-state volume of distribu-
are developed to calculate how much drug is in the tion. The product of the steady-state concentration,
vascular fluid, as well as outside the vascular fluid (Cp)ss, and the (VD)ss yields the amount of drug in
(ie, extravascular or in the tissue pool). Hypotheti- the body at steady state. The amount of drug in the
cal refers to an unproven model. The assumptions body at steady state is considered vital information
in the compartmental models simply imply that the in dosing drugs clinically. Students should realize
model simulates the mass transfer of drug between that tissue drug concentrations are not predicted
the circulatory system and the tissue pool. The mass by the model. However, plasma drug concentra-
balance of drug moving out of the plasma fluid is tion is fully predictable after any given dose once
described even though we know the tissue pool is the parameters become known. Initial pharmacoki-
not real (the tissue pool represents the virtual tissue netic parameter estimation may be obtained from the
mass that receives drug from the blood). While the literature using comparable age and weight for a
model is a less-than-perfect representation, we can specific individual.
interpret it, knowing its limitations. All pharmaco-
kinetic models need interpretation. We use a model If physiologic models are better than compartment

when there are no simple ways to obtain needed models, why not just use physiologic models?

information. As long as we know the model limi- • A physiologic model is a detailed representation of
tations (ie, that the tissue compartment is not the drug disposition in the body. The model requires
brain or the muscle!) and stay within the bounds of blood flow, extraction ratio, and specific tissue
the model, we can extract useful information from and organ size. This information is not often avail-
it. For example, we may determine the amount of able for the individual. Thus, the less sophisticated
drug that is stored outside the plasma compartment compartment models are used more often.
at any desired time point. After an IV bolus drug
injection, the drug distributes rapidly throughout Since clearance is the term most often used in clini-

the plasma fluid and more slowly into the fluid- cal pharmacy, why is it necessary to know the other

filled tissue spaces. Drug distribution is initially pharmacokinetic parameters?

rapid and confined to a fixed fluid volume known • Clearance is used to calculate the steady-state drug
as the Vp or the initial volume. As drug distribution concentration and to calculate the maintenance
expands into other tissue regions, the volume of the dose. However, clearance alone is not useful in
penetrated spaces increases, until a critical point determining the maximum and minimum drug
(steady state) is obtained when all penetrable tissue concentrations in a multiple-dosing regimen.
regions are equilibrated with the drug. Knowing
that there is heterogenous drug distribution within What is the significance of the apparent volume of

and between tissues, the tissues are grouped into distribution?

compartments to determine the amount of drugs in • Apparent volumes of distribution are not real tis-
them. Mass balance, including drug inside and out- sue volumes, but rather reflect the volume in which
side the vascular pool, accounts for all body drug the drug is contained. For example,

 

Multicompartment Models: Intravenous Bolus Administration 127

V ion for the curve:
p = initial or plasma volume 2. Equat

Vt = tissue volume C 28e−0.63t 077t

p = +10.5e−0.46t
+14e−0.

(VD)ss = steady-state volume of distribution (most Note: When feathering curves by hand, a
often listed in the literature). minimum of three points should be used to

The steady-state drug concentration multiplied determine the line. Moreover, the rate constants
by (VD)ss yields the amount of drug in the body. and y intercepts may vary according to the indi-
(VD)b is a volume usually determined from area un- vidual’s skill. Therefore, values for Cp should
der the curve (AUC), and differs from (VD)ss some- be checked by substitution of various times for
what in magnitude. (VD)b multiplied by b gives t, using the derived equation. The theoretical
clearance of the drug. curve should fit the observed data.

3. C
t is the error assumed in a one-compartment p = 11.14 mg/mL.

Wha
4. The initial decline in the plasma drug concen-

model compared to a two-compartment or multicom-
tration is due mainly to uptake of drug into

partment model?
tissues. During the initial distribution of drug,

• If the two-compartment model is ignored and the some drug elimination also takes place. After
data are treated as a one-compartment model, the the drug has equilibrated with the tissues, the
estimated values for the pharmacokinetic param- drug declines at a slower rate because of drug
eters are distorted. For example, during the dis- elimination.
tributive phase, the drug declines rapidly according 5. A third compartment may indicate that the
to distribution a half-life, while in the elimina- drug has a slow elimination component. If
tion (terminal) part of the curve, the drug declines the drug is eliminated by a very slow elimina-
according to a b elimination half-life. tion component, then drug accumulation may

occur with multiple drug doses or long IV drug
What kind of improvement in terms of patient care

infusions. Depending on the blood sampling,
or drug therapy is made using the compartment

a third compartment may be missed. However,
model?

some data may fit both a two-compartment and
• Compartment models have been used to develop a three-compartment model. In this case, if the

dosage regimens and pharmacodynamic models. fit for each compartment model is very close
Compartment models have improved the dosing of statistically, the simpler compartment model
drugs such as digoxin, gentamicin, lidocaine, and should be used.
many others. The principal use of compartment 6. Because of the heterogeneity of the tissues,
models in dosing is to simulate a plasma drug con- drug equilibrates into the tissues at different
centration profile based on pharmacokinetic (PK) rates and different drug concentrations are
parameters. This information allows comparison usually observed in the different tissues. The
of PK parameters in patients with only two or three drug concentration in the “tissue” compartment
points to a patient with full profiles using gener- represents an “average” drug concentration and
ated PK parameters. does not represent the drug concentration in

any specific tissue.

Learning Questions 7. C = Ae−α t + Be−βt
p

After substitution,
1. Equation for the curve:

Cp 4.62e−8.94t
= + 0.64e−019t

C 52e–1.39t 18e–0.135tp = +

D0 75,000
k = 0.41 h–1 a. V

k12 = 0.657 h–1 k21 = 0.458 h–1 p = = = 14,259 mL
A + B 4.62 + 0.64

 

128 Chapter 5

Vp k12 (14,259)(6.52) A (1) = 2.0049 A (2) = 6.0057 (two preexponential
b. Vt = = = 74,375 mL

k21 (1.25) values)

AB(β −α )2
k (1) = 0.15053 k (2) = 7.0217 (two exponential

c. k12 = values)

(A+ B)(Aβ + Bα )
The equation that describes the data is:

(4.62)(064)(0.19−8.94)2
k12 =

(4.62+ 0.64)[(4.62)(0.19) + (0.64)(8.94)] C e−

p 2.0049 0.15053t 6.0057e−7.0217t
= +

k = −1
12 6.52 h The coefficient of correlation = 0.999 (very

good fit).
Aβ + Bα (4.62)(0.19)(4.64)(8.94) The model selection criterion = 11.27 (good

k21 = =
A+ B 4.62 + 0.64 model).

The sum of squared deviations = 9.3 × 10−5
k = −1
21 1.25 h

(there is little deviation between the observed

αβ( data and the theoretical value).
A + B)

d. k =
Aβ + Bα α = 7.0217 h–1, β = 0.15053 h–1.

(8.94)(0.19)(4.62 + 0.64) 10. a. Late-time samples were taken in some
=
(4.62)(0.19) + (0.64)(8.94) patients, yielding data that resulted in a

monoexponential elimination profile. It is
1.35 h−1

=
also possible that a patient’s illness contrib-

8. The tissue compartments may not be sampled utes to impaired drug distribution.
directly to obtain the drug concentration. b. The range of distribution half-lives is
Theoretical drug concentration, Ct, represents 30–45 minutes.
the average concentration in all the tissues c. None. Tissue concentrations are not generally
outside the central compartment. The amount well predicted from the two-compartment
of drug in the tissue, Dt, represents the total model. Only the amount of drug in the tissue
amount of drug outside the central or plasma compartment may be predicted.
compartment. Occasionally Ct may be equal d. No. At steady state, the rate in and the rate
to a particular tissue drug concentration in an out of the tissues are the same, but the drug
organ. However, this Ct may be equivalent by concentrations are not necessarily the same.
chance only. The plasma and each tissue may have differ-

9. The data were analyzed using computer soft- ent drug binding.
ware called RSTRIP, and found to fit a two- e. None. Only the pooled tissue is simulated
compartment model: by the tissue compartment.

REFERENCES
Avery JK: Routine procedure—bad outcome. Tenn Med 91(7): Eichler HG, Müller M: Drug distribution; the forgotten relative in

280–281, 1998. clinical pharmacokinetics. Clin Pharmacokinet 34(2): 95–99,
Butler TC: The distribution of drugs. In LaDu BN, et al (eds). Fun- 1998.

damentals of Drug Metabolism and Disposition. Baltimore, Greenspon AJ, Mohiuddin S, Saksena S, et al: Comparison of
Williams & Wilkins, 1972. intravenous tocainide with intravenous lidocaine for treat-

Eger E: In Papper EM and Kitz JR (eds). Uptake and Distribution ing ventricular arrhythmias. Cardiovasc Rev Rep 10:55–59,
of Anesthetic Agents. New York, McGraw-Hill, 1963, p. 76. 1989.

 

Multicompartment Models: Intravenous Bolus Administration 129

Harron DWG: Digoxin pharmacokinetic modelling—10 years Rochdi M, Sabouraud A, Baud FJ, Bismuth C, Scherrmann JM:
later. Int J Pharm 53:181–188, 1989. Toxicokinetics of colchicine in humans: Analysis of tis-

Hill HF, Coda BA, Tanaka A, Schaffer R: Multiple-dose evalua- sue, plasma and urine data in ten cases. Hum Exp Toxicol
tion of intravenous hydromorphone pharmacokinetics in nor- 11(6):510–516, 1992.
mal human subjects. Anesth Analg 72:330–336, 1991. Schentag JJ, Jusko WJ, Plaut ME, Cumbo TJ, Vance JW, Abutyn E:

Jambhekar SS, Breen JP: Two compartment model. Basic Phar- Tissue persistence of gentamicin in man. JAMA 238:327–329,
macokinetics. London, Chicago, Pharmaceutical Press, 2009, 1977.
p. 269. Schumacher GE: Therapeutic Drug Monitoring. Norwalk, CT,

Mitenko PA, Ogilvie RI: Pharmacokinetics of intravenous theoph- Appleton & Lange, 1995.
ylline. Clin Pharmacol Ther 14:509, 1973. Swanson DJ, Reitberg DP, Smith IL, Wels PB, Schentag JJ: Steady-

Müller M: Monitoring tissue drug levels by clinical microdialysis. state moxalactam pharmacokinetics in patients: Noncompart-
Altern Lab Anim 37(suppl 1):57–59, 2009. mental versus two-compartmental analysis. J Pharmacokinet-

Pantuck AJ, Goldsmith JW, Kuriyan JB, Weiss RE: Seizures Biopharm 11(4):337–353, 1983.
after ureteral stone manipulation with lidocaine. J Urol Vallner JJ, Stewart JT, Kotzan JA, Kirsten EB, Honiger IL: Phar-
157(6):2248, 1997. macokinetics and bioavailability of hydromorphone following

Parab PV, Ritschel WA, Coyle DE, Gree RV, Denson DD: Phar- intravenous and oral administration to human subjects. J Clin
macokinetics of hydromorphone after intravenous, peroral and Pharmacol 21:152–156, 1981.
rectal administration to human subjects. Biopharm Drug Dispos Winters ME: Basic Clinical Pharmacokinetics, 3rd ed. Vancouver,
9:187–199, 1988. WA, Applied Therapeutics, 1994, p. 23.

Park GP, Kershner RP, Angellotti J, et al: Oral bioavailability
and intravenous pharmacokinetics of amrinone in humans.
J Pharm Sci 72:817, 1983.

BIBLIOGRAPHY
Dvorchick BH, Vessell ES: Significance of error associated with Mayersohn M, Gibaldi M: Mathematical methods in pharmacoki-

use of the one-compartment formula to calculate clearance of netics, II: Solution of the two compartment open model. Am J
38 drugs. Clin Pharmacol Ther 23:617–623, 1978. Pharm Ed 35:19–28, 1971.

Jusko WJ, Gibaldi M: Effects of change in elimination on various Riegelman S, Loo JCK, Rowland M: Concept of a volume of dis-
parameters of the two-compartment open model. J Pharm Sci tribution and possible errors in evaluation of this parameter.
61:1270–1273, 1972. J Pharm Sci 57:128–133, 1968.

Loughman PM, Sitar DS, Oglivie RI, Neims AH: The two- Riegelman S, Loo JCK, Rowland M: Shortcomings in pharmaco-
compartment open-system kinetic model: A review of its clini- kinetics analysis by conceiving the body to exhibit properties
cal implications and applications. J Pediatr 88:869–873, 1976. of a single compartment. J Pharm Sci 57:117–123, 1968.

 

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Intravenous Infusion

6 HaiAn Zheng

Chapter Objectives Drugs may be administered to patients by oral, topical, parenteral,
or other various routes of administration. Examples of parenteral

»» Describe the concept of steady
routes of administration include intravenous, subcutaneous, and

state and how it relates to
intramuscular. Intravenous (IV) drug solutions may be either

continuous dosing.
injected as a bolus dose (all at once) or infused slowly through a

»» Determine optimum dosing for vein into the plasma at a constant rate (zero order). The main
an infused drug by calculating advantage for giving a drug by IV infusion is that it allows precise
pharmacokinetic parameters control of plasma drug concentrations to fit the individual needs of
from clinical data. the patient. For drugs with a narrow therapeutic window (eg, hepa-

»» Calculate loading doses to rin), IV infusion maintains an effective constant plasma drug con-

be used with an intravenous centration by eliminating wide fluctuations between the peak

infusion. (maximum) and trough (minimum) plasma drug concentration.
Moreover, the IV infusion of drugs, such as antibiotics, may be

»» Describe the purpose of a given with IV fluids that include electrolytes and nutrients.
loading dose. Furthermore, the duration of drug therapy may be maintained or

»» Compare the pharmacokinetic terminated as needed using IV infusion.
outcomes and clinical The plasma drug concentration−time curve of a drug given by
implications after giving a constant IV infusion is shown in Fig. 6-1. Because no drug was
loading dose for a drug that present in the body at zero time, drug level rises from zero drug
follows a one-compartment concentration and gradually becomes constant when a plateau or
model to a drug that follows a steady-state drug concentration is reached. At steady state, the rate
two-compartment model. of drug leaving the body is equal to the rate of drug (infusion rate)

entering the body. Therefore, at steady state, the rate of change in
the plasma drug concentration dCp/dt = 0, and

Rate of drug input = rate of drug output
(infusion rate) (elimination rate)

Based on this simple mass balance relationship, a pharmaco-
kinetic equation for infusion may be derived depending on whether
the drug follows one- or two-compartment kinetics.

ONE-COMPARTMENT MODEL DRUGS
The pharmacokinetics of a drug given by constant IV infusion fol-
lows a zero-order input process in which the drug is directly
infused into the systemic blood circulation. For most drugs,

131

 

132 Chapter 6

elimination of drug from the plasma is a first-order Steady-State Drug Concentration (Css) and
process. Therefore, in this one-compartment model, Time Needed to Reach Css
the infused drug follows zero-order input and first- Once the steady state is reached, the rate of drug
order output. The change in the amount of drug in leaving the body is equal to the rate of drug entering
the body at any time (dDB/dt) during the infusion is the body (infusion rate). In other words, there is no
the rate of input minus the rate of output.

net change in the amount of drug in the body, DB, as
dD

B a function of time during steady state. Drug elimina-
= R − kD (6.1)

dt B
tion occurs according to first-order elimination

where DB is the amount of drug in the body, R is the kinetics. Whenever the infusion stops, either before
infusion rate (zero order), and k is the elimination or after steady state is reached, the drug concentra-
rate constant (first order). tion always declines according to first-order kinetics.

Integration of Equation 6.1 and substitution of The slope of the elimination curve equals to −k/2.3
DB = CpVD gives: (Fig. 6-2). Even if the infusion is stopped before

R steady state is reached, the slope of the elimination
C (1 e−kt

p = − ) (6.2)
VDk

curve remains the same (Fig. 6-2B).
Mathematically, the time to reach true steady-

Equation 6.2 gives the plasma drug concentration at state drug concentrations, Css, would take an infinite
any time during the IV infusion, where t is the time time. The time required to reach the steady-state
for infusion. The graph for Equation 6.2 appears in drug concentration in the plasma is dependent on the
Figs. 6-1 and 6-2. As the drug is infused, the value elimination rate constant of the drug for a constant
for certain time (t) increases in Equation 6.2. At infi- volume of distribution, as shown in Equation 6.4.
nite time t = ∞, e−kt approaches zero, and Equation 6.2 Because drug elimination is exponential (first order),
reduces to Equation 6.4, as the steady-state drug the plasma drug concentration becomes asymptotic
concentration (Css). to the theoretical steady-state plasma drug concen-

R
tration. For zero-order elimination processes, if rate

Cp = (1− e−∞ ) (6.3)
VDk of input is greater than rate of elimination, plasma

R drug concentrations will keep increasing and no
Css = (6.4)

V steady state will be reached. This is a potentially
Dk

dangerous situation that will occur when saturation
The body clearance, Cl, is equal to VDk, therefore: of metabolic process occurs.

R R
Css = = (6.5)

VDk Cl
80

Steady state A

B
60

Steady-state level
40

20

0
0 4 8 12 16 20 24 28 32 36

Time (hours)

FIGURE 6-2 Plasma drug concentration−time profiles
Time

after IV infusion. IV infusion is stopped at steady state (A) or prior
FIGURE 6-1 Plasma level−time curve for constant to steady state (B). In both cases, plasma drug concentrations
IV infusion. decline exponentially (first order) according to a similar slope.

Plasma level

Plasma drug level

 

Intravenous Infusion 133

In clinical practice, a plasma drug concentration
2R

prior to, but asymptotically approaching, the theo-
retical steady state is considered the steady-state
plasma drug concentration (Css). In a constant IV R

infusion, drug solution is infused at a constant or
zero-order rate, R. During the IV infusion, the
plasma drug concentration increases and the rate of

Time
drug elimination increases because rate of elimina-
tion is concentration dependent (ie, rate of drug FIGURE 6-3 Plasma level−time curve for IV infusions

elimination = kCp). Cp keeps increasing until steady given at rates of R and 2R, respectively.

state is reached at which time the rate of drug input
(IV infusion rate) equals rate of drug output (elimi-
nation rate). The resulting plasma drug concentra-
tion at steady state (Css) is related to the rate of An increase in the infusion rate will not shorten

infusion and inversely related to the body clearance the time to reach the steady-state drug concentration.

of the drug as shown in Equation 6.5. If the drug is given at a more rapid infusion rate, a

In clinical practice, the drug activity will be higher steady-state drug level will be obtained, but

observed when the drug concentration is close to the time to reach steady state is the same (Fig. 6-3).

the desired plasma drug concentration, which is This equation may also be obtained with the fol-

usually the target or desired steady-state drug con- lowing approach. At steady state, the rate of infu-

centration. For therapeutic purposes, the time for sion equals the rate of elimination. Therefore, the

the plasma drug concentration to reach more than rate of change in the plasma drug concentration is

95% of the steady-state drug concentration in the equal to zero.

plasma is often estimated. The time to reach 90%,
95%, and 99% of the steady-state drug concentra-
tion, Css, may be calculated. As detailed in dCp

= 0
Table 6-1, after IV infusion of the drug for 5 half- dt

lives, the plasma drug concentration will be
between 95% (4.32 t1/2) and 99% (6.65 t1/2) of the dCp R

= − kC
dt V p = 0

steady-state drug concentration. Thus, the time for a D

drug whose t1/2 is 6 hours to reach 95% of the steady-
state plasma drug concentration will be approxi- (Ratein )− (Rateout ) = 0
mately 5 t1/2, or 5 × 6 hours = 30 hours. The calculation
of the values in Table 6-1 is given in the example R

= kC
that follows. V p

D

R
C = (6.6)

ss
TABLE 6-1 Number of t1/2 to Reach a VDk

Fraction of Css

Percent of Css Reacheda Number of Half-Lives
Equation 6.6 is the same as Equation 6.5 that

90 3.32 shows that the steady-state concentration (Css) is

95 4.32 dependent on the volume of distribution, the elimi-
nation rate constant, and the infusion rate. Altering

99 6.65
any one of these factors can affect steady-state

aCss is the steady-state drug concentration in plasma. concentration.

Plasma level

 

134 Chapter 6

EXAMPLES »» » Take the natural logarithm on both sides:
−kt = ln 0.01

1. An antibiotic has a volume of distribution of 10 L ln0.01 −4.61 4.61
t

and a k of 0.2 h−1. A steady-state plasma concen- 99%ss = = =

−k −k k
tration of 10 μg/mL is desired. The infusion rate

Substituting (0.693/t1/2) for k,
needed to maintain this concentration can be
determined as follows: 4.61 4.61

t99%ss = = t
(0.693/t ) 1/2

Equation 6.6 can be rewritten as 1/2 0.693

R = t99%ss = 6.65t
C 1/2
ssVDk

= (10 µg/mL)(10)(1000 mL)(0.2 h−1 Notice that in the equation directly above,
)

the time needed to reach steady state is not
=20 mg/h dependent on the rate of infusion, but only

on the elimination half-life. Using similar cal-
Assume the patient has a uremic condi-

culations, the time needed to reach any per-
tion and the elimination rate constant has

centage of the steady-state drug concentra-
decreased to 0.1 h−1. To maintain the steady-

tion may be obtained (Table 6-1).
state concentration of 10 μg/mL, we must

IV infusion may be used to determine
determine a new rate of infusion as follows:

total body clearance if the infusion rate and

R = (10 mg/mL)(10)(1000 mL)(0.1 h−1) = 10 mg/h the steady-state level are known, as with
Equation 6.6 repeated here:

When the elimination rate constant decreases,
then the infusion rate must decrease propor- R

Css = (6.6)
tionately to maintain the same Css. However, VDk

because the elimination rate constant is R
smaller (ie, the elimination t1/2 is longer), the VDk =

Css
time to reach Css will be longer.

Because total body clearance, ClT, is equal
2. An infinitely long period of time is needed to

to VDk,
reach steady-state drug levels. However, in
practice it is quite acceptable to reach 99% Css R

ClT = (6.7)
(ie, 99% steady-state level). Using Equation 6.6, Css

we know that the steady-state level is
3. A patient was given an antibiotic (t1/2 = 6 hours)

R
C = by constant IV infusion at a rate of 2 mg/h. At
ss

VDk the end of 2 days, the serum drug concentra-
and 99% steady-state level would be equal to tion was 10 mg/L. Calculate the total body

R clearance ClT for this antibiotic.
99%

VDk
Solution

Substituting into Equation 6.2 for Cp, we can
find out the time needed to reach steady state The total body clearance may be estimated from

by solving for t. Equation 6.7. The serum sample was taken after
2 days or 48 hours of infusion, which time repre-

R R
99% (1 e−kt

= − ) sents 8 × t1/2; therefore, this serum drug concen-
VDk VDk

tration approximates the Css.

99% 1 e−kt
= − R 2 mg/h

ClT = = =200 mL/h
e−kt

=1% Css 10 mg/L

 

Intravenous Infusion 135

Frequently Asked Questions EXAMPLE »» »

»»How does one determine whether a patient has
1. An antibiotic has an elimination half-life of

reached steady state during an IV infusion?
3−6 hours in the general population. A patient

»»What is the clinical relevance of steady state? was given an IV infusion of an antibiotic at an

»»How can the steady-state drug concentration be infusion rate of 15 mg/h. Blood samples were

achieved more quickly? taken at 8 and 24 hours, and plasma drug con-
centrations were 5.5 and 6.5 mg/L, respectively.
Estimate the elimination half-life of the drug in
this patient.

INFUSION METHOD FOR
Solution

CALCULATING PATIENT
Because the second plasma sample was taken

ELIMINATION HALF-LIFE
at 24 hours, or 24/6 = 4 half-lives after infusion,

The Cp-versus-time relationship that occurs during the plasma drug concentration in this sample is

an IV infusion (Equation 6.2) may be used to calcu- approaching 95% of the true plasma steady-state

late k, or indirectly the elimination half-life of the drug concentration, assuming the extreme case of

drug in a patient. Some information about the elimi- t1/2 = 6 hours.

nation half-life of the drug in the population must be By substitution into Equation 6.8:

known, and one or two plasma samples must be 6 − .5 (8)
lo  .5 5  k
g  = −

taken at a known time after infusion. Knowing the 6.5 2.3
half-life in the general population helps determine if

k = 0.234 h−1
the sample is taken at steady state in the patient. To
simplify calculation, Equation 6.2 is arranged to t1/2 = 0.693/0.234 =2.96 hours

solve for k:
The elimination half-life calculated in this

R
manner is not as accurate as the calculation of t1/2

C (1 e−kt
p = − ) (6.2)

VDk using multiple plasma drug concentration time
points after a single IV bolus dose or after stop-

Since
ping the IV infusion. However, this method may

R be sufficient in clinical practice. As the second
Css =

VDk blood sample is taken closer to the time for steady
state, the accuracy of this method improves. At the

substituting into Equation 6.2: 30th hour, for example, the plasma concentration
would be 99% of the true steady-state value (cor-

C (1 −k
p C t

= ss − e )
responding to 30/6 or 5 elimination half-lives), and

Rearranging and taking the log on both sides: less error would result in applying Equation 6.8.
When Equation 6.8 was used in the example

C above to calculate the drug t1/2 of the patient, the
ss −C 

p kt
log  = − and

 Css  2.3 second plasma drug concentration was assumed to
be the theoretical Css. As demonstrated below, when

−2.3 Css −C  k and the corresponding values are substituted,
=  p

k log  (6.8)
t  Css  Css −5.5 (0.234)(8)

log  = −
 Css  2.3

where Cp is the plasma drug concentration taken at
time t, and Css is the approximate steady-state plasma Css −5.5

= 0.157
drug concentration in the patient. Css

 

136 Chapter 6

LOADING DOSE PLUS IV INFUSION—
Css = 6.5 mg/L ONE-COMPARTMENT MODEL

(Note that Css is in fact the same as the concentra- The loading dose DL, or initial bolus dose of a drug,
tion at 24 hours in the example above.) is used to obtain desired concentrations as rapidly as

possible. The concentration of drug in the body for a
one-compartment model after an IV bolus dose is

In practice, before starting an IV infusion, an
described by

appropriate infusion rate (R) is generally calculated
from Equation 6.8 using literature values for Css, k, D

C C e−kt L e−kt
= = (6.9)

and VD or ClT. Two plasma samples are taken and the 1 0 V
D

sampling times recorded. The second sample should
be taken near the theoretical time for steady state. and concentration by infusion at the rate R is
Equation 6.8 would then be used to calculate a k and
then t1/2. If the elimination half-life calculated con- R

C (1 e−kt

firms that the second sample was taken at steady 2 = − ) (6.10)
VDk

state, the plasma concentration is simply assumed as
the steady-state concentration and a new infusion

Assume that an IV bolus dose DL of the drug is given
rate may be calculated.

and that an IV infusion is started at the same time.
The total concentration Cp at t hours after the start of

EXAMPLE »» » infusion would be equal to C1 + C2 due to the sum
contributions of bolus and infusion, or

1. If the desired therapeutic plasma concentration
is 8 mg/L for the above patient (Example 1),

Cp =C1 +C2
what is the suitable infusion rate for the patient?

D R
Solution C = L

e−kt
p + (1− e−kt )

VD VDk
From Example 1, the trial infusion rate was 15 mg/h.

Assuming the second blood sample is the steady- D R R

= L
e−kt + − e−kt

state level, 6.5 mg/mL, the clearance of the patient is VD VDk VDk

Css = R/Cl R D R 
= +  L

e−kt − e−kt (6.11)
Cl = R/C VDk VD VDk 

ss =15/6.5=2.31 L/h

The new infusion rate should be

R = Css ×Cl = 8×2.31=18.48 mg/h Let the loading dose (DL) equal the amount of drug
in the body at steady state

In this example, the t1/2 of this patient is a lit-
tle shorter, about 3 hours compared to 3−6 hours DL =CssVD
reported for the general population. Therefore, the
infusion rate should be a little greater in order to

From Equation 6.4, CssVD = R/k. Therefore,
maintain the desired steady-state level of 15 mg/L.

Equation 6.7 or the steady-state clearance
method has been applied to the clinical infusion DL = R/k (6.12)

of drugs. The method was regarded as simple and
accurate compared with other methods, including Substituting DL = R/k in Equation 6.11 makes the

the two-point method (Hurley and McNeil, 1988). expression in parentheses cancel out. Equation 6.11
reduces to Equation 6.13, which is the same

 

Intravenous Infusion 137

expression for Css or steady-state plasma concentra- By differentiating this equation at steady state, we
tions (Equation 6.14 is identical to Equation 6.6): obtain:

R dC
p −D k

Cp = (6.13) Rk
= 0 = L

e−kt + e−kt (6.16)
VDk dt VD VDk

R
Css = (6.14) −D 

0 = −  Lk R
e kt

VDk
+ 

 VD VD 
Therefore, if an IV loading dose of R/k is given, fol-

DLk R
lowed by an IV infusion, steady-state plasma drug = (6.17)

V
D VD

concentrations are obtained immediately and main-

tained (Fig. 6-4). In this situation, steady state is R
DL = = loading dose

also achieved in a one-compartment model, since the k

rate in = rate out (R = dDB/dt). In order to maintain instant steady-state level
The loading dose needed to get immediate ([dCp/dt] = 0), the loading dose should be equal to R/k.

steady-state drug levels can also be found by the fol- For a one-compartment drug, if the DL and infu-
lowing approach. sion rate are calculated such that C0 and Css are the

Loading dose equation: same and both DL and infusion are started concur-
D rently, then steady state and C

L ss will be achieved
C1 e−kt

=

V immediately after the loading dose is administered
D

(Fig. 6-4). Similarly, in Fig. 6-5, curve b shows the
Infusion equation:

blood level after a single loading dose of R/k plus
R

C infusion from which the concentration desired at
2 (1 e−kt

= − )
VDk steady state is obtained. If the DL is not equal to R/k,

Adding up the two equations yields Equation 6.15, then steady state will not occur immediately. If the
an equation describing simultaneous infusion after a loading dose given is larger than R/k, the plasma drug
loading dose. concentration takes longer to decline to the concentra-

D tion desired at steady state (curve a). If the loading
L R

C e−kt t
p = + (1− e−k ) (6.15)

VD VDk
dose is lower than R/k, the plasma drug concentrations
will increase slowly to desired drug levels (curve c),
but more quickly than without any loading dose.

IV infusion plus loading dose combined

Steady-
state a

IV infusion level

b

IV bolus loading dose

c

Time (hours)
d

FIGURE 6-4 IV infusion with loading dose DL. The loading
dose is given by IV bolus injection at the start of the infusion.
Plasma drug concentrations decline exponentially after DL

Time
whereas they increase exponentially during the infusion. The
resulting plasma drug concentration−time curve is a straight FIGURE 6-5 Intravenous infusion with loading doses a, b,
line due to the summation of the two curves. and c. Curve d represents an IV infusion without a loading dose.

Plasma drug concentration (µg/mL)

Plasma level

 

138 Chapter 6

Another method for the calculation of loading 3. Calculate the drug concentration in the blood
dose DL is based on knowledge of the desired steady- after infusion has been stopped.
state drug concentration Css and the apparent volume of
distribution VD for the drug, as shown in Equation 6.18. Solution

This concentration can be calculated in two
D ( 8

L =C .
ssV

6 1 )
D parts (see Fig. 6-2A). First, calculate the con-

For many drugs, the desired Css is reported in the centration of drug during infusion, and second,

literature as the effective therapeutic drug concentra- calculate the concentration after the stop of

tion. The VD and the elimination half-life are also the infusion, C. Then use the IV bolus dose

available for these drugs. equation (C = C0e
−kt) for calculations for any

further point in time. For convenience, the two
equations can be combined as follows:

PRACTICE PROBLEMS
R

C −

p = (1 e−kb
− )e k (t−b) (6.19)

V
1. A physician wants to administer an anesthetic Dk

agent at a rate of 2 mg/h by IV infusion. The where b = length of time of infusion period, t =
elimination rate constant is 0.1 h−1 and the volume total time (infusion and postinfusion), and t − b =
of distribution (one compartment) is 10 L. How length of time after infusion has stopped. Here,
much is the drug plasma concentration at the we assume no bolus loading dose was given.
steady state? What loading dose should be recom-

4. A patient was infused for 6 hours with a drug
mended to reach steady state immediately?

(k = 0.01 h−1; VD = 10 L) at a rate of 2 mg/h.

Solution What is the concentration of the drug in the
body 2 hours after cessation of the infusion?

R 2000
Css = = = 2 µg/mL

V 3
Dk (10×10 )(0.1) Solution

Using Equation 6.19,
To reach Css instantly,

2000
R 2 mg/h Cp = (1− e−0.01(6) )e−0.01(8−6)

D (0.01)(10,000)
L = = DL = 20 mg

k 0.1/h
C

2. What is the concentration of a drug at 6 hours p =1.14 µg/mL

after infusion administration at 2 mg/h, with an
Alternatively, when infusion stops, Cp′ is

initial loading dose of 10 mg (the drug has a t1/2 calculated:
of 3 hours and a volume of distribution of 10 L)?

Solution R
Cp′ (1 −kt

= − e )
VDk

0.693
k = 2000

3 h C′ −
p = (1− e 0.01(6) )

0.01×10,000
D

= L R
C e−kt

p + (1− e−kt ) C C e−0.01(2)
=

V p′
D VDk

10,000 C =1.14 µg/mL
C = (e−(0.693/3)(6)

p )
10,000

The two approaches should give the same answer.
2000

+ (1− e−(0.693/3)(6) ) 5. An adult male asthmatic patient (78 kg, 48 years
(10,000)(0.693/3)

old) with a history of heavy smoking was given
Cp = 0.90 µg/mL an IV infusion of aminophylline at a rate of

 

Intravenous Infusion 139

0.75 mg/kg/h. A loading dose of 6 mg/kg was (equivalent to 0.75 × 0.8 = 0.6 mg theoph-
given by IV bolus injection just prior to the ylline) per kg was given to the patient, the
start of the infusion. Two hours after the start plasma theophylline concentration of 5.8 mg/L
of the IV infusion, the plasma theophylline con- is the steady-state Css. Total clearance may be
centration was measured and found to contain estimated by
5.8 mg/mL of theophylline. The apparent VD
for theophylline is 0.45 L/kg. (Aminophylline R (0.6 mg/h/kg)(78 kg)

Cl
is the ethylenediamine salt of theophylline and T = =

Css,present 5.8 mg/L
contains 80% of theophylline base.)

Because the patient was responding poorly ClT = 8.07 L/h or 1.72 mL/min/kg

to the aminophylline therapy, the physician
wanted to increase the plasma theophylline

The usual ClT for adult, nonsmoking patients
concentration in the patient to 10 mg/mL. What

with uncomplicated asthma is approximately
dosage recommendation would you give the

0.65 mL/min/kg. Heavy smoking is known to
physician? Would you recommend another

increase ClT for theophylline.
loading dose?

The new IV infusion rate, R′ in terms of

Solution theophylline, is calculated by

If no loading dose is given and the IV infu- R′ = Css,desired ClT

sion rate is increased, the time to reach R′ = 10 mg/L × 8.07 L/h = 80.7 mg/h or
steady-state plasma drug concentrations 1.03 mg/h/kg of theophylline, which is equiva-
will be about 4 to 5 t1/2 to reach 95% of Css. lent to 1.29 mg/h/kg of aminophylline.
Therefore, a second loading dose should be 6. An adult male patient (43 years, 80 kg) is to be
recommended to rapidly increase the plasma given an antibiotic by IV infusion. According to
theophylline concentration to 10 mg/mL. The the literature, the antibiotic has an elimination
infusion rate must also be increased to main- t1/2 of 2 hours and VD of 1.25 L/kg, and is effec-
tain this desired Css. tive at a plasma drug concentration of 14 mg/L.

The calculation of loading dose DL must The drug is supplied in 5-mL ampuls contain-
consider the present plasma theophylline ing 150 mg/mL.
concentration. a. Recommend a starting infusion rate in milli-

grams per hour and liters per hour.
VD (Cp,desired −Cp,present )

DL = (6.20)
(S)(F) Solution

where S is the salt form of the drug and F is Assume the effective plasma drug concentra-

the fraction of drug bioavailable. For amino- tion is the target drug concentration or Css.

phylline S is equal to 0.80 and for an IV bolus
R =C

injection F is equal to 1. sskVD

= (14 mg/L)(0.693/2 h)(1.5 L/kg)(80 kg)
(0.45 L/kg)(78 kg)(10−5.8 mg/L)

DL =

(0.8)(1) = 485.1 mg/h

DL =184 mg aminophylline
Because the drug is supplied at a concentration

of 150 mg/mL,
The maintenance IV infusion rate may be

calculated after estimation of the patient’s (485.1 mg)(mL/150 mg) = 3.23 mL
clearance, ClT. Because a loading dose and
an IV infusion of 0.75 mg/h aminophylline Thus, R = 3.23 mL/h.

 

140 Chapter 6

b. Blood samples were taken from the patient d. After reviewing the pharmacokinetics of the
at 12, 16, and 24 hours after the start of the antibiotic in this patient, should the infusion
infusion. Plasma drug concentrations were rate for the antibiotic be changed?
as shown below:

Solution

To properly decide whether the infusion rate
t (hours) Cp (mg/L)

should be changed, the clinical pharmacist must
12 16.1 consider the pharmacodynamics and toxicity of

16 16.3 the drug. Assuming the drug has a wide thera-
peutic window and shows no sign of adverse

24 16.5
drug toxicity, the infusion rate of 485.1 mg/h,
calculated according to pharmacokinetic litera-

From these additional data, calculate the ture values for the drug, appears to be correct.
total body clearance ClT for the drug in this

R
patient. Cl V t

Cp (1 −( / )
= −e D )
Cl

Solution

Because the plasma drug concentrations at 12, ESTIMATION OF DRUG CLEARANCE
16, and 24 hours were similar, steady state has

AND VD FROM INFUSION DATA
essentially been reached. (Note: The continu-
ous increase in plasma drug concentrations The plasma concentration of a drug during constant
could be caused by drug accumulation due to a infusion was described in terms of volume of distri-
second tissue compartment, or could be due to bution VD and elimination constant k in Equation 6.2.
variation in the drug assay.) Assuming a Css of Alternatively, the equation may be described in terms
16.3 mg/mL, ClT is calculated. of clearance by substituting for k into Equation 6.2

with k = Cl/V
R 485.1 mg/h D:

ClT = = = 29.8 L/h
Css 16.3 mg/L R

(1 −(Cl /V t
C e D )

p = − ) (6.21)
Cl

c. From the above data, estimate the elimi- The drug concentration in this physiologic
nation half-life for the antibiotics in this model is described in terms of volume of distribution
patient. VD and total body clearance Cl. The independent

Solution parameters are clearance and volume of distribution;
k is viewed as a dependent variable that depends on

Generally, the apparent volume of distribution
Cl and VD. In this model, the time for steady state

(VD) is less variable than t1/2. Assuming that
and the resulting steady-state concentration will be

the literature value for VD is 1.25 L/kg, then t1/2 dependent on both clearance and volume of distribu-
may be estimated from the ClT.

tion. When a constant volume of distribution is evi-

Cl dent, the time for steady state is then inversely
T 29.9 L/h

k 0.299 h−1
= = =

VD (1.25 L/kg)(80 kg) related to clearance. Thus, drugs with small clear-
ance will take a long time to reach steady state.

0.693 Although this newer approach is preferred by some
t1/2 = = 2.32 h

0.299 h−1
clinical pharmacists, the alternative approach to

parameter estimation was known for some time in
Thus the t1/2 for the antibiotic in this patient is classical pharmacokinetics. Equation 6.21 has been

2.32 hours, which is in good agreement with applied in population pharmacokinetics to estimate
the literature value of 2 hours. both Cl and VD in individual patients with one or

 

Intravenous Infusion 141

more data points. However, clearance in patients By rearranging this equation, the infusion rate for a
may differ greatly from subjects in the population, desired steady-state plasma drug concentration may
especially subjects with different renal functions. be calculated.
Unfortunately, the plasma samples taken at time

R =CssVpk (6.24)
equivalent to less than 1 half-life after infusion was
started may not be very discriminating due to the
small change in the drug concentration. Blood sam- Loading Dose for Two-Compartment

ples taken at 3−4 half-lives later are much more Model Drugs

reflective of their difference in clearance. Drugs with long half-lives require a loading dose to
more rapidly attain steady-state plasma drug levels. It
is clinically desirable to achieve rapid therapeutic

INTRAVENOUS INFUSION OF TWO- drug levels by using a loading dose. However, for a

COMPARTMENT MODEL DRUGS drug that follows the two-compartment pharmacoki-
netic model, the drug distributes slowly into extravas-

Many drugs given by IV infusion follow two- cular tissues (compartment 2). Thus, drug equilibrium
compartment kinetics. For example, the respective is not immediate. The plasma drug concentration of a
distributions of theophylline and lidocaine in humans drug that follows a two-compartment model after
are described by the two-compartment open model. various loading doses is shown in Fig. 6-6. If a load-
With two-compartment-model drugs, IV infusion ing dose is given too rapidly, the drug may initially
requires a distribution and equilibration of the drug give excessively high concentrations in the plasma
before a stable blood level is reached. During a con- (central compartment), which then decreases as drug
stant IV infusion, drug in the tissue compartment is equilibrium is reached (Fig. 6-6). It is not possible to
in distribution equilibrium with the plasma; thus, maintain an instantaneous, stable steady-state blood
constant Css levels also result in constant drug con- level for a two-compartment model drug with a zero-
centrations in the tissue, that is, no net change in the order rate of infusion. Therefore, a loading dose
amount of drug in the tissue occurs during steady produces an initial blood level either slightly higher
state. Although some clinicians assume that tissue or lower than the steady-state blood level. To over-
and plasma concentrations are equal when fully come this problem, several IV bolus injections given
equilibrated, kinetic models only predict that the as short intermittent IV infusions may be used as a
rates of drug transfer into and out of the compart-
ments are equal at steady state. In other words, drug
concentrations in the tissue are also constant, but
may differ from plasma concentrations.

The time needed to reach a steady-state blood d

level depends entirely on the distribution half-life of
c Css

the drug. The equation describing plasma drug con-
b

centration as a function of time is as follows:

R  k − b − a − k  
Cp = 1−  − 

 −  e at  e−bt

P    −  
(6.22) a

V k a b a b

where a and b are hybrid rate constants and R is the rate
Time

of infusion. At steady state (ie, t = ∞), Equation 6.22
reduces to FIGURE 6-6 Plasma drug level after various loading

doses and rates of infusion for a drug that follows a two-

R compartment model: a, no loading dose; b, loading dose = R/k
Css = (6.23) (rapid infusion); c, loading dose = R/b (slow infusion); and d,

Vpk loading dose = R/b (rapid infusion).

Plasma level

 

142 Chapter 6

method for administering a loading dose to the patient which reduces to
(see Chapter 9).

k
(VD )

12
ss =Vp + V (6 30)

k p .
Apparent Volume of Distribution at Steady 21

State, Two-Compartment Model In practice, Equation 6.30 is used to calculate

After administration of any drug that follows two- (VD)ss. The (VD)ss is a function of the transfer con-

compartment kinetics, plasma drug levels will decline stants, k12 and k21, which represent the rate constants

due to elimination, and some redistribution will occur of drug going into and out of the tissue compartment,

as drug in tissue diffuses back into the plasma fluid. respectively. The magnitude of (VD)ss is dependent

The volume of distribution at steady state, (V on the hemodynamic factors responsible for drug
D)ss, is the

“hypothetical space” in which the drug is assumed to distribution and on the physical properties of the

be distributed. The product of the plasma drug concen- drug, properties which, in turn, determine the rela-

tration with (VD)ss will give the total amount of drug in tive amount of intra- and extravascular drug.

the body at that time period, such that (Cp)ss × (VD) Another volume term used in two-compartment
ss =

amount of drug in the body at steady state. At steady- modeling is (VD)b (see Chapter 5). (VD)b is often

state conditions, the rate of drug entry into the tissue calculated from total body clearance divided by b,

compartment from the central compartment is equal to unlike the steady-state volume of distribution, (VD)ss,

the rate of drug exit from the tissue compartment into (VD)b is influenced by drug elimination in the beta

the central compartment. These rates of drug transfer “b ” phase. Reduced drug clearance from the body

are described by the following expressions: may increase AUC, such that (VD)b is either reduced
or unchanged depending on the value of b as shown

Dtk21 = Dpk12 (6.25) in Equation 5.37 (see Chapter 5):

k12Dp D0
D V

t = (6.26) ( D )β = (VD )area = (5.37)
b ∞

k21 [AUC]0

Because the amount of drug in the central compart- Unlike (VD)b, (VD)ss is not affected by changes in

ment Dp is equal to VpCp, by substitution in the above drug elimination. (VD)ss reflects the true distributional

equation, volume occupied by the plasma and the tissue pool when
steady state is reached. Although this volume is not use-

k12CpVp
Dt = (6.27) ful in calculating the amount of drug in the body during

k21 pre-steady state, (VD)ss multiplied by the steady-state

The total amount of drug in the body at steady plasma drug concentration, Css, yields the amount of

state is equal to the sum of the amount of drug in the drug in the body. This volume is often used to determine

tissue compartment, Dt, and the amount of drug in the loading dose necessary to upload the body to a desired

the central compartment, Dp. Therefore, the apparent plasma drug concentration. As shown by Equation 6.30,

volume of drug at steady state (VD)ss may be calcu- (VD)ss is several times greater than Vp, which represents

lated by dividing the total amount of drug in the the volume of the plasma compartment, but differs

body by the concentration of drug in the central somewhat in value depending on the transfer constants.

compartment at steady state:

Dp +Dt
(VD )ss = (6.28) PRACTICAL FOCUS

Cp
Questions

Substituting Equation 6.27 into Equation 6.28, and
expressing Dp as VpCp, a more useful equation for 1. Do you agree with the following statements for

the calculation of (VD)ss is obtained: a drug that is described by a two-compartment
pharmacokinetic model? At steady state, the drug

C is well equilibrated between the plasma and the
pVp+ k12VpCp /k21

(V (6 2
D ) . 9)

ss =
Cp tissue compartment, Cp = Ct, and the rates of drug

 

Intravenous Infusion 143

diffusion into and from the plasma compartment biopharmaceutic studies, the factors that account
are equal. for high tissue concentrations include diffusion

2. Azithromycin may be described by a plasma constant, lipid solubility, and tissue binding to
and a tissue compartment model (refer to cell components. A ratio measuring the relative
Chapter 5). The steady-state volume of distribu- drug concentration in tissue and plasma is the
tion is much larger than the initial volume, Vi, partition coefficient, which is helpful in predict-
or the original plasma volume, Vp, of the central ing the distribution of a drug into tissues. Ulti-
compartment. Why? mately, studies of tissue drug distribution using

3. “Rapid distribution of azithromycin into cells radiolabeled drug are much more useful.
causes higher concentration in the tissues than in The real tissue drug level will differ from
the plasma. …” Does this statement conflict with the plasma drug concentration depending on
the steady-state concept? Why is the loading dose the partitioning of drug in tissues and plasma.
often calculated using the (VD)ss instead of Vp. (VD)b is a volume of distribution often calculated

4. Why is a loading dose used? because it is easier to calculate than (VD)ss. This
volume of distribution, (VD)b, allows the area
under the curve to be calculated, an area which

Solutions
has been related to toxicities associated with

1. For a drug that follows a multiple-compartment many cancer chemotherapy agents. Many values
model, the rates of drug diffusion into the tis- for apparent volumes of distribution reported
sues from the plasma and from the tissues into in the clinical literature are obtained using the
the plasma are equal at steady state. However, area equation. Some early pharmacokinetic
the tissue drug concentration is generally not literature only includes the steady-state volume
equal to the plasma drug concentration. of distribution, which approximates the (VD)b

2. When plasma drug concentration data are used but is substantially smaller in many cases. In
alone to describe the disposition of the drug, general, both volume terms reflect extravascular
no information on tissue drug concentration is drug distribution. (VD)b appears to be much more
known, and no model will predict actual tissue affected by the dynamics of drug disposition
drug concentrations. To account for the mass in the beta phase, whereas (VD)ss reflects more
balance (drug mass/volume = body drug concen- accurately the inherent distribution of the drug.
tration) of drug present in the body (tissue and 4. When drugs are given in a multiple-dose regi-
plasma pool) at any time after dosing, the body men, a loading dose may be given to achieve
drug concentration is assumed to be the plasma steady-state drug concentrations more rapidly.
drug concentration. In reality, azithromycin tis-
sue concentration is much higher. Therefore, the

Frequently Asked Questions
calculated volume of the tissue compartment is

»»What is the main reason for giving a drug by slow
much bigger (31.1 L/kg) than its actual volume.

IV infusion?
3. The product of the steady-state apparent (VD)ss

and the steady-state plasma drug concentration »»Why do we use a loading dose to rapidly achieve

C therapeutic concentration for a drug with a long elim-
ss estimates the amount of drug present in the

body. The amount of drug present in the body ination half-life instead of increasing the rate of drug

may be important information for toxicity con- infusion or increasing the size of the infusion dose?

siderations, and may also be used as a therapeutic »»Explain why the application of a loading dose as a
end point. In most cases, the therapeutic drug at single IV bolus injection may cause an adverse event
the site of action accounts for only a small frac- or drug toxicity in the patient if the drug follows a two-

tion of total drug in the tissue compartment. The compartment model with a slow elimination phase.

pharmacodynamic profile may be described as a »»What are some of the complications involved with
separate compartment (see effect compartment IV infusion?
in Chapter 21). Based on pharmacokinetic and

 

144 Chapter 6

CHAPTER SUMMARY
An IV bolus injection puts the drug into the systemic A loading dose given as an IV bolus injection may
circulation almost instantaneously. For some drugs, be used at the start of an infusion to quickly achieve
IV bolus injections can result in immediate high the desired steady-state plasma drug concentration.
plasma drug concentrations and drug toxicity. An IV For drugs that follow a two-compartment model,
drug infusion slowly inputs the drug into the circula- multiple small loading doses or intermittent IV infu-
tion and can provide stable drug concentrations in sions may be needed to prevent plasma drug concen-
the plasma for extended time periods. Constant IV trations from becoming too high. Pharmacokinetic
drug infusions are considered to have zero-order parameters may be calculated from samples taken
drug absorption because of direct input. Once the during the IV infusion and after the infusion is
drug is infused, the drug is eliminated by first-order stopped, regardless of whether steady state has been
elimination. Steady state is achieved when the rate of achieved. These calculated pharmacokinetic param-
drug infusion (ie, rate of drug absorption) equals the eters are then used to optimize dosing for that patient
rate of drug elimination. Four to five elimination when population estimates do not provide outcomes
half-lives are needed to achieve 95% of steady state. suitable for the patient.

LEARNING QUESTIONS
1. A female patient (35 years old, 65 kg) with The serum drug concentrations are as presented

normal renal function is to be given a drug by in Table 6-2.
IV infusion. According to the literature, the a. What is the steady-state plasma drug level?
elimination half-life of this drug is 7 hours and b. What is the time for 95% steady-state
the apparent VD is 23.1% of body weight. The plasma drug level?
pharmacokinetics of this drug assumes a first- c. What is the drug clearance?
order process. The desired steady-state plasma d. What is the plasma concentration of the drug
level for this antibiotic is 10 mg/mL. 4 hours after stopping infusion (infusion was
a. Assuming no loading dose, how long after stopped after 24 hours)?

the start of the IV infusion would it take to
reach 95% of the Css? TABLE 6-2 Serum Drug Concentrations for a

b. What is the proper loading dose for this Hypothetical Anticonvulsant Drug
antibiotic?

c. What is the proper infusion rate for this TIME Single IV dose Constant IV Infusion
(hour) (1 mg/kg) (0.2 mg/kg per hour)

drug?
d. What is the total body clearance? 0 10.0 0

e. If the patient suddenly develops partial renal 2 6.7 3.3
failure, how long would it take for a new

4 4.5 5.5
steady-state plasma level to be established
(assume that 95% of the Css is a reasonable 6 3.0 7.0

approximation)? 8 2.0 8.0
f. If the total body clearance declined 50% due

10 1.35 8.6
to partial renal failure, what new infusion rate
would you recommend to maintain the desired 12 9.1

steady-state plasma level of 10 mg/mL. 18 9.7
2. An anticonvulsant drug was given as (a) a single

24 9.9
IV dose and then (b) a constant IV infusion.

 

Intravenous Infusion 145

e. What is the infusion rate for a patient weigh- 6. Calculate the excretion rate at steady state for a
ing 75 kg to maintain a steady-state drug drug given by IV infusion at a rate of 30 mg/h.
level of 10 mg/mL? The Css is 20 mg/mL. If the rate of infusion

f. What is the plasma drug concentration were increased to 40 mg/h, what would be
4 hours after an IV dose of 1 mg/kg the new steady-state drug concentration, Css?
followed by a constant infusion of Would the excretion rate for the drug at the
0.2 mg/kg/h? new steady state be the same? Assume first-order

3. An antibiotic is to be given by IV infusion. elimination kinetics and a one-compartment
How many milliliters per hour should a sterile model.
25 mg/mL drug solution be given to a 75-kg 7. An antibiotic is to be given to an adult male
adult male patient to achieve an infusion rate patient (58 years, 75 kg) by IV infusion. The
of 1 mg/kg/h? elimination half-life is 8 hours and the apparent

4. An antibiotic drug is to be given to an adult volume of distribution is 1.5 L/kg. The drug is
male patient (75 kg, 58 years old) by IV supplied in 60-mL ampules at a drug concen-
infusion. The drug is supplied in sterile vials tration of 15 mg/mL. The desired steady-state
containing 30 mL of the antibiotic solution drug concentration is 20 mg/mL.
at a concentration of 125 mg/mL. What rate a. What infusion rate in mg/h would you rec-
in milliliters per hour would you infuse this ommend for this patient?
patient to obtain a steady-state concentration b. What loading dose would you recommend
of 20 mg/mL? What loading dose would you for this patient? By what route of admin-
suggest? Assume the drug follows the pharma- istration would you give the loading dose?
cokinetics of a one-compartment open model. When?
The apparent volume of distribution of this c. Why should a loading dose be
drug is 0.5 L/kg and the elimination half-life recommended?
is 3 hours. d. According to the manufacturer, the recom-

5. According to the manufacturer, a steady- mended starting infusion rate is 15 mL/h. Do
state serum concentration of 17 mg/mL was you agree with this recommended infusion
measured when the antibiotic, cephradine rate for your patient? Give a reason for your
(Velosef®) was given by IV infusion to 9 adult answer.
male volunteers (average weight, 71.7 kg) at a e. If you were to monitor the patient’s serum
rate of 5.3 mg/kg/h for 4 hours. drug concentration, when would you request a
a. Calculate the total body clearance for this blood sample? Give a reason for your answer.

drug. f. The observed serum drug concentration is
b. When the IV infusion was discontinued, the higher than anticipated. Give two possible

cephradine serum concentration decreased reasons based on sound pharmacokinetic
exponentially, declining to 1.5 mg/mL at principles that would account for this
6.5 hours after the start of the infusion. Cal- observation.
culate the elimination half-life. 8. Which of the following statements (a−e) is/are

c. From the information above, calculate the true regarding the time to reach steady-state for
apparent volume of distribution. the three drugs below?

d. Cephradine is completely excreted
unchanged in the urine, and studies have Drug A Drug B Drug C

shown that probenecid given concurrently Rate of infusion 10 20 15
causes elevation of the serum cephradine (mg/h)

concentration. What is the probable mecha-
k (h−1) 0.5 0.1 0.05

nism for this interaction of probenecid with
cephradine? Cl (L/h) 5 20 5

 

146 Chapter 6

a. Drug A takes the longest time to reach 9. If the steady-state drug concentration of a
steady state. cephalosporin after constant infusion of 250 mg/h

b. Drug B takes the longest time to reach is 45 mg/mL, what is the drug clearance of this
steady state. cephalosporin?

c. Drug C takes the longest time to reach 10. Some clinical pharmacists assumed that, at
steady state. steady state when equilibration is reached

d. Drug A takes 6.9 hours to reach steady state. between the plasma and the tissue, the tissue
e. None of the above is true. drug concentration would be the same as the

plasma. Do you agree?

ANSWERS

Frequently Asked Questions administered if the initial steady-state drug level is

What is the main reason for giving a drug by slow inadequate for the patient.

IV infusion? What are some of the complications involved with

• IV infusion?
Slow IV infusion may be used to avoid side effects
due to rapid drug administration. For example, • The common complications associated with intra-
intravenous immune globulin (human) may cause venous infusion include phlebitis and infections at
a rapid fall in blood pressure and possible ana- the infusion site caused by poor intravenous tech-
phylactic shock in some patients when infused niques or indwelling catheters.
rapidly. Some antisense drugs also cause a rapid
fall in blood pressure when injected via rapid IV Learning Questions
into the body. The rate of infusion is particularly
important in administering antiarrhythmic agents 1. a. To reach 95% of Css:

in patients. The rapid IV bolus injection of many 4.32t1/2 = (4.32)(7) = 30.2 hours
drugs (eg, lidocaine) that follow the pharmacoki-
netics of multiple-compartment models may cause
an adverse response due to the initial high drug b. DL = CssVD

concentration in the central (plasma) compartment = (10)(0.231)(65,000) = 150 mg
before slow equilibration with the tissues.

c. R = CssVDk = (10)(15,000)(0.099)
Why do we use a loading dose to rapidly achieve
therapeutic concentration for a drug with a long elimi- = 14.85 mg/h
nation half-life instead of increasing the rate of drug
infusion or increasing the size of the infusion dose? d. ClT =VDk = (15,000)(0.099) =1485 mL/h

• The loading drug dose is used to rapidly attain the e. To establish a new Css will still take 4.32t1/2.

target drug concentration, which is approximately However, the t1/2 will be longer in renal

the steady-state drug concentration. However, the failure.

loading dose will not maintain the steady-state f. If ClT is decreased by 50%, then the infusion

level unless an appropriate IV drug infusion rate rate R should be decreased proportionately:

or maintenance dose is also used. If a larger IV R =10(0.50)(1485) = 7.425 mg/h
drug infusion rate or maintenance dose is given,
the resulting steady-state drug concentration will 2. a. The steady-state level can be found by
be much higher and will remain sustained at plotting the IV infusion data. The plasma
the higher level. A higher infusion rate may be drug−time curves plateau at 10 mg/mL.

 

Intravenous Infusion 147

Alternatively, VD and k can be found from 3. Infusion rate R for a 75-kg patient:
the single IV dose data:

R = (1 mg/kg ⋅ h)(75 kg) = 75 mg/h

V 100 mL/kg k 0.2 h−1
D = =

Sterile drug solution contains 25 mg/mL.

b. Using equations developed in Example 2 in Therefore, 3 mL contains (3 mL) × (25 mg/mL),

the first set of examples in this chapter: or 75 mg. The patient should receive 3 mL
(75 mg/h) by IV infusion.

R R
0.95 (1 e−kt

= − )
VDk VDk

R
4. Css = R = C V k

V k ss D
D

0.95 1 e−0.2t
= −

 0.693
0.05 e−0.2t R = (20 mg/L)(0.5L/kg)(75 kg) 3 h =

ln 0.05 = 173.25 mg/h
t95% = =15 hours

SS −0.2
Drug is supplied as 125 mg/mL. Therefore,

D
c. 0

ClT =VDk VD =
C0 173.25 mg

P 125 mg/mL = X = 1.386 mL
X

1000 100 mL
ClT =100× 0.2 VD = = R = 1.386 mL/h

10 kg
DL = CssVD = (20 mg/L)(0.5 L/kg)(75 kg)

ClT = 20 mL/kg ⋅ h
= 750 mg

d. The drug level 4 hours after stopping the IV
infusion can be found by considering the drug
concentration at the termination of infusion R R

5. Css = =
as C0

p . At the termination of the infusion, the kVD ClT

drug level will decline by a first-order process. R 5.3 mg/kg ⋅h × 71.71 kg
a. ClT = =

C = C0e− Css 17 mg/L
kt

p p

= 22.4 L/h
C = 9.9 −(0.2)(4)

p e

b. At the end of IV infusion, Cp = 17 mg/mL.
Cp = 4.5 µg/mL

Assuming first-order elimination kinetics:

e. The infusion rate to produce a Css of
C C0 −

p = pe
kt

10 mg/mL is 0.2 mg/kg/h. Therefore, the
infusion rate needed for this patient is 1.5 −

= 17e kt(2.5)

0.2 mg/kg ⋅h× 75 kg =15 mg/h 0.0882 e−2.5k
=

f. From the data shown, at 4 hours after the start ln 0.0882 = −2.5 k
of the IV infusion, the drug concentration is
5.5 mg/mL; the drug concentration after an −2.43 = −2.5 k

IV bolus of 1 mg/kg is 4.5 mg/mL. Therefore,
k 0.971 h−1

=

if a 1-mg dose is given and the drug is then
infused at 0.2 mg/kg/h, the plasma drug con- 0.693

t1/2 = = 0.714 hour
centration will be 4.5 + 5.5 = 10 mg/mL. 0.971

 

148 Chapter 6

Cl 7. a.
c. T

ClT = kVD VD =
k R = CsskVD

22.4
VD = = 23.1 L R = (20 mg/L)(0.693/8 h)(1.5 L/kg)(75 kg)

0.971
= 194.9 mg/h

d. Probenecid blocks active tubular secretion of
cephradine. 195 mg/h

R = = 13 mL/h
6. At steady state, the rate of elimination should 15 mg/mL

equal the rate of absorption. Therefore, the rate
of elimination would be 30 mg/h. The C b. DL

ss is = CssVD = (20)(1.5)(75) = 2250 mg given
directly proportional to the rate of infusion R, by IV bolus injection.
as shown by c. The loading dose is given to obtain steady-state

drug concentrations as rapidly as possible.

R R d. 15 mL of the antibiotic solution contains
Css = kVD =

kV 225 mg of drug. Thus, an IV infusion rate of
D Css

15 mL/h is equivalent to 225 mg/h. The Css
Rold Rnew achieved by the manufacturer’s recommen-

=
Css,old Css, new dation is

30 mg/h 40 mg/h R 225
= Css = = = 23.1 mg/L

20 kV
µg/mL C D (0.0866)(112.5)

ss,new

The theor
Css,new = 26.7 µg/mL etical Css of 23.1 mg/L is close to the desired

Css of 20 mg/L. Assuming a reasonable therapeutic
window, the manufacturer’s suggested starting infusion

The new elimination rate will be 40 mg/h. rate is satisfactory.

REFERENCE
Hurley SF, McNeil JJ: A comparison of the accuracy of a least-

squares regression, a Bayesian, Chiou’s and the steady-state
clearance method of individualizing theophylline dosage.
Clin Pharmacokinet 14:311−320, 1988.

BIBLIOGRAPHY
Gibaldi M: Estimation of the pharmacokinetic parameters of the Mitenko P, Ogilvie R: Rapidly achieved plasma concentration

two-compartment open model from postinfusion plasma con- plateaus, with observations on theophylline kinetics. Clin
centration data. J Pharm Sci 58:1133−1135, 1969. Pharmacol Ther 13:329−335, 1972.

Koup J, Greenblatt D, Jusko W, et al: Pharmacokinetics of digoxin Riegelman JS, Loo JC: Assessment of pharmacokinetic constants
in normal subjects after intravenous bolus and infusion dose. from postinfusion blood curves obtained after IV infusion.
J Pharmacokinet Biopharm 3:181−191, 1975. J Pharm Sci 59:53, 1970.

Loo J, Riegelman S: Assessment of pharmacokinetic constants Sawchuk RJ, Zaske DE: Pharmacokinetics of dosing regimens
from postinfusion blood curves obtained after IV infusion. which utilize multiple intravenous infusions: Gentamicin
J Pharm Sci 59:53−54, 1970. in burn patients. J Pharmacokinet Biopharm 4:183−195,

Loughnam PM, Sitar DS, Ogilvie RI, Neims AH: The two- 1976.
compartment open system kinetic model: A review of its clini- Wagner J: A safe method for rapidly achieving plasma concentra-
cal implications and applications. J Pediatr 88:869−873, 1976. tion plateaus. Clin Pharmacol Ther 16:691−700, 1974.

 

Drug Elimination,

7 Clearance, and
Renal Clearance
Murray P. Ducharme

Chapter Objectives DRUG ELIMINATION
»» Describe the main routes of drug Drugs are removed from the body by various elimination pro-

elimination from the body. cesses. Drug elimination refers to the irreversible removal of drug
»» Understand the importance from the body by all routes of elimination. The declining plasma

of the role of clearance as a PK drug concentration observed after systemic drug absorption shows
parameter. that the drug is being eliminated from the body but does not neces-

sarily differentiate between distribution and elimination, and does
»» Define clearance and its

not indicate which elimination processes are involved.
relationship to a corresponding

Drug elimination is usually divided into two major components:
half-life and a volume of

excretion and biotransformation. Drug excretion is the removal of
distribution.

the intact drug. Nonvolatile and polar drugs are excreted mainly by
»» Differentiate between clearance renal excretion, a process in which the drug passes through the

and renal clearance. kidney to the bladder and ultimately into the urine. Other pathways

»» Describe the processes for renal for drug excretion may include the excretion of drug into bile,

drug excretion and explain sweat, saliva, milk (via lactation), or other body fluids. Volatile

which renal excretion process drugs, such as gaseous anesthetics, alcohol, or drugs with high

predominates in the kidney for volatility, are excreted via the lungs into expired air.

a specific drug, given its renal Biotransformation or drug metabolism is the process by

clearance. which the drug is chemically converted in the body to a metabolite.
Biotransformation is usually an enzymatic process. A few drugs

»» Describe the renal clearance may also be changed chemically1 by a nonenzymatic process
model based on renal blood (eg, ester hydrolysis). The enzymes involved in the biotransforma-
flow, glomerular filtration, and tion of drugs are located mainly in the liver (see Chapter 12). Other
drug reabsorption. tissues such as kidney, lung, small intestine, and skin also contain

»» Describe organ drug clearance biotransformation enzymes.

in terms of blood flow and Drug elimination in the body involves many complex rate

extraction. processes. Although organ systems have specific functions, the
tissues within the organs are not structurally homogeneous, and

»» Calculate clearance
elimination processes may vary in each organ. In Chapter 4, drug

using different methods
elimination was modeled by an overall first-order elimination rate

including the physiological,
process. In this chapter, drug elimination is described in terms of

noncompartmental, and
clearance from a hypothetical well-stirred compartment containing

compartmental approaches.

1 Nonenzymatic breakdown of drugs may also be referrered to as degradation. For
example, some drugs such as aspirin (acetylsalicylic acid) may break down in the
stomach due to acid hydrolysis at pH 1–3.

149

 

150 Chapter 7

uniform drug distribution. The term clearance or more generally
describes the process of drug elimination from the

DOSE = Cl/F × AUC0-inf (7.2)
body or from a single organ without identifying the
individual processes involved. Clearance may be in which Cl/F can be called the “apparent clearance”
defined as the volume of fluid removed of the drug when the absolute bioavailability (F) is unknown or
from the body per unit of time. The units for clearance simply not specified or assumed.
are sometimes in milliliters per minute (mL/min) but
most often reported in liters per hour (L/h). The vol-
ume concept is simple and convenient, because all Frequently Asked Question
drugs are dissolved and distributed in the fluids of

»»Why is clearance a useful pharmacokinetic
the body. parameter?

Clearance is even more important clinically than
a half-life for several reasons. First and foremost,
clearance directly relates to the systemic exposure of
a drug (eg, AUCinf), making it the most useful PK DRUG CLEARANCE
parameter clinically as it will be used to calculate Drug clearance is a pharmacokinetic term for
doses to administer in order to reach a therapeutic describing drug elimination from the body without
goal in terms of exposure. While the terminal half- identifying the mechanism of the process. Drug
life gives information only on the terminal phase of clearance (also called body clearance or total body
drug disposition, clearance takes into account all clearance, and abbreviated as Cl or ClT) considers
processes of drug elimination regardless of their the entire body as a single drug-eliminating system
mechanism. When the PK behavior of the drug fol- from which many unidentified elimination processes
lows linear PK, clearance is a constant, whereas the may occur. Instead of describing the drug elimina-
rate of drug elimination is not. For example, first- tion rate in terms of amount of drug removed per unit
order elimination processes consider that a certain of time (eg, mg/h), drug clearance is described in
portion or fraction (percent) of the distribution vol- terms of volume of fluid removed from the drug per
ume is cleared of drug over a given time period. This unit of time (eg, L/h).
basic concept (see also Chapter 3) will be elaborated There are several definitions of clearance, which
along with a review of the anatomy and physiology are similarly based on a volume removed from the
of the kidney. drug per unit of time. The simplest concept of clear-

As will be seen later on in this chapter and in the ance regards the body as a space that contains a defi-
noncompartmental analysis chapter (Chapter 25), nite volume of apparent body fluid (apparent volume
the clearance of a drug (Cl) is directly related to of distribution, V or VD) in which the drug is dis-
the dose administered and to the overall systemic solved. Drug clearance is defined as the fixed volume
exposure achieved with that dose as per the equation of fluid (containing the drug) removed from the drug
Cl = DOSE/AUC0-inf. The overall systemic exposure per unit of time. The units for clearance are volume/
(AUC0-inf) of a drug resulting from an administered time (eg, mL/min, L/h). For example, if the Cl of
dose correlates with its efficacy and toxicity. The penicillin is 15 mL/min in a patient and penicillin
drug clearance (Cl) is therefore the most important has a VD of 12 L, then from the clearance definition,
PK parameter to know in a given patient. If the thera- 15 mL of the 12 L will be removed from the drug
peutic goal in terms of AUC0-inf is known for a drug, per minute.
then the dose to administer to this patient is com- Alternatively, Cl may be defined as the rate of
pletely dictated by the clearance value (Cl). drug elimination divided by the plasma drug concen-

tration. This definition expresses drug elimination in
Hence, after IV administration

terms of the volume of plasma eliminated of drug
DOSE = Cl × AUC0-inf (7.1) per unit time. This definition is a practical way to

 

Drug Elimination, Clearance, and Renal Clearance 151

calculate clearance based on plasma drug concentra-
tion data. From Equation 7.4, the rate of drug elimination is

dD
Elimination rate E =10 µg/mL×15 mL/min=150 µg/min

Cl = dt
Plasma concentration (Cp ) (7.3)

Thus, 150 mg/min of penicillin is eliminated from
dD  µg/min the body when the plasma penicillin concentra-

Cl =  E /dt = = min (7.4)
 Cp  µ mL/

g/mL tion is 10 mg/mL.
Clearance may be used to estimate the rate

where DE is the amount of drug eliminated and of drug elimination at any given concentration.
dDE/dt is the rate of elimination. Using the same example, if the elimination rate of

Rearrangement of Equation 7.4 gives Equation 7.5. penicillin was measured as 150 mg/min when the

dD plasma penicillin concentration was 10 mg/mL,
Elimination rate E

= = C Cl (7.5)
dt p then the clearance of penicillin is calculated from

Equation 7.4:
The two definitions for clearance are similar because

150 µg/min
dividing the elimination rate by the Cp yields the Cl = 15 mL/min

10 µ =
g/mL

volume of plasma cleared of drug per minute, as
shown in Equation 7.4.

As discussed in previous chapters, a first-order
Just as the elimination rate constant (k or k

elimination rate, dD el) represents
E/dt, is equal to kDB or kCpVD.

the total sum of all of the different rate constants for
Based on Equation 7.3, substituting elimination rate

drug elimination, including for example the renal (k
for kC R)

pVD,
and liver (kH) elimination rate constants, Cl is the total

kCpVD sum of all of the different clearance processes in the
Cl = = kVD (7.6)

C body that are occurring in parallel in terms of cardiac
p

blood flow (therefore excepting lung clearance),
Equation 7.6 shows that clearance is the product of a

including for example clearance through the kidney
volume of distribution, VD, and a rate constant, k,

(renal clearance abbreviated as Cl
both of which are constants when the PK is linear. As R), and through the

liver (hepatic clearance abbreviated as Cl
the plasma drug concentration decreases during H):

elimination, the rate of drug elimination, dDE/dt, Elimination rate constant:
decreases accordingly, but clearance remains con- k or kel where k = kR + kH + kother (7.7)
stant. Clearance is constant as long as the rate of

Clearance:
drug elimination is a first-order process.

Cl where Cl = ClR + ClH + Clother (7.8)

EXAMPLE »» » where

Penicillin has a Cl of 15 mL/min. Calculate the elim- Renal clearance: ClR = kR × V (7.9)
ination rate for penicillin when the plasma drug

Hepatic clearance: Cl
concentration, C H = kH × V (7.10)

p, is 2 mg/mL.
Total clearance:

Solution
Cl = k × V = (kR + kH + kother) × V (7.11)

Elimination rate = Cp × Cl (from Equation 7.5)

dD From Equation 7.11, for a one-compartment model
E =2 µg/mL×15 mL/min=30 µg/min

dt (ie, where V = Vss and where k = lz), the total body
clearance Cl of a drug is the product of two con-

Using the previous penicillin example, assume that
stants, l

the plasma penicillin concentration is 10 mg/mL. z and Vss, which reflect all the distribution
and elimination processes of the drug in the body.

 

152 Chapter 7

Distribution and elimination are affected by blood
flow, which will be considered below (and in 0.693 0.693

k 1h−1
= = = 0.23

Chapter 11) using a physiologic model. t1/2 3

For a multicompartment model (eg, where the Cl = 0.23 h−1 × 100 mL/kg = 23.1 mL/(kg⋅h)
total volume of distribution [Vss] includes a central
volume of distribution [Vc], and one [Vp] or more For a 70-kg patient, Cl = 23.1 × 70 = 1617 mL/h

peripheral volumes of distributions), the total body
clearance of a drug will be the product of the elimi- CLEARANCE MODELS
nation rate constant from the central compartment
(k10) and Vc. The equations become: The calculation of clearance from a rate constant

(eg, k or k10) and a volume of distribution (eg, V or Vc)
Renal clearance: ClR = kR × VC (7.12) assumes (sometimes incorrectly) a defined compart-

Hepatic clearance: ClH = kH × VC (7.13) mental model, whereas clearance estimated directly
from the plasma drug concentration−time curve using

Total clearance:
noncompartmental PK approaches does not need one

Cl = k10 × VC = (kR + kH + kother) × VC (7.14) to specify the number of compartments that would
describe the shape of the concentration−time curve.

Clearance values are often adjusted on a per-kilogram- Although clearance may be regarded as the product of
of-actual-body-weight (ABW) or on a per-meter- a rate constant k and a volume of distribution V,
square-of-surface-area basis, such as L/h per kilogram Equation 7.11 is far more general because the reaction
or per m2, or normalized for a “typical” adult of 72 kg order for the rate of drug elimination, dDE/dt, is not
or 1.72 m2. This approach is similar to the method for specified, and the elimination rate may or may not
expressing V, because both pharmacokinetic param- follow first-order kinetics. The various approaches for
eters vary with body weight or body size. It has estimating a drug clearance are described in Fig. 7-1
been found, however, that when expressing clearance and will be explored one by one below:
between individuals of varying ABW, such as predict-
ing Cl between children and adults, Cl varies best allo- Compartmental model

metrically with ABW, meaning that Cl is best expressed IV k10
Vc (Cp)with an allometric exponent (most often 0.75 is rec-

ommended) relating it to ABW as per the following Static volume and
rst-order processes are assumed in
expression (see also Chapter 25): simpler models. Here Cl = k10 x Vc.

Cl (predicted in a patient) Physiologic model

= Cl(population value for a 72-kg patient) × (ABW/72)0.75 Q Ca Q Cv

(7.15)

EXAMPLE »» » Elimination
Clearance is the product of the ow through an organ (Q)
and the extraction ratio of that organ (E). For example, the

Determine the total body clearance for a drug in a hepatic clearance is ClH = QH x EH.

70-kg male patient. The drug follows the kinetics Noncompartmental approach
of a one-compartment model and has an elimina- Cp
tion half-life of 3 hours with an apparent volume of AUC0-inf

distribution of 100 mL/kg.
Time (h)

Solution
Volume of distribution does not need to be de
ned.

First determine the elimination rate constant (k) Cl = DOSE/AUCinf.

and then substitute properly into Equation 7.11. FIGURE 71 General approaches to clearance. Volume
and elimination rate constant not defined.

 

Drug Elimination, Clearance, and Renal Clearance 153

Q Ca Elimination Q C Equation 7.16 adapted for the liver as an organ yields
v

organ the hepatic clearance (ClH)

ClH = QH × EH (7.19)

Elimination Therefore, if Cl = ClH + ClNH (where ClNH is the
drug

nonhepatic clearance), then
FIGURE 72 Drug clearance model. (Q = blood flow,
Ca = incoming drug concentration [usually arterial drug con- Cl = (QH × EH) + ClNH (7.20)
centration], Cv = outgoing drug concentration [venous drug
concentration].) For some drugs Cl ~ ClH, and so Cl ~ QH × EH.

The physiologic approach to organ clearance
Physiologic/Organ Clearance

shows that the clearance from an organ depends on
Clearance may be calculated for any organ involved its blood flow rate and its ability at eliminating the
in the irreversible removal of drug from the body. drug, whereas the total clearance is that of a constant
Many organs in the body have the capacity for drug or static fraction of the volume in which the drug is
elimination, including drug excretion and biotrans- distributed or is removed from the drug per unit of
formation. The kidneys and liver are the most com- time. Organ clearance measurements using the phys-
mon organs involved in excretion and metabolism, iologic approach require invasive techniques to
respectively. Physiologic pharmacokinetic models obtain measurements of blood flow and extraction
are based on drug clearance through individual ratio. The physiologic approach has been used to
organs or tissue groups (Fig. 7-2). describe hepatic clearance, which is discussed fur-

For any organ, clearance may be defined as the ther under hepatic elimination (Chapter 12). More
fraction of blood volume containing drug that flows classical definitions of clearance have been applied
through the organ and is eliminated of drug per unit to renal clearance because direct measurements of
time. From this definition, clearance is the product plasma drug concentration and urinary drug excre-
of the blood flow (Q) to the organ and the extraction tion may be obtained. Details will be presented in the
ratio (E). The E is the fraction of drug extracted by Renal Clearance section of this chapter.
the organ as drug passes through.

Cl (organ) = Q (organ) × E (organ) (7.16) Noncompartmental Methods

If the drug concentration in the blood (Ca) entering Clearance is commonly used to describe first-order

the organ is greater than the drug concentration of drug elimination from compartment models such as

blood (C the one-compartment model, C(t) = Cp = C0e−kt
p in

v) leaving the organ, then some of the drug
has been extracted by the organ (Fig. 7-2). The E is which the distribution volume and elimination rate

C constant are well defined. Clearance estimated directly
a − Cv divided by the entering drug concentration

(Ca), as shown in Equation 7.17. from the area under the plasma drug concentration−

time curve using the noncompartmental method is
C −C

a v
E = (7.17) often called a “model-independent” approach as it

C
a does not need any assumption to be set in terms of

E is a ratio with no units. The value of E may range the number of compartments describing the kinetics

from 0 (no drug removed by the organ) to 1 (100% of or concentration−time profile of the drug under study.

the drug is removed by the organ). An E of 0.25 indi- It is not exactly true that this method is a “model-

cates that 25% of the incoming drug concentration is independent” one, though, as this method still assumes

removed by the organ as the drug passes through. that the terminal phase decreases in a log-linear fash-

Substituting for E into Equation 7.16 yields ion that is model dependent, and many of its parame-
ters can be calculated only when one assumes PK

C −C
Cl (organ) =Q (organ) a v

(7.18) linearity. Referring to this method as “noncompart-
Ca mental” is therefore more appropriate.

 

154 Chapter 7

The noncompartmental approach is based on described as the “observed” AUC and calculated
statistical moment theory and is presented in more using the linear or mixed log-linear trapezoidal rule,
details in Chapter 25. The main advantages of this while the AUC that needs to be extrapolated from
approach are that (1) clearance can be easily calcu- time t to infinity (AUCt-inf) is often described as the
lated without making any assumptions relating to “extrapolated” AUC. It is good pharmacokinetic
rate constants (eg, distribution vs. elimination rate practice for the clearance to be calculated robustly to
constants), (2) volume of distribution is presented in never extrapolate the AUC0-t by more than 20%. In
a clinically useful context as it is related to systemic addition, it is also good pharmacokinetic practice for
exposure and the dose administered, and (3) its esti- the AUC0-t to be calculated using a rich sampling
mation is robust in the context of rich sampling data strategy, meaning a minimum of 12 concentration−

as very little modeling is involved, if any (eg, no time points across the concentration−time curve
modeling at steady-state data, and only very limited from zero to Ct.
modeling by way of linear regression of the terminal At steady state, when the concentration−time
phase after single dose administration). profiles between administered doses become con-

Clearance can be determined directly from the stant, the amount of drug administered over the dos-
time−concentration curve by ing interval is exactly equal to the amount eliminated

over that dosing interval (t). The formula for clear-

Cl = ∫ D × F /C(t)dt (7.21) ance therefore becomes:
0

where D is the dose administered, F is the bioavail- F × D
Cl or Cl = (7.22)

(ss)
ability factor associated with the administration route AUC

τ (ss)

used of the drug product, and C(t) is an unknown If the drug exhibits linear pharmacokinetics in terms
function that describes the changing plasma drug of time, then the clearance calculated after single
concentrations. dose administration (Cl) using Equation 7.2 and the

Using the noncompartmental approach, the gen- clearance calculated from steady-state data (Cl(ss))
eral equation therefore uses the area under the drug using Equation 7.22 will be the same.
concentration curve, [AUC]∞0 , for the calculation of From Equation 7.22, it can be derived that follow-
clearance. ing a constant intravenous infusion (see Chapter 6), the

F × D steady-state concentration (Css) will then be equal to
Cl = (as presented before

AUC “rate in,” the administration dosing rate (R0), divided
0-inf in Equation 7.2)

by “rate out” or the clearance:

where AUC0-inf = [AUC]∞0 = ∫ C t nd s t e t tal
0 p d a i h o F × R

C 0 F × R0
(ss) = or Cl = (7.23)

systemic exposure obtained after a single dose (D) Cl Css

until infinity.
where R0 is the constant dosing rate (eg, in mg/h), C

Because [AUC]∞0 is calculated from the drug ss
is the steady-state concentration (eg, in mg/L), and

concentration−time curve from zero to infinity using
Cl is the total body clearance (eg, in L/h).

the trapezoidal rule, no model is assumed until the
terminal phase after the last detectable concentration
is obtained (Ct). To extrapolate the data to infinity to Compartmental Methods
obtain the residual [AUC]∞ or (Cp /k), first-order Clearance is a direct measure of elimination from the

t t

elimination is usually assumed. central compartment, regardless of the number of
Equation 7.2 is used to calculate clearance after compartments. The central compartment consists of

administration of a single dose, and where concen- the plasma and highly perfused tissues in which drug
trations would be obtained in a rich sampling fashion equilibrates rapidly (see Chapter 5). The tissues for
until a last detectable concentration time point, Ct. drug elimination, namely, kidney and liver, are con-
The AUC from time zero to t (AUC0-t) is often sidered integral parts of the central compartment.

 

Drug Elimination, Clearance, and Renal Clearance 155

Clearance is always the product of a rate con- rate constant instead of assuming it is the “elimina-
stant and a volume of distribution. There are different tion” rate constant.
clearance formulas depending on the pharmacoki- Cl V

ss
netic model that would describe appropriately the = λ ×

F z F
concentration-versus-time profiles of a drug product.

Relationship with the noncompartmental approach
The clearance formulas depend upon whether the

after IV administration:
drug is administered intravenously or extravascularly

Dose
and range from simple to more complicated scenarios: Cl = λz ×Vss and Cl =

AUC
0-inf

Drug that is well described pharmacokinetically

with a one-compartment model and Vss = Cl × MRT

After intravenous administration, such a drug Therefore, MRT (mean residence time2) = 1/lz and
will exhibit a concentration−time profile that Vss = Dose/(AUC0-inf × lz).
decreases in a straight line when viewed on a semilog Relationship with the noncompartmental approach
plot and would therefore be well described by a after extravascular administration:
monoexponential decline. This is the simplest model

Cl V
ss Cl Dose

that can be used and often will describe well the phar- = λz × and =
F F F AUC

macokinetics of drugs that are very polar and that are 0-inf

readily eliminated in the urine. Clinically, aminogly- MRT and Vss /F are not computable directly using
coside antibiotics are relatively well characterized noncompartmental methods after extravascular
and predicted by a one-compartment model. administration, but only MTT (mean transit time),

which is the sum of MAT (mean absorption time)
Cl = lz × Vss and MRT.

But we have seen that MRT = 1/lz and Vss/F =
where lz is the only rate constant describing the fate

Dose/(AUC0-inf × lz). MAT can then be calculated by
of the concentration−time profile and dividing 0.693

subtracting MRT from the MTT.
by its value, therefore, estimates the terminal half-
life. Vss is the total volume of distribution, and in this Drug that is well described pharmacokinetically

case, there is only one volume that is describing the with a two-compartment model

pharmacokinetic behavior of the drug. After intravenous administration, such a drug will

Calculated parameters: exhibit a concentration−time profile that decreases
in a profile that can be characterized by two different

The terminal half-life of the drug is T1/2 = 0.693/l exponentials or two different straight lines when
z

viewed on a semilog plot (see Chapter 5). This

After oral administration the formula for clearance is model will describe well the pharmacokinetics of

exactly the same but a Cl/F is calculated. There is drugs that are not so polar and distribute in a second

also an absorption process in addition to an elimina- compartment that is not so well perfused by blood or

tion one. If the absorption process is faster than the plasma. Clinically, the antibiotic vancomycin is rela-

elimination, the terminal rate constant, lz, will tively well characterized and predicted by a two-

describe the elimination of the drug. If the drug compartment model.

exhibits a “flip-flop” profile because the absorption
Cl = k10 × Vc (7.24)

of the drug is much slower than the elimination pro-
cess (eg, often the case with modified release formu- where k10 is the rate constant describing the disap-
lations), then the terminal rate constant, l pearance of the drug from its central volume of dis-

z, will be
reflective of the absorption and not the elimination. tribution (Vc).
It is sometimes not possible to know if a drug exhib-
its a slower absorption than elimination. In these 2MRT is mean residence time and is discussed more fully in
cases, it is always best to refer to lz as the “terminal” Chapter 25.

 

156 Chapter 7

The distributional clearance (Cld) describes the It is often stated that clearances and volumes are
clearance occurring between the central (Vc) and the “independent” parameters, while rate constants are
peripheral compartment (Vp), and where the central “dependent” parameters. This assumption is made in
compartment includes the plasma and the organs that PK models to facilitate data analysis of the underly-
are very well perfused, while the peripheral compart- ing kinetic processes. Stated differently, a change in
ment includes organs that are less well perfused. a patient in its drug clearance may not result in a

The concentration−time curve profile will fol- change in its volume of distribution or vice versa,
low a biexponential decline on a semilog graph and while a change in clearance or in the volume of dis-
the distributional rate constant (l1) will be describ- tribution will result in a change in the appropriate
ing the rapid decline after IV administration that rate constant (eg, k10, lz). While mostly true, this
describes the distribution process, and the second statement can be somewhat confusing, as there are
and last exponential (lz) will describe the terminal clinical instances where a change can lead to both
elimination phase. volume of distribution and clearance changes, without

The distribution (l1) and terminal elimination a resulting change in the rate constant (eg, k10, lz).
(lz) rate constants can be obtained with the follow- A common example is a significant abrupt change
ing equations: in actual body weight (ABW) as both clearances

and volumes of distribution correlate with ABW.
° l1 = [((Cl + Cld)/Vc + Cld/Vp) + SQRT (((Cl +

A patient becoming suddenly edematous will not
Cld)/Vc + Cld/Vp)

2 − 4 × Cl/Vc*Cld/Vp))]/2 see his or her liver or renal function necessarily
° lz = [((Cl + Cld)/Vc + Cld/Vp) − SQRT (((Cl +

affected. In that example, both the patient’s clear-
Cld)/Vc + Cld/Vp)

2 − 4 × Cl/Vc*Cld/Vp))]/2 ance and volume of distribution will be increased,
The distribution and terminal elimination half-lives while half-life or half-lives will remain relatively
are therefore: unchanged. In that situation the dosing interval will

not need to be changed, as the half-life will stay
° T1/2(l1) = 0.693/l1 constant, but the dose to be given will need to be
° T1/2(lz) = 0.693/lz increased due to the greater volume of distribution

The total volume of distribution Vss will be the sum and clearance.
of Vc and Vp:

Summary Regarding Clearance Calculations
Vss = Vc + Vp (7.25)

Clearance can be calculated using physiologic, com-
Relationship with the noncompartmental approach partmental, or noncompartmental methods. What is
after IV administration: important to remember is that all methods will lead

Dose to the same results if they are applied correctly and

Cl = and Vss = Cl × MRT
AUC if there are enough data supporting the calculations.

0-inf
Clearance can therefore be calculated:

(noncompartmental equations)
• After a single dose administration using the area

Cl = k10 × Vc and Vss = Vc + Vp under the concentration−time curve from time zero
(compartmental equations) to infinity using a noncompartmental approach:

Cl = (Dose × F)/AUC
Therefore, MRT = (Vc + Vp)/(k10 × Vc)

0-inf.
• At steady-state conditions using the area under the

concentration−time curve during a dosing interval
Relationship between Rate Constants, using a noncompartmental approach: Cl = (Dose ×
Volumes of Distribution, and Clearances F)/AUCt (ss).
As seen previously in Equation 7.24, Cl = k10 × Vc, • When a constant infusion is administered until
which for a drug well described by a one-compart- steady-state concentrations (Css) are achieved:
ment model can be simplified to Cl = lz × Vss. Cl = F × R0 /Css.

 

Drug Elimination, Clearance, and Renal Clearance 157

• At any time using a compartmental approach with Medulla
Renal artery Cortex

the appropriate volume(s) of distribution and rate
Renal

constant(s): vein

° Cl = k10 × Vc when the PK of a drug is well Renal
described by any compartment model when the pelvis LEFT
drug displays linear pharmacokinetics. RIGHT KIDNEY

KIDNEY Ureters (cut
° Which equation can be simplified to Cl = lz × Vss surface)

when the PK of a drug is well described by only
a one-compartment model as lz is then equal to
k10, and Vss to Vc.

• For an organ using its blood ow and its extraction
Urinary

ratio. For example, the hepatic clearance could be bladder
Direction of

calculated as ClH = QH × EH. For a drug that would urine ow
be only eliminated via the liver, then Cl would be

FIGURE 73 The general organizational plan of the
equal to ClH.

urinary system. (Reproduced with permission from Guyton
AC: Textbook of Medical Physiology, 8th ed. Philadelphia,
Saunders, 1991.)

THE KIDNEY
The liver (see Chapter 12) and the kidney are the two nephrons have long loops of Henle that extend into
major drug-eliminating organs in the body, though the medulla (Fig. 7-5). The longer loops of Henle
drug elimination can also occur almost anywhere in allow for a greater ability of the nephron to reabsorb
the body. The kidney is the main excretory organ for water, thereby producing more concentrated urine.
the removal of metabolic waste products and plays a
major role in maintaining the normal fluid volume Blood Supply
and electrolyte composition in the body. To maintain

The kidneys represent about 0.5% of the total body
salt and water balance, the kidney excretes excess

weight and receive approximately 20%−25% of the
electrolytes, water, and waste products while con-

cardiac output. The kidney is supplied by blood via
serving solutes necessary for proper body function.

the renal artery, which subdivides into the interlobar
In addition, the kidney has two endocrine functions:
(1) secretion of renin, which regulates blood pres-
sure, and (2) secretion of erythropoietin, which Medulla Cortex

Capsule
stimulates red blood cell production.

Renal pyramid

Papilla
Anatomic Considerations Interlobar arteries

Minor calyx
The kidneys are located in the peritoneal cavity. A Major calyx Arcuate arteries
general view is shown in Fig. 7-3 and a longitudinal Renal

artery Interlobular
view in Fig. 7-4. The outer zone of the kidney is arteries

HILUS Renal vein
called the cortex, and the inner region is called the Renal Segmental

pelvis arteries
medulla. The nephrons are the basic functional units,

Column of Bertin
collectively responsible for the removal of metabolic
waste and the maintenance of water and electrolyte
balance. Each kidney contains 1−1.5 million neph- Ureter

rons. The glomerulus of each nephron starts in the FIGURE 74 Longitudinal section of the kidney, illustrat-
cortex. Cortical nephrons have short loops of Henle ing major anatomical features and blood vessels. (From West,
that remain exclusively in the cortex; juxtamedullary 1985, with permission.)

 

158 Chapter 7

JUXTAMEDULLARY CORTICAL Interlobular Afferent Efferent Peritubular
NEPHRON NEPHRON artery arteriole arteriole capillaries

Bowman’s capsule
Proximal Distal
tubule tubule Afferent arteriole

Glomerular
Glomerular capillaries
capillaries

Arcuate vein
(outer Arcuate artery
stripe)
(inner Efferent
stripe) arteriole

Loop of Collecting
Henle tubule/duct

Vasa recta

Interlobar
A B artery and vein

FIGURE 75 Cortical and juxtamedullary nephrons and their vasculature. (From West, 1985, p. 452, with permission.)

arteries penetrating within the kidney and branching The relationship of RBF to RPF is given by a rear-
further into the afferent arterioles. Each afferent arteri- rangement of Equation 7.26:
ole carries blood toward a single nephron into the glo-

RPF = RBF (1 − Hct) (7.27)
merular portion of the nephron (Bowman’s capsule).
The filtration of blood occurs in the glomeruli in Assuming a hematocrit of 0.45 and an RBF of 1.2 L/min
Bowman’s capsule. From the capillaries (glomerulus) and using the above equation, RPF = 1.2 − (1.2 ×
within Bowman’s capsule, the blood flows out via the 0.45) = 0.66 L/min or 660 mL/min, or approximately
efferent arterioles and then into a second capillary 950 L/d. The average glomerular filtration rate (GFR)
network that surrounds the tubules (peritubule capil- is about 120 mL/min in an average adult,3 or about
laries and vasa recti), including the loop of Henle, 20% of the RPF. The ratio GFR/RPF is the filtration
where some water is reabsorbed. fraction.

The renal blood flow (RBF) is the volume of
blood flowing through the renal vasculature per unit Regulation of Renal Blood Flow
of time. RBF exceeds 1.2 L/min or 1700 L/d. Renal

Blood flow to an organ is directly proportional to the
plasma flow (RPF) is the RBF minus the volume of

arteriovenous pressure difference (perfusion pressure)
red blood cells present. RPF is an important factor in

across the vascular bed and indirectly proportional to
the rate of drug filtration at the glomerulus.

the vascular resistance. The normal renal arterial pres-

RPF = RBF − (RBF × Hct) (7.26) sure (Fig. 7-6) is approximately 100 mm Hg and falls
to approximately 45−60 mm Hg in the glomerulus

where Hct is the hematocrit.
Hct is the fraction of blood cells in the blood, 3GFR is often based on average body surface, 1.73 m2. GFR is less

about 0.45 or 45% of the total blood volume. in women and also decreases with age.

Medulla Cortex

Inner zone Outer
(papilla) zone

 

Drug Elimination, Clearance, and Renal Clearance 159

18 mm Hg autoregulation refers to the maintenance of a con-
10 mm Hg stant blood flow in the presence of large fluctuations

60 mm Hg in arterial blood pressure. Because autoregulation
maintains a relatively constant blood flow, the filtra-

18 mm Hg
tion fraction (GFR/RPF) also remains fairly constant

13 mm Hg in this pressure range.

10 mm Hg Glomerular Filtration and Urine Formation

100 mm Hg A normal adult subject has a GFR of approxi-
mately 120 mL/min. About 180 L of fluid per day are
filtered through the kidneys. In spite of this large fil-
tration volume, the average urine volume is 1−1.5 L.
Up to 99% of the fluid volume filtered at the glom-

8 mm Hg
Intersitial uid erulus is reabsorbed. Besides fluid regulation, the

pressure 6 mm Hg 0 mm Hg
kidney also regulates the retention or excretion of

FIGURE 76 Approximate pressures at different points in various solutes and electrolytes (Table 7-1). With the
the vessels and tubules of the functional nephron and in the exception of proteins and protein-bound substances,
interstitial fluid. (Reproduced with permission from Guyton most small molecules are filtered through the glom-
AC: Textbook of Medical Physiology, 8th ed. Philadelphia, erulus from the plasma. The filtrate contains some
Saunders, 1991.)

ions, glucose, and essential nutrients as well as waste
products, such as urea, phosphate, sulfate, and other

(glomerular capillary hydrostatic pressure). This pres- substances. The essential nutrients and water are
sure difference is probably due to the increasing vas- reabsorbed at various sites, including the proximal
culature resistance provided by the small diameters of tubule, loops of Henle, and distal tubules. Both active
the capillary network. Thus, the GFR is controlled by reabsorption and secretion mechanisms are involved.
changes in the glomerular capillary hydrostatic The urine volume is reduced, and the urine generally
pressure. contains a high concentration of metabolic wastes

In the normal kidney, RBF and GFR remain and eliminated drug products. Advances in molec-
relatively constant even with large differences in ular biology have shown that transporters such as
mean systemic blood pressure (Fig. 7-7). The term P-glycoprotein and other efflux proteins are pres-

ent in the kidney, and can influence urinary drug
excretion. Further, CYP enzymes are also present

1000
RPF in the kidney, and can impact drug clearance by

800 metabolism.

600 Renal Drug Excretion

Renal excretion is a major route of elimination for
400

many drugs. Drugs that are nonvolatile, are water

200 GFR soluble, have a low molecular weight (MW), or are
slowly biotransformed by the liver are eliminated by

0 renal excretion. The processes by which a drug is
0 80 160 240 excreted via the kidneys may include any combination

Mean arterial pressure (mm Hg) of the following:
FIGURE 77 Schematic representation of the effect

• Glomerular filtration
of mean arterial pressure on GFR and RPF, illustrating the
phenomenon of autoregulation. (From West, 1985, p. 465, with • Active tubular secretion
permission.) • Tubular reabsorption

GFR or RPF (mL/min)

 

160 Chapter 7

TABLE 71 Quantitative Aspects of Urine Formationa

Per 24 Hours

Substance Filtered Reabsorbed Secreted Excreted Percent Reabsorbed

Sodium ion (mEq) 26,000 25,850 150 99.4

Chloride ion (mEq) 18,000 17,850 150 99.2

Bicarbonate ion (mEq) 4,900 4,900 0 100

Urea (mM) 870 460b 410 53

Glucose (mM) 800 800 0 100

Water (mL) 180,000 179,000 1000 99.4

Hydrogen ion Variable Variablec

Potassium ion (mEq) 900 900d 100 100 100d

aQuantity of various plasma constituents filtered, reabsorbed, and excreted by a normal adult on an average diet.

bUrea diffuses into, as well as out of, some portions of the nephron.

cpH or urine is on the acid side (4.5−6.9) when all bicarbonate is reabsorbed.

dPotassium ion is almost completely reabsorbed before it reaches the distal nephron. The potassium ion in the voided urine is actively secreted into
the urine in the distal tubule in exchange for sodium ion.

From Levine (1990), with permission.

Glomerular filtration is a unidirectional process Active tubular secretion is an active transport
that occurs for most small molecules (MW < 500), process. As such, active renal secretion is a carrier-
including undissociated (nonionized) and dissoci- mediated system that requires energy input,
ated (ionized) drugs. Protein-bound drugs behave as because the drug is transported against a concen-
large molecules and do not get filtered at the glom- tration gradient. The carrier system is capacity
erulus. The major driving force for glomerular filtra- limited and may be saturated. Drugs with similar
tion is the hydrostatic pressure within the glomerular structures may compete for the same carrier sys-
capillaries. The kidneys receive a large blood supply tem. Among the active renal secretion systems that
(approximately 25% of the cardiac output) via the have been identified, there are some for weak acids
renal artery, with very little decrease in the hydro- (organic anion transporter, OAT) and some for
static pressure. weak bases (organic cation transporter, OCT).

Glomerular filtration rate (GFR) is measured Active tubular secretion rate is dependent on RPF.
by using a drug that is eliminated primarily by filtra- Drugs commonly used to measure active tubular
tion only (ie, the drug is neither reabsorbed nor secretion include p-amino-hippuric acid (PAH)
secreted). Clinically inulin and creatinine are often and iodopyracet (Diodrast). These substances are
used for this purpose, although creatinine is also both filtered by the glomeruli and secreted by the
secreted. The clearance of inulin is approximately tubular cells. Active secretion is extremely rapid
equal to the GFR, which can equal 120 mL/min. The for these drugs, and practically all the drug carried
value for the GFR correlates fairly well with body to the kidney is eliminated in a single pass. The
surface area. Glomerular filtration of drugs is directly clearance for these drugs therefore reflects the
related to the free or nonprotein-bound drug concen- effective renal plasma flow (ERPF), which varies
tration in the plasma. As the free drug concentration from 425 to 650 mL/min. The ERPF is determined by
in the plasma increases, the glomerular filtration for both RPF and the fraction of drug that is effectively
the drug increases proportionately, thus increasing extracted by the kidney relative to the concentration
renal drug clearance for some drugs. in the renal artery.

 

Drug Elimination, Clearance, and Renal Clearance 161

For a drug that is excreted solely by glomerular may decrease (acidify) or increase (alkalinize) the
filtration, the elimination half-life may change mark- urinary pH, respectively, when administered in large
edly in accordance with the binding affinity of the quantities. By far the most important changes in
drug for plasma proteins. In contrast, drug protein urinary pH are caused by fluids administered intra-
binding has very little effect on the elimination half- venously. Intravenous fluids, such as solutions of
life of the drug excreted mostly by active secretion. bicarbonate or ammonium chloride, are used in
Because drug protein binding is reversible, drug acid−base therapy to alkalinize or acidify the urine,
bound to plasma protein rapidly dissociates as free respectively. Excretion of these solutions may drasti-
drug is secreted by the kidneys. For example, some cally change urinary pH and alter drug reabsorption
of the penicillins are extensively protein bound, but and drug excretion by the kidney.
their elimination half-lives are short due to rapid The percentage of ionized weak acid drug cor-
elimination by active secretion. responding to a given pH can be obtained from the

Tubular reabsorption occurs after the drug is Henderson−Hasselbalch equation.
filtered through the glomerulus and can be an active
or a passive process involving transporting back into Ionized

pH = pKa + log (7.28)
the plasma. If a drug is completely reabsorbed (eg, Nonionized

glucose), then the value for the clearance of the drug
Rearrangement of this equation yields:

is approximately zero. For drugs that are partially
reabsorbed without being secreted, clearance values Ionized
are less than the GFR of 120 mL/min. 10pH−pKa

= (7.29)
Nonionized

The reabsorption of drugs that are acids or weak
bases is influenced by the pH of the fluid in the renal Fraction of drug ionized
tubule (ie, urine pH) and the pKa of the drug. Both of
these factors together determine the percentage of [Ionized]

=

dissociated (ionized) and undissociated (nonionized) [Ionized]+ [Nonionized]

drug. Generally, the undissociated species is more 10pH−pKa

lipid soluble (less water soluble) and has greater = (7.30)
1 10pH−pK

+ a

membrane permeability. The undissociated drug is
easily reabsorbed from the renal tubule back into the The fraction or percent of weak acid drug ionized in
body. This process of drug reabsorption can signifi- any pH environment may be calculated with Equation
cantly reduce the amount of drug excreted, depend- 7.30. For acidic drugs with pKa values from 3 to 8, a
ing on the pH of the urinary fluid and the pKa of the change in urinary pH affects the extent of dissocia-
drug. The pKa of the drug is a constant, but the nor- tion (Table 7-2). The extent of dissociation is more
mal urinary pH may vary from 4.5 to 8.0, depending greatly affected by changes in urinary pH for drugs
on diet, pathophysiology, and drug intake. In addi- with a pKa of 5 than with a pKa of 3. Weak acids with
tion, the initial morning urine generally is more
acidic and becomes more alkaline later in the day.
Vegetable and fruit diets (alkaline residue diet4) TABLE 72 Effect of Urinary pH and pKa on
result in higher urinary pH, whereas diets rich in the lonization of Drugs
protein result in lower urinary pH. Drugs such as Percent of Drug Percent of Drug
ascorbic acid and antacids such as sodium carbonate pH of Urine Ionized: pKa53 Ionized: pKa55

4The alkaline residue diet (also known as the alkaline ash diet) is a 7.4 100 99.6

diet composed of foods, such as fruits and vegetables, from which 5 99 50.0
the carbohydrate portion of the diet is metabolized in the body
leaving an alkaline residue containing cations such as sodium, 4 91 9.1
potasium, calcium, etc. These cations are excreted through the

3 50 0.99
kidney and cause the urine to become alkaline.

 

162 Chapter 7

TABLE 73 Properties of Renal Drug Elimination Processes

Active/Passive Location in Drug Protein
Process Transport Nephron Drug Ionization Binding Influenced by

Filtration Passive Glomerulus Either Only free drug Protein binding

Secretion Active Proximal tubule Mostly weak acids No effect Competitive inhibitors
and weak bases

Reabsorption Passive/Active Distal tubule Nonionized Not applicable Urinary pH and flow

pKa values of less than 2 are highly ionized at all acid), acidification of the urine causes greater reab-
urinary pH values and are only slightly affected by sorption of the drug and alkalinization of the urine
pH variations. causes more rapid excretion of the drug.

For a weak base drug, the Henderson−Hasselbalch In summary, renal drug excretion is a composite
equation is given as of passive filtration at the glomerulus, active secretion

in the proximal tubule, and passive and/or active
Nonionized

pH = pKa + log (7.31) reabsorption in the distal tubule (Table 7-3). Active
Ionized

secretion is an enzyme (transporter)-mediated pro-
and cess that is saturable. Although reabsorption of drugs

is mostly a passive process, the extent of reabsorp-
10pKa −pH

Percent of drug ionized = (7.32) tion of weak acid or weak base drugs is influenced
1 10pKa −pH

+
by the pH of the urine and the degree of ionization

The greatest effect of urinary pH on reabsorption of the drug. In addition, an increase in blood flow to
occurs for weak base drugs with pKa values of the kidney, which may be due to diuretic therapy or
7.5−10.5. large alcohol consumption, decreases the extent of

From the Henderson−Hasselbalch relationship, drug reabsorption in the kidney and increases the
a concentration ratio for the distribution of a weak rate of drug excreted in the urine.
acid or basic drug between urine and plasma may be
derived. The urine−plasma (U/P) ratios for these CLINICAL APPLICATION
drugs are as follows.

For weak acids, Both sulfisoxazole (Gantrisin) tablets and the com-
bination product, sulfamethoxazole/trimethoprim

U 1 p −
+10 Hurine pKa

= (7.33) (Bactrim) tablets, are used for urinary tract infec-
P pHp −pK

1+10 lasma a

tions. Sulfisoxazole and sulfamethoxazole are sul-
For weak bases, fonamides that are well absorbed after oral

administration and are excreted in high concentra-
1 1 pK − H

U + 0 a p urine

= (7.34) tions in the urine. Sulfonamides are N-acetylated to
P pK −pH

1+10 a plasma

a less water-soluble metabolite. Both sulfonamides
For example, amphetamine, a weak base, will be reab- and their corresponding N-acetylated metabolite are
sorbed if the urine pH is made alkaline and more less water soluble in acid and more soluble in alka-
lipid-soluble nonionized species are formed. In con- line conditions. In acid urine, renal toxicity can
trast, acidification of the urine will cause the amphet- occur due to precipitation of the sulfonamides in the
amine to become more ionized (form a salt). The salt renal tubules. To prevent crystalluria and renal com-
form is more water soluble, less likely to be reab- plications, patients are instructed to take these drugs
sorbed, and tends to be excreted into the urine more with a high amount of fluid intake and to keep the
quickly. In the case of weak acids (such as salicylic urine alkaline.

 

Drug Elimination, Clearance, and Renal Clearance 163

a constant fraction of the central volume of distribu-
Frequently Asked Questions tion in which the drug is contained that is excreted
»»Which renal elimination processes are influenced by by the kidney per unit of time. More simply, renal

protein binding? clearance is defined as the urinary drug excretion

»»Is clearance a first-order process? Is clearance a rate (dDu/dt) divided by the plasma drug concentra-

better parameter to describe drug elimination and tion (Cp).

exposure than half-life? Why is it necessary to use
both parameters in the literature? Excretion rate dDu /dt Cl = = (7.35)

R Plasma concentration Cp

PRACTICE PROBLEMS As seen earlier in this chapter, most clearances
besides that of the lung are additive, and therefore,

Let pKa = 5 for an acidic drug. Compare the U/P at the total body clearance can be defined as the sum of
urinary pH (a) 3, (b) 5, and (c) 7. the renal clearance (ClR) and the nonrenal clearance

(ClNR), whatever it may consist of (eg, hepatic or
Solution other):

a. At pH = 3,
Cl = ClR + ClNR (7.36)

U 1 −
+103 5 1.01 1.01 1

= = = =
P 1 107.4−5

+ 1+102.4 252 252 Therefore, ClR = fe × Cl (7.37)

b. At pH = 5, where fe is the proportion of the bioavailable dose
that is eliminated unchanged in the urine. Using the

U 1 105−5
+ 2 2 noncompartmental formula for Cl studied earlier

= = =
P 1 −

+107.4 5 1+102.4 252 (Equation 7.2), we obtain

c. At pH = 7, fe × F × Dose
ClR =

AUC0-inf
U 1 07−5

+1 101 101
= = =

P 1 1 −
+ 07.4 5 1+102.4 252

and consequently

In addition to the pH of the urine, the rate of urine flow
Ae

influences the amount of filtered drug that is reabsorbed. 0-inf
ClR = (7.38)

AUC
The normal flow of urine is approximately 1−2 mL/min. 0-inf

Nonpolar and nonionized drugs, which are normally well where Ae0-inf is the amount of drug eliminated
reabsorbed in the renal tubules, are sensitive to changes in unchanged in the urine from time 0 to infinity after a
the rate of urine flow. Drugs that increase urine flow, such single dose. In practice it is not possible to measure
as ethanol, large fluid intake, and methylxanthines (such the amount of drug excreted unchanged in the urine
as caffeine or theophylline), decrease the time for drug until infinity, and so in order to get a reasonable
reabsorption and promote their excretion. Thus, forced estimate of the renal clearance with this noncompart-
diuresis through the use of diuretics may be a useful mental approach formula using the amount excreted
adjunct for removing excessive drug in an intoxicated unchanged in the urine and the systemic exposure,
patient, by increasing renal drug excretion. one has to collect the urine and observe the AUC for

the longest time period possible, ideally more than

RENAL CLEARANCE 3−4 terminal half-lives, so that the error made using
this formula is less than 10%. So if, for example, a

Renal clearance, ClR, is defined as the volume that is drug product has a terminal half-life of 12 hours,
removed from the drug per unit of time through the then one may need to collect the urine for 48 hours
kidney. Similarly, renal clearance may be defined as and calculate the ratio of Ae0-48 divided by AUC0-48.

 

164 Chapter 7

In essence for that particular drug product one could It can therefore be appreciated that the nonrenal
say that: clearance can be readily calculated when the drug

Ae product is administered intravenously, as ClNR =
0-inf Ae

R ~ 0-48
Cl =

AUC Cl − ClR. However, this calculation is not possible
0-inf AUC0-48

after extravascular administration if the exact rela-
At steady-state conditions it is easier to calculate tive bioavailability is not known or assumed as the
renal clearance, as at steady state all of the excreted exact renal clearance can be calculated (ClR), but
drug eliminated unchanged in the urine from one only the apparent clearance can (Cl/F). The non-
dose occurs over one dosing interval. Equation 7.38 renal clearance can only be estimated if the relative
therefore becomes: bioavailability is assumed. For example, if the rela-

Ae tive bioavailability is estimated to be hypothetically
τ (ss)

ClR(ss) = (7.39)
AUC between 75% and 100%, then the nonrenal clearance

τ (ss)
could be presented in the following manner:

where t is the dosing interval at which the drug is
administered until steady state (ss) conditions are Cl

= 10 L/h and ClR = 5 L/h
seen, and Ae F

t (ss) is the amount of drug excreted
unchanged in the urine during a dosing interval at Therefore,

steady state and AUCt (ss) is the area under the sys-
If F~100%, then ClNR = 5 L/h (eg, ClNR =

temic concentration−time curve over the same dos-
(Cl/F × 1) − ClR)

ing interval at steady state.
One important note is that by virtue of its method But if F ~ 75%, then ClNR = 2.5 L/h (eg, ClNR =

of calculation, the relative bioavailability (F) of the (Cl/F × 0.75) − ClR)

drug is not present in the renal clearance calculations
An alternative approach to obtaining Equation 7.38

while it always is for the total body clearance. So this
is to consider the mass balance of drug cleared by

means that if systemic concentrations and collected
the kidney and ultimately excreted in the urine. For

urinary excretion are only obtained after a drug prod-
any drug cleared through the kidney, the rate of the

uct is administered extravascularly, for example orally,
drug passing through kidney (via filtration, reabsorp-

then only an apparent clearance will be calculated
tion, and/or active secretion) must equal the rate of

(eg, Cl/F and not Cl) while the true renal clearance
drug excreted in the urine.

will be (eg, ClR and not ClR/F).
Rate of drug passing through kidney = rate of

Total clearance will be reported as an “apparent”
drug excreted:

clearance:

ClR × Cp = Qu × Cu (7.40)
Cl Dose

= (after single dose administration)
F AUC0-inf where ClR is renal clearance, Cp is plasma drug con-

centration, Qu is the rate of urine flow, and Cu is the
Cl Dose (at steady state during a dosing

= urine drug concentration. Rearrangement of
F AUC

τ (ss) interval) Equation 7.40 gives

While the renal clearance will not be “apparent”: Qu ×Cu Excretion rate
ClR = = (7.41)

ClR = Ae0-x/AUC0-x (after single dose adminis- Cp Cp

tration and where x is the maximum length of time
during which both urinary excreted amounts and the Because the excretion rate = QuCu = dDu/dt,

AUC can be observed; as mentioned earlier it should Equation 7.41 is the equivalent of Equation 7.38.

be a minimum of 3−4 terminal half-lives) Renal clearance can also be obtained using data
modeling and fitting with compartmental methods.

Ae
τ (ss)

Cl (at steady state during a dosing The most accurate method to obtain renal clearance
R =

AUC
τ (ss) interval) as well as total clearance with this method will be to

 

Drug Elimination, Clearance, and Renal Clearance 165

• The apparent total volume of distribution, Vss/F,
ka
T would be the addition of Vc/F to the Vp/Flag Cld/F

• The distribution (l1) and terminal elimination (lz)
Vc/F Vp/F rate constants would be:

Cl/F – Cl ° l1 = [((Cl + Cld)/Vc + Cld/Vp) + SQRT(((Cl +
R

ClR Cld)/Vc + Cld/Vp)
2 − 4 × Cl/Vc*Cld/Vp))]/2

° lz = [((Cl + Cld)/Vc + Cld/Vp) − SQRT(((Cl +
Urine Cld)/Vc + Cld/Vp)

2 − 4 × Cl/Vc*Cld/Vp))]/2

• The distribution and terminal elimination half-

FIGURE 78 Schematic description of a hypothetical lives would be:
two-compartment PK model in which plasma concentrations

° T1/2(l1) = 0.693/l1
and urinary excreted data would be simultaneously fitted and

° T1/2(lz) = 0.693/l
explained. z

model simultaneously observed systemic concentra- Comparison of Drug Excretion Methods
tions with observed excreted urinary amounts over a Renal clearance may be measured without regard to the
period of time that allows for robust estimates, so physiologic mechanisms involved in the process. From
ideally over 3−4 terminal half-lives or longer. As a physiologic viewpoint, however, renal clearance may
with any data modeling exercise, it is critical to use be considered the ratio of the sum of the glomerular
the simplest model that can explain all the data filtration and active secretion rates less the reabsorption
appropriately and to use a model that is identifiable. rate divided by the plasma drug concentration:

So using the example of a drug administered via Filtration rate + Secretion rate − Reabsorption rate
the oral route and where the plasma concentration C lR =

Cp
profile is fitted to a two-compartment model and
where the excreted urinary amounts are fitted simul- (7.42)

taneously, a typical model would look like Fig. 7-8,
The renal clearance of a drug is often related to the

where the “fitted” pharmacokinetic parameters by
renal glomerular filtration rate, GFR, when reabsorp-

the model would be:
tion is negligible and the drug is not actively secreted.

• T The renal clearance value for the drug is compared to
lag would be the time elapsed after dosing before

the beginning of the absorption process that of a standard reference, such as inulin, which is
• ka would be the first-order absorption rate constant cleared completely through the kidney by glomerular
• Vc/F would be the apparent central volume of filtration only. The clearance ratio, which is the ratio

distribution of drug clearance to inulin clearance, may give an
• (Cl/F − ClR) would be the apparent total clearance indication for the mechanism of renal excretion of the

that does not include the renal clearance drug (Table 7-4). However, further renal drug excre-
• Cl tion studies are necessary to confirm unambiguously

R would be the renal clearance
• Cld/F would be the apparent distributional clear- the mechanism of excretion.

ance between the central and peripheral volumes
of distribution Filtration Only

• Vp/F would be the apparent peripheral volume of If glomerular filtration is the sole process for drug
distribution excretion, the drug is not bound to plasma proteins,

and is not reabsorbed, then the amount of drug filtered
And where the subsequently “derived” or “calculated”

at any time (t) will always be C
pharmacokinetic parameters would be: p × GFR (Table 7-5).

Likewise, if the ClR of the drug is by glomerular filtra-
• The apparent total clearance, Cl/F, would be the tion only, as in the case of inulin, then ClR = GFR.

addition of ClR to the (Cl/F − ClR) Otherwise, ClR represents all the processes by which

 

166 Chapter 7

TABLE 74 Comparison of Clearance of a Total excretion
Sample Drug to Clearance of a Reference
Drug, Inulin

Probable Mechanism of Renal Filtration only

Clearance Ratio Excretion

Cl Drug is partially reabsorbed
drug <1 Active

Clinulin secretion only

Cl Drug is filtered only
drug

=1
Clinulin

Cl Drug is actively secreted
drug

>1
Clinulin Plasma level (Cp)

FIGURE 79 Excretion rate−plasma level curves for a drug

the drug is cleared through the kidney, including any that demonstrate active tubular secretion and a drug that is
secreted by glomerular filtration only.

combination of filtration, reabsorption, and active
secretion. Using compartmental PK even when lacking

any knowledge of GFR, active secretion, or the reab-
Filtration and Active Secretion

sorption process, modeling the data allows the pro-
For a drug that is primarily filtered and secreted, with cess of drug elimination to be described quantitatively.
negligible reabsorption, the overall excretion rate will If a change to a higher-order elimination rate process
exceed GFR (Table 7-4). At low drug plasma concen- occurs, then an additional process besides GFR may
trations, active secretion is not saturated, and the drug be involved. The compartmental analysis aids the
is excreted by filtration and active secretion. At high ultimate development of a model consistent with
concentrations, the percentage of drug excreted by physiologic functions of the body.
active secretion decreases due to saturation. Clearance We often relate creatinine clearance (CrCl) to the
decreases because excretion rate decreases (Fig. 7-9). overall clearance of a drug in clinical practice. This
Clearance decreases because the total excretion rate allows clinicians to adjust dosage of drugs depending
of the drug increases to the point where it is approxi- on a patient’s observed renal function. As the renal
mately equal to the filtration rate (Fig. 7-10). clearance is the summation of filtration, secretion, and

reabsorption, it can be simplified to:
TABLE 75 Urinary Drug Excretion Ratea

ClR = Slope × CrCl + Intercept (7.43)
Excretion Rate ( lg/min)

Time (Drug Filtered by Active secretion plus
(minutes) Cp ( lg/mL) GFR per Minute) passive ltration

0 (Cp)0 (Cp)0 × 125

1 (Cp)1 (Cp)1 × 125

2 (Cp)2 (Cp)2 × 125 Filtration
only

T (Cp)t (Cp)t = 125

aAssumes that the drug is excreted by filtration only, is not plasma Plasma drug concentration, Cp

protein bound, and that the GFR is 125 mL/min.
FIGURE 710 Graph representing the decline of renal

Note that the quantity of drug excreted per minute is always the
plasma concentration (Cp) multiplied by a constant (eg, 125 mL/min), clearance. As the drug plasma level increases to a concentra-

which in this case is also the renal clearance for the drug. The glomeru- tion that saturates the active tubular secretion, glomerular
lar filtration rate may be treated as a first-order process relating to Cp. filtration becomes the major component for renal clearance.

Excretion rate (dDu/dt)
Renal clearance, ClR

 

Drug Elimination, Clearance, and Renal Clearance 167

where the intercept reflects the reabsorption and
secretion processes, assuming that the CrCl only helps identify the mechanism of drug elimination. In

reflects GFR. this example, both drugs have the same clearance.

Because Cl = ClR + ClNR, then Basing the calculation on the elimination con-
cept and applying Equation 7.14, kR and lz are eas-

Cl = (Slope × CrCl + Intercept) + ClNR ily determined, resulting in an obvious difference

An assumption that is often made when adjusting in the elimination t1/2 between the two drugs—in

doses based on differing renal function is that spite of similar drug clearance.

decreasing renal function does not change the nonre- Cl

nal clearance (eg, hepatic and/or other clearances). kR(drug A) = k10(drug A) = λZ(drug A) =
Vss

This is a reasonable assumption to make until quite-
125

severe renal impairment is observed at which point = = 0.0125 min−1

10×1000
changes in protein binding capacity and affinity as
well as changes in enzymatic and transporter affinity Cl

kR(drug B) = k10(drug B) = λZ(drug B) =
and/or activity may be seen. Because ClNR and the Vss

intercept are both constants, then overall clearance 125
= = 0.00625 min−1

formula can therefore be simplified to: 20×1000

Cl = (Slope × CrCl) + Intercept2 (7.44) In spite of identical drug clearances, the lz for drug

The intercept2 is often simplified to ClNR, but in A is twice that of drug B. Drug A has an elimina-

reality if CrCl is assumed to only reflect GFR func- tion half-life of 55.44 minutes, while that of drug

tion, then it is really representative of the clearance B is 110.88 minutes—much longer because of the

from kidney secretion and reabsorption as well as bigger volume of distribution.

from nonrenal routes. 2. In a subject with a normal GFR (eg, a CrCl of
125 mL/min), the renal clearance of a drug is

EXAMPLES »» » 10 L/h while the nonrenal clearance is 5 L/h.
Assuming no significant secretion and reab-

1. Two drugs, A and B, are entirely eliminated sorption, how should we adjust the dosing regi-
through the kidney by glomerular filtration men of the drug if the renal function and the
(125 mL/min), with no reabsorption, and are GFR decrease in half (eg, CrCl = 62.5 mL/min)?
well described by a one-compartment model.

Solution
Drug A has half the distribution volume of drug
B, and the Vss of drug B is 20 L. What are the For a patient with “normal GFR”:

drug clearances for each drug using both the
Cl = ClR + ClNR, so Cl = 15 L/h

compartmental and physiologic approaches?
ClR = Slope × CrCl, therefore,

Solution slope = 10/(125 × 60/1000) = 1.33

Since glomerular filtration of the two drugs is the For a patient with a GFR that decreases in half:
same, and both drugs are not eliminated by other
means, clearance for both drugs depends on renal ClR = Slope × CrCl = 1.33 × (62.5 × 60/1000)

plasma flow and extraction by the kidney only. = 5 L/h

Basing the clearance calculation on the physi- Cl = ClR + ClNR = 5 + 5 = 10 L/h
ologic definition and using Equation 7.18 results in

The clearance therefore decreased by 33%. In
Q(Ca −Cv ) order to reach the same target exposure of the

Cl = =125 mL/min
Ca drug (AUCinf), the dose per day will need to be

Interestingly, known drug clearance tells little about decreased by 33% as Dose = Cl/AUCinf.
the dosing differences of the two drugs, although it

 

168 Chapter 7

Frequently Asked Question Slope = renal clearance A
(ClR)

»»What is the relationship between drug clearance and
creatinine clearance?

DETERMINATION OF RENAL
CLEARANCE

Graphical Methods B

Clearance is given by the slope of the curve obtained
Plasma level (Cp)by plotting the rate of drug excretion in urine

(dDu/dt) against Cp (Equation 7.45). For a drug that FIGURE 712 Rate of drug excretion versus concentra-
is excreted rapidly, dDu/dt is large, the slope is tion of drug in the plasma. Drug A has a higher clearance than

steeper, and clearance is greater (Fig. 7-11, line A). drug B, as shown by the slopes of line A and line B.

For a drug that is excreted slowly through the kidney,
the slope is smaller (Fig. 7-11, line B). estimated by the trapezoidal rule or by other measure-

From Equation 7.35, ment methods. The disadvantage of this method is

dD that if a data point is missing, the cumulative amount
u /dt

ClR =
C of drug excreted in the urine is difficult to obtain.

p
However, if the data are complete, then the determina-

Multiplying both sides by Cp gives tion of clearance is more accurate by this method.

ClR × C By plotting cumulative drug excreted in the urine
p = dDu/dt (7.45)

t t
from t1 to t2, (D

2
u ) versus (AUC) 2 , one ob

1 t tains an
t

By rearranging Equation 7.45 and integrating, one 1

equation similar to that presented previously:
obtains

[Du] [D t1−t2 = ClR × AUCt1−t2 (7.47)
u]0-t = ClR × AUC0-t (7.46)

The slope is equal to the renal clearance (Fig. 7-13).
A graph is then plotted of cumulative drug excreted in
the urine versus the area under the concentration−time

Midpoint Method
curve (Fig. 7-12). Renal clearance is obtained from
the slope of the curve. The area under the curve can be From Equation 7.35,

dDu /dt Cl
R =

Cp

Slope = renal clearance (ClR)

t (AUC)t2(AUC)0 t1

FIGURE 711 Cumulative drug excretion versus AUC. FIGURE 713 t
Drug excreted versus (AUC) 2 . The slope is

t1
The slope is equal to ClR. equal to ClR.

Drug excreted in urine (Du)

Rate of drug excretion (dDu/dt)
(Du)

t2
t1

 

Drug Elimination, Clearance, and Renal Clearance 169

which can be simplified to 1000 L. From the information given, find (a) the

X apparent clearance and the clearance, (b) the renal and
u(0-24) /Cp12

ClR = (7.48) nonrenal clearance, (c) the formation clearance of the
24

drug to the metabolite, and (d) if the drug undergoes
where Xu(0-24) is the 24-hour excreted urinary amount

another systemic metabolic or elimination route.
of the drug obtained by multiplying the collected
24-hour urine volume (Vu(0-24)) by the measured uri- Solution
nary concentration (Cu(0-24)) and Cp12 is the midpoint a. Apparent clearance and clearance:
plasma concentration of the drug measured at the

Cl V
midpoint of the collected interval, here at 12 hours. = K ×

F F
This equation is obviously not very robust as it is
on only one measured plasma concentration, Cl 0.693

based = ×1000 = 210 L/h
F 3.3

but it is often very useful in the clinic when very few
plasma concentrations of drugs can be collected and Cl

Cl = × F = 210 × 0.9 = 189 L/h
measured. The overall duration of urinary collection F

is typically 24 hours, but different collection intervals b. Renal and nonrenal clearance:

can obviously be used. Ae0-inf
ClR =

AUC0-inf

PRACTICE PROBLEM and,
DOSE 100

Consider a drug that is eliminated by first-order renal AUC0-inf = = = 0.4762 mg ⋅h/L
Cl /F 210

excretion and hepatic metabolism. The drug follows a
Therefore,

one-compartment model and is given in a single intra-
60

venous or oral dose (Fig. 7-14). Working with the ClR = = 126 L/h
odel presented, assume that a single dose (100 mg) 0.4762

m

of this drug is given orally. The drug has a 90% oral ClNR = 189 −126 = 63 L/h
bioavailability. The total amount of unchanged drug c. Formation clearance of the parent drug to the
recovered in the urine is 60 mg, and the total amount metabolite:
of metabolite recovered in the urine is 30 mg (expressed Ae0-inf 30
as milligram equivalents to the parent drug). According Clf = = = 63 L/h

AUC0-inf 0.4762
to the literature, the elimination half-life for this drug

d. Does the drug undergo other elimination or
is 3.3 hours and its apparent volume of distribution is

metabolic routes?
Cl

= ClR +ClNR = ClR + (Cl
F f +Clother )

Dose Clf
Vss (Cp) Vss(m) (Cm)

F Then, Clother = Cl − ClR − Clf = 189 − 126 − 63 =

Cl-ClR-Clf Cl 0 L/h
R

The drug does not undergo additional elimina-
tion or metabolic routes.

Urine Urine
(parent) (metabolite)

PRACTICE PROBLEM
FIGURE 714 Model of a drug eliminated by first-order
renal excretion and hepatic transformation into a metabolite also An antibiotic is given by IV bolus injection at a dose of
excreted in the urine. (ClR = renal clearance of parent drug, Clf = 500 mg. The drug follows a one-compartment model.
formation clearance of parent drug to metabolite, Cm = plasma The total volume of distribution was 21 L and the elimi-
concentration of the metabolite, Cp = plasma concentration of nation half-life was 6 hours. Urine was collected for
the parent drug, Vss = total volume of distribution of parent drug,
V 48 hours, and 400 mg of unchanged drug was recov-

ss(m) = apparent volume of distribution of metabolite,
(Cl − Cl − Cl of parent drug minus the renal and ered. What is the fraction of the dose excreted unchanged

R f) clearance
formation clearances, F = absolute bioavailability of parent drug.) in the urine? Calculate k, kR, Cl, ClR, and ClNR.

 

170 Chapter 7

Solution Cl = k × Vss = 0.1155 × 21 = 2.43 L/h

Since the elimination half-life, t1/2, for this drug is ClR = kR × Vss = 0.0924 × 21 = 1.94 L/h
6 hours, a urine collection for 48 hours represents

ClNR = Cl − ClR = 2.43 − 1.94 = 0.49 L/h
8 × t1/2, which allows for greater than 99% of the
drug to be eliminated from the body. The fraction of
drug excreted unchanged in the urine, fe, is obtained
by using Equation 7.37 and recalling that F = 1 for RELATIONSHIP OF CLEARANCE
drugs given by IV bolus injection. TO ELIMINATION HALF-LIFE AND

400 VOLUME OF DISTRIBUTION
fe = = 0.8

500 A common area of confusion for students is the
relationship between half-lives, volumes of distri-

Therefore, 80% of the bioavailable dose is excreted
bution, clearances, and noncompartmental-versus-

in the urine unchanged. Calculations for k, kR, ClT,
compartmental approaches.

ClR, and ClNR are given here:
As seen previously, clearances are always

related to a rate constant (k) and a volume of distri-
0.693

k = 0.1155 h−1 bution (Vd) but these will vary according to the math-
=

6 ematical model that describes appropriately the PK
kR = fe × k = 0.8 × 0.1155 = 0.0924 h−1 of the drug. Table 7-6 aims at reconciling this.

TABLE 76 Relationships between Clearance, Volumes of Distribution, and Half-Life

Appearance of
Cp Versus Time Compartmental Method Noncompartmental Method

Monoexponen- Model after IV administration: Single dose IV administration:
tial decline

Cl = k10 × Vc

AUC
V 0-t typically calculated with linear or mixed

ss = Vc as there is only one compartment
linear/log-linear trapezoidal rule

lz = k10 as there is only one compartment

Cl = ClR + ClNR
C

Cl t is the last detectable concentration time point.
R = kR × Vc

T1/2 = 0.693/lz
lz is the negative slope using linear regression of

Biexponential Model after IV administration: the terminal elimination log-linear phase of the
decline

Cl = k10 × V concentration-versus-time profile.
c

Vp = k12 × Vc/k21

Vss = Vc + V Cl = DOSE/AUC
p 0-inf

l1 = [((Cl + Cl AUC
d)/Vc + Cld/Vp) + SQRT(((Cl + Cld)/Vc 0-inf = AUC0-t + Ct/lz

+ Cld/Vp)2 − 4 × Cl/Vc∗Cld/Vp))]/2 MRT = AUMC0-inf/AUC0-inf − (Duration of infusion/2)
lz = [((Cl + Cld)/Vc + Cld/Vp) − SQRT(((Cl+Cld)/Vc Vss = Cl × MRT

+ Cld/Vp)2 − 4 × Cl/Vc∗Cld/Vp))]/2
T1/2 (elimination) = 0.693/lz

T1/2 (distribution) = 0.693/l1

T1/2 (elimination) = 0.693/lz

 

Drug Elimination, Clearance, and Renal Clearance 171

CHAPTER SUMMARY
Clearance refers to the irreversible removal of drug the clearance will be the product of the terminal
from the systemic circulation of the body by all elimination rate constant and the total volume of
routes of elimination. Clearance may be defined as distribution. Clearance is therefore inversely related
the volume of fluid removed from the drug per unit to the elimination half-life of a drug. Organ clear-
of time. The clearance of a drug is a very clinically ances are additive, except for lung, and so the total
useful parameter as it is related to the systemic expo- body clearance is often described in terms of renal
sure of a drug, which dictates efficacy and safety, and nonrenal clearance. The renal clearance is depen-
and its administered dose. Clearance is a constant dent on renal blood flow, glomerular filtration, drug
when the PK behavior of a drug is linear in terms of secretion, and reabsorption. Reabsorption of drugs is
time and dose. Clearance can be calculated by many often a passive process and the extent of reabsorp-
different methods, including noncompartmental, tion of weak acid or weak base drugs is influenced
compartmental, and physiological. Assuming a spe- by the pH of the urine and the degree of ionization
cific compartment model, clearance will be the prod- of the drug. In addition, an increase in blood flow to
uct of an elimination rate constant and a volume of the kidney, which may be due to diuretic therapy or
distribution. In the simplest case, a one-compartment large beer consumption, decreases the extent of drug
model for drugs whose concentration−time profile reabsorption in the kidney and increases the rate of
decreases according to a monoexponential decline, drug excreted in the urine.

LEARNING QUESTIONS
1. Theophylline is effective in the treatment of b. What is the renal clearance for this drug?

bronchitis at a blood level of 10−20 mg/mL. At c. What is the probable mechanism for renal
therapeutic range, theophylline follows linear clearance of this drug?
pharmacokinetics. The average t1/2 is 3.4 hours, 3. A drug with an elimination half-life of 1 hour
and the range is 1.8−6.8 hours. The average was given to a male patient (80 kg) by intrave-
volume of distribution is 30 L. nous infusion at a rate of 300 mg/h. At 7 hours
a. What are the average upper and lower after infusion, the plasma drug concentration

clearance limits for theophylline assuming a was 11 mg/mL.
one-compartment model? a. What is the total body clearance for this drug?

b. The renal clearance of theophylline is 0.36 L/h. b. What is the apparent Vss for this drug assum-
What are the kNR and kR? ing a one-compartment model?

2. A single 250-mg oral dose of an antibiotic c. If the drug is not metabolized and is elimi-
is given to a young man (age 32 years, nated only by renal excretion, what is the
creatinine clearance CrCl = 122 mL/min, renal clearance of this drug?
ABW = 78 kg). From the literature, the d. What would then be the probable mecha-
drug is known to have an apparent Vss equal nism for renal clearance of this drug?
to 21% of body weight and an elimination 4. In order to rapidly estimate the renal clearance
half-life of 2 hours. The dose is normally of a drug in a patient, a 2-hour postdose urine
90% bioavailable and is not bound signifi- sample was collected and found to contain
cantly to plasma proteins. Urinary excretion 200 mg of drug. A midpoint plasma sample
of the unchanged drug is equal to 70% of the was taken (1 hour postdose) and the drug con-
bioavailable dose. centration in plasma was found to be 2.5 mg/L.
a. What is the total body clearance for this Estimate the renal clearance for this drug in

drug assuming a one-compartment model? this patient.

 

172 Chapter 7

5. According to the manufacturer, after the the drug using urinary data. (b) Determine the
antibiotic cephradine (Velosef), given by IV clearance using the noncompartmental method.
infusion at a rate of 5.3 mg/kg/h to 9 adult (c) Is there any nonrenal clearance of the drug in
male volunteers (average weight, 71.7 kg), a this patient? What would be the nonrenal clear-
steady-state serum concentration of 17 μg/mL ance, if any? How would you determine clear-
was measured. Calculate the average clearance ance using a compartmental approach and com-
for this drug in adults. pare that with the noncompartmental method?

6. Cephradine is completely excreted unchanged 9. Ciprofloxacin hydrochloride (Cipro) is a
in the urine, and studies have shown that pro- fluoroquinolone antibacterial drug used to
benecid given concurrently causes elevation of treat urinary tract infections. Ciprofloxacin
the serum cephradine concentration. What is contains several pKas (basic amine and car-
the probable mechanism for the interaction of boxylic group) and may be considered a weak
probenecid with cephradine? acid and eliminated primarily by renal excre-

7. When deciding on a dosing regimen of a drug tion, although about 15% of a drug dose is
to administer to a patient, what information can metabolized. The serum elimination half-life in
be obtained from knowing only the elimination subjects with normal renal function is approxi-
half life? The clearance? mately 4 hours. The renal clearance of cip-

8. A patient was given 2500 mg of a drug by rofloxacin is approximately 300 mL/min. By
IV bolus dose, and periodic urinary data were what processes of renal excretion would you
collected. (a) Determine the renal clearance of conclude that ciprofloxacin is excreted? Why?

ANSWERS

Frequently Asked Questions saturated, then the clearance cannot be described
by a constant.

Why is clearance a useful pharmacokinetic parameter? Clearance is related to the administered dose

• Clearance is very useful clinically as it is the and the overall exposure of a drug as per the formula

only PK parameter that relates to dose and the Cl/F = DOSE/AUC0-inf. As the exposure of a drug

overall exposure of a drug, for example, Cl/F correlates with its efficacy and toxicity, clearance is
=

DOSE/AUC a much more useful parameter clinically than the
0-inf.

terminal half-life as it will directly dictate what dose
Which renal elimination processes are influenced by to administer to a patient in order to reach a cer-
protein binding? tain systemic exposure. Although it will not dictate

• what dose to administer, the terminal half-life will
Only the free drug can be filtered by the kidney, so

be important in deciding how often to administer a
protein binding influences the filtration of drugs,

drug. Both parameters are therefore important.
but it has no significant influences on secretion
and reabsorption.

What is the relationship between drug clearance and
Is clearance a first-order process? Is clearance a creatinine clearance?
better parameter to describe drug elimination and

• The Cl of a drug is composed of the renal (ClR)
exposure than half-life? Why is it necessary to use

and of the nonrenal (ClNR) components. The ClR
both parameters in the literature?

is composed of filtration, reabsorption, and secre-
• The clearance of a drug is a constant only if the tion components. Creatinine is mostly filtrated but

drug exhibits linear pharmacokinetic characteris- also secreted, so the creatinine clearance (CrCl),
tics. If the clearance changes with drug concen- whether estimated by the Cockcroft and Gault
trations, for example, when metabolism becomes formula or calculated by collecting its urinary

 

Drug Elimination, Clearance, and Renal Clearance 173

excretion, is used in clinical practice to give us an R0 300
indication of the filtration capacity (eg, GFR) of Cl = = = 27.27 mg/L

Css 11
the kidney in a given patient.

Because Cl = ClR + ClNR, and because the Cl = λz ×Vss
CrCl directly correlates with ClR, the clearance
of a drug can often be expressed as Cl = (Slope × b. Cl 27.27

Vss = = = 39.354 L
CrCl) + Intercept, where the intercept can often be λ 0.693/1

z

assumed to mostly reflect the nonrenal clearance
component. c. ClR ~ Cl = 27.27 L/h

d. ClR = 27.27 × 1000/60 = 454.54 mL/min
Learning Questions

The binding to plasma protein is unknown

1. a. Cl = k × V, where V = 30 L and k = 0.693/T (eg, only free drug is filtered), the renal
1/2

function of the patient is unknown, and the
Average Cl = 30 × 0.693/3.4 = 6.11 L/h molecular weight of the drug is unknown
Upper Cl = 30 × 0.693/1.8 = 11.55 L/h (drugs with large molecular weight are not

Lower Cl = 30 × 0.693/6.8 = 3.06 L/h filtered). So at this point, this drug is likely
filtered but we cannot be sure based on the

b. ClR = kR × V limited information available.
Because the ClR > GFR, we know for

kR = ClR/V = 0.36/30 = 0.36 L/h
sure, though, that the drug is actively

Cl = ClR + ClNR secreted. It could also be reabsorbed, but
ClNR = Cl − ClR = 6.11 × 0.36 = 5.75 L/h we cannot be sure based on the information

available.
kNR = ClNR/V = 5.75/30 = 0.192 h−1

4. The renal clearance can be calculated using the
2. a. Cl = lz × Vss as the drug PK is well described midpoint clearance formula,

by a one-compartment model
Curine × Volume urine

lz = 0.693/2 = 0.3465 h−1 ClR =
Cp(midpoint)

Vss = 0.21 × 78 = 16.38 L
where (Curine × Volume urine) = 200 mg.

Cl = 0.3465 × 16.38 = 5.68 L/h
200

b. fe = 70% ClR = = 80 L per 2 hours, or 40 L/h
2.5

ClR = fe × Cl = 0.7 × 5.68 = 3.97 L/h
R

C 0
5. ss =

c. ClR = 3.97 L/h = 66.2 mL/min Cl
This man has a CrCl of 122 mL/min. Because

R0 5.3× 71.7
the ClR is less than the CrCl, and because the Cl = = = 22.4 L/h

Css 17
drug is not bound to plasma protein, then we
can expect that the drug is filtered but also

6. Probenecid is likely decreasing the renal secre-
reabsorbed with or without being secreted.

tion of cephradine.
3. a. During intravenous infusion, the drug levels

will reach more than 99% of the plasma steady- 7. Cl/F = DOSE/AUC0-inf, so if the target AUC0-inf
state concentration after 7 half-lives of the is known in order to achieve a desired level of
drug, 7 hours in this case. So we can assume efficacy without significant toxicity, then the
that steady-state conditions are reached. At dose to administer per day to a patient will be
steady state, dictated by its Cl/F value.

 

174 Chapter 7

For example, if the targeted AUC per day
is 100 mg/L and the Cl/F in a patient is 400 y = –1.4824 + 1493.4x

R2 = 1.000
1 L/h, then the drug has to be adminis-
tered at a dose of 100 mg per day. 300

The elimination half-life will not help us under-
stand what dose per day to administer, but will help

200
us decide how frequently to administer the drug.

For example, if the minimum level of effi-
cacy of the previous drug is seen at 1 mg/L, 100

if its Cmax at steady state after 100-mg dose
per day is 4 mg/L, then the drug can be 0
given every 2 half-lives in order to reach 0 100 200 300

a Cmax of 4 and a minimum concentration Cp between time points (average)

of 1 mg/L at steady state. If the half-life in FIGURE A1
a patient is 12 hours, then the drug can be
administered as 100 mg every 24 hours. concentration curve [AUC] must be calculated

and summed. The tailpiece is extrapolated
8. because the data are not taken to the end. A plot

of log Cp versus t (Fig. A-2) yields a slope of
Plasma Urinary Urinary Urinary

k = 0.23 h−1. The tailpiece of area is extrapo-
Time Concentration Volume Concentration
(hours) (lg/mL) (mL) (lg/mL) lated using the last data point divided by k or

31.55/0.23 = 137.17 mg/mL/h.
0 250.00 100.00 0.00

1 198.63 125.00 2880.00 1000

2 157.82 140.00 1901.20

3 125.39 100.00 2114.80

4 99.63 80.00 2100.35

5 79.16 250.00 534.01 100 k = –2.3x slope

6 62.89 170.00 623.96

7 49.97 160.00 526.74

8 39.70 90.00 744.03

9 31.55 400.00 133.01 10
0 2 4 6 8 10 12

10 25.06 240.00 176.13 Time

From the data, determine urinary rate of drug FIGURE A2
excretion per time period by multiplying
urinary volume by the urinary concentration Subtotal area (0−9 h) 953.97

for each point. Average Cp for each period by
Tailpiece (9−∞ h) 137.17

taking the mean of two consecutive points (see
table). Plot dDu/dt versus Cp to determine renal Total area (0−∞) 1091.14

clearance from the slope. The renal clearance
FD 2,500,000

from the slope is 1493.4 mL/h (Fig. A-1). Total clearance 0
= ClT = =

[AUC]∞ 1091.14
To determine the total body clearance by 0

the area method, the area under the plasma
= 2291.2 mL/h

Cp dDu/dt (thousands)

 

Drug Elimination, Clearance, and Renal Clearance 175

Time Plasma Concentration Urinary Urinary Concentration Urinary Rate,
(hours) (mg/mL) Volume (mL) (lg/mL) dDu/dt (lg/h) Average Cp

0 250.00 100.00 0 0

1 198.63 125.00 2680.00 334,999.56 224.32

2 157.82 140.00 1901.20 266,168.41 178.23

3 125.39 100.00 2114.80 211,479.74 141.61

4 99.63 80.00 2100.35 168,027.76 112.51

5 79.16 250.00 534.01 133,503.70 89.39

6 62.89 170.00 623.96 106,073.18 71.03

7 49.97 160.00 526.74 84,278.70 56.43

8 39.70 90.00 744.03 66,962.26 44.84

9 31.55 400.00 133.01 53,203.77 35.63

Because total body clearance is much larger from the graph. VD is 10 L and k is 0.23 h−1. Total
than renal clearance, the drug is probably also clearance is 2300 mL/min (a slightly different
excreted by a nonrenal route. value when compared with the area method).

Nonrenal clearance = 2291.2−1493.4 9. The ClR of Ciprofloxacin is larger than the GFR
(eg, 300 mL/min) and so the drug is at least

= 797.8 mL/h secreted in addition to be filtered. Weak acids
The easiest way to determine clearance by a are known to be secreted.

compartmental approach is to estimate k and VD

REFERENCES
Guyton AC: Textbook of Medical Physiology, 8th ed. Philadelphia, West JB (ed): Best and Taylor’s Physiological Basis of Medical

Saunders, 1991. Practice, 11th ed. Baltimore, Williams & Wilkins, 1985.
Levine RR: Pharmacology: Drug Actions and Reactions, 4th ed.

Boston, Little, Brown, 1990.

BIBLIOGRAPHY
Benet LZ: Clearance (née Rowland) concepts: A downdate and Smith H: The Kidney: Structure and Function in Health and Disease.

an update. J Pharmacokinet Pharmacodyn 37:529−539, 2010. New York, Oxford University Press, 1951.
Cafruny EJ: Renal tubular handling of drugs. Am J Med 62: Thomson P, Melmon K, Richardson J, et al: Lidocaine pharma-

490−496, 1977. cokinetics in advanced heart failure, liver disease and renal
Hewitt WR, Hook JB: The renal excretion of drugs. In Bridges VW, failure in humans. Ann Intern Med 78:499−508, 1973.

Chasseaud LF (eds.), Progress in Drug Metabolism, vol. 7. Tucker GT: Measurement of the renal clearance of drugs. Br J
New York, Wiley, 1983, chap 1. Clin Pharm 12:761−770, 1981.

Holford N, Heo YA, Anderson B. A pharmacokinetic standard for Weiner IM, Mudge GH: Renal tubular mechanisms for excretion
babies and adults. J Pharm Sci 102(9):2941−2952, 2013. and organic acids and bases. Am J Med 36:743−762, 1964.

Renkin EM, Robinson RR: Glomerular filtration. N Engl J Med Wilkinson GR: Clearance approaches in pharmacology. Pharmacol
290:785−792, 1974. Rev 39:1−47, 1987.

Rowland M, Benet LZ, Graham GG: Clearance concepts in phar-
macokinetics. J Pharm Biopharm 1:123−136, 1973.

 

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Pharmacokinetics of

8 Oral Absorption
John Z. Duan

Chapter Objectives INTRODUCTION
»» Define oral drug absorption Extravascular delivery routes, particularly oral dosing, are impor-

and describe the absorption tant and popular means of drug administration. Unlike intravenous
process. administration, in which the drug is injected directly into the gen-

»» Introduce two general eral circulation (see Chapters 4–7), pharmacokinetic models after
approaches used for studying extravascular drug administration must consider drug absorption
absorption kinetics and their from the site of administration, for example, the gut, the lung, etc.
similarities and differences. The aim of this chapter is to study the kinetics of absorption.

Before delving into the details, it is important to clarify the defini-
»» Understand the basic principles

tion of absorption.
for physiologically based

There are three different definitions of absorption in exis-
absorption kinetics.

tence. Traditionally, absorption occurs when drug reaches the
»» Describe the oral one- systemic circulation, or sometimes when it reaches the portal vein

compartment model and blood stream. In recent years, a new definition is presented, in which
explain how this model drug is assumed to be absorbed when it leaves the lumen and
simulates drug absorption from crosses the apical membrane of the enterocytes lining the intestine
the gastrointestinal tract. (GastroPlus manual). It is important to distinguish among these

»» Calculate the pharmacokinetic definitions when the kinetics study is performed, especially during

parameters of a drug comparisons of the study results.

that follows the oral one- Drug absorption from the gastrointestinal (GI) tract or any

compartment model. other extravascular site is dependent on (1) the physicochemical
properties of the drug and the environment in the small intestine,

»» Calculate the fraction of drug (2) the dosage form used, and (3) the anatomy and physiology of
absorbed in a one-compartment the absorption site, such as surface area of the GI tract, stomach-
model using the Wagner–Nelson emptying rate, GI mobility, and blood flow to the absorption site.
method. Extravascular drug delivery is further complicated by variables at

»» Calculate the fraction of drug the absorption site, including possible drug degradation and sig-

absorbed in a two-compartment nificant inter- and intrapatient differences in the rate and extent

model using the Loo–Riegelman of absorption. The variability in drug absorption can be mini-

method. mized to some extent by proper biopharmaceutical design of the
dosage form to provide predictable and reliable drug therapy

»» Describe the conditions that
(Chapters 15–18). Although this chapter will focus primarily on

may lead to flip-flop of ka and k
oral dosing, the concepts discussed here may be easily extrapo-

during pharmacokinetics (PK)
lated to other extravascular routes.

data analysis.
There are generally two methodologies to study the kinetics of

absorption. Pharmacokinetic models can be built based mainly on

177

 

178 Chapter 8

»» Describe the model parameters the observed clinical data (“top-down” approach) or based on the
that form the foundation of drug broader understanding of the human body and its mechanisms
absorption and bioavailability of (“bottom-up” approach) (Jamei et al, 2009). A top-down model is
oral dosage forms. often specified with the assistance of “black boxes” (such as the

compartment model). In a bottom-up approach the elements of the
»» Discuss how ka and k may

system are first specified in great detail. These elements are then
influence Cmax, tmax, and AUC

linked together to form larger subsystems, which in turn are
and how changes in these

linked, sometimes in many levels, until a complete top-level sys-
parameters may affect drug

tem is formed. The goals of the two approaches are the same: to
safety in a clinical situation.

make physiologically plausible predictions.
This chapter will introduce the basic concept of the physiolog-

ically based absorption kinetics (the bottom-up approach) with
some examples followed by the detailed explanation of the tradi-
tional top-down approach, and finally, the combination of the two
approaches is proposed.

BASIC PRINCIPLES OF PHYSIOLOGICALLY
BASED ABSORPTION KINETICS
(BOTTOM-UP APPROACH)
The physiologically based absorption models provide a quantita-
tive mechanistic framework by which scaled drug-specific param-
eters can be used to predict the plasma and, importantly, tissue
concentration–time profiles of drugs following oral administra-
tion. The main advantage of physiology-based pharmacokinetic
(PBPK) models is that they can be used to extrapolate outside the
studied population and experimental conditions. For example,
PBPK can be used to extrapolate the absorption process in healthy
volunteers to that in a disease population if the relevant physiologi-
cal properties of the target population are available. The trade-off for
this advantage is a complex system of differential equations with a
considerable number of model parameters. When these parameters
cannot be informed from in vitro or in silico1 experiments, PBPK
models are usually optimized with respect to observed clinical data.
Parameter estimation in complex models is a challenging task asso-
ciated with many methodological issues.

Historically, PBPK approach stemmed from a natural thinking
for elucidating the kinetics of absorption. The first pharmacoki-
netic model described in the scientific literature was in fact a
PBPK model (Teorell, 1937). However, this model led to great
difficulty in computations due to lack of computers. Additionally,
the in vitro science was not advanced enough to obtain the neces-
sary key information. Therefore, the lack of in vitro and in silico
techniques hindered the development of PBPK approach for many

1In silico refers to computer-based models.

 

Pharmacokinetics of Oral Absorption 179

years. Recently, PBPK development has been accel- two directions, indicating the drug transit among
erated mainly due to the explosion of computer sci- these compartments. Each transit process, repre-
ence and the increasing availability of in vitro sented by an arrow in Fig. 8-1, can be expressed by
systems that act as surrogates for in vivo reactions a differential equation. The model equations follow
relevant to absorption. the principles of mass transport, fluid dynamics, and

Parameter estimation in PBPK models is chal- biochemistry in order to simulate the fate of a sub-
lenging because of the large number of parameters stance in the body. Most of the equations involve
involved and the relative small amount of observed linear kinetics. For example, for non-eliminating
data usually available. An absorption model consists tissues, the following principles are followed: the
of a set of values for the absorption scale factors, “rate of change of drug in the tissue” is equal to the
transit times, pH assignments, compartment geome- “rate in” (QT · CA) minus the “rate out” (QT · CvT) as
tries (individual compartment radii and lengths, and shown in Equation 8.1.
volume), and pharmacokinetic parameters that pro-
vide the best predictions for a compound in human. dC

V T
 = Q C −Q C (8.1)

For example, an advanced absorption transit model T dt T A T vT

developed in GastroPlus™2 contains nine compart-
ments, which represent the five segments of the GI where Q = blood flow (L/h), C = concentration
tract—stomach, duodenum, jejunum, ileum, and (mg/L), V = volume (L), T = tissues, A = arterial, v =
colon. The fluid content, carrying dissolved and venous, CvT = CT/(Kp/B:P), B:P = blood-to-plasma
undissolved compound, passes from one compart- ratio. On the other hand, Michaelis–Menten nonlin-
ment to the next, simulating the action of peristaltic ear kinetics is used to describe saturable metabolism
motion. Within each compartment, the dynamic and carrier-mediated transport.
interconversion between dissolved and undissolved The PBPK approach can specifically define the
compound is modeled. Dissolved compound can be absorption for a specific drug product. Figure 8-2
absorbed across the GI tract epithelium. The volume shows the simulation results using PBPK software
of each compartment, which represents the fluid GastroPlus for several drugs with different physico-
content, is modeled dynamically, simulating the fol- chemical properties. The first column lists the drug
lowing processes: names and the second column is the pKa of the com-

pound. The solubility factor (Sol Factor) is the ratio
• Transit of the fluid with characteristic rate con-

of the solubility of the completely ionized form of an
stants through each compartment

ionizable group to the completely unionized form.
• Gastric secretion into the stomach, and biliary and

The figure also lists the solubility and logD pH pro-
pancreatic secretions into the duodenum

files for each drug (two green vertical lines indicate
• Absorption of uid from duodenum, jejunum, ileum,

pH 1.2 and 7.5, respectively). Notice that the color of
and large intestine

the cells for dose number (Dose No), absorption
Figure 8-1 shows the graphic representation of number (Abs No), and dissolution number (Dis No)

this model. As seen, each of the nine compartments changes depending on the physicochemical and bio-
is divided into four subcompartments: unreleased, pharmaceutical properties of the drug selected. The
undissolved, dissolved, and enterocyte. colors approximate the four Biopharmaceutical

In the figure, the compartments and subcom- Classification System (BCS) categories. All green
partments in GI tract are connected to each other by indicates high permeability, high solubility, and
arrows. These arrows are of either one direction or rapid dissolution (BCS Class I). Red absorption

number and green dose number may indicate low
permeability and high solubility (BCS Class III). All

2GastroPlus is a mechanistically based simulation software package
red may indicate low permeability and low solubility

that simulates absorption, pharmacokinetics, and pharmacodynamics
in human and animals (http://www.simulations-plus.com/Products (BCS Class VI). These color systems are not perfect
.aspx?GastroPlus&grpID=3&cID=16&pID=11). cutoffs for the BCS, but they represent most drugs.

 

ch
Stoma

Asc C
olon

Cecu
m

Ileu
m 3

Ileu
m 2

Ileu
m 1

Jeju
num 2

Jeju
num 1

Duodenum

Hepatic Hepatic Unreleased
Vein Artery

Liver Undissolved

Dissolved
Portal
Vein

Enterocyte
GI Tract

Venous Arterial
Portal Vein

Flow Flow

FIGURE 81 A graphic representation of drug absorption from the GI tract.

180

 

Solubility pH LogD pH Pro„le Absorption & Dissolution Compartmental
absorption Plasma Concentration

pro„le

Metoprolol
1.0 Metoprolol

AmtDiss AmtPV
150

0.5 AmtAbs Total SC Metoprolol
150 93.0% 93% 0.25

150
0.0 0.20

100 100
100 0.15

36.7%
–0.5 50 20.4% 17.6% 0.10

50 –1.0 50 8.9% 4.7%
0% 2.6% 0.4% 1.7% 0.05

0.00
–1.5 0

0 0 5 10 15 20 25
5 10 15 20

0 2 4 6 8 10 0 2 4 6 8 10 Time (h)
pH Time (h)

pH

3 Ketoprofen
Ketoprofen

AmtDiss AmtPV
60 AmtAbs Total SC Ketoprofen

2.5
2 50 50 99.9% 99.9%

2.0
40 40

40
30 55.7%

30 1.5
1 20 31.6% 1.0

20
20 10 6.8%

0% 1.6% 0.4% 0.2% 2.7% 0.8% 0.5
10

0 0.0
0

0 0 5 10 15 20 25
5 10 15 20

0 2 4 6 8 10 0 2 4 6 8 10 Time (h)
Time (h)

pH pH

Carbamazepine
1.5 Carbamazepine

20 AmtDiss AmtPV
AmtAbs Total SC 99.5% Carbamazepine

200 99.3% 1.5
15 200

1.0 150
150 1.0

10 100
100 31.6%

50 21.4%
0.5 10.5% 12.7% 0.5

5 7.1% 4% 8.6%
50 0% 3.6%

0.0
0

0
0.0

0 2 4 6 8 10 0 2 4 6 8 10 5 10 15 20 0 5 10 15 20 25

pH pH Time (h) Time (h)

–0.5 Atenolol Atenolol
150 AmtDiss AmtPV

Atenolol
–1.0 AmtAbs Total SC

100 40 37.9% 0.3
37.9%

100 80 30 0.2
–1.5

60 20
11.9%

40 10 8.5% 0.1
3.8% 6%

50 –2.0 4.1%
0% 2.8% 0.1% 0.6%

20
0.0

0
–2.5 0 5 10 15 20 25

5 10 15 20
0 2 4 6 8 10 0 2 4 6 8 10 Time (h)

pH pH Time (h)

Furosemide Furosemide
1.5

4 AmtDiss AmtPV Furosemide
1.0 AmtAbs Total SC 60

3 100 .1%
50 52.1% 52

0.5 80 40 1.0
2 60 30

0.0
20 16.4%

40 11.3%
10 7.7% 0.5

4% 5.1%
1 –0.5 3.3% 3.2%

20 0% 1%

–1.0 0 0.0
0

5 10 15 20 0 5 10 15 20 25
0 2 4 6 8 10 0 2 4 6 8 10 Time (h) Time (h)

pH pH

FIGURE 82 The modeling results for several drugs using GastroPlus software.

Furosemide Atenolol Carbamazepine Ketoprofen Metoprolol tartrate Drug

–0.59, 3.88, 9.37 9.33 11.83 4.39 9.39 pka
36.7 16.2 503 35.3 35.9 Sol Factor

64.9565 0.0025 6.8376 0.9792 0.0064 Dose No
0.261 5.761 × 103 5.299 9.075 6.257 × 103 Dis No
0.596 0.367 8.546 17.29 2.663 Abs No

Solubility (mg/mL) Solubility (mg/mL) Solubility (mg/mL) Solubility (mg/mL) Solubility (mg/mL)

logD logD logD logD logD

Mass (mg) Mass (mg) Mass (mg) Mass (mg) Mass (mg)

Amount (mg) Amount (mg) Amount (mg) Amount (mg) Amount (mg)

Stomach Stomach Stomach Stomach Stomach

Duodenum Duodenum Duodenum Duodenum Duodenum

Jejunum 1 Jejunum 1 Jejunum 1 Jejunum 1 Jejunum 1

Jejunum 2 Jejunum 2 Jejunum 2 Jejunum 2 Jejunum 2

IIeum 1 IIeum 1 IIeum 1 IIeum 1 IIeum 1

IIeum 2 IIeum 2 IIeum 2 IIeum 2 IIeum 2

IIeum 3 IIeum 3 IIeum 3 IIeum 3 IIeum 3

Caecum Caecum Caecum Caecum Caecum

Asc colon Asc colon Asc colon Asc colon Asc colon

AmtAbs AmtAbs AmtAbs AmtAbs AmtAbs

Concentration (µg/mL) Concentration (µg/mL) Concentration (µg/mL) Concentration (µg/mL) Concentration (µg/mL)

181

 

182 Chapter 8

Based on the in vitro properties and assuming a ABSOROPTION KINETICS
set of general physiological conditions, the absorp-

(THE TOP-DOWN APPROACH)
tion profiles, the absorption amount in each of the
nine compartments, and the plasma concentration The top-down approach is a traditional methodology
profiles are predicted in the last three columns, to study the kinetics of drug absorption. With the
respectively. In the “Absorption & Dissolution” col- advances of statistical methods and computer sci-
umn, the profiles for the total dissolved (red), the ence, many software packages are available to calcu-
absorbed (cyan, the absorption is defined as the drug late the pharmacokinetic parameters. The following
leaves the lumen and crosses the apical membrane of sections provide the basic concepts and rationales.
the enterocytes lining the intestine), the cumulative
amount entering portal vein (blue), and the cumula-
tive amount entering systemic circulation (green) are PHARMACOKINETICS
characterized. These profiles along with the informa-

OF DRUG ABSORPTION
tion about the amount absorbed in each compartment
give the plasma concentration profiles as shown in In pharmacokinetics, the overall rate of drug absorp-
the last column. As seen, due to the physicochemical tion may be described as either a first-order or a zero-
property differences, the rate and the extent of order input process. Most pharmacokinetic models
absorption vary among the drugs listed. assume first-order absorption unless an assumption

Drug absorption from the gastrointestinal tract of zero-order absorption improves the model signifi-
is a highly complex process dependent upon numer- cantly or has been verified experimentally.
ous factors. In addition to the physicochemical The rate of change in the amount of drug in the
properties of the drug as shown in Fig. 8-2 (with body, dDB/dt, is dependent on the relative rates of
limited extents), characteristics of the formulation drug absorption and elimination (Fig. 8-3). The net
and interplay with the underlying physiological rate of drug accumulation in the body at any time is
properties of the GI tract play important roles. In equal to the rate of drug absorption less the rate of
GastroPlus, the formulation types that can be drug elimination, regardless of whether absorption
selected include both immediate release (IR) formu- rate is zero-order or first-order.
lations (solution, suspension, tablet, and capsule)
and controlled release (CR) formulations (enteric- dD dD dD

B GI E
= − (8.2)

coated or other form of delayed release [DR]). For dt dt dt
CR, release of either dissolved material (drug in

where DGI is the amount of drug in the gastrointestinal
solution) or undissolved material (solid particles,

tract and DE is the amount of drug eliminated. A
which then dissolve according to the selected dis-

plasma level–time curve showing drug absorption and
solution model) can be evoked.

elimination rate processes is given in Fig. 8-4. During
In addition to GastroPlus, there are several other

the absorption phase of a plasma level–time curve
physiologically based softwares available for studying

(Fig. 8-4), the rate of drug absorption3 is greater than
absorption kinetics, such as SimCyp (http://www
.simcyp.com/) and PK-Sim (http://www.systems
-biology.com/products/pk-sim.html).

Absorption Elimination
The major advantage of the PBPK approach is D D V D

GI B D E

that if adequate information of physicochemical
properties of a drug is available, a reasonable predic-

FIGURE 83 Model of drug absorption and elimination.
tion for the performance of the drug product can be
made with certain assumptions according to previ-
ous experience. With little or no human PK data 3The rate of drug absorption is dictated by the product of the drug
generated, the predictions would be very valuable in the gastrointestinal tract, DGI times the rst-order absorption
for further drug development. rate constant, ka.

 

Pharmacokinetics of Oral Absorption 183

or dDGI/dt = 0. The plasma level–time curve (now the
elimination phase) then represents only the elimina-
tion of drug from the body, usually a first-order pro-

Postabsorption
phase cess. Therefore, during the elimination phase the rate

of change in the amount of drug in the body is
described as a first-order process:

Elimination dDB
phase = −kD

dt B (8.6)
Absorption
phase

where k is the first-order elimination rate constant.

Clinical Application
Peak

concentration, Cmax Time Manini et al (2005) reported a case of adverse drug
reaction in a previously healthy young man who

FIGURE 84 Plasma level–time curve for a drug given in
ingested a recommended dose of an over-the-counter

a single oral dose. The drug absorption and elimination phases
of the curve are shown. (OTC) cold remedy containing pseudoephedrine.

Forty-five minutes later, he had an acute myocardial
infarction (MI). Elevations of cardiac-specific creatinine

the rate of drug elimination.4 Note that during the kinase and cardiac troponin I confirmed the diagnosis.
absorption phase, elimination occurs whenever drug Cardiac magnetic resonance imaging (MRI) confirmed
is present in the plasma, even though absorption a regional MI. Cardiac catheterization 8 hours later
predominates. revealed normal coronary arteries, suggesting a mech-

dD dD anism of vasospasm.
GI E

> (8.3)
dt dt 1. Could rapid drug absorption (large ka) contrib-

At the peak drug concentration in the plasma ute to high-peak drug concentration of pseudo-
(Fig. 8-4), the rate of drug absorption just equals the ephedrine in this subject?
rate of drug elimination, and there is no net change 2. Can an adverse drug reaction (ADR) occur
in the amount of drug in the body. before absorption is complete or, before Cmax

dD dD is reached?
GI = E (8.4)

dt dt 3. What is the effect of a small change in k on the
time and magnitude of Cmax (maximum plasma

Immediately after the time of peak drug absorp- concentration)? (Remember to correctly assign ka
tion, some drug may still be at the absorption site and k values when computing ka and k from patient
(ie, in the GI tract or other site of administration). data. See Flip-flop in oral absorption model in
However, the rate of drug elimination at this time is the next section.) In addition, see Chapter 13 for
faster than the rate of absorption, as represented by reasons why some subjects may have a smaller k.
the postabsorption phase in Fig. 8-4. 4. Do you believe that therapeutic drug concentra-

dD dD tion and toxic plasma concentration are always
GI < E (8.5)

dt dt clearly defined for individual subjects as intro-
duced in Fig. 1-2 (see Chapter 1)?

When the drug at the absorption site becomes
depleted, the rate of drug absorption approaches zero,

Discussion

4 From past experience, generally transient high plasma
The rate of drug elimination is dictated by the product of the

amount of drug in the body, DB times the rst-order elimination drug concentrations are not considered unsafe as long
rate constant, k. as the steady-state plasma concentration is within a

Plasma drug level

 

184 Chapter 8

recommended range. This is generally true for OTC plasma drug concentrations following multiple dos-
drugs. This case highlights a potential danger of some ing. In bioequivalence studies, drug products are
sympathomimetic drugs such as pseudoephedrine and given in chemically equivalent (ie, pharmaceutical
should alert the pharmacist that even drugs with a equivalents) doses, and the respective rates of sys-
long history of safe use may still exhibit dangerous temic absorption may not differ markedly. Therefore,
ADRs in some susceptible subjects. for these studies, tmax, or time of peak drug concen-

Do you believe that pseudoephedrine can be tration, can be very useful in comparing the respec-
sold safely without advice from a pharmacist? What tive rates of absorption of a drug from chemically
other types of medication are important to monitor equivalent drug products.
where a large ka may present transient high drug
concentrations in the blood?

A small elimination rate constant, k may be ZERO-ORDER ABSORPTION MODEL
caused by reduced renal drug excretion as discussed in Zero-order drug absorption from the dosing site into the
Chapter 7, but a small k may also be due to reduced plasma usually occurs when either the drug is absorbed
hepatic clearance caused by relatively inactive meta- by a saturable process or a zero-order controlled-release
bolic enzymes such as CYPs for some patients (see delivery system is used (see Chapter 19). The pharma-
Chapter 12). What are the kinetic tools that will allow cokinetic model assuming zero-order absorption is
one to make this differentiation? described in Fig. 8-5. In this model, drug in the gastro-

The pharmacokinetic concepts presented in this intestinal tract, DGI, is absorbed systemically at a con-
chapter will allow you to decide whether an unusual stant rate, k0. Drug is simultaneously and immediately
peak plasma drug concentration, Cmax is caused by a eliminated from the body by a first-order rate process
large ka, a small k (or Cl), both, or neither. defined by a first-order rate constant, k. This model is

analogous to that of the administration of a drug by
intravenous infusion (see Chapter 6).

SIGNIFICANCE OF ABSORPTION The rate of first-order elimination at any time is

RATE CONSTANTS equal to DBk. The rate of input is simply k0.
Therefore, the net change per unit time in the body

The overall rate of systemic drug absorption from an can be expressed as
orally administered solid dosage form encompasses
many individual rate processes, including dissolution dD

B
= k − kD

dt 0 B (8.7)
of the drug, GI motility, blood flow, and transport of
the drug across the capillary membranes and into the Integration of this equation with substitution of VDCp
systemic circulation. The rate of drug absorption rep- for DB produces
resents the net result of all these processes. The selec-
tion of a model with either first-order or zero-order k

C 0 (1 e−kt
p = − )

V (8.8)
absorption is generally empirical. Dk

The actual drug absorption process may be zero-
The rate of drug absorption is constant until the

order, first-order, or a combination of rate processes
amount of drug in the gut, DGI, is depleted. The time

that is not easily quantitated. For many immediate-
for complete drug absorption to occur is equal to

release dosage forms, the absorption process is first-
DGI/k0. After this time, the drug is no longer available

order due to the physical nature of drug diffusion.
For certain controlled-release drug products, the rate
of drug absorption may be more appropriately k k

0

described by a zero-order rate constant. D D V
GI B D

The calculation of ka is useful in designing a
multiple-dosage regimen. Knowledge of the ka and k FIGURE 85 One-compartment pharmacokinetic model
values allows for the prediction of peak and trough for zero-order drug absorption and first-order drug elimination.

 

Pharmacokinetics of Oral Absorption 185

for absorption from the gut, and Equation 8.7 no FIRST-ORDER ABSORPTION MODEL
longer holds. The drug concentration in the plasma
subsequently declines in accordance with a first- Although zero-order drug absorption can occur, sys-

order elimination rate process. temic drug absorption after oral administration of a
drug product (eg, tablet, capsule) is usually assumed
to be a first-order process. This model assumes a
first-order input across the gut wall and first-order

CLINICAL APPLICATION—
elimination from the body (Fig. 8-7). This model

TRANSDERMAL DRUG DELIVERY applies mostly to the oral absorption of drugs in
solution or rapidly dissolving dosage (immediate

The stratum corneum (horny layer) of the epidermis
release) forms such as tablets, capsules, and supposi-

of the skin acts as a barrier and rate-limiting step for
tories. In addition, drugs given by intramuscular or

systemic absorption of many drugs. After applica-
subcutaneous aqueous injections may also be

tion of a transdermal system (patch), the drug dis-
described using a first-order process.

solves into the outer layer of the skin and is absorbed
After oral administration of a drug product, the

by a pseudo first-order process due to high concen-
drug is relased from the drug product and dissolves

tration and is eliminated by a first-order process.
into the fluids of the GI tract. In the case of an

Once the patch is removed, the residual drug concen-
immediate-release compressed tablet, the tablet first

trations in the skin continues to decline by a first-
disintegrates into fine particles from which the drug

order process.
then dissolves into the fluids of the GI tract. Only

Ortho Evra is a combination transdermal contra-
drug in solution is absorbed into the body. The rate

ceptive patch with a contact surface area of 20 cm2.
of disappearance of drug from the gastrointestinal

Each patch contains 6.00 mg norelgestromin
tract is described by

(NGMN) and 0.75 mg ethinyl estradiol (EE) and is
designed to deliver 0.15 mg of NGMN and 0.02 mf
EE to the systemic circulation daily. As shown in dD

GI
= −kaD F (8.9)

Fig. 8-6, serum EE (ethinyl estradiol) is absorbed dt GI

from the patch at a zero-order rate.
where ka is the first-order absorption rate constant
from the GI tract, F is the fraction absorbed, and

70 DGI is the amount of drug in solution in the GI

60 tract at any time t. Integration of the differential
Equation (8.8) gives

50

40
dDGI = D e−kat

0 (8.10)
30

20 where D0 is the dose of the drug.

10 The rate of drug elimination is described by a
first-order rate process for most drugs and is equal

0
0 24 48 72 96 120 144 158 192 216 240 to −kDB. The rate of drug change in the body, dDB/dt,

Time (hours)

Cycle 1 week 1 Cycle 3 week 2
Cycle 3 week 1 Cycle 3 week 3

k k
FIGURE 86 Mean serum EE concentrations (pg/mL) in a

D D V
GI B D

healthy female volunteers following application of Ortho Evra
on the buttock for three consecutive cycles (vertical arrow
indicates time of patch removal). (Adapted from approved label for FIGURE 87 One-compartment pharmacokinetic model
Ortho Evra, September, 2009.) for first-order drug absorption and first-order elimination.

EE Concentration (pg/mL)

 

186 Chapter 8

is therefore the rate of drug in, minus the rate of The maximum plasma concentration after oral
drug out—as given by the differential equation, dosing is Cmax, and the time needed to reach maximum
Equation 8.10: concentration is tmax. The tmax is independent of dose

and is dependent on the rate constants for absorption
dDB

= rate in − rate out (k and elimination (k) (Equation 8.13). At C
dt a) max, some-

(8.11) times called peak concentration, the rate of drug
dDB absorbed is equal to the rate of drug eliminated.

= FkaDGI − kD
dt B Therefore, the net rate of concentration change is equal

to zero. At Cmax, the rate of concentration change can be
where F is the fraction of drug absorbed systemi-

obtained by differentiating Equation 8.11, as follows:
cally. Since the drug in the gastrointestinal tract
also follows a first-order decline (ie, the drug is dCp Fk D
absorbed across the gastrointestinal wall), the = a 0 (−ke−kt + k e−kat

a ) = 0 (8.13)
dt VD (ka − k)

amount of drug in the gastrointestinal tract at any
time t is equal to D e−kat. This can be simplified as follows:

0

dD kt −k
B at −k

Fk D e−kat − +
= a = 0 or ke−kt

= k e at
a

dt a 0 − kD ke− k e
B

ln k − kt = ln ka − kat
The value of F may vary from 1 for a fully

absorbed drug to 0 for a drug that is completely ln ka − ln k ln (ka /k)
tmax = =

unabsorbed. This equation can be integrated to give ka − k ka − k

the general oral absorption equation for calculation 2.3 log ( .14)
ka /k)

(8
of the drug concentration (Cp) in the plasma at any tmax =

ka − k
time t, as shown below.

Fk shown in Equation 8.13, the time for maxi-
aD

As
0 kt

C = (e−
− e−kat

p )
VD (ka k) (8.12) mum drug concentration, tmax, is dependent only on

the rate constants ka and k. In order to calculate Cmax,

A typical plot of the concentration of drug in the the value for tmax is determined via Equation 8.13

body after a single oral dose is presented in Fig. 8-8. and then substituted into Equation 8.11, solving for
Cmax. Equation 8.11 shows that Cmax is directly pro-
portional to the dose of drug given (D0) and the frac-
tion of drug absorbed (F). Calculation of tmax and

Cmax

Cmax is usually necessary, since direct measurement
of the maximum drug concentration may not be pos-
sible due to improper timing of the serum samples.

The first-order elimination rate constant may be
determined from the elimination phase of the plasma
level–time curve (Fig. 8-4). At later time intervals,
when drug absorption has been completed, that is,
e−kat

AUC ≈ 0, Equation 8.11 reduces to

FkaD0
Cp = e−kt (8.15)

VD (ka − k)

tmax Taking the natural logarithm of this expression,
Time

Fk D
FIGURE 88 Typical plasma level–time curve for a drug lnCp ln a 0

= − kt (8.16)
given in a single oral dose. VD (ka − k)

Plasma level

 

Pharmacokinetics of Oral Absorption 187

Substitution of common logarithms gives A

Fk D kt
logCp log a 0

= − (8.17)
VD (ka − k) 2.3 20

With this equation, a graph constructed by plotting 15

log Cp versus time will yield a straight line with a
slope of −k/2.3 (Fig. 8-9A). 10

With a similar approach, urinary drug excretion
5

data may also be used for calculation of the first-
order elimination rate constant. The rate of drug

0
excretion after a single oral dose of drug is given by 0 5 10 15 20 25

Time (hours)
dDu FkakeD0

(e−kt e−kat
= − − ) (8.18) B

dt ka − k

where dDu/dt = rate of urinary drug excretion, ke = 20

first-order renal excretion constant, and F = fraction
of dose absorbed. 15

10

A

5
100

FD
I 0kntercept = a

VD(ka – k) 0
0 5 10 15 20 25

10 Time (hours)

Slope = –k FIGURE 810 A. Plasma drug concentration versus time,
2.3 single oral dose. B. Rate of urinary drug excretion versus time,

1
single oral dose.

0.1
0 5 10 15 20 25 A graph constructed by plotting dDu/dt versus

Time (hours) time will yield a curve identical in appearance to the
B plasma level–time curve for the drug (Fig. 8-10B).

After drug absorption is virtually complete, −e−kat
100

FkekaDIntercept = 0 approaches zero, and Equation 8.18 reduces to
ka – k

10 dD Fk k
u eDa 0

e−kt
= (8.19)

dt k − k
a

Slope = –k
2.3

1 Taking the natural logarithm of both sides of this
expression and substituting for common logarithms,
Equation 8.19 becomes

0.1
0 5 10 15 20 25 dD Fk k D kt

log u
= log a e 0

− (8.20)
Time (hours) dt ka − k 2.3

FIGURE 89 A. Plasma drug concentration versus time,
single oral dose. B. Rate of urinary drug excretion versus time, When log(dDu/dt) is plotted against time, a
single oral dose. graph of a straight line is obtained with a slope of

Rate of drug excretion (dDu/dt) Concentration (mg/mL)

Rate of drug excretion, dDu/dt (mg/h) Plasma drug concentration, Cp (µg/mL)

 

188 Chapter 8

−k/2.3 (Fig. 8-9B). Because the rate of urinary drug virtually complete. Equation 8.12 then reduces to
excretion, dDu/dt, cannot be determined directly for Equation 8.23.
any given time point, an average rate of urinary drug
excretion is obtained (see also Chapter 4), and this Fk

C aD0 e−kt
p = (8.23)

value is plotted against the midpoint of the collection VD (ka − k)
period for each urine sample.

To obtain the cumulative drug excretion in the From this, one may also obtain the intercept of
urine, Equation 8.18 must be integrated, as shown the y axis (Fig. 8-12).
below.

Fk
aD0

Fk k D −e−kat e−kt  = A
Fk D

D a e 0
= VD (ka − k)

−  −  + e 0
u (8.21)

k k  k k k
a a 

where A is a constant. Thus, Equation 8.23 becomes
A plot of Du versus time will give the urinary

drug excretion curve described in Fig. 8-11. When all C Ae−kt
p = (8.24)

of the drug has been excreted, at t = ∞, Equation 8.21
reduces to This equation, which represents first-order drug

elimination, will yield a linear plot on semilog paper.
Fk

∞ eDD 0 . 2
u = (8 2 ) The slope is equal to − k/2.3. The value for ka can be

k obtained by using the method of residuals or a feath-
ering technique, as described in Chapter 5. The value

where D∞

u is the maximum amount of active or par-
of ka is obtained by the following procedure:

ent drug excreted.
1. Plot the drug concentration versus time on

Determination of Absorption Rate Constants semilog paper with the concentration values on

from Oral Absorption Data the logarithmic axis (Fig. 8-12).
2. Obtain the slope of the terminal phase (line BC,

Method of Residuals Fig. 8-12) by extrapolation.
Assuming ka >> k in Equation 8.12, the value for the
second exponential will become insignificantly
small with time (ie, e−kat ≈ 0) and can therefore be A X1′
omitted. When this is the case, drug absorption is 40 X2′

X3′

20 Cp = 43(e–0.40t – e–1.5t)

X
250 3

200 10 X2 B
150

100 5
X1

50

0
0 5 10 15 20 25 2

C
Time (hours)

FIGURE 811 Cumulative urinary drug excretion versus 1
0 2 4 6 8 10

time, single oral dose. Urine samples are collected at various
Time

time periods after the dose. The amount of drug excreted in
each sample is added to the amount of drug recovered in the FIGURE 812 Plasma level–time curve for a drug dem-
previous urine sample (cumulative addition). The total amount onstrating first-order absorption and elimination kinetics. The
of drug recovered after all the drug is excreted is D∞

u . equation of the curve is obtained by the method of residuals.

Cumulative drug
excretion (Du)

Plasma level

 

Pharmacokinetics of Oral Absorption 189

3. Take any points on the upper part of line BC
(eg, x′1, x′2, x′3, …) and drop vertically to obtain
corresponding points on the curve (eg, x1, x2,
x3, …).

4. Read the concentration values at x1 and x′1, x2
and x′2, x3 and x′3, and so on. Plot the values of
the differences at the corresponding time points
∆1, ∆2, ∆3, … . A straight line will be obtained
with a slope of −ka/2.3 (Fig. 8-12).

When using the method of residuals, a mini-
mum of three points should be used to define the
straight line. Data points occurring shortly after tmax Lag time

Time
may not be accurate, because drug absorption is
still continuing at that time. Because this portion of FIGURE 813 The lag time can be determined graphi-

the curve represents the postabsorption phase, only cally if the two residual lines obtained by feathering the plasma
level–time curve intersect at a point where t > 0.

data points from the elimination phase should be
used to define the rate of drug absorption as a first-
order process. The lag time, t0, represents the beginning of drug

If drug absorption begins immediately after absorption and should not be confused with the phar-
oral administration, the residual lines obtained by macologic term onset time, which represents latency,
feathering the plasma level–time curve (as shown in that is, the time required for the drug to reach mini-
Fig. 8-12) will intersect on the y axis at point A. The mum effective concentration.
value of this y intercept, A, represents a hybrid constant Two equations can adequately describe the curve
composed of ka, k, VD, and FD0. The value of A has no in Fig. 8-13. In one, the lag time t0 is subtracted from
direct physiologic meaning (see Equation 8.24). each time point, as shown in Equation 8.25.

FkaD0 Fk
A = C aD0 (e−k (t−t0 ) e−ka (t−t0 )

p = − ) (8.25)
VD (ka − k) VD (ka − k)

The value for A, as well as the values for k and ka, where FkaD0/VD(ka − k) is the y value at the point of
may be substituted back into Equation 8.11 to obtain intersection of the residual lines in Fig. 8-13.
a general theoretical equation that will describe the The second expression that describes the curve
plasma level–time curve. in Fig. 8-13 omits the lag time, as follows:

C Be−kt A −kat
p = − e (8.26)

Lag Time

In some individuals, absorption of drug after a single where A and B represent the intercepts on the y axis
oral dose does not start immediately, due to such after extrapolation of the residual lines for absorp-
physiologic factors as stomach-emptying time and tion and elimination, respectively.
intestinal motility. The time delay prior to the com-
mencement of first-order drug absorption is known
as lag time. Frequently Asked Question

The lag time for a drug may be observed if the »»If drug absorption is simulated using the oral one-
two residual lines obtained by feathering the oral compartment model, would a larger absorption
absorption plasma level–time curve intersect at a point rate constant result in a greater amount of drug
greater than t = 0 on the x axis. The time at the point of absorbed?

intersection on the x axis is the lag time (Fig. 8-13).

Plasma level

 

190 Chapter 8

Flip-Flop of ka and k 1.38 h−1 or 0.69 h−1). Because most of the drugs used

In using the method of residuals to obtain estimates orally have longer elimination half-lives compared to

of k rption half-lives, the assumption that the smaller
a and k, the terminal phase of an oral absorption abso

curve is usually represented by k, whereas the steeper slope or smaller rate constant (ie, the terminal phase

slope is represented by k ed as the elimi-
a (Fig. 8-14). In a few cases, of the curve in Fig. 8-14) should be us

the elimination rate constant k obtained from oral nation constant is generally correct.

absorption data does not agree with that obtained For drugs that have a large elimination rate

after intravenous bolus injection. For example, the k constant (k > 0.69 h−1), the chance for flip-flop of ka

obtained after an intravenous bolus injection of a and k is much greater. The drug isoproterenol, for

bronchodilator was 1.72 h−1, whereas the k calculated example, has an oral elimination half-life of only a

after oral administration was 0.7 h−1 (Fig. 8-14). few minutes, and flip-flop of ka and k has been noted

When ka was obtained by the method of residuals, the (Portmann, 1970). Similarly, salicyluric acid was

rather surprising result was that the ka was 1.72 h−1. flip-flopped when oral data were plotted. The k for

Apparently, the k was much larger than its ka (Levy
a and k obtained by the method salicyluric acid

of residuals have been interchanged. This phenome- et al, 1969). Many experimental drugs show flip-

non is called flip-flop of the absorption and elimina- flop of k and ka, whereas few marketed oral drugs

tion rate constants. Flip-flop, or the reversal of the do. Drugs with a large k are usually considered to be

rate constants, may occur whenever ka and k are unsuitable for an oral drug product due to their large

estimated from oral drug absorption data. Use of elimination rate constant, corresponding to a very

computer methods does not ensure against flip-flop short elimination half-life. An extended-release

of the two constants estimated. drug product may slow the absorption of a drug,

In order to demonstrate unambiguously that the such that the ka is smaller than the k and producing

steeper curve represents the elimination rate for a a flip-flop situation.

drug given extravascularly, the drug must be given
by intravenous injection into the same patient. After Frequently Asked Question
intravenous injection, the decline in plasma drug »»How do you explain that ka is often greater than k
levels over time represents the true elimination rate. with most drugs?
The relationship between ka and k on the shape of the
plasma drug concentration–time curve for a constant
dose of drug given orally is shown in Fig. 8-14. Determination of ka by Plotting Percent

Most of the drugs observed to have flip-flop char- of Drug Unabsorbed Versus Time
acteristics are drugs with fast elimination (ie, k > ka). (Wagner–Nelson Method)
Drug absorption of most drug solutions or fast-

The Wagner–Nelson method may be used as an
dissolving products is essentially complete or at

alternative means of calculating k his method
least half-complete within an hour (ie, absorption a. T

estimates the loss of drug from the GI over time,
half-life of 0.5 or 1 hour, corresponding to a ka of

whose slope is inversely proportional to ka. After a
single oral dose of a drug, the total dose should be
completely accounted for for the amount present in
the body, the amount present in the urine, and the
amount present in the GI tract. Therefore, dose (D0)

k = 0.7 h–1 ka = 0.7 h–1 is expressed as follows:
ka = 1.72 h–1

k = 1.72 h–1

Time Time D0 = DGI +DB +Du (8.27)
A. Incorrect B. Correct

FIGURE 814 Flip-flop of k Let Ab = DB + Du = amount of drug absorbed and let
a and k. Because k > ka, the

right-hand figure and slopes represent the correct values for Ab∞ = amount of drug absorbed at t = ∞. At any
ka and k. given time the fraction of drug absorbed is Ab/Ab∞,

log Cp

log Cp

 

Pharmacokinetics of Oral Absorption 191

and the fraction of drug unabsorbed is 1 − (Ab/Ab∞). 10

The amount of drug excreted at any time t can be
calculated as 1

Du = kVD[AUC]t0 (8.28)
0.1

The amount of drug in the body (DB) at any time =
CpVD. At any time t, the amount of drug absorbed
(Ab) is 0

0 5 10 15
Time (hours)

Ab =CpVD + kVD[AUC]t0 (8.29)
FIGURE 815 Semilog graph of data in Table 8-2, depict-

At t = C∞
∞, p = 0 (ie, plasma concentration is negli- ing the fraction of drug unabsorbed versus time using the

gible), and the total amount of drug absorbed is Wagner–Nelson method.

Ab∞ k ∞
= 0+ VD[AUC]0 (8.30)

5. Find k by adding up all the [AUC] pieces, from
The fraction of drug absorbed at any time is

t = 0 to t = ∞.
C 6. Determine the 1 − (Ab/Ab∞) value correspond-

pV kV t
Ab D + D[AUC]0

= (8.31)
Ab∞ kVD[AUC]∞ ing to each time point t by using Table 8-1.

0
7. Plot 1 − (Ab/Ab∞) versus time on semilog paper,

C k t
Ab p + [AUC] with 1 − (Ab/Ab∞) on the logarithmic axis.

0
= (8.32)

Ab∞ k[AUC]∞0 If the fraction of drug unabsorbed, 1 − Ab/Ab∞,
gives a linear regression line on a semilog graph,

The fraction unabsorbed at any time t is
then the rate of drug absorption, dDGI/dt, is a first-

C + k[ t
Ab p AUC] order process. Recall that 1 − Ab/Ab∞ is equal to

0
1− =1− (8.33)

Ab∞ k[AUC]∞ dDGI/dt (Fig. 8-15).
0

As the drug approaches 100% absorption, Cp
The drug remaining in the GI tract at any time t is becomes very small and difficult to assay accurately.

Consequently, the terminal part of the line described
D D e−kat

GI = 0 (8.34) by 1 − Ab/Ab∞ versus time tends to become scattered
or nonlinear. This terminal part of the curve is excluded,

Therefore, the fraction of drug remaining is
and only the initial linear segment of the curve is used

D for the estimate of the slope.
GI D −k t

= e−kat log GI a
= (8.35)

D0 D0 2.3

Because DGI/D0 is actually the fraction of drug PRACTICE PROBLEM
unabsorbed—that is, 1 − (Ab/Ab∞)—a plot of 1 − (Ab/
Ab∞) versus time gives −ka/2.3 as the slope (Fig. 8-15). Drug concentrations in the blood at various times are

The following steps should be useful in determi- listed in Table 8-1. Assuming the drug follows a one-

nation of ka:
compartment model, find the ka value, and compare it
with the ka value obtained by the method of residuals.

1. Plot log concentration of drug versus time.
2. Find k from the terminal part of the slope when

Solution
the slope = −k/2.3.

3. Find [AUC]t0 by plotting C The AUC is approximated by the trapezoidal rule.
p versus t.

4. Find k [AUC]t by multiplying each [AUC]t This method is fairly accurate when there are suffi-
0 0

by k. cient data points. The area between each time point

Ion of drug unabsorbed
[1 – (Ab/Ab∞)]

 

192 Chapter 8

TABLE 81 Blood Concentrations and Associated Data for a Hypothetical Drug

Time tn Concentration Ab  Ab 
(h) C tn [AUC]t k[AUC t + [AU ]

p (lg/mL) [AUC]
t 0 ] 1−

C t ∞ 
0 p k C 

0
– Ab∞  Ab 

n 1

0 0 0 0 1.000

1 3.13 1.57 1.57 0.157 3.287 0.328 0.672

2 4.93 4.03 5.60 0.560 5.490 0.548 0.452

3 5.86 5.40 10.99 1.099 6.959 0.695 0.305

4 6.25 6.06 17.05 1.705 7.955 0.794 0.205

5 6.28 6.26 23.31 2.331 8.610 0.856 0.140

6 6.11 6.20 29.51 2.951 9.061 0.905 0.095

7 5.81 5.96 35.47 3.547 9.357 0.934 0.066

8 5.45 5.63 41.10 4.110 9.560 0.955 0.045

9 5.06 5.26 46.35 4.635 9.695 0.968 0.032

10 4.66 4.86 51.21 5.121

12 3.90 8.56 59.77 5.977

14 3.24 7.14 66.91 6.691

16 2.67 5.92 72.83 7.283

18 2.19 4.86 77.69 7.769

24 1.20 10.17 87.85 8.785

28 0.81 4.02 91.87 9.187

32 0.54 2.70 94.57 9.457

36 0.36 1.80 96.37 9.637

48 0.10 2.76 99.13 9.913

k = 0.1 h–1.

is calculated as has fallen to an insignificant drug concentration,
0.1 μg/mL. The rest of the needed information is

t C
[AU n n +C

C] −1 n
t = (t − )

1 2 n t
n n 1 (8.36)

− given in Table 8-1. Notice that k is obtained from the

plot of log Cp versus t; k was found in this example to
where Cn and Cn−1 are concentrations. For example, be 0.1 h−1. The plot of 1− (Ab/Ab∞) versus t on semi-
at n = 6, the [AUC] is log paper is shown in Fig. 8-15.

A more complete method of obtaining ka is to
6.28+6.11

(6− 5) = 6.20 estimate the residual area from the last observed
2

plasma concentration, Cp at tn to time equal to infinity.
This equation for the residual AUC from Cp to time

To obtain [AUC]∞0 , add all the area portions
equal to infinity is

under the curve from zero to infinity. In this case,
48 hours is long enough to be considered infinity, C

p
[AUC]∞ n

t = (8.37)
because the blood concentration at that point already k

 

Pharmacokinetics of Oral Absorption 193

The total [AUC]∞0 is the sum of the areas obtained Substituting for dCp/dt into Equation 8.42 and kDu/ke
by the trapezoidal rule, [AUC]t0 , and the residual for DE,
area [AUC]∞t , as described in the following

dAb d(dDu /dt) k dD 
expression:  u

= +  (8.45)
dt ke dt ke  dt 

[AUC]∞ t
0 = [AUC]0 + [AUC]∞t (8.38)

When the above expression is integrated from zero
to time t,

Estimation of ka from Urinary Data
1 dD k

The absorption rate constant may also be estimated Abt = ( u ) + (D )
ke dt t k u t (8.46)

from urinary excretion data, using a plot of percent of e

drug unabsorbed versus time. For a one-compartment At t = ∞, all the drug that is ultimately absorbed is
model: expressed as Ab∞ and dDu/dt = 0. The total amount

Ab = total amount of drug absorbed—that is, the of drug absorbed is

amount of drug in the body plus the amount of
k

drug excreted Ab∞ = D∞

k u
DB = amount of drug in the body e

Du = amount of unchanged drug excreted in the urine where D∞ is the total amount of unchanged drug
u

Cp = plasma drug concentration excreted in the urine.
DE = total amount of drug eliminated (drug and The fraction of drug absorbed at any time t is

metabolites) equal to the amount of drug absorbed at this time, Abt,
divided by the total amount of drug absorbed, Ab∞.

Ab = DB + DE (8.39)
Abt (dDu /dt)t + k(Du )t

The differential of Equation 8.39 with respect to = (8.47)
Ab∞ kD∞

time gives u

dAb dD dD A plot of the fraction of drug unabsorbed, 1 −
B E

= + (8.40)
dt dt dt Ab/Ab∞, versus time gives −ka/2.3 as the slope

from which the absorption rate constant is obtained
Assuming first-order elimination kinetics with renal

(Fig. 8-15; refer to Equation 8.35).
elimination constant ke, When collecting urinary drug samples for the

dD determination of pharmacokinetic parameters, one
u = k .

eDB = keVDC (8 41)
dt p should obtain a valid urine collection as discussed in

Chapter 4. If the drug is rapidly absorbed, it may be
Assuming a one-compartment model, difficult to obtain multiple early urine samples to

V describe the absorption phase accurately. Moreover,
DCp = DB

drugs with very slow absorption will have low con-
Substituting VDCp into Equation 8.40, centrations, which may present analytical problems.

dAb dCp dD
=V E

+ (8.42) Effect of ka and k on Cmax, tmax, and AUC
dt D dt dt

Changes in ka and k may affect tmax, Cmax, and AUC as
And rearranging Equation 8.41, shown in Table 8-2. If the values for ka and k are

1 dD  reversed, then the same tmax is obtained, but the Cmax
C u

p =  
k V 4
e D  dt  (8. 3) and AUC are different. If the elimination rate constant

is kept at 0.1 h−1 and the ka changes from 0.2 to 0.6 h−1
dCp d(dDu /dt) (absorption rate increases), then the tmax becomes

=
dt dt keV

(8.44)
D shorter (from 6.93 to 3.58 hours), the Cmax increases

 

194 Chapter 8

TABLE 82 Effects of the Absorption Rate Constant and Elimination Ratea

Absorption Rate Elimination Rate
Constant, ka (h–1) Constant, k (h–1) tmax (h) Cmax (lg/mL) AUC (lg . h/mL)

0.1 0.2 6.93 2.50 50

0.2 0.1 6.93 5.00 100

0.3 0.1 5.49 5.77 100

0.4 0.1 4.62 6.29 100

0.5 0.1 4.02 6.69 100

0.6 0.1 3.58 6.99 100

0.3 0.1 5.49 5.77 100

0.3 0.2 4.05 4.44 50

0.3 0.3 3.33 3.68 33.3

0.3 0.4 2.88 3.16 25

0.3 0.5 2.55 2.79 20

atmax = peak plasma concentration, Cmax = peak drug concentration, AUC = area under the curve. Values are based on a single oral dose (100 mg) that
is 100% bioavailable (F = 1) and has an apparent VD of 10 L. The drug follows a one-compartment open model. tmax is calculated by Equation 8.14 and
Cmax is calculated by Equation 8.12. The AUC is calculated by the trapezoidal rule from 0 to 24 hours.

(from 5.00 to 6.99 μg/mL), but the AUC remains con- (from 5.77 to 2.79 μg/mL), and the AUC decreases
stant (100 μg h/mL). In contrast, when the absorption (from 100 to 20 μg h/mL). Graphical representations
rate constant is kept at 0.3 h−1 and k changes from 0.1 for the relationships of ka and k on the time for peak
to 0.5 h−1 (elimination rate increases), then the tmax absorption and the peak drug concentrations are
decreases (from 5.49 to 2.55 hours), the Cmax decreases shown in Figs. 8-16 and 8-17.

8 2.8

2.4 0.2/h
0.5/h

6 2.0
0.3/h

1.5
4 0.2/h

1.2

0.3/h 0.8 0.5/h
2

0.4

0 0
0 4 8 12 15 20 0 4 8 12 15 20

Time (hours) Time (hours)

FIGURE 816 Effect of a change in the absorption rate FIGURE 817 Effect of a change in the elimination rate

constant, ka, on the plasma drug concentration–time curve. constant, k, on the plasma drug concentration–time curve.

Dose of drug is 100 mg, VD is 10 L, and k is 0.1 h–1. Dose of drug is 100 mg, VD is 10 L, and ka is 0.1 h–1.

Concentration (mg/mL)

Concentration (mg/mL)

 

Pharmacokinetics of Oral Absorption 195

Modified Wagner–Nelson Method Models for Estimation of Drug Absorption

Hayashi et al (2001) introduced a modified Wagner– There are many models and approaches that have
Nelson method to study the subcutaneous absorption been used to predict drug absorption since the intro-
of a drug with nonlinear kinetics from the central duction of the classical approaches by John Wagner
compartment. Nonlinear kinetics occurs in some (1967) and Jack Loo. Deconvolution and convolu-
drugs where the kinetic parameter such as k change tion approaches are used to predict plasma drug
with dose. The method was applicable to a biotech- concentration of oral dosage forms. Several com-
nological drug (recombinant human granulocyte- mercial software (eg, GastroPlus, iDEA, Intellipharm
colony stimulating factors, rhG-CSF) which is PK, and PK-Sim) are now available for formulation
eliminated nonlinearly. The drug was absorbed into and drug development or to determine the extent of
the blood from the dermal site after subcutaneous drug absorption. The new software allows the char-
injection. Because of nonlinear kinetics the extent of acteristics of the drug, physiologic factors, and the
absorption was not easily determined. The amount dosage form to be inputed into the software. An impor-
of drug absorbed, Ab for each time sample, tn, is tant class of programs involves the Compartmental
given by Equation 8.48. V1 and Vss are central com- Absorption and Transit (CAT) models. This model
partment and steady-state volume of distribution, integrates the effect of solubility, permeability, as well
respectively. as gastric emptying and GI transit time in the estima-

Vmax and Km are Michaelis–Menten parameters tion of in vivo drug absorption. CAT models were
that describe the saturable elimination (see Chapter 10) successfully used to predict the fraction of drug oral
of the drug. ti is the sample time which = 0,1,2,4… absorption of 10 common drugs based on a small
48 hours in this example, and C(t) is the average intestine transit time (Yu, 1999). The CAT models
serum drug concentraton between time points, that is, compared well overall with other plausible models
ti and ti+1. such as the dispersion model, the single mixing tank

model, and some flow models. It is important to note
n−1 that the models discussed earlier in this chapter are

ti+1
( ) = ∑  VmaxC(t) 

Aab tn ∫ +
= 

kV
C t K 1C(t) dt +VssC(tn )

t ( ) + used to compute extent of absorption after the plasma
i 1 i m drug concentrations are measured. In contrast, the

(8.48) later models/software allow a comprehensive way to
simulate or predict drug (product) performance in vivo.

From the mass balance of the above equation, The subjects of dissolution, dosage form design, and
the authors did account for the amount of drug pres- drug absorption will be discussed in more detail in
ent in the tissue compartment. (Note the authors Chapters 14 and 15.
stated that the central compartment V1 is 4.56 L and
that of Vss is 4.90 L.) To simplify the model, the

Determination of ka from Two-Compartment
authors used convolution to show that the contribu-

Oral Absorption Data (Loo–Riegelman
tion of the tissue compartment is not significant and
therefore may be neglected. Thus, the Loo– Method)

Riegelman method which requires a tissue compart- Plotting the percent of drug unabsorbed versus time
ment was not used by the authors. Convolution is an to determine ka may also be calculated for a drug
analytical method that predicts plasma time drug exhibiting a two-compartment kinetic model. As in
concentration using input and disposition functions the method used previously to obtain an estimate of
for drugs with linear kinetics. The disposition func- the ka, no limitation is placed on the order of the
tion may be first obtained by deconvolution of sim- absorption process. However, this method does
ple IV plasma drug concentration data or from the require that the drug be given intravenously as well as
terminal phase of an oral solution. Alternatively, the orally to obtain all the necessary kinetic constants.
method of Lockwood and Gillespie (1996) abbrevi- After oral administration of a dose of a drug that
ated the need for the simple solution. exhibits two-compartment model kinetics, the amount

 

196 Chapter 8

k
k 12 A plot of the fraction of drug unabsorbed, 1 −

a Central compartment Tissue compartment
Dp Vp Cp Dt Vt Ct Ab/Ab∞, versus time gives −ka/2.3 as the slope from

k21 which the value for the absorption rate constant is
k obtained (refer to Equation 8.35).

The values for k[AUC]t0 are calculated from a plot
FIGURE 818 Two-compartment pharmacokinetic mode. of Cp versus time. Values for (Dt/Vp) can be approxi-
Drug absorption and elimination occur from the central mated by the Loo–Riegelman method, as follows:
compartment.

k12∆Cp ∆ t k
(C ) = + 12 k t k t

t t (Cp )2 t (1− e− 21∆ )+ (C 21
1 t )t e− ∆

n k n− n−1

of drug absorbed is calculated as the sum of the 21

amounts of drug in the central compartment (Dp), in (8.57)

the tissue compartment (Dt), and the amount of drug
where Ct is Dt/Vp, or apparent tissue concentration;

eliminated by all routes (Du) (Fig. 8-18).
t = time of sampling for sample n; tn−1 = time of

Ab = Dp + Dt + Du (8.49) sampling for the sampling point preceding sample n;
and (Cp )t = concentration of drug at central com-

Each of these terms may be expressed in terms of n−1

partment for sample n − 1.
kinetics constants and plasma drug concentrations, Calculation of Ct values is shown in Table 8-3,
as follows: using a typical set of oral absorption data. After calcu-

lation of Ct values, the percent of drug unabsorbed is
Dp = VpCp (8.50)

calculated with Equation 8.56, as shown in Table 8-4.
Dt = VtCt (8.51) A plot of percent of drug unabsorbed versus time

on semilog graph paper gives a ka of approximately
dD

u = kV (8.52)
dt pCp 0.5 h−1.

For calculation of ka by this method, the drug

D must be given intravenously to allow evaluation of the
u = kVp[AUC]t0

distribution and elimination rate constants. For drugs
Substituting the above expression for Dp and Du into that cannot be given by the IV route, the ka cannot be
Equation 8.49, calculated by the Loo–Riegelman method. For drugs

that are given by the oral route only, the Wagner–
Ab = VpCp +Dt + kVp[AUC]t0 (8.53)

Nelson method, which assumes a one-compartment

By dividing this equation by Vp to express the equation model, may be used to provide an initial estimate of

on drug concentrations, we obtain ka. If the drug is given intravenously, there is no way
of knowing whether there is any variation in the values

Ab D
C t for the elimination rate constant, k and the distributive

= p + + k[AUC]t (8 54
p V 0 . )
V p rate constants, k12 and k21. Such variations alter the

rate constants. Therefore, a one-compartment model
At t = ∞, this equation becomes

is frequently used to fit the plasma curves after an oral
Ab or intramuscular dose. The plasma level predicted

= k[AUC]∞0 (8.55)
Vp from the ka obtained by this method does deviate from

the actual plasma level. However, in many instances,
Equation 8.54 divided by Equation 8.55 gives the

this deviation is not significant.
fraction of drug absorbed at any time as shown in
Equation 8.56.

Cumulative Relative Fraction Absorbed
D 

C t
p +  + k[AUC]t The fraction of drug absorbed at any time t (Equation

0
Ab Vp  (8.56) 8.32) may be summed or cumulated for each time

∞ =
Ab k[AUC]∞0 period for which a plasma drug sample was obtained.

 

Pharmacokinetics of Oral Absorption 197

TABLE 83 Calculation of Ct Valuesa

(k12 /k21)× (Cp )t (k12 / k21)×
(k n−1

12∆Cp∆t)
−k ∆t

(C 2 e−k21∆t ) (1−e−k21∆t ) (C ) e 21
t tn−1

p)tn (t)tn D(Cp) Dt (Cp) (1−
t (C
n-1 t)tn

3.00 0.5 3.0 0.5 0.218 0 0.134 0 0 0.218

5.20 1.0 2.2 0.5 0.160 3.00 0.134 0.402 0.187 0.749

6.50 1.5 1.3 0.5 0.094 5.20 0.134 0.697 0.642 1.433

7.30 2.0 0.8 0.5 0.058 6.50 0.134 0.871 1.228 2.157

7.60 2.5 0.3 0.5 0.022 7.30 0.134 0.978 1.849 2.849

7.75 3.0 0.15 0.5 0.011 7.60 0.134 1.018 2.442 3.471

7.70 3.5 –0.05 0.5 –0.004 7.75 0.134 1.039 2.976 4.019

7.60 4.0 –0.10 0.5 –0.007 7.70 0.134 1.032 3.444 4.469

7.10 5.0 –0.50 1.0 –0.073 7.60 0.250 1.900 3.276 5.103

6.60 6.0 –0.50 1.0 –0.073 7.10 0.250 1.775 3.740 5.442

6.00 7.0 –0.60 1.0 –0.087 6.60 0.250 1.650 3.989 5.552

5.10 9.0 –0.90 2.0 –2.261 6.00 0.432 2.592 2.987 5.318

4.40 11.0 –0.70 2.0 –0.203 5.10 0.432 2.203 2.861 4.861

3.30 15.0 –1.10 4.0 –0.638 4.40 0.720 3.168 1.361 3.891

aCalculated with the following rate constants: k12 = 0.29 h–1, k21 = 0.31 h–1.

Adapted with permission from Loo and Riegelman (1968).

From Equation 8.32, the term Ab/Ab∞ becomes the To determine the real percent of drug absorbed,
cumulative relative fraction absorbed (CRFA). a modification of the Wagner–Nelson equation was

suggested by Welling (1986). A reference drug prod-
Cp + k[AUC]t0

CRFA = (8.58) uct was administered and plasma drug concentra-
k[AUC]∞0 tions were determined over time. CRFA was then

estimated by dividing Ab/Ab∞ , where Ab is the
where Cp is the plasma concentration at time t. ref

cumulative amount of drug absorbed from the drug
In the Wagner–Nelson equation, Ab/Ab∞ or CRFA

product and Ab∞

will eventually equal unity, or 100%, even though the ref is the cumulative final amount of
drug absorbed from a reference dosage form. In this

drug may not be 100% systemically bioavailable. The
case, the denominator of Equation 8.58 is modified

percent of drug absorbed is based on the total amount
as follows:

of drug absorbed (Ab∞) rather than the dose D0.
Because the amount of drug ultimately absorbed, Ab∞ C ∞

p + k[AUC]0
in fractional term, is analogous to CRFA = (8.59)

k[AUC]∞0 , the k ∞

ref [AUC]ref
numerator will always equal the denominator at time
infinity, whether the drug is 10%, 20%, or 100% where kref and [AUC]∞ref are the elimination constant
bioavailable. The percent of drug absorbed based on and the area under the curve determined from the
Ab/Ab∞ is therefore different from the real percent of reference product, respectively. The terms in the
drug absorbed unless F = 1. However, for the calcula- numerator of Equation 8.59 refer to the product, as
tion of k in Equation 8.58.

a, the method is acceptable.

 

198 Chapter 8

TABLE 84 Calculation of Percentage Unabsorbeda

100% –
t

Time (h) (Cp)t t
[AUC] n [AUC]tn k[AUC] n

n (Ct)tn Ab/Vp %Ab/V
t p Ab/Vp%
n–1 t0 t0

0.5 3.00 0.750 0.750 0.120 0.218 3.338 16.6 83.4

1.0 5.20 2.050 2.800 0.448 0.749 6.397 31.8 68.2

1.5 6.50 2.925 5.725 0.916 1.433 8.849 44.0 56.0

2.0 7.30 3.450 9.175 1.468 2.157 10.925 54.3 45.7

2.5 7.60 3.725 12.900 2.064 2.849 12.513 62.2 37.8

3.0 7.75 3.838 16.738 2.678 3.471 13.889 69.1 30.9

3.5 7.70 3.863 20.601 3.296 4.019 15.015 74.6 25.4

4.0 7.60 3.825 24.426 3.908 4.469 15.977 79.4 20.6

5.0 7.10 7.350 31.726 5.084 5.103 17.287 85.9 14.1

6.0 6.60 6.850 38.626 6.180 5.442 18.222 90.6 9.4

7.0 6.00 6.300 44.926 7.188 5.552 18.740 93.1 6.9

9.0 5.10 11.100 56.026 8.964 5.318 19.382 96.3 3.7

11.0 4.40 9.500 65.526 10.484 4.861 19.745 98.1 1.9

15.0 3.30 15.400 80.926 12.948 3.891 20.139 100.0 0

a t
Ab/V = (C )+ k[AUC] n

p p + (C )
t0 t tn

(C ) = k 21 21
12∆Cp∆t /2+ k12 /k21 (Cp ) (1− e−k ∆t )+ (C ) e−k ∆t

t tn tn t t
−1 n−1

k = 0.16; k12 = 0.29; k21 = 0.31

Each fraction of drug absorbed is calculated and is shown in Fig. 8-20. The data for Fig. 8-21 were
plotted against the time interval in which the plasma obtained from the serum tolazamide levels–time
drug sample was obtained (Fig. 8-19). An example of curves in Fig. 8-20. The CRFA–time graph provides
the relationship of CRFA versus time for the absorp- a visual image of the relative rates of drug absorp-
tion of tolazamide from four different drug products tion from various drug products. If the CRFA–time

1.2
1.0 A

0.8 D
0.8

0.6

0.4 C

0.4
0.2

B

0
0 0 2 4 6 8 12 16
0 5 10 15 20 25 30 Time (hours)

Time (hours)
FIGURE 820 Mean cumulative relative fractions of

FIGURE 819 Fraction of drug absorbed. (Wagner–Nelson tolazamide absorbed as a function of time. (From Welling et al,
method.) 1982, with permission.)

Fraction of drug absorbed
(Ab/Ab)

Cumulative relative
fraction absorbed

 

Pharmacokinetics of Oral Absorption 199

30 absorption may not differ markedly. Therefore, for
A these studies, tmax, or time of peak drug concentra-

20 tion, can be very useful in comparing the respective
D

rates of absorption of a drug from chemically equiv-
C alent drug products.

10

B

0 Frequently Asked Questions
0 2 4 6 8 12 16 24

Time (hours) »»Can the Wagner–Nelson method be used to calculate
ka for an orally administered drug that follows the

FIGURE 821 Mean serum tolazamide levels as a func- pharmacokinetics of a two-compartment model?
tion of time. (From Welling et al, 1982, with permission.)

»»What is the absorption half-life of a drug and how is
it determined?

curve is a straight line, then the drug was absorbed
from the drug product at an apparent zero-order »»In switching a drug from IV to oral dosing, what is the

most important consideration?
absorption rate.

The calculation of ka is useful in designing a »»Drug clearance is dependent on dose and area under
multiple-dosage regimen. Knowledge of the ka and k the time–drug concentration curve. Would drug

allows for the prediction of peak and trough plasma clearance be affected by the rate of absorption?

drug concentrations following multiple dosing. In »»Does a larger absorption rate constant affect Cmax,
bioequivalence studies, drug products are given in tmax, and AUC if the dose and elimination rate con-
chemically equivalent (ie, pharmaceutical equiva- stant, k remains constant?
lents) doses, and the respective rates of systemic

CHAPTER SUMMARY
Pharmacokinetic absorption models range from two-compartment model using the Loo–Riegelman
being entirely “exploratory” and empirical, to semi- method. The determination of the fraction of drug
mechanistic and ultimately complex physiologically absorbed is an important tool in evaluating drug dos-
based pharmacokinetic (PBPK) models. This choice age form and design. The Wagner–Nelson method
is conditional on the modeling purpose as well as the and Loo–Reigelman method are classical methods for
amount and quality of the available data. determinating absorption rate constants and fraction

Empirically, the pharmacokinetics of drug of drug absorbed. Convolution and deconvolution are
absorption may be described by zero-order or first- powerful alternative tools used to predict a plasma
order kinetics. Drug elimination from the body is drug concentration–time profile from dissolution of
generally described by first-order kinetics. Using the data during drug development.
compartment model, various important pharmacoki- The empirical models presented in this chapter
netics parameters about drug absorption such as ka, are very basic with simple assumptions. More
k, Cmax, tmax, and other parameters may be computed sophisticated methods based on these basic con-
from data by the method of residuals (feathering) or cepts may be extended to include physiological
by computer modeling. The pharmacokinetic param- factors such as GI transit in the physiologically
eters are important in evaluating drug absorption and based models that represent the advance drug
understanding how these parameters affect drug absorption model development. These models are
concentrations in the body. The fraction of drug useful to predict drug absorption over time curves
absorbed may be computed in a one-compartment in designing oral dosage forms (see Chapters 14
model using the Wagner–Nelson method or in a and 15).

Serum tolazamide
concentration (mg/mL)

 

200 Chapter 8

Although the current development of in vitro increases. Although such an approach has limita-
studies and computer science have allowed rapid tions, further methodology research in this field and
advances of PBPK models, the combination of the advances in computer science can address many
physiologically based modeling with parameter esti- of them. It is apparent that “bottom-up” and “top-
mation techniques seems to be the way forward and down” modeling strategies need to approach and
its impact on the drug development progressively borrow skills from each other.

ANSWERS

Frequently Asked Questions with oral solutions and immediate-release drug
products such as compressed tablets or capsules.

If drug absorption is simulated using the oral one- The determination of the absorption rate constant,
compartment model, would a larger absorption rate ka, is most often calculated by the Wagner–Nelson
constant result in a greater amount of drug absorbed? method for drugs, which follows a one-compartment

• The fraction of drug absorbed, F, and the absorption model with first-order absorption and first-order

rate constant, ka, are independent parameters. A drug elimination.

in an oral solution may have a more rapid rate of In switching a drug from IV to oral dosing, what is
absorption compared to a solid drug product. If the the most important consideration?
drug is released from the drug product slowly or is
formulated so that the drug is absorbed slowly, the • The fraction of drug absorbed may be less than 1

drug may be subjected to first-pass effects, degraded (ie, 100% bioavailable) after oral administration.

in the gastrointestinal tract, or eliminated in the feces In some cases, there may be a different salt form

so that less drug (smaller F) may be absorbed sys- of the drug used for IV infusion compared to the

temically compared to the same drug formulated to salt form of the drug used orally. Therefore, a cor-

be absorbed more rapidly from the drug product. rection is needed for the difference in MW of the
two salt forms.

How do you explain that ka is often greater than k
with most drugs? Drug clearance is dependent on dose and area under

the time–drug concentration curve. Would drug
• A drug with a rate of absorption slower than its rate clearance be affected by the rate of absorption?

of elimination will not be able to obtain optimal
systemic drug concentrations to achieve efficacy. • Total body drug clearance and renal drug clear-

Such drugs are generally not developed into prod- ance are generally not affected by drug absorp-

ucts. However, the apartment ka for drugs absorbed tion from most absorption sites. In the gastroin-

from controlled-release products (Chapter 18) may testinal tract, a drug is absorbed via the hepatic

be smaller, but the initial rate of absorption from portal vein to the liver and may be subject to

the GI tract is faster than the rate of drug elimina- hepatic clearance.

tion since, dDGI/dt = − kaDGI.
Learning Questions

What is the absorption half-life of a drug and how is
1. a. T he elimination rate constant is 0.1 h−1

it determined? (t1/2 =
6.93 h).

• For drugs absorbed by a first-order process, the b. The absorption rate constant, ka, is 0.3 h−1
absorption half-life is 0.693/ka. Although drug (absorption half-life = 2.31 h).
absorption involves many stochastic (system-based
random) steps, the overall rate process is often ln(k /k)

The calculated a
tmax = = 5.49 h

approximated by a first-order process, especially ka − k

 

Pharmacokinetics of Oral Absorption 201

c. The y intercept was observed to be 60 ng/mL. 5. The equations for a drug that follows the
Therefore, the equation that fits the observed kinetics of a one-compartment model with
data is first-order absorption and elimination are

FD k ln(k /k)
C = 60(e−0.1t e−0.3t

p − ) C 0 a
p = (ekt −kat

− e ) t a
=

VD(k k max
a − ) ka − k

Note: Answers obtained by “hand” feather- As shown by these equations:

ing the data on semilog graph paper may vary a. tmax is influenced by ka and k and not by F,

somewhat depending on graphing skills and D0, or VD.

skill in reading data from a graph. b. Cp is influenced by F, D0, VD, ka, and k.

2. By direct observation of the data, the tmax
6. A drug product that might provide a zero-order

is 6 hours and the Cmax is 23.01 ng/mL. input is an oral controlled-release tablet or a trans-

The apparent volume of distribution, VD, is dermal drug delivery system (patch). An IV drug

obtained from the intercept, I, of the terminal infusion will also provide a zero-order drug input.

elimination phase, and substituting F = 0.8, 7. The general equation for a one-compartment

D = 10,000,000 ng, ka = 0.3 h−1, k = 0.1 h−1: open model with oral absorption is

FD
C 0ka (e−kt e−kat

p = − )
Fk D

I a 0 VD (ka − k)
=

VD (ka − k)
From Cp = 45(e−0.17t − e−1.5t)

(0.8)(0.3)(10,000,000)
60 =

VD (0.3− 0.1) FD0ka
= 45

VD (ka − k)
VD = 200 L

k −1
= 0.17 h

3. The percent-of-drug-unabsorbed method k 1.5 h−1
a =

is applicable to any model with first-order

elimination, regardless of the process of drug ln(ka /k) ln(1.5/0.17)
input. If the drug is given by IV injection, the a. tmax = = =1.64 h

ka − k 1.5− 0.17
elimination rate constant, k, may be deter-

b. Cma = 45(e−(0.17)(1.64) − e−(1.5)(1.64)
mined accurately. If the drug is administered x )

orally, k and ka may flip-flop, resulting in an = 30.2 µg/mL
error unless IV data are available to determine
k. For a drug that follows a two-compartment 0.693 0.693

c. t1/2 = = = 4.08 h
model, an IV bolus injection is used to deter- k 0.17
mine the rate constants for distribution and
elimination. In(1.0/0.2)

8. a. Drug A tmax = = 2.01 h
4. After an IV bolus injection, a drug such as 1.0 −1.2

theophylline follows a two-compartment In(0.2/1.0)
Drug B tmax = = 2.01 h

model with a rapid distribution phase. During 0.2−1.0
oral absorption, the drug is distributed dur-

FD k
ing the absorption phase, and no distribution b. C 0 a ( − tmax −katmax

max = e k
− e )

VD (ka − k)
phase is observed. Pharmacokinetic analy-
sis of the plasma drug concentration data (1)(500)(1)

Drug A Cma = (e−(0.2)(2)− e−(1)(2)
obtained after oral drug administration will x − )

(10)(1 0.2)
show that the drug follows a one-compartment

model. Cmax = 33.4 µg/mL

 

202 Chapter 8

(1)(500)(0.2) c. The Loo–Riegelman method requires IV
Drug B Cmax =

(20)(0.2 −1.0) data. Therefore, only the Wagner–Nelson
method may be used on these data.

= (e−1(2) − e−(0.2)(2) ) d. Observed tmax and Cmax values are taken

Cmax = 33.4 µ directly from the experimental data. In
g/mL

this example, Cmax is 85.11 ng/mL, which
occurred at a tmax of 1 hour. The theoretical

9. a. The method of residuals using manual
tmax and Cmax are obtained as follows:

graphing methods may give somewhat dif-
ferent answers depending on personal skill 2.3log(k /k)

t a
and the quality of the graph paper. Values max =

ka − k
obtained by the computer program ESTRIP
gave the following estimates: 2.3log(2.84/0.186)

= =1.03 h
2.84 − 0.186

ka = 2.84 h−1 k = 0.186 h−1 t1/2 = 3.73 h
FD k

b. A drug in an aqueous solution is in the most C 0 a − max − a max
max = (e kt e k t

− )
VD (ka − k)

absorbable form compared to other oral dosage
forms. The assumption that k where FD0ka/VD(ka − k) is the y intercept

a > k is generally
true for drug solutions and immediate-release equal to 110 ng/mL and tmax = 1.03 h.
oral dosage forms such as compressed tablets

C (1 −(1.186)(1.0) −(2
max = 10)(e − e .84)(1.03) )

and capsules. Drug absorption from extended-
release dosage forms may have ka < k. To dem- Cmax = 85 ng/mL
onstrate unequivocally which slope represents

e. A more complete model-fitting program, such
the true k, the drug must be given by IV bolus

as WINNONLIN, is needed to fit the data
or IV infusion, and the slope of the elimination

statistically to a one-compartment model.
curve obtained.

APPLICATION QUESTIONS
1. Plasma samples from a patient were collected From the given data:

after an oral bolus dose of 10 mg of a new a. Determine the elimination constant of the
benzodiazepine solution as follows: drug.

b. Determine ka by feathering.
Time (hours) Concentration (ng/mL) c. Determine the equation that describes

0.25 2.85 the plasma drug concentration of the new
benzodiazepine.

0.50 5.43
2. Assuming that the drug in Question 1 is

0.75 7.75 80% absorbed, find (a) the absorption con-
1.00 9.84 stant, ka; (b) the elimination half-life, t1/2;

(c) the tmax, or time of peak drug concentra-
2.00 16.20

tion; and (d) the volume of distribution of
4.00 22.15 the patient.
6.00 23.01 3. Contrast the percent of drug-unabsorbed methods

10.00 19.09 for the determination of rate constant for
absorption, ka, in terms of (a) pharmacokinetic

14.00 13.90
model, (b) route of drug administration, and

20.00 7.97 (c) possible sources of error.

 

Pharmacokinetics of Oral Absorption 203

4. What is the error inherent in the measure- subjects. The following data represent the
ment of ka for an orally administered drug mean blood phenylpropanolamine hydrochlo-
that follows a two-compartment model when ride concentrations (ng/mL) after the oral
a one-compartment model is assumed in the administration of a single 25-mg dose of
calculation? phenylpropanolamine hydrochloride solution:

5. What are the main pharmacokinetic parameters
that influence (a) time for peak drug concen-

Concen- Concen-
tration and (b) peak drug concentration?

Time tration Time tration
6. Name a method of drug administration that (hours) (ng/mL) (hours) (ng/mL)

will provide a zero-order input.
7. A single oral dose (100 mg) of an antibiotic 0 0 3 62.98

was given to an adult male patient (43 years, 0.25 51.33 4 52.32

72 kg). From the literature, the pharmacokinetics 0.5 74.05 6 36.08
of this drug fits a one-compartment open model.
The equation that best fits the pharmacokinetics 0.75 82.91 8 24.88

of the drug is 1.0 85.11 12 11.83

C 45(e−0.17t e−1.5t
p = − ) 1.5 81.76 18 3.88

2 75.51 24 1.27
From the equation above, calculate (a) tmax,
(b) Cmax, and (c) t1/2 for the drug in this patient.
Assume Cp is in μg/mL and the first-order rate a. From the above data, obtain the rate constant
constants are in h−1. for absorption, ka, and the rate constant for

8. Two drugs, A and B, have the following phar- elimination, k, by the method of residuals.
macokinetic parameters after a single oral dose b. Is it reasonable to assume that ka > k for a
of 500 mg: drug in a solution? How would you deter-

mine unequivocally which rate constant

Drug ka (h−1) k (h−1) VD (mL) represents the elimination constant k?
c. From the data, which method, Wagner–

A 1.0 0.2 10,000
Nelson or Loo–Riegelman, would be more

B 0.2 1.0 20,000 appropriate to determine the order of the rate
constant for absorption?

Both drugs follow a one-compartment pharma- d. From your values, calculate the theoreti-
cokinetic model and are 100% bioavailable. cal tmax. How does your value relate to the
a. Calculate the tmax for each drug. observed tmax obtained from the subjects?
b. Calculate the Cmax for each drug. e. Would you consider the pharmacokinetics of

9. The bioavailability of phenylpropanolamine phenylpropanolamine HCl to follow a one-
hydrochloride was studied in 24 adult male compartment model? Why?

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Thérapie 57:205–240, 1937.

BIBLIOGRAPHY
Boxenbaum HG, Kaplan SA: Potential source of error in absorption Veng-Pedersena P, Gobburub JVS, Meyer MC, Straughn AB:

rate calculations. J Pharmacokinet Biopharm 3:257–264, 1975. Carbamazepine level-A in vivo–in vitro correlation (IVIVC):
Boyes R, Adams H, Duce B: Oral absorption and disposition A scaled convolution based predictive approach. Biopharm

kinetics of lidocaine hydrochloride in dogs. J Pharmacol Exp Drug Dispos 21:1–6, 2000.
Ther 174:1–8, 1970. Wagner JG, Nelson E: Kinetic analysis of blood levels and urinary

Dvorchik BH, Vesell ES: Significance of error associated with use excretion in the absorptive phase after single doses of drug.
of the one-compartment formula to calculate clearance of 38 J Pharm Sci 53:1392, 1964.
drugs. Clin Pharmacol Ther 23:617–623, 1978.

 

Multiple-Dosage Regimens

9 Rodney C. Siwale and Shabnam N. Sani

Chapter Objectives Earlier chapters of this book discussed single-dose drug and
constant-rate drug administration. By far though, most drugs are

»» Define the index for measuring
given in several doses, for example, multiple doses to treat chronic

drug accumulation.
disease such as arthritis, hypertension, etc. After single-dose drug

»» Define drug accumulation and administration, the plasma drug level rises above and then falls
drug accumulation t1/2. below the minimum effective concentration (MEC), resulting in a

»» Explain the principle of decline in therapeutic effect. To treat chronic disease, multiple-

superposition and its dosage or IV infusion regimens are used to maintain the plasma

assumptions in multiple-dose drug levels within the narrow limits of the therapeutic window

regimens. (eg, plasma drug concentrations above the MEC but below the
minimum toxic concentration or MTC) to achieve optimal clinical

»» Calculate the steady-state Cmax effectiveness. These drugs may include antibacterials, cardioton-
and Cmin after multiple IV bolus ics, anticonvulsants, hypoglycemics, antihypertensives, hormones,
dosing of drugs. and others. Ideally, a dosage regimen is established for each drug

»» Calculate k and V to provide the correct plasma level without excessive fluctuation
D of

aminoglycosides in multiple- and drug accumulation outside the therapeutic window.
dose regimens. For certain drugs, such as antibiotics, a desirable MEC can be

determined. For drugs that have a narrow therapeutic range
»» Adjust the steady-state Cmax and

(eg, digoxin and phenytoin), there is a need to define the therapeu-
Cmin in the event the last dose

tic minimum and maximum nontoxic plasma concentrations
is given too early, too late, or

(MEC and MTC, respectively). In calculating a multiple-dose regi-
totally missed following multiple

men, the desired or target plasma drug concentration must be
IV dosing.

related to a therapeutic response, and the multiple-dose regimen
must be designed to produce plasma concentrations within the
therapeutic window.

There are two main parameters that can be adjusted in
developing a dosage regimen: (1) the size of the drug dose and
(2) t, the frequency of drug administration (ie, the time interval
between doses).

DRUG ACCUMULATION
To calculate a multiple-dose regimen for a patient or patients,
pharmacokinetic parameters are first obtained from the plasma
level–time curve generated by single-dose drug studies. With these
pharmacokinetic parameters and knowledge of the size of the dose
and dosage interval (t), the complete plasma level–time curve or

205

 

206 Chapter 9

the plasma level may be predicted at any time after concentration obtained by adding the residual drug
the beginning of the dosage regimen. concentration obtained after each previous dose. The

For calculation of multiple-dose regimens, it is superposition principle may be used to predict drug
necessary to decide whether successive doses of concentrations after multiple doses of many drugs.
drug will have any effect on the previous dose. The Because the superposition principle is an overlay
principle of superposition assumes that early doses method, it may be used to predict drug concentra-
of drug do not affect the pharmacokinetics of subse- tions after multiple doses given at either equal or
quent doses. Therefore, the blood levels after the unequal dosage intervals. For example, the plasma
second, third, or nth dose will overlay or superim- drug concentrations may be predicted after a drug
pose the blood level attained after the (n−1)th dose. dose is given every 8 hours, or 3 times a day before

In addition, the AUC = ( ∫ C AM, 12 noon, and 6 PM.
p dt) for the first dose is meals at 8

0
There are situations, however, in which the

equal to the steady-state area between doses, that is,
superposition principle does not apply. In these

t
∫ 2

( Cp dt) as shown in Fig. 9-1. cases, the pharmacokinetics of the drug change after
t1

The principle of superposition allows the pharma- multiple dosing due to various factors, including
cokineticist to project the plasma drug concentration– changing pathophysiology in the patient, saturation
time curve of a drug after multiple consecutive doses of a drug carrier system, enzyme induction, and
based on the plasma drug concentration–time curve enzyme inhibition. Drugs that follow nonlinear phar-
obtained after a single dose. The basic assumptions are macokinetics (see Chapter 10) generally do not have
(1) that the drug is eliminated by first-order kinetics predictable plasma drug concentrations after multi-
and (2) that the pharmacokinetics of the drug after a ple doses using the superposition principle.
single dose (first dose) are not altered after taking mul- If the drug is administered at a fixed dose and a
tiple doses. fixed dosage interval, as is the case with many mul-

The plasma drug concentrations after multiple tiple-dose regimens, the amount of drug in the body
doses may be predicted from the plasma drug con- will increase and then plateau to a mean plasma level
centrations obtained after a single dose. In Table 9-1, higher than the peak Cp obtained from the initial
the plasma drug concentrations from 0 to 24 hours dose (Figs. 9-1 and 9-2). When the second dose is
are measured after a single dose. A constant dose given after a time interval shorter than the time
of drug is given every 4 hours and plasma drug con- required to “completely” eliminate the previous
centrations after each dose are generated using the dose, drug accumulation will occur in the body. In
data after the first dose. Thus, the predicted plasma other words, the plasma concentrations following the
drug concentration in the patient is the total drug second dose will be higher than corresponding

plasma concentrations immediately following the
first dose. However, if the second dose is given after
a time interval longer than the time required to elimi-
nate the previous dose, drug will not accumulate
(see Table 9-1).

AUC = As repetitive equal doses are given at a constant
t

∫ 2
t Cpdt frequency, the plasma level–time curve plateaus and
1

a steady state is obtained. At steady state, the plasma

AUC = ∫0C pdt drug levels fluctuate between C∞

max and C∞

min. Once
Time (hours) t steady state is obtained, C∞

t max and C∞

min are constant
1 t2

and remain unchanged from dose to dose. In addi-
t2

Doses tion, the AUC between ( ∫ Cp dt) is constant during
t1

FIGURE 91 a dosing interval at steady state (see Fig. 9-1). The
Simulated data showing blood levels after

administration of multiple doses and accumulation of blood C∞

max is important in determining drug safety. The
levels when equal doses are given at equal time intervals. C∞

max should always remain below the MTC. The C∞

max

Blood level

 

Multiple-Dosage Regimens 207

TABLE 91 Predicted Plasma Drug Concentrations for Multiple-Dose Regimen Using the
Superposition Principlea

Plasma Drug Concentration ( lg/mL)
Dose

Number Time (h) Dose 1 Dose 2 Dose 3 Dose 4 Dose 5 Dose 6 Total

1 0 0 0

1 21.0 21.0

2 22.3 22.3

3 19.8 19.8

2 4 16.9 0 16.9

5 14.3 21.0 35.3

6 12.0 22.3 34.3

7 10.1 19.8 29.9

3 8 8.50 16.9 0 25.4

9 7.15 14.3 21.0 42.5

10 6.01 12.0 22.3 40.3

11 5.06 10.1 19.8 35.0

4 12 4.25 8.50 16.9 0 29.7

13 3.58 7.15 14.3 21.0 46.0

14 3.01 6.01 12.0 22.3 43.3

15 2.53 5.06 10.1 19.8 37.5

5 16 2.13 4.25 8.50 16.9 0 31.8

17 1.79 3.58 7.15 14.3 21.0 47.8

18 1.51 3.01 6.01 12.0 22.3 44.8

19 1.27 2.53 5.06 10.1 19.8 38.8

6 20 1.07 2.13 4.25 8.50 16.9 0 32.9

21 0.90 1.79 3.58 7.15 14.3 21.0 48.7

22 0.75 1.51 3.01 6.01 12.0 22.3 45.6

23 0.63 1.27 2.53 5.06 10.1 19.8 39.4

24 0.53 1.07 2.13 4.25 8.50 16.9 33.4

aA single oral dose of 350 mg was given and the plasma drug concentrations were measured for 0–24 h. The same plasma drug concentrations are
assumed to occur after doses 2–6. The total plasma drug concentration is the sum of the plasma drug concentrations due to each dose. For this
example, VD = 10 L, t1/2 = 4 h, and ka = 1.5 h−1. The drug is 100% bioavailable and follows the pharmacokinetics of a one-compartment open model.

 

208 Chapter 9

is also a good indication of drug accumulation. If a 50
drug produces the same C∞

max at steady state, com-
pared with the (C 1) f e h i

n max a t r t e f rst dose, then 45
− 600 mg every 24 h

there is no drug accumulation. If C∞

max is much larger
than (C 1) , t e h r

n max h n t e e is significant accumulation 40

during the multiple-dose regimen. Accumulation is
affected by the elimination half-life of the drug and 35

200 mg
the dosing interval. The index for measuring drug every 8 h
accumulation R is 30 IV infusion

(25 mg/h)

(C∞ )
R = max (9.1) 25 300 mg

(C every 12 h
n=1)max

20

Substituting for Cmax after the first dose and at steady
state yields 15

D0 /VD[1/(1− e−kτ )] 10
R = (9.2)

D0 /VD
5

1
R =

1− e−kτ
0

0 10 20 30 40 50
Time (hours)

Equation 9.2 shows that drug accumulation mea-
sured with the R index depends on the elimination FIGURE 93 Simulated plasma drug concentration–time

constant and the dosing interval and is independent curves after IV infusion and oral multiple doses for a drug with an
elimination half-life of 4 hours and apparent VD of 10 L. IV infusion

of the dose. For a drug given in repetitive oral doses,
given at a rate of 25 mg/h, oral multiple doses are 200 mg every

the time required to reach steady state is dependent 8 hours, 300 mg every 12 hours, and 600 mg every 24 hours.
on the elimination half-life of the drug and is inde-
pendent of the size of the dose, the length of the
dosing interval, and the number of doses. For exam-

Furthermore, if the drug is given at the same
ple, if the dose or dosage interval of the drug is

dosing rate but as an infusion (eg, 25 mg/h), the aver-
altered as shown in Fig. 9-2, the time required for the

age plasma drug concentrations will (C∞

drug to reach steady state is the same, but the final av ) be the
same but the fluctuations between C∞

max and C∞

min will
steady-state plasma level changes proportionately.

vary (Fig. 9-3). An average steady-state plasma drug
concentration is obtained by dividing the area under

t
the curve (AUC) for a dosing period (ie, ∫ 2

C dt) by
t p

1000 1
the dosing interval t, at steady state.

800 An equation for the estimation of the time to
600 Max reach one-half of the steady-state plasma levels or

Min
400 the accumulation half-life has been described by van

Rossum and Tomey (1968).
200

0
0 20 40 60 80 100  k 

Accumulation 1 3. a
t1/2 = t1/2 + 3log  (9.3)

Time (hours)  ka − k
FIGURE 92 Amount of drug in the body as a function of
time. Equal doses of drug were given every 6 hours (upper curve) For IV administration, ka is very rapid (approaches ∞);
and every 8 hours (lower curve). ka and k remain constant. k is very small in comparison to ka and can be omitted

Amount of drug in body (mg)

Plasma level (mg/mL)

 

Multiple-Dosage Regimens 209

in the denominator of Equation 9.3. Thus, Equation 9.3 of the drug and the dosage interval t (see Table 9-3).
reduces to If the drug is given at a dosage interval equal to the

half-life of the drug, then 6.6 doses are required to
 k 

Accumulation a
t1/2 = t1/2 1+ 3.3log  (9.4) reach 99% of the theoretical steady-state plasma

 ka  drug concentration. The number of doses needed to
reach steady state is 6.6t1/2/t, as calculated in the far

Since ka/ka = 1 and log 1 = 0, the accumulation t1/2 of
right column of Table 9-3. As discussed in Chapter 6,

a drug administered intravenously is the elimination
Table 6-1, it takes 4.32 half-lives to reach 95% of

t1/2 of the drug. From this relationship, the time to
steady state.

reach 50% steady-state drug concentrations is depen-
dent only on the elimination t1/2 and not on the dose
or dosage interval. CLINICAL EXAMPLE

As shown in Equation 9.4, the accumulation
t1/2 is directly proportional to the elimination t1/2. Paroxetine (Prozac) is an antidepressant drug with a
Table 9-2 gives the accumulation t1/2 of drugs with long elimination half-life of 21 hours. Paroxetine is
various elimination half-lives given by multiple well absorbed after oral administration and has a tmax
oral doses (see Table 9-2). of about 5 hours, longer than most drugs. Slow

From a clinical viewpoint, the time needed to elimination may cause the plasma curve to peak
reach 90% of the steady-state plasma concentration is slowly. The tmax is affected by k and ka, as discussed
3.3 times the elimination half-life, whereas the time in Chapter 8. The Cmax for paroxetine after multiple
required to reach 99% of the steady-state plasma dosing of 30 mg of paroxetine for 30 days in one
concentration is 6.6 times the elimination half-life study ranged from 8.6 to 105 ng/mL among 15 sub-
(Table 9-3). It should be noted from Table 9-3 that at jects. Clinically it is important to achieve a stable
a constant dose size, the shorter the dosage interval, steady-state level in multiple dosing that does not
the larger the dosing rate (mg/h), and the higher the “underdose” or overdose the patient. The pharmacist
steady-state drug level. should advise the patient to follow the prescribed

The number of doses for a given drug to reach dosing interval and dose as accurately as possible.
steady state is dependent on the elimination half-life Taking a dose too early or too late contributes to

TABLE 92 Effect of Elimination Half-Life and Absorption Rate Constant on Accumulation
Half-Life after Oral Administrationa

Elimination Elimination Rate Absorption Rate Accumulation
Half-life (h) constant (1/h) Constant (1/h) Half-life (h)

4 0.173 1.50 4.70

8 0.0866 1.50 8.67

12 0.0578 1.50 12.8

24 0.0289 1.50 24.7

4 0.173 1.00 5.09

8 0.0866 1.00 8.99

12 0.0578 1.00 13.0

24 0.0289 1.00 25.0

aAccumulation half-life is calculated by Equation 8.3, and is the half-time for accumulation of the drug to 90% of the steady-state plasma drug
concentration.

 

210 Chapter 9

TABLE 93 Interrelation of Elimination Half-Life, Dosage Interval, Maximum Plasma
Concentration, and Time to Reach Steady-State Plasma Concentrationa

Elimination Dosage Interval, Time for NO. Doses to Reach 99%
Half-Life (h) s (h) C∞ (lg/mL) C∞b (h) Steady State

max av

0.5 0.5 200 3.3 6.6

0.5 1.0 133 3.3 3.3

1.0 0.5 341 6.6 13.2

1.0 1.0 200 6.6 6.6

1.0 2.0 133 6.6 3.3

1.0 4.0 107 6.6 1.65

1.0 10.0 100c 6.6 0.66

2.0 1.0 341 13.2 13.2

2.0 2.0 200 13.2 6.1

aA single dose of 1000 mg of three hypothetical drugs with various elimination half-lives but equal volumes of distribution (VD = 10 L) were given by
multiple IV doses at various dosing intervals. All time values are in hours; C∞ = maximum steady-state concentration; (C∞b

max av ) = average steady-state
plasma concentration; the maximum plasma drug concentration after the first dose of the drug is (Cn

=1)max = 100 mg/mL.

bTime to reach 99% of steady-state plasma concentration.

cSince the dosage interval, t, is very large compared to the elimination half-life, no accumulation of drug occurs.

variation. Individual variation in metabolism rate The fraction ( f ) of the dose remaining in the body is
can also cause variable blood levels, as discussed related to the elimination constant (k) and the dosage
later in Chapter 13. interval (t) as follows:

D
f B −kτ

= = e (9.7)
D

REPETITIVE INTRAVENOUS 0

INJECTIONS With any given dose, f depends on k and t. If t is
large, f will be smaller because DB (the amount of

The maximum amount of drug in the body follow-
drug remaining in the body) is smaller.

ing a single rapid IV injection is equal to the dose of
the drug. For a one-compartment open model, the
drug will be eliminated according to first-order

EXAMPLES »» »
kinetics.

1. A patient receives 1000 mg every 6 hours by
D D e−kτ

= (9.5)
B 0 repetitive IV injection of an antibiotic with an

elimination half-life of 3 hours. Assume the drug
If t is equal to the dosage interval (ie, the time between is distributed according to a one-compartment
the first dose and the next dose), then the amount of model and the volume of distribution is 20 L.
drug remaining in the body after several hours can be a. Find the maximum and minimum amounts of
determined with drug in the body.

b. Determine the maximum and minimum

D τ plasma concentrations of the drug.
= D e−k (9.6)

B 0

 

Multiple-Dosage Regimens 211

TABLE 94 Fraction of the Dose in the Body
Solution before and after Intravenous Injections of a

1000-mg Dosea
a. The fraction of drug remaining in the body is

estimated by Equation 9.7. The concentration of Amount of Drug in Body
the drug declines to one-half after 3 hours (t1/2 =

Number of Doses Before Dose After Dose
3 h), after which the amount of drug will again
decline by one-half at the end of the next 3 hours. 1 0 1000

Therefore, at the end of 6 hours, only one-quarter,
2 250 1250

or 0.25, of the original dose remains in the body.
Thus f is equal to 0.25. To use Equation 9.7, we 3 312 1312

must first find the value of k from the t1/2. 4 328 1328

0.693 0.693 5 332 1332
k = = = 0.231h−1

t1/2 3 6 333 1333

The time interval τ is equal to 6 hours. From 7 333 1333

Equation 9.7, ∞ 333 1333

f = e–(0.231)(6)
af = 0.25.

f = 0.25

Substituting known data, we obtain
In this example, 1000 mg of drug is given

intravenously, so the amount of drug in the body ∞ 1000
D =
max − =1333 mg

is immediately increased by 1000 mg. At the end 1 0.25

of the dosage interval (ie, before the next dose), Then, from Equation 9.8,

the amount of drug remaining in the body is 25%
D∞

of the amount of drug present just after the previ-
min =1333 − 1000 = 333 mg

ous dose, because f = 0.25. Thus, if the value of f
The average amount of drug in the body at steady

is known, a table can be constructed relating the
state, D∞

av, can be found by Equation 9.10 or
fraction of the dose in the body before and after

Equation 9.11. F is the fraction of dose absorbed. For
rapid IV injection (Table 9-4).

an IV injection, F is equal to 1.0.
From Table 9-4 the maximum amount of

drug in the body is 1333 mg and the minimum FD
0

D∞

a = (9.10)
v

amount of drug in the body is 333 mg. The differ- kτ

ence between the maximum and minimum val- FD01.44t ∞ 1/2
D (9.1

ues, D0, will always equal the injected dose. av = 1)
τ

Equations 9.10 and 9.11 can be used for repetitive
Dmax −Dmin =D0 (9.8)

dosing at constant time intervals and for any route
of administration as long as elimination occurs from

In this example, the central compartment. Substitution of values
from the example into Equation 9.11 gives

1333 − 333 = 1000 mg

(1)(1000)(1.44)(3)
D∞

av = = 720 mg
D∞ can also be calculated directly by the
max 6

relationship Since the drug in the body declines exponentially
D (ie, first-order drug elimination), the value D∞

av is not
D∞ 0

max = (9.9)
1− f the arithmetic mean of D∞ and D∞

min . The limitation
max

 

212 Chapter 9

of using D∞

av is that the fluctuations of D∞ and D∞ is dependent on both AUC and t. The C∞

max min av reflects
are not known. drug exposure after multiple doses. Drug expo-

b. To determine the concentration of drug in the sure is often related to drug safety and efficacy as

body after multiple doses, divide the amount discussed later in Chapter 21. For example, drug

of drug in the body by the volume in which it is exposure is closely monitored when a cytotoxic

dissolved. For a one-compartment model, the or immunosuppressive, anticancer drug is admin-

maximum, minimum, and steady-state concen- istered during therapy. AUC may be estimated by

trations of drug in the plasma are found by the sampling several plasma drug concentrations over

following equations: time. Theoretically, AUC is superior to sampling
just the Cmax or Cmin. For example, when cyclospo-

D∞

C∞ = max (9.12) rine dosing is clinically evaluated using AUC, the
max V

D AUC is approximately estimated by two or three

D∞ points. Dosing error is less than using AUC com-
min

C∞ (9.13)
min =

V pared to the trough method alone (Primmett et al,
D

1998). In general, Cmin or trough level is more fre-
D∞

av
C∞

av = (9.14) quently used than C∞ . C is the drug concentra-
V max min

D
tion just before the next dose is given and is less

A more direct approach to finding C∞ , and C∞ ,
max min variable than peak drug concentration, C∞ . The

max

is C∞

av: sample time for C∞ is approximated and the true
max

C 0 C∞ may not be accurately estimated. In some
max

∞ p
Cmax = (9.15)

1 e−kτ cases, the plasma trough level, C∞

− min is considered
by some investigators as a more reliable sample

where C 0
p is equal to D0/VD. since the drug is equilibrated with the surround-

ing tissues, although this may also depend on
0

C e−kτ

∞ p (9.16)
C other factors.

min =
1− e−kτ

The AUC is related to the amount of drug

FD absorbed divided by total body clearance (Cl), as
∞ 0

Cav = (9.17)
V shown in the following equation:

Dkτ

For this example, the values for C∞ , C∞ C∞

max min, and av t FD FD
[AUC] 2 = 0 = 0 (9.19)

t1
are 66.7, 16.7, and 36.1 μg/mL, respectively. Cl kVD

As mentioned, C∞

av is not the arithmetic mean
Substitution of FD0/kVD for AUC in Equation 9.18

of C∞ and C∞

max min because plasma drug concentra-
gives Equation 9.17. Equation 9.17 or 9.18 can be

tion declines exponentially. The C∞

av is equal to
used to obtain C∞

t t2 av after a multiple-dose regimen
[AUC] 2 or ( al at steady

t ∫ Cp dt ) for a dosage interv
1 t regardless of the route of administration.

1

state divided by the dosage interval t. It is sometimes desirable to know the

t plasma drug concentration at any time after the
[AUC] 2

t
∞ 1

C administration of n doses of drug. The general
av = (9.18)

τ
expression for calculating this plasma drug con-

C∞ centration is
av gives an estimate of the mean plasma drug

concentration at steady state. The C∞

av is often the
D0 1− e−nkτ 

target drug concentration for optimal therapeu- Cp = e−kt
VD 

(9.20)
1− e−kτ 

tic effect and gives an indication as to how long
this plasma drug concentration is maintained dur- where n is the number of doses given and t is the

ing the dosing interval (between doses). The C∞ time after the nth dose.
av

 

Multiple-Dosage Regimens 213

Problem of a Missed Dose
At steady state, e−nkt approaches zero and

Equation 9.22 describes the plasma drug concentra-
Equation 9.20 reduces to

tion t hours after the nth dose is administered; the
D0  1 

C∞ −
p =  τ  e kt

VD 1− e−k  (9.21) doses are administered t hours apart according to a
multiple-dose regimen:

where C∞

p is the steady-state drug concentration at
D 1− e−nkτ 

time t after the dose. C = 0  e−kt
p V 1− e− (9.22)

kτ 
D 

2. The patient in the previous example received 1000
mg of an antibiotic every 6 hours by repetitive IV Concentration contributed by the missing dose is
injection. The drug has an apparent volume of dis-

D
tribution of 20 L and elimination half-life of 3 hours. C′ = 0 e−ktmiss

p (9.23)
Calculate (a) the plasma drug concentration, Cp at VD

3 hours after the second dose, (b) the steady-state
in which tmiss = time elapsed since the scheduled

plasma drug concentration, C∞

p at 3 hours after the
dose was missed. Subtracting Equation 9.23 from

last dose, (c) C∞ , (d) C∞

ma min, and (e) CSS.
x Equation 9.20 corrects for the missing dose as shown

Solution in Equation 9.24.

a. The Cp at 3 hours after the second dose—use D 1− − τ  
C 0 e nk

Equation 9.20 and let n = 2, t = 3 hours, and make =  e−kt − e ktmiss
p (9

V  − −kτ  −  .24)
D  1 e  

other appropriate substitutions.

1000 1− e−(2)(0.231)(6) Note: If steady state is reached (ie, either n = large
C = e−0.231(3)
p 20  1− e−(0.231)(6)  or after many doses), the equation simplifies to

Equation 9.25. Equation 9.25 is useful when steady

Cp = 31.3 mg/L state is reached.

b. The C∞

p at 3 hours after the last dose—because D  −k 
C = 0 e t

 −
steady state is reached, use Equation 9.21 and mis

p 1 −  − e kt s (9.25)
VD  − e kt 

perform the following calculation:

1000 1
C∞
p =  

  e−0.231(3) Generally, if the missing dose is recent, it will affect
20 1− e−0.231(6) the present drug level more. If the missing dose is

several half-lives later (>5t1/2), the missing dose
C∞
p = 33.3 mg/L

may be omitted because it will be very small.

c. The C∞ is calculated from Equation 9.15. Equation 9.24 accounts for one missing dose, but
max

several missing doses can be subtracted in a similar
1000/20

C∞ =
max − = 66.7 mg/L way if necessary.

1− e (0.231)(6)

d. The C∞

min may be estimated as the drug concen-
tration after the dosage interval t, or just before

EXAMPLE »» »
the next dose.

C∞ = C∞ e−kt 66.7e (0.231)(6)
min max = − =16.7 mg/L A cephalosporin (k = 0.2 h−1, VD = 10 L) was admin-

istered by IV multiple dosing; 100 mg was injected
e. The C∞

av is estimated by Equation 9.17—because
every 6 hours for 6 doses. What was the plasma

the drug is given by IV bolus injections, F = 1.
drug concentration 4 hours after the sixth dose

1000
C∞

av = = 36.1mg/L (ie, 40 hours later) if (a) the fifth dose was omitted,
(0.231)(20)(6)

(b) the sixth dose was omitted, and (c) the fourth
C∞

av is represented as CSS in some references. dose was omitted?

 

214 Chapter 9

the late or early dose added back to take into account
Solution the actual time of dosing, using Equation 9.26.

Substitute k = 0.2 h−1, VD = 10 L, D = 100 mg, n = 6,
D 

0 1− e−nkt
t = 4 hours, and t = 6 hours into Equation 9.20 and C e kt e− 

=  − ktmiss e−ktactual
p V e− − +

kt
D  −  (9.26)

evaluate: 1 
Cp = 6.425 mg/L

in which tmiss = time elapsed since the dose (late or
If no dose was omitted, then 4 hours after the sixth early) is scheduled, and tactual = time elapsed since the
injection, Cp would be 6.425 mg/L. dose (late or early) is actually taken. Using a similar
a. Missing the fifth dose, its contribution must be approach, a second missed dose can be subtracted

subtracted off, tmiss = 6 + 4 = 10 hours (the time from Equation 9.20. Similarly, a second late/early
elapsed since missing the dose) using the steady- dose may be corrected by subtracting the scheduled
state equation: dose followed by adding the actual dose. Similarly, if

D0 100 a different dose is given, the regular dose may be
C e−kt

′ = miss = e−(0.2×10)
p V subtracted and the new dose added back.

D 10

Drug concentration correcting for the missing
dose = 6.425 − 1.353 = 5.072 mg/L.

EXAMPLE »» »
b. If the sixth dose is missing, tmiss = 4 hours:

D0 100 Assume the same drug as above (ie, k = 0.2 h−1
C ′ = e−ktmiss = e−(0.2×4) , VD =

p = 4.493 mg/L
VD 10 10 L) was given by multiple IV bolus injections and

Drug concentration correcting for the missing that at a dose of 100 mg every 6 hours for 6 doses.

dose = 6.425 − 4.493 = 1.932 mg/L. What is the plasma drug concentration 4 hours
after the sixth dose, if the fifth dose were given an

c. If the fourth dose is missing, tmiss = 12 + 4 =
hour late?

16 hours:
Substitute into Equation 9.26 for all unknowns:

D0 100
C ′ = e−ktmiss = e−(0.2×16)

p = 0.408 mg/L k = 0.2 h−1, VD = 10 L, D = 100 mg, n = 6, t = 4 h, t = 6 h,
VD 10

tmiss = 6 + 4 = 10 hours, tactual = 9 hours (taken 1 hour
The drug concentration corrected for the missing late, ie, 5 hours before the sixth dose).

dose = 6.425 − 0.408 = 6.017 mg/L.
Note: The effect of a missing dose becomes D k

0 1− e−n τ
− 

C = e kτ
 − − e−kt miss + e−kt actual

p
VD  1− e kτ less pronounced at a later time. A strict dose

regimen compliance is advised for all drugs.
Cp = 6.425 – 1.353 + 1.653 = 6.725 mg/L

With some drugs, missing a dose can have a
serious effect on therapy. For example, compli- Note: 1.353 mg/L was subtracted and 1.653 mg/mL
ance is important for the anti-HIV1 drugs such was added because the fifth dose was not given as
as the protease inhibitors. planned, but was given 1 hour later.

INTERMITTENT INTRAVENOUS
Early or Late Dose Administration during
Multiple Dosing INFUSION

When one of the drug doses is taken earlier or later Intermittent IV infusion is a method of successive
than scheduled, the resulting plasma drug concentra- short IV drug infusions in which the drug is given by
tion can still be calculated based on the principle of IV infusion for a short period of time followed by a
superposition. The dose can be treated as missing, with drug elimination period, then followed by another

 

Multiple-Dosage Regimens 215

4.0
EXAMPLE »» »

3.5

3.0
An antibiotic was infused with a 40-mg IV dose

2.5 over 2 hours. Ten hours later, a second dose of
2.0 40 mg was infused, again over 2 hours. (a) What
1.5 is the plasma drug concentration 2 hours after the
1.0 start of the first infusion? (b) What is the plasma

0.5 drug concentration 5 hours after the second dose

0.0 infusion was started? Assume k = 0.2 h−1 and VD =
0 2 4 6 8 10 12 14 16 18 20 10 L for the antibiotic.

Time (hours)
Solution

FIGURE 94 Plasma drug concentration after two doses
by IV infusion. Data from Table 9-5. The predicted plasma drug concentrations after

the first and second IV infusions are shown in
Table 9-5. Using the principle of superposition, the

short IV infusion (Fig. 9-4). In drug regimens
total plasma drug concentration is the sum of the

involving short IV infusion, the drug may not reach
residual drug concentrations due to the first IV infu-

steady state. The rationale for intermittent IV infu- sion (column 3) and the drug concentrations due to
sion is to prevent transient high drug concentrations the second IV infusion (column 4). A graphical rep-
and accompanying side effects. Many drugs are better resentation of these data is shown in Fig. 9-4.
tolerated when infused slowly over time compared to

a. The plasma drug concentration at 2 hours after
IV bolus dosing.

the first IV infusion starts is calculated from Equa-
tion 9.28.

Administering One or More Doses by 40/2
Constant Infusion: Superposition of Several Cp (1 e−0.2/2

= − )= 3.30 mg/L
10× 0.2

IV Infusion Doses
b. From Table 9-5, the plasma drug concentration

For a continuous IV infusion (see Chapter 7):
at 15 hours (ie, 5 hours after the start of the sec-

R R ond IV infusion) is 2.06 μg/mL. At 5 hours after
C = (1− −

p e kt ) = (1− e−kt ) (9.27)
Cl kVD the second IV infusion starts, the plasma drug

concentration is the sum of the residual plasma
Equation 9.27 may be modified to determine drug drug concentrations from the first 2-hour infu-
concentration after one or more short IV infusions sion according to first-order elimination and the
for a specified time period (Equation 9.28). residual plasma drug concentrations from the

D second 2-hour IV infusion as shown in the fol-
Cp = (1− e−kt ) (9.28)

t lowing scheme:
infVDk

where R = rate of infusion = D/tinf, D = size of infu- 10 hours 10 hours
sion dose, and tinf = infusion period.

First Stopped Second Stopped
After the infusion is stopped, the drug concen- infusion (no infusion infusion (no infusion

tration post-IV infusion is obtained using the first- for 2 hours for 8 hours) for 2 hours for 8 hours)
order equation for drug elimination:

The plasma drug concentration is calculated
C = −

p Cstope
kt (9.29) using the first-order elimination equation, where

Cstop is the plasma drug concentration at the stop
where Cstop = concentration when infusion stops, and

of the 2-hour IV infusion.
t = time elapsed since infusion stopped.

mcg/mL

 

216 Chapter 9

TABLE 95 Drug Concentration after Two Intravenous Infusionsa

Plasma Drug Plasma Drug Total Plasma
Concentration Concentration Drug

Time(h) after Infusion 1 after Infusion 2 Concentration

Infusion 1 begins 0 0 0

1 1.81 1.81

Infusion 1 stopped 2 3.30 3.30

3 2.70 2.70

4 2.21 2.21

5 1.81 1.81

6 1.48 1.48

7 1.21 1.21

8 0.99 0.99

9 0.81 0.81

Infusion 2 begins 10 0.67 0 0.67

11 0.55 1.81 2.36

Infusion 2 stopped 12 0.45 3.30 3.74

13 0.37 2.70 3.07

14 0.30 2.21 2.51

15 0.25 1.81 2.06

aDrug is given by a 2-hour infusion separated by a 10-hour drug elimination interval. All drug concentrations are in lg/mL. The declining
drug concentration after the first infusion dose and the drug concentration after the second infusion dose give the total plasma drug
concentration.

CLINICAL EXAMPLE
The plasma drug concentration after the com-

pletion of the first IV infusion when t = 15 hours is Gentamicin sulfate was given to an adult male
patient (57 years old, 70 kg) by intermittent IV infu-

C = C e–kt = 3.30e–0.2×15
p stop = 0.25 µg/L sions. One-hour IV infusions of 90 mg of gentami-

cin was given at 8-hour intervals. Gentamicin
The plasma drug concentration 5 hours after the

clearance is similar to creatinine clearance and was
second IV infusion is

estimated as 7.2 L/h with an elimination half-life of

C = C e–kt = 3.30e–0.2×3
p stop =1.81µg/mL 3 hours.

a. What is the plasma drug concentration after the
The total plasma drug concentration 5 hours after

first IV infusion?
the start of the second IV infusion is

b. What is the peak plasma drug concentration,

0.25 mg/L + 1.81 mg/L = 2.06 mg/L. Cmax, and the trough plasma drug concentration,
Cmin, at steady state?

 

Multiple-Dosage Regimens 217

Solution ESTIMATION OF k AND VD OF
a. The plasma drug concentration directly after the AMINOGLYCOSIDES IN CLINICAL

first infusion is calculated from Equation 9.27,
SITUATIONS

where R = 90 mg/h, Cl = 7.2 L/h, and k = 0.231 h−1.
The time for infusion, tint, is 1 hour. As illustrated above, antibiotics are often infused

90 intravenously by multiple doses, so it is desirable to
C = (1− e−(0.231)(1)

p ) = 2.58 mg/L
7.2 adjust the recommended starting dose based on the

patient’s individual k and VD values. According to
b. The C∞

max at steady state may be obtained from Sawchuk and Zaske (1976), individual parameters
Equation 9.30. for aminoglycoside pharmacokinetics may be deter-

∞ R(1− e−ktinf ) 1 mined in a patient by using a limited number of
Cmax = (9.30)

Cl (1− e−kt ) plasma drug samples taken at appropriate time inter-
vals. The equation was simplified by replacing an

where Cmax is the peak drug concentration fol-
elaborate model with the one-compartment model to

lowing the nth infusion, at steady state, tinf is
describe drug elimination and appropriately avoid-

the time period of infusion, and t is the dosage
ing the distributive phase. The plasma sample should

interval. The term 1/(1 − e−kt) is the accumula-
be collected 15–30 minutes postinfusion (with infu-

tion factor for repeated drug administration.
sion lasting about 60 minutes) and, in patients with

Substitution in Equation 9.30 gives
poor renal function, 1–2 hours postinfusion, to allow

90(1 e−(0.231)(1)
− ) 1

C∞ adequate tissue distribution. The second and third
max = ×

7.2 (1 −(0.231)(8)
− e ) blood samples should be collected about 2–3 half-

lives later, in order to get a good estimation of the
= 3.06 mg/L

slope. The data may be determined graphically or by
The plasma drug concentration C∞

p at any time t regression analysis using a scientific calculator or
after the last infusion ends when steady state computer program.
is obtained by Equation 9.31 and assumes that
plasma drug concentrations decline according R(1− e−ktinf )

VD = (9.32)
to first-order elimination kinetics. [C∞ f

max −C∞ e−ktin
min

R(1 −
− e ktinf ) 1

C∞ e−k (t)
p = × × (9.31)

Cl (1 e−kt The dose of aminoglycoside is generally fixed by
− )

the desirable peak, C∞

max, and trough plasma concen-
where tinf is the time for infusion and t is the tration, C∞

min. For example, C∞

max for gentamicin may
time period after the end of the infusion. be set at 6–10 mg/mL with the steady-state trough

The trough plasma drug concentration, level, C∞

min, generally about 0.5–2 mg/mL, depending
C∞

min, at steady state is the drug concentration on the severity of the infection and renal consider-
just before the start of the next IV infusion or ations. The upper range is used only for life-threat-
after a dosage interval equal to 8 hours after ening infections. The infusion rate for any desired
the last infusion stopped. Equation 9.31 can peak drug concentration may be calculated using
be used to determine the plasma drug con- Equation 9.33.
centration at any time after the last infusion is
stopped (after steady state has been reached). V kC∞ (1 −kτ

D max − e )
R = (9.33)

(1 e−kt
− inf

90(1 e−(0.231)(1) −(0.231)(8) )
− ) e

C∞

min = ×
7.2 (1 e−(0.231)(8)

− )
The dosing interval t between infusions may be

= 0.48 mg/L adjusted to obtain a desired concentration.

 

218 Chapter 9

the dosage interval (t) is shortened, then the value
Frequently Asked Questions for C∞ will increase. The C∞ will be predictably

av av

»»Is the drug accumulation index (R) applicable to any higher for drugs distributed in a small VD (eg, plasma
drug given by multiple doses or only to drugs that are water) or that have long elimination half-lives than
eliminated slowly from the body? for drugs distributed in a large VD (eg, total body

»»What are the advantages/disadvantages for giving water) or that have very short elimination half-lives.
a drug by a constant IV infusion, intermittent IV Because body clearance (ClT) is equal to kVD, substi-
infusion, or multiple IV bolus injections? What drugs tution into Equation 9.17 yields
would most likely be given by each route of adminis- FD
tration? Why? C∞ 0

av = (9.36)
ClTτ

»»Why is the accumulation index, R, not affected by the
Thus, if ClT decreases, C∞ will increase.

dose or clearance of a drug? Would it be possible for av

a drug with a short half-life to have R much greater The C∞ does not give information concerning the
av

than 1? fluctuations in plasma concentration (C∞

max and C∞

min).
In multiple-dose regimens, Cp at any time can be
obtained using Equation 9.34, where n = nth dose. At
steady state, the drug concentration can be determined

MULTIPLE-ORAL-DOSE REGIMEN by letting n equal infinity. Therefore, e−nkt becomes

Figures 9-1 and 9-2 present typical cumulation approximately equal to zero and Equation 9.22 becomes

curves for the concentration of drug in the body after k 
∞ = aFD0  1 k 1

C 
   

− − a
p − e t e

VD (k

a k) 1− e k  −
− τ  τ  k t

multiple oral doses given at a constant dosage inter- 1− eka  
val. The plasma concentration at any time during an
oral or extravascular multiple-dose regimen, assum- (9.37)

ing a one-compartment model and constant doses The maximum and minimum drug concentrations
and dose interval, can be determined as follows: (C∞

max and C∞

min) can be obtained with the following

Fk D 1 e− aτ
 − nk   n

 t 1− e− kτ   equations:
C a 0  e−ka −  e−kt

p =
V (k − ka ) 1− e−kaτ  1− e−kτ
D  

 ∞ FD
= 0  1 

 −kt
C p

max − τ e (9.38)
VD 1− e k 

where n = number of doses, t = dosage interval, F =
fraction of dose absorbed, and t = time after admin- ∞ k  1 

C = aFD0
min e k (9.39)

V (  −
τ  τ

k − k) 1− e−k
istration of n doses. 

D a

The mean plasma level at steady state, C∞ , is
av The time at which maximum (peak) plasma concen-

determined by a similar method to that employed for
tration (or tmax) occurs following a single oral dose is

repeat IV injections. Equation 9.17 can be used for
finding C∞ for any route of administration. 2.3 k

av = a
tmax − log (9.40)

FD ka k k

C∞ 0
av = (9.17)

VDkτ whereas the peak plasma concentration, tp, following

Because proper evaluation of F and V multiple doses is given by Equation 9.41.
D requires IV

data, the AUC of a dosing interval at steady state 1 k (1− e−kτ )
tp = − ln a

may be substituted in Equation 9.17 to obtain (9.41)
k k 

k(1− e−k τ 
a

a )

∫ C dt ∞
∞ 0 p [AUC] ( . 5)

C 0 9
av = = 3 Large fluctuations between C∞

max and C∞

min can be
τ τ hazardous, particularly with drugs that have a narrow

One can see from Equation 9.17 that the magnitude therapeutic index. The larger the number of divided
of C∞ is directly proportional to the size of the dose doses, the smaller the fluctuations in the plasma drug

av

and the extent of drug absorbed. Furthermore, if concentrations. For example, a 500-mg dose of drug

 

Multiple-Dosage Regimens 219

given every 6 hours will produce the same C∞ value as
av

a 250-mg dose of the same drug given every 3 hours, c. Cp at 4 hours after the seventh dose may be calcu-

while the C∞ d using Equation 9.34, letting n = 7, t = 4, t = 8,
max and C∞

min fluctuations for the latter dose late

will be decreased by one-half (see Fig. 9-3). With drugs and making the appropriate substitutions.

that have a narrow therapeutic index, the dosage inter- (0.75)(250)(0.9)
val should not be longer than the elimination half-life. Cp =

(121.5)(0.07−0.9)
 (7)(0.9)(8)   −(7)(0.07)(8)  

EXAMPLE »» » 1−e− 1−e
×  e−0.9(4) −  e−0.07(4)

 1−e−0.9(8) 
  1−e− 

 (0.07)(8)  
An adult male patient (46 years old, 81 kg) was given

Cp =2.86 mg/L
250 mg of tetracycline hydrochloride orally every
8 hours for 2 weeks. From the literature, tetracycline d. C ∞ at steady state: t at steady state is obtained

max p
hydrochloride is about 75% bioavailable and has an from Equation 9.41.
apparent volume of distribution of 1.5 L/kg. The elim-

1 ka(1−e−kτ )
ination half-life is about 10 hours. The absorption rate tp = ln

k − 
k k − −kaτ 

a (1 e )
constant is 0.9 h−1. From this information, calculate
(a) C (0.07)(8)

max after the first dose, (b) Cmin after the first dose,
1 0.9(1−e− )

t = ln
(c) plasma drug concentration C p

p at 4 hours after the −  
0.9 0.07 0.07(1−e−(0.9)(8) )

seventh dose, (d) maximum plasma drug concentra-
tion at steady state, C∞ , (e) minimum plasma drug tp =2.05 hours

max

concentration at steady state, C∞

min, and (f) average
Then C∞ is obtained using Equation 9.38.

plasma drug concentration at steady state, max
C∞

av.

Solution ∞ 0.75(250)  1 
Cmax =  e−0.07(2.05)

121.5 1−e−0.07(8) 
a. Cmax after the first dose occurs at tmax—therefore,

using Equation 9.40, C∞
min =3.12 mg/L

2.3  0.9 
tmax = log  e. C∞

0.9−0.07 0.07 min at steady state is calculated from
Equation 9.39.

tmax =3.07
(0.9)(0.75)(250)  1 

C∞ =
Then substitute t  e−(0.7)(8)

min
max into the following equa- (121.5)(0.9−0.07) 1−e−0.07(8) 

tion for a single oral dose (one-compartment
model) to obtain C C∞

max =2.23 mg/L
max.

FD k f. C∞

C = 0 a (e−ktmax −e−katmax av at steady state is calculated from Equation 9.17.
max )

VD(ka −k)
(0.75)(250)

C∞

av =
(0.75)(250)(0.9)

C (121.5)(0.07)(8)
max = (e−0.07(3.07) −e−0.9(3.07) )

(121.5)(0.9−0.07)
C∞

av =2.76 mg/L
Cmax =1.28 mg/L

b. Cmin after the first dose occurs just before the
administration of the next dose of drug—there-
fore, set t = 8 hours and solve for Cmin. LOADING DOSE

(0.75)(250)(0.9) Since extravascular doses require time for absorption
C = (e−0.07(8) −e−0.9(8)
min )

(121.5)(0.9−0.07) into the plasma to occur, therapeutic effects are

C delayed until sufficient plasma concentrations are
min = 0.95 mg/L

achieved. To reduce the onset time of the drug—that is,

 

220 Chapter 9

the time it takes to achieve the minimum effective
concentration (assumed to be equivalent to the C∞ )—a D

A L D
av = 3 C L = 1.5

Dm Dm
loading (priming) or initial dose of drug is given. The A

main objective of the loading dose is to achieve desired D
B L D

= 2 D L = 1
Dm D

plasma concentrations, C∞ , as quickly as possible. If m

av

the drug follows one-compartment pharmacokinetics,
B

then in theory, steady state is also achieved immedi-
ately following the loading dose. Thereafter, a mainte- C

nance dose is given to maintain C∞ and steady state so MEC
av

that the therapeutic effect is also maintained. In prac- D

tice, a loading dose may be given as a bolus dose or a
short-term loading IV infusion.

As discussed earlier, the time required for the
drug to accumulate to a steady-state plasma level is 0

0 Time (hours)
dependent mainly on its elimination half-life. The
time needed to reach 90% of C∞ is approximately

av D
3.3 half-lives, and the time required to reach 99% of Doses

C∞ is equal to approximately 6.6 half-lives. For a
av FIGURE 95 Concentration curves for dosage regimens

drug with a half-life of 4 hours, it will take approxi- with equal maintenance doses (D) and dosage intervals (τ)

mately 13 and 26 hours to reach 90% and 99% of and different dose ratios. (From Kruger-Thiemer, 1968, with
permission.)

C∞ , respectively.
av

For drugs absorbed rapidly in relation to elimi-
nation (ka >> k) and that are distributed rapidly, the different loading doses. A rapid approximation of
loading dose DL can be calculated as follows: loading dose, DL, may be estimated from

D V C∞

L 1
= (9.42) D = D av

L (9.45)
D − (S)(F)

0 (1− e kaτ )(1 e−kτ
− )

For extremely rapid absorption, as when the product of where C∞ is the desired plasma drug concentration,
av

k S is the salt form of the drug, and F is the fraction of
at is large or in the case of IV infusion, −k

e aτ becomes
approximately zero and Equation 9.42 reduces to drug bioavailability.

Equation 9.45 assumes very rapid drug absorp-
DL 1

tion from an immediate-release dosage form. The DL
=

D k (9.43)
0 1− e− τ

calculated by this method has been used in clinical

The loading dose should approximate the amount of situations for which only an approximation of the DL

drug contained in the body at steady state. The dose is needed.

ratio is equal to the loading dose divided by the main- These calculations for loading doses are not

tenance dose. applicable to drugs that demonstrate multicompart-
ment kinetics. Such drugs distribute slowly into extra-

D
Dose ratio = L (9.44) vascular tissues, and drug equilibration and steady

D0 state may not occur until after the apparent plateau is

As a general rule of thumb, if the selected dosage reached in the vascular (central) compartment.

interval is equal to the drug’s elimination half-life,
then the dose ratio calculated from Equation 9.44 DOSAGE REGIMEN SCHEDULES
should be equal to 2.0. In other words, the loading
dose will be equal to double the initial drug dose. Predictions of steady-state plasma drug concentra-
Figure 9-5 shows the plasma level–time curve for tions usually assume the drug is given at a constant
dosage regimens with equal maintenance doses but dosage interval throughout a 24-hour day. Very often,

Plasma level

 

Multiple-Dosage Regimens 221

20 procainamide given with a 1.0-g loading dose on the
first day followed by maintainence doses of 0.5-g four
times a day. On the second, third, and subsequent days,

15 the procainamide plasma levels did not reach the thera-
peutic range until after the second dose of drug.

Ideally, drug doses should be given at evenly
10

spaced intervals. However, to improve patient com-
pliance, dosage regimens may be designed to fit

5 with the lifestyle of the patient. For example, the
patient is directed to take a drug such as amoxicillin
four times a day (QID), before meals and at bed-

0 time, for a systemic infection. This dosage regimen
0 20 40 60 80

Time (hours) will produce unequal dosage intervals during the
day, because the patient takes the drug before

FIGURE 96 Plasma level–time curve for theophylline
breakfast, at 0800 hours (8 AM); before lunch, at

given in doses of 160 mg 3 times a day. Dashed lines indicate
the therapeutic range. (From Niebergall et al, 1974, with 1200 hours (12 noon); before dinner, at 1800 hours
permission.) (6 PM); and before bedtime, at 2300 hours (11 PM).

For these drugs, evenly spaced dosage intervals are not
that critical to the effectiveness of the antibiotic as long

however, the drug is given only during the waking
as the plasma drug concentrations are maintained

hours (Fig. 9-6). Niebergall et al (1974) discussed the
above the minimum inhibitory concentration (MIC) for

problem of scheduling dosage regimens and particu-
the microorganism. In some cases, a drug may be given

larly warned against improper timing of the drug
at a larger dose allowing for a longer duration above

dosage. For drugs with a narrow therapeutic index
MIC if fluctuation is less critical. In Augmentin Bid-875

such as theophylline (Fig. 9-6), large fluctuation
(amoxicillin/clavulanate tablets), the amoxicillin/

between the maximum and minimum plasma levels
clavulanate tablet is administered twice daily.

are undesirable and may lead to subtherapeutic
Patient compliance with multiple-dose regimens

plasma drug concentrations and/or to high, possibly
may be a problem for the patient in following the

toxic, drug concentrations. These wide fluctuations
prescribed dosage regimen. Occasionally, a patient

occur if larger doses are given at wider dosage inter-
may miss taking the drug dose at the prescribed

vals (see Fig. 9-3). For example, Fig. 9-7 shows
dosage interval. For drugs with long elimination half-
lives (eg, levothyroxine sodium or oral contraceptives),

10 the consequences of one missed dose are minimal, since
only a small fraction of drug is lost between daily dos-

8 ing intervals. The patient should either take the next
drug dose as soon as the patient remembers or continue

6 the dosing schedule starting at the next prescribed dos-
ing period. If it is almost time for the next dose, then the

4
skipped dose should not be taken and the regular dosing

2 schedule should be maintained. Generally, the patient
should not double the dose of the medication. For spe-

0 cific drug information on missed doses, USP DI II,
0 20 40 60 80

Time (hours) Advice for the Patient, published annually by the United
States Pharmacopeia, is a good source of information.

FIGURE 97 Plasma level–time curve for procainamide
The problems of widely fluctuating plasma drug

given in an initial dose of 1.0 g followed by doses of 0.5 g 4 times
a day. Dashed lines indicate the therapeutic range. (From concentrations may be prevented by using a con-
Niebergall et al, 1974, with permission.) trolled-release formulation of the drug, or a drug in

Plasma level (mg/mL)
Plasma level (mg/mL)

 

222 Chapter 9

the same therapeutic class that has a long elimination The effective concentration of this drug is
half-life. The use of extended-release dosage forms 15 mg/mL. After the patient is given a single
allows for less frequent dosing and prevents under- IV dose, the elimination half-life for the drug
medication between the last evening dose and the first is determined to be 3.0 hours and the apparent
morning dose. Extended-release drug products may VD is 196 mL/kg. Determine a multiple IV dose
improve patient compliance by decreasing the number regimen for this drug (assume drug is given
of doses within a 24-hour period that the patient needs every 6 hours).
to take. Patients generally show better compliance with
a twice-a-day (BID) dosage regimen compared to a Solution
three-times-a-day (TID) dosage schedule. FD

C∞ 0
av =

VDkτ

CLINICAL EXAMPLE For IV dose, F = 1,
Bupropion hydrochloride (Wellbutrin) is a noradren-  
ergic/dopaminergic antidepressant. Jefferson et al, D = (15 µ 0.693

0 g/mL) (196 mL/kg)(6 h)
 3 h 

2005, have reviewed the pharmacokinetic properties
of bupropion and its various various formulations and D0 = 4.07 mg/kg every 6 hours
clinical applications, the goal of which is optimization
of major depressive disorder treatment. Bupropion Since patient C.S. weighs 76.6 kg, the dose should

hydrochloride is available in three oral formulations. be as shown:

The immediate-release (IR) tablet is given three times D0 = (4.07 mg/kg)(76.6 kg)
a day, the sustained-release tablet (Wellbutrin SR) is D0 = 312 mg every 6 hours
given twice a day, and the extended-release tablet
(Wellbutrin XL) is given once a day. After the condition of this patient has stabilized,

The total daily dose was 300 mg bupropion HCl. the patient is to be given the drug orally for con-
The area under the curve, AUC, for each dose treatment venience of drug administration. The objective is
was similar showing that the formulations were bio- to design an oral dosage regimen that will produce
equivalent based on extent of absorption. The fluctua- the same steady-state blood level as the mul-
tions between peak and trough levels were greatest for tiple IV doses. The drug dose will depend on the
the IR product given three times a day and least for the bioavailability of the drug from the drug product,
once-a-day XL product. According to the manufac- the desired therapeutic drug level, and the dosage
turer, all three dosage regimens provide equivalent interval chosen. Assume that the antibiotic is 90%
clinical efficacy. The advantage of the extended-release bioavailable and that the physician would like to
product is that the patient needs only to take the drug continue oral medication every 6 hours.
once a day. Often, immediate-release drug products are The average or steady-state plasma drug level
less expensive compared to an extended-release drug is given by
product. In this case, the fluctuating plasma drug levels FD
for buproprion IR tablet given three times a day are not C∞ = 0

av VDkτ
a safety issue and the tablet is equally efficacious as the
150-mg SR tablet given twice a day or the 300-mg XL (15 µg/mL)(193 mL/kg)(0.693)(6 h)

D
tablet given once a day. The patient may also consider 0 =

(0.9)(3 h)
the cost of the medication.

D0 = 454 mg/kg

PRACTICE PROBLEMS Because patient C.S. weighs 76.6 kg, he should
be given the following dose:

1. Patient C.S. is a 35-year-old man weighing
76.6 kg. The patient is to be given multiple D0 = (4.54 mg/kg)(76.6 kg)
IV bolus injections of an antibiotic every 6 hours. D0 = 348 mg every 6 hours

 

Multiple-Dosage Regimens 223

For drugs with equal absorption but slower absorp- Solution
tion rates (F is the same but ka is smaller), the initial

(0.9)(500,000)(3)
dosing period may show a lower blood level; however, C∞

av =
(196)(76.6)(8)(0.693)

the steady-state blood level will be unchanged.

2. In practice, drug products are usually commer- C∞
av =16.2 µg/mL

cially available in certain specified strengths.
Using the information provided in the pre- Notice that a larger dose is necessary if the drug is
ceding problem, assume that the antibiotic is given at longer intervals.
available in 125-, 250-, and 500-mg tablets. In designing a dosage regimen, one should
Therefore, the pharmacist or prescriber must consider a regimen that is practical and con-
now decide which tablets are to be given to the venient for the patient. For example, for good
patient. In this case, it may be possible to give compliance, the dosage interval should be spaced
the patient 375 mg (eg, one 125-mg tablet and conveniently for the patient. In addition, one
one 250-mg tablet) every 6 hours. However, should consider the commercially available dosage
the C∞ should be calculated to determine if

av strengths of the prescribed drug product.
the plasma level is approaching a toxic value. The use of Equation 9.17 to estimate a dosage
Alternatively, a new dosage interval might be regimen initially has wide utility. The C∞ is equal

av

appropriate for the patient. It is very important to the dosing rate divided by the total body clear-
to design the dosage interval and the dose to be ance of the drug in the patient:
as simple as possible, so that the patient will
not be confused and will be able to comply FD

0 1
C∞

with the medication program properly. av = (9.47)
τ ClT

a. What is the new C∞ if the patient is given
av

375 mg every 6 hours? where FD0/t is equal to the dosing rate R, and
1/ClT is equal to 1/kVD.

Solution In designing dosage regimens, the dosing rate
(0.9)(375,000)(3)

C∞ D0/t is adjusted for the patient’s drug clearance
av =

(196)(76.6)(6)(0.693) to obtain the desired C∞ . For an IV infusion, the
av

zero-order rate of infusion (R) is used to obtain
C∞

av =16.2 µg/mL
the desired steady-state plasma drug concentration

Because the therapeutic objective was to achieve CSS. If R is substituted for FD0/t in Equation 9.47,

a minimum effective concentration (MEC) of then the following equation for estimating CSS

15 mg/mL, a value of 16.2 mg/mL is reasonable. after an IV infusion is obtained:

b. The patient has difficulty in distinguishing
R

tablets of different strengths. Can the patient C = (9.48)
ss Cl

take a 500-mg dose (eg, two 250-mg tablets)? T

Solution From Equations 9.47 and 9.48, all dosage sched-
ules having the same dosing rate D0/t, or R, will

The dosage interval (t) for the 500-mg tablet
have the same C∞ or C

a SS, whether the drug is given
would have to be calculated as follows: v

by multiple doses or by IV infusion. For example,
(0.9)(500,000)(3) dosage schedules of 100 mg every 4 hours, 200 mg

τ =
(196)(76.6)(15)(0.693) every 8 hours, 300 mg every 12 hours, and 600 mg

every 24 hours will yield the same C∞ in the
av

τ = 8.63 h
patient. An IV infusion rate of 25 mg/h in the same

c. A dosage interval of 8.63 hours is difficult patient will give a CSS equal to the C∞ obtained
av

to remember. Is a dosage regimen of 500 mg with the multiple-dose schedule (see Fig. 9-3;
every 8 hours reasonable? Table 9-6).

 

224 Chapter 9

TABLE 96 Effect of Dosing Schedule on Predicted Steady-State Plasma Drug Concentrationsa

Dosing Schedule Steady-State Drug Concentration (lg/mL)

Dosing Rate, D0/τ
C∞ C∞ C∞

Dose (mg) 1 (h) (mg/h) max av min

— — 25b 14.5 14.5 14.5

100 4 25 16.2 14.5 11.6

200 8 25 20.2 14.5 7.81

300 12 25 25.3 14.5 5.03

600 24 25 44.1 14.5 1.12

400 8 50 40.4 28.9 15.6

600 8 75 60.6 43.4 23.4

aDrug has an elimination half-life of 4 hours and an apparent VD of 10 L.

bDrug given by IV infusion. The first-order absorption rate constant ka is 1.2 h−1 and the drug follows a one-compartment open model.

Frequently Asked Questions »»Why is the Cmin value at steady state less variable

»»Why is the steady-state peak plasma drug concen- than the Cmax value at steady state?

tration measured sometime after an IV dose is given
»»Is it possible to take a single blood sample to mea-

in a clinical situation? sure the Cav value at steady state?

CHAPTER SUMMARY
The purpose of giving a loading dose is to achieve multiple dosing. A clinical example of multiple dos-
desired (therapeutic) plasma concentrations as ing using short, intermittent intravenous infusions
quickly as possible. For a drug with long elimination has been applied to the aminoglycosides and is based
half-life, it may take a long time (several half-lives) on pharmacokinetics and clinical factors for safer
to achieve steady-state levels. The loading dose must dosing. The index for measuring drug accumulation
be calculated appropriately based on pharmacoki- during multiple dosing, R, is related to the dosing
netic parameters to avoid overdosing. When several interval and the half-life of the drug, but not the
doses are administered for a drug with linear kinetics, dose. This parameter compares the steady-state con-
drug accumulation may occur according to the prin- centration with drug concentration after the initial
ciple of superposition. Superposition allows the deri- dose. The plasma concentration at any time during
vation of equations that predict the plasma drug peak an oral or extravascular multiple-dose regimen, for a
and trough concentrations of a drug at steady state one-compartment model and constant doses and
and the theoretical drug concentrations at any time dose interval, is dependent on n = number of doses,
after the dose is given. The principle of superposition t = dosage interval, F = fraction of dose absorbed,
is used to examine the effect of an early, late, or miss- and t = time after administration of n doses.
ing dose on steady-state drug concentration.

C∞ ∞ ∞ or Fk  −nkaτ    
max, C n, and C are useful parameters f

av aD0 1− e 1 nkaτ
mi C =   e−k − e−

ring the safety and efficacy of a drug during p − e−kt
monito VD (k − k τ 

  1− e−kτ  
a ) 1− e−ka  

 

Multiple-Dosage Regimens 225

The trough steady-state concentration after multiple The relationship between average steady-state con-
oral dosing is centration, the AUC, and dosing interval is


∫ C d

∞ 0 p t [AUC]∞
C 0

∞ kaFD0  1 
C k

min = e
V ( k
D ka −  − τ av = =

− τ  τ τ
k) 1− e  This parameter is a good measure of drug exposure.

LEARNING QUESTIONS
1. Gentamicin has an average elimination half- is given at a dose of 200 mg every 4 hours

life of approximately 2 hours and an apparent by multiple IV bolus injections. Predict the
volume of distribution of 20% of body weight. plasma drug concentration at 1 hour after the
It is necessary to give gentamicin, 1 mg/kg third dose.
every 8 hours by multiple IV injections, to 9. The elimination half-life of an antibiotic is
a 50-kg woman with normal renal function. 3 hours and the apparent volume of distribution
Calculate (a) Cmax, (b) Cmin, and (c) C∞ . is 20% of the body weight. The therapeutic

av

2. A physician wants to give theophylline to a window for this drug is from 2 to 10 mg/mL.
young male asthmatic patient (age 29 years, Adverse toxicity is often observed at drug
80 kg). According to the literature, the elimina- concentrations above 15 mg/mL. The drug will
tion half-life for theophylline is 5 hours and be given by multiple IV bolus injections.
the apparent VD is equal to 50% of the body a. Calculate the dose for an adult male patient
weight. The plasma level of theophylline (68 years old, 82 kg) with normal renal func-
required to provide adequate airway ventilation tion to be given every 8 hours.
is approximately 10 mg/mL. b. Calculate the anticipated C∞

max and C∞

min
a. The physician wants the patient to take med- values.

ication every 6 hours around the clock. What c. Calculate the C∞ value.
av

dose of theophylline would you recommend d. Comment on the adequacy of your dosage
(assume theophylline is 100% bioavailable)? regimen.

b. If you were to find that theophylline is avail- 10. Tetracycline hydrochloride (Achromycin V,
able to you only in 225-mg capsules, what Lederle) is prescribed for a young adult male
dosage regimen would you recommend? patient (28 years old, 78 kg) suffering from

3. What pharmacokinetic parameter is most gonorrhea. According to the literature, tetra-
important in determining the time at which the cycline HCl is 77% orally absorbed, is 65%
steady-state plasma drug level (C∞ ) is reached? bound to plasma proteins, has an apparent

av

4. Name two ways in which the fluctuations of volume of distribution of 0.5 L/kg, has an
plasma concentrations (between C∞

max and C∞

min) elimination half-life of 10.6 hours, and is 58%
can be minimized for a person on a multiple-dose excreted unchanged in the urine. The minimum
drug regimen without altering the C∞ . inhibitory drug concentration (MIC) for gonor-

av

5. What is the purpose of giving a loading dose? rhea is 25–30 mg/mL.
6. What is the loading dose for an antibiotic (k = a. Calculate an exact maintenance dose for this

0.23 h−1) with a maintenance dose of 200 mg patient to be given every 6 hours around the
every 3 hours? clock.

7. What is the main advantage of giving a potent b. Achromycin V is available in 250- and
drug by IV infusion as opposed to multiple 500-mg capsules. How many capsules (state
IV injections? dose) should the patient take every 6 hours?

8. A drug has an elimination half-life of 2 hours c. What loading dose using the above capsules
and a volume of distribution of 40 L. The drug would you recommend for this patient?

 

226 Chapter 9

11. The body clearance of sumatriptan (Imitrex) is 12. Cefotaxime has a volume of distribution
250 mL/min. The drug is about 14% bioavail- of 0.17 L/kg and an elimination half-life
able. What would be the average plasma drug of 1.5 hours. What is the peak plasma drug
concentration after 5 doses of 100 mg PO concentration in a patient weighing 75 kg after
every 8 hours in a patient? (Assume steady receiving 1 g IV of the drug 3 times daily for
state was reached.) 3 days?

ANSWERS

Frequently Asked Questions are more suitable to be administered as an IV bolus

Is the drug accumulation index (R) applicable to any injection. For example, some reports show that an

drug given by multiple doses or only to drugs that aminoglycoside given once daily resulted in fewer

are eliminated slowly from the body? side effects compared with dividing the dose into
two or three doses daily. Due to drug accumulation

• Accumulation index, R, is a ratio that indicates in the kidney and adverse toxicity, aminoglycosides
steady-state drug concentration to the drug concen- are generally not given by prolonged IV infusions.
tration after the first dose. The accumulation index In contrast, a prolonged period of low drug level for
does not measure the absolute size of overdosing; penicillins and tetracyclines may not be so effica-
it measures the amount of drug cumulation that can cious and may result in a longer cure time for an
occur due to frequent drug administration. Factors infection. The pharmacodynamics of the individual
that affect R are the elimination rate constant, k, and drug must be studied to determine the best course of
the dosing interval, t. If the first dose is not chosen action. (2) Drugs such as nitroglycerin are less likely
appropriately, the steady-state level may still be to produce tolerance when administered intermit-
incorrect. Therefore, the first dose and the dosing tently versus continuously.
interval must be determined correctly to avoid any
significant drug accumulation. The accumulation Why is the steady-state peak plasma drug concentra-

index is a good indication of accumulation due to tion often measured sometime after an IV dose is

frequent drug dosing, applicable to any drug, re- given in a clinical situation?

gardless of whether the drug is bound to tissues. • After an IV bolus drug injection, the drug is well

What are the advantages/disadvantages for giving a distributed within a few minutes. In practice, how-

drug by constant IV infusion, intermittent IV infu- ever, an IV bolus dose may be administered slowly

sion, or multiple IV bolus injections? What drugs over several minutes or the drug may have a slow

would most likely be given by each route of adminis- distribution phase. Therefore, clinicians often pre-

tration? Why? fer to take a blood sample 15 minutes or 30 minutes
after IV bolus injection and refer to that drug con-

• Some of the advantages of administering a drug by centration as the peak concentration. In some cases,
constant IV infusion include the following: (1) A a blood sample is taken an hour later to avoid the
drug may be infused continuously for many hours fluctuating concentration in the distributive phase.
without disturbing the patient. (2) Constant infusion The error due to changing sampling time can be
provides a stable blood drug level for drugs that have large for a drug with a short elimination half-life.
a narrow therapeutic index. (3) Some drugs are bet-

Is a loading dose always necessary when placing a
ter tolerated when infused slowly. (4) Some drugs

patient on a multiple-dose regimen? What are the
may be infused simultaneously with electrolytes

determining factors?
or other infusion media in an acute-care setting.
Disadvantages of administering a drug by constant • A loading or priming dose is used to rapidly raise
IV infusion include the following: (1) Some drugs the plasma drug concentration to therapeutic drug

 

Multiple-Dosage Regimens 227

levels to obtain a more rapid pharmacodynamic FD 1.44
c. C∞ 0 t1/2

response. In addition, the loading dose along with av =
VDτ

the maintenance dose allows the drug to reach
steady-state concentration quickly, particularly for (50)(1.44)(2)

= =1.8 µg/mL
drugs with long elimination half-lives. (10,000)(8)

An alternative way of explaining the load-
C∞

2. a. D avVDτ
ing dose is based on clearance. After multiple IV

0 =
1.44t

dosing, the maintenance dose required is based on 1/2

Cl, Css, and t. (10)(40,000)(6)
=

(1.44)(5)
Dose

CSS =
τCl = 333 mg every 6 h

Dose =CSS τCl FD01.44tb. 1/2

τ =
VDC

av

If Css and t are fixed, a drug with a smaller clear- (225,000)(1.44)(5)
ance requires a smaller maintenance dose. In prac- = = 4.05 h

(40,000)(10)
tice, the dosing interval is adjustable and may be
longer for drugs with a small Cl if the drug does 6. Dose the patient with 200 mg every 3 hours.

not need to be dosed frequently. The steady-state D
0 200

DL = = = 400 mg
drug level is generally determined by the desired 1 − τ

− e k 1 −
− e (0.23)(3)

therapeutic drug. Notice that DL is twice the maintenance dose,

Does a loading dose significantly affect the steady- because the drug is given at a dosage interval

state concentration of a drug given by a constant equal approximately to the t1/2 of 3 hours.

multiple-dose regimen? 8. The plasma drug concentration, Cp, may be cal-
culated at any time after n doses by Equation 9.21

• The loading dose will affect only the initial drug and proper substitution.
concentrations in the body. Steady-state drug
levels are obtained after several elimination half- D 

0 1− e−nkτ 
Cp =

V  − τ  e−kt

D 1− e k
lives (eg, 4.32t1/2 for 95% steady-state level). 
Only 5% of the drug contributed by the loading

200 1− e−(3)(0.347)(4) 
dose will remain at 95% steady state. At 99% Cp = e−(0.347)(1)

40 
 1− e−(0.347)(4) 

steady-state level, only 1% of the loading dose 
will remain. = 4.63 mg/L

Alternatively, one may conclude that for a drug
Learning Questions whose elimination t1/2 is 2 hours, the predicted

plasma drug concentration is approximately at
1. VD = 0.20(50 kg) =10,000 mL

steady state after 3 doses or 12 hours. Therefore,
D 50 mg the above calculation may be simplified to the

a. D 0
max = − = =

− − 53.3 mg
1 f 1 e (0.693/2)(8) following:

D D0  1 

C max 53.3 mg Cp =  e−k
max = = = 5.33 µg/mL

VD 10,000 mL VD 1− e kτ  τ
− 

b. Dmin = 53.3− 50 = 3.3 mg 200 1 
C =   e−(0.347)(1)

p −
 40 1− e (0.347)(4) 

3.3 mg
Cmin = = 0.33 µg/mL

10,000 mL = 4.71 mg/L

 

228 Chapter 9

D / FD
9. C∞ 0 VD

max = 10. a. C∞ = 0

1 e−kτ av
− kVDτ

where Let C∞
av = 27.5 mg/L

VD = 20% of 82 kg = (0.2)(82) = 16.4 L
C∞

a kV τ (27.5)(0.693/10.6)(0.5)(78)(6)
k −

= (0.693/3) = 0.231 h 1 D v D
0 = =

F 0.77

D V C∞ e−kτ e−(0.231)(8)
0 = D max (1− ) = (16.4)(10)(1− = 546.3 mg

a. D0 = 138.16 mg to be given every 8 hours D0 = 546.3 mg every 6 h
b. C∞ =C∞ (e−kτ ) = (10)(e−(0.231)(8)

min max )

= b. If a 500-mg capsule is given every 6 hours,
1.58 mg/L

D
c. 138.16 FD (0.77)(500)

C∞ 0
av = = C∞ 0

kV av = =

Dτ (0.231)(16.4)(8) kVDτ (0.693/10.6)(0.5)(78)(6)

= 4.56 mg/L = 25.2 mg/L

d. In the above dosage regimen, the C∞

min of
1.59 mg/L is below the desired C∞ D 500

min of 2 mg/L. c. D = M

Alternatively, the dosage interval, L
t, could − = =1543 mg

1− e kτ 1− e(0.654)(6)
be changed to 6 hours.

DL = 3× 500 mg capsules =1500 mg

D = V C∞ (1− e−kτ ) = (16.4)(10)(1− e−(0.231)(6)
0 D max )

D0 = 123 mg to be given every 6 h

C∞ = C∞ (e−kτ ) = (10)(e−(0.231)(6)

min max ) = 2.5 mg/L

D
∞ 123

C = 0
av = = 5.41 mg/L

kVDτ (0.231)(16.4)(6)

REFERENCES
Jefferson JW, Pradko JF, Muir KT: Bupropion for major depressive three-point methods that estimate area under the curve are

disorder: Pharmacokinetic and formulation considerations. Clin superior to trough levels in predicting drug exposure, therapeutic
Ther 27(11):1685–1695, 2005. drug monitoring 20(3):276–283, June 1998.

Kruger-Thiemer E: Pharmacokinetics and dose-concentration Sawchuk RJ, Zaske DE: Pharmacokinetics of dosing regimens
relationships. In Ariens EJ (ed.), Physico-Chemical Aspects of which utilize multiple intravenous infusions: Gentamycin in
Drug Action. New York, Pergamon, 1968, p 97. burn patients. J Pharmacokin Biopharm 4(2):183–195, 1976.

Niebergall PJ, Sugita ET, Schnaare RC: Potential dangers of com- van Rossum JM, Tomey AHM: Rate of accumulation and plateau
mon drug dosing regimens. Am J Hosp Pharm 31:53–59, 1974. concentration of drugs after chronic medication. J Pharm

Primmett D, Levine M, Kovarik, J, Mueller E, Keown, P: Cyclo- Pharmacol 30:390–392, 1968.
sporine monitoring in patients with renal transplants: Two- or

BIBLIOGRAPHY
Gibaldi M, Perrier D: Pharmacokinetics, 2nd ed. New York, Wagner JG: Kinetics of pharmacological response, I: Proposed

Marcel Dekker, 1962, pp 451–457. relationship between response and drug concentration in the
Levy G: Kinetics of pharmacologic effect. Clin Pharmacol Ther intact animal and man. J Theor Biol 20:173, 1968.

7:362, 1966. Wagner JG: Relations between drug concentrations and response.
van Rossum JM: Pharmacokinetics of accumulation. J Pharm Sci J Mond Pharm 14:279–310, 1971.

75:2162–2164, 1968.

 

Nonlinear Pharmacokinetics

10 Andrew B.C. Yu and Leon Shargel

Chapter Objectives Previous chapters discussed linear pharmacokinetic models using
simple first-order kinetics to describe the course of drug disposi-

»» Describe the differences between
tion and action. These linear models assumed that the pharmaco-

linear pharmacokinetics and
kinetic parameters for a drug would not change when different

nonlinear pharmacokinetics.
doses or multiple doses of a drug were given. With some drugs,

»» Illustrate nonlinear pharmaco- increased doses or chronic medication can cause deviations from
kinetics with drug disposition the linear pharmacokinetic profile previously observed with single
examples. low doses of the same drug. This nonlinear pharmacokinetic

»» Discuss some potential risks behavior is also termed dose-dependent pharmacokinetics.
in dosing drugs that follow Many of the processes of drug absorption, distribution, bio-
nonlinear kinetics. transformation, and excretion involve enzymes or carrier-mediated

systems. For some drugs given at therapeutic levels, one of
»» Explain how to detect nonlinear

these specialized processes may become saturated. As shown in
kinetics using AUC-versus-doses

Table 10-1, various causes of nonlinear pharmacokinetic behavior
plots.

are theoretically possible. Besides saturation of plasma protein-
»» Apply the appropriate equation binding or carrier-mediated systems, drugs may demonstrate non-

and graphical methods, to calculate linear pharmacokinetics due to a pathologic alteration in drug
the Vmax and KM parameters after absorption, distribution, and elimination. For example, aminogly-
multiple dosing in a patient. cosides may cause renal nephrotoxicity, thereby altering renal drug

»» Describe the use of the Michaelis– excretion. In addition, gallstone obstruction of the bile duct will alter
Menten equation to simulate biliary drug excretion. In most cases, the main pharmacokinetic
the elimination of a drug by a outcome is a change in the apparent elimination rate constant.
saturable enzymatic process. A number of drugs demonstrate saturation or capacity-limited

metabolism in humans. Examples of these saturable metabolic
»» Estimate the dose for a nonlinear

processes include glycine conjugation of salicylate, sulfate conju-
drug such as phenytoin in

gation of salicylamide, acetylation of p-aminobenzoic acid, and
multiple-dose regimens.

the elimination of phenytoin (Tozer et al, 1981). Drugs that dem-
»» Describe chronopharmaco- onstrate saturation kinetics usually show the following

kinetics, time-dependent characteristics:
pharmacokinetics, and its
influence on drug disposition. 1. Elimination of drug does not follow simple first-order kinetics—

that is, elimination kinetics are nonlinear.
»» Describe how transporters may

2. The elimination half-life changes as dose is increased. Usually,
cause uneven drug distribution at

the elimination half-life increases with increased dose due
cellular level; and understand that

to saturation of an enzyme system. However, the elimination
capacity-limited or concentration-

half-life might decrease due to “self”-induction of liver bio-
dependent kinetics may occur at

transformation enzymes, as is observed for carbamazepine.
the local level within body organs.

229

 

230 Chapter 10

TABLE 101 Examples of Drugs Showing Nonlinear Kinetics

Causea Drug

Gl Absorption

Saturable transport in gut wall Riboflavin, gebapentin, L-dopa, baclofen, ceftibuten

Intestinal metabolism Salicylamide, propranolol

Drugs with low solubility in GI but relatively Chorothiazide, griseofulvin, danazol
high dose

Saturable gastric or GI decomposition Penicillin G, omeprazole, saquinavir

Distribution

Saturable plasma protein binding Phenylbutazone, lidocaine, salicylic acid, ceftriaxone,
diazoxide, phenytoin, warfarin, disopyramide

Cellular uptake Methicillin (rabbit)

Tissue binding Imiprimine (rat)

CSF transport Benzylpenicillins

Saturable transport into or out of tissues Methotrexate

Renal Elimination

Active secretion Mezlocillin, para-aminohippuric acid

Tubular reabsorption Riboflavin, ascorbic acid, cephapirin

Change in urine pH Salicylic acid, dextroamphetamine

Metabolism

Saturable metabolism Phenytoin, salicyclic acid, theophylline, valproic acidb

Cofactor or enzyme limitation Acetaminophen, alcohol

Enzyme induction Carbamazepine

Altered hepatic blood flow Propranolol, verapamil

Metabolite inhibition Diazepam

Biliary Excretion

Biliary secretion Iodipamide, sulfobromophthalein sodium

Enterohepatic recycling Cimetidine, isotretinoin

aHypothermia, metabolic acidosis, altered cardiovascular function, and coma are additional causes of dose and time dependencies in drug overdose.

bIn guinea pig and probably in some younger subjects.

Data from Evans et al (1992).

3. The area under the curve (AUC) is not propor- the same enzyme or carrier-mediated system
tional to the amount of bioavailable drug. (ie, competition effects).

4. The saturation of capacity-limited processes 5. The composition and/or ratio of the metabolites of
may be affected by other drugs that require a drug may be affected by a change in the dose.

 

Nonlinear Pharmacokinetics 231

Because these drugs have a changing apparent
A

elimination constant with larger doses, prediction
of drug concentration in the blood based on a
single small dose is difficult. Drug concentrations C
in the blood can increase rapidly once an elimina-
tion process is saturated. In general, metabolism
(biotransformation) and active tubular secretion of
drugs by the kidney are the processes most usually
saturated. Figure 10-1 shows plasma level–time Dose

curves for a drug that exhibits saturable kinetics.
FIGURE 102 Area under the plasma level–time curve

When a large dose is given, a curve is obtained versus dose for a drug that exhibits a saturable elimination
with an initial slow elimination phase followed by process. Curve A represents dose-dependent or saturable
a much more rapid elimination at lower blood elimination kinetics. Curve C represents dose-independent

concentrations (curve A). With a small dose of the kinetics.

drug, apparent first-order kinetics is observed,
because no saturation kinetics occurs (curve B). If
the pharmacokinetic data were estimated only
from the blood levels described by curve B, then a SATURABLE ENZYMATIC
twofold increase in the dose would give the blood

ELIMINATION PROCESSES
profile presented in curve C, which considerably
underestimates the drug concentration as well as The elimination of drug by a saturable enzymatic
the duration of action. process is described by Michaelis–Menten kinetics.

In order to determine whether a drug is follow- If Cp is the concentration of drug in the plasma, then
ing dose-dependent kinetics, the drug is given at
various dosage levels and a plasma level–time curve dCp VmaxCp

Elimination rate = = (10.1)
is obtained for each dose. The curves should exhibit dt KM +Cp

parallel slopes if the drug follows dose-independent
kinetics. Alternatively, a plot of the areas under the where Vmax is the maximum elimination rate and KM
plasma level–time curves at various doses should be is the Michaelis constant that reflects the capacity of
linear (Fig. 10-2). the enzyme system. It is important to note that KM is

not an elimination constant, but is actually a hybrid
rate constant in enzyme kinetics, representing both
the forward and backward reaction rates and equal to

100
the drug concentration or amount of drug in the body
at 0.5Vmax. The values for KM and Vmax are dependent

A on the nature of the drug and the enzymatic process

10 C involved.
The elimination rate of a hypothetical drug with

B
a KM of 0.1 mg/mL and a Vmax of 0.5 mg/mL per hour
is calculated in Table 10-2 by using Equation 10.1.

1 Because the ratio of the elimination rate to drug con-
Time

centration changes as the drug concentration changes
FIGURE 101 Plasma level–time curves for a drug (ie, dCp/dt is not constant, Equation 10.1), the rate of
that exhibits a saturable elimination process. Curves A and B drug elimination also changes and is not a first-order
represent high and low doses of drug, respectively, given in a

or linear process. In contrast, a first-order elimina-
single IV bolus. The terminal slopes of curves A and B are the
same. Curve C represents the normal first-order elimination of tion process would yield the same elimination rate
a different drug. constant at all plasma drug concentrations. At drug

Plasma level

Area under curve

 

232 Chapter 10

TABLE 102 Effect of Drug Concentration on PRACTICE PROBLEM
the Elimination Rate and Rate Constanta

Using the hypothetical drug considered in Table 10-2
Drug Elimination Elimination Rate/ (Vmax = 0.5 mg/mL per hour, KM = 0.1 mg/mL), how
Concentration Rate Concentrationb

long would it take for the plasma drug concentration
( lg/mL) ( lg/mL/h) (h−1)

to decrease from 20 to 12 mg/mL?
0.4 0.400 1.000

0.8 0.444 0.556 Solution
1.2 0.462 0.385 Because 12 mg/mL is above the saturable level, as

1.6 0.472 0.294 indicated in Table 10-2, elimination occurs at a zero-
order rate of approximately 0.5 mg/mL per hour.

2.0 0.476 0.238 Time needed for the drug to decrease to
2.4 0.480 0.200

12 µ 20 −12 µg
2.8 0.483 0.172 g/mL = µ =16 h

0.5 g/h
3.2 0.485 0.152

A saturable process can also exhibit linear elimination
10.0 0.495 0.0495 when drug concentrations are much less than enzyme
10.4 0.495 0.0476 concentrations. When the drug concentration Cp is

10.8 0.495 0.0459 small in relation to the KM, the rate of drug elimina-
tion becomes a first-order process. The data generated

11.2 0.496 0.0442 from Equation 10.2 (Cp ≤ 0.05 mg/mL, Table 10-3)
11.6 0.496 0.0427 using KM = 0.8 mg/mL and Vmax = 0.9 mg/mL per hour

shows that enzymatic drug elimination can change
aKM = 0.1 mg/mL, Vmax = 0.5 mg/mL/h.

from a nonlinear to a linear process over a restricted
bThe ratio of the elimination rate to the concentration is equal to the
rate constant.

concentrations of 0.4–10 mg/mL, the enzyme system TABLE 103 Effect of Drug Concentration on
the Elimination Rate and Rate Constanta

is not saturated and the rate of elimination is a mixed
or nonlinear process (Table 10-2). At higher drug Drug Elimination Elimination Rate
concentrations, 11.2 mg/mL and above, the elimina- Concentration Rate Concentration
tion rate approaches the maximum velocity (Vmax) of (Cp) ( lg/mL) ( lg/mL/h) (h−1)b

approximately 0.5 mg/mL per hour. At Vmax, the 0.01 0.011 1.1
elimination rate is a constant and is considered a
zero-order process. 0.02 0.022 1.1

Equation 10.1 describes a nonlinear enzyme 0.03 0.033 1.1

process that encompasses a broad range of drug 0.04 0.043 1.1
concentrations. When the drug concentration Cp is
large in relation to KM (Cp >> KM), saturation of the 0.05 0.053 1.1

enzymes occurs and the value for KM is negligible. 0.06 0.063 1.0

The rate of elimination proceeds at a fixed or con- 0.07 0.072 1.0
stant rate equal to Vmax. Thus, elimination of drug
becomes a zero-order process and Equation 10.1 0.08 0.082 1.0

becomes: 0.09 0.091 1.0

dC V C aKM = 0.8 mg/mL, Vmax = 0.9 mg/mL/h.
p max p

− = =V (10.2)
dt C max bThe ratio of the elimination rate to the concentration is equal to the

p rate constant.

 

Nonlinear Pharmacokinetics 233

concentration range. This is evident because the rate Because C0
p = 0.05 µg/mL, k =1.1 h−1, and C =

p
constant (or elimination rate/drug concentration) 0.005 mg/mL.
values are constant. At drug concentrations below
0.05 mg/mL, the ratio of elimination rate to drug
concentration has a constant value of 1.1 h−1 2.3(log 0.05− log 0.005)

. t =
1.1

Mathematically, when Cp is much smaller than KM,
Cp in the denominator is negligible and the elimina- 2.3(−1.30+ 2.3)

=
tion rate becomes first order. 1.1

2.3
dCp VmaxCp V = = 2.09 h

max
− = = C 1.1

dt C +K K p
p M M

dC When given in therapeutic doses, most drugs pro-
p

− = k′C (
dt p 10.3) duce plasma drug concentrations well below KM for

all carrier-mediated enzyme systems affecting the
The first-order rate constant for a saturable process, pharmacokinetics of the drug. Therefore, most drugs
k¢, can be calculated from Equation 10.3: at normal therapeutic concentrations follow first-

order rate processes. Only a few drugs, such as
Vmax 0.9 salicylate and phenytoin, tend to saturate the hepatic

k′ = = = ∼ 1.1 h−1
KM 0.8 mixed-function oxidases at higher therapeutic doses.

With these drugs, elimination kinetics is first order
This calculation confirms the data in Table 10-3, with very small doses, is mixed order at higher
because enzymatic drug elimination at drug con- doses, and may approach zero order with very high
centrations below 0.05 mg/mL is a first-order rate therapeutic doses.
process with a rate constant of 1.1 h−1. Therefore,
the t1/2 due to enzymatic elimination can be
calculated: Frequently Asked Questions

»»What kinetic processes in the body can be considered
0.693 saturable?

t1/2 = = 0.63 h
1.1

»»Why is it important to monitor drug levels carefully
for dose dependency?

PRACTICE PROBLEM
How long would it take for the plasma concentration
of the drug in Table 10-3 to decline from 0.05 to DRUG ELIMINATION BY CAPACITY-
0.005 mg/mL? LIMITED PHARMACOKINETICS:

ONE-COMPARTMENT MODEL,
Solution

Because drug elimination is a first-order process for IV BOLUS INJECTION
the specified concentrations, The rate of elimination of a drug that follows capacity-

limited pharmacokinetics is governed by the Vmax
C = 0 −kt

p Cpe and KM of the drug. Equation 10.1 describes the
elimination of a drug that distributes in the body as a

kt
logC = 0

p Cp −
2.3 single compartment and is eliminated by Michaelis–

Menten or capacity-limited pharmacokinetics. If a
logC − logC0

= p single IV bolus injection of drug (D0) is given at t =
t

k 0, the drug concentration (Cp) in the plasma at any

 

234 Chapter 10

time t may be calculated by an integrated form of 1000
Equation 10.1 described by

C0 −Cp K
M C

l 0
=V 0

t max − n (1 .4)
t Cp

100

Alternatively, the amount of drug in the body after an
IV bolus injection may be calculated by the follow-
ing relationship. Equation 10.5 may be used to simu-
late the decline of drug in the body after various size 10

0 0.5 1.0 1.5 2.0 2.5 3.0
doses are given, provided the KM and Vmax of drug Time (hours)
are known.

FIGURE 104 Amount of drug in the body versus time for

D a capacity-limited drug following an IV dose. Data generated
0 −D

t KM D0
=Vma − ln (

t x 10.5) using KM of 38 mg/L () and 76 mg/L (O). Vmax is kept constant.
t D

t

where D0 is the amount of drug in the body at t = 0.
In order to calculate the time for the dose of the drug was calculated for a drug with a KM of 38 mg/L and a

to decline to a certain amount of drug in the body, Vmax that varied from 200 to 100 mg/h (Table 10-4).

Equation 10.5 must be rearranged and solved for With a Vmax of 200 mg/h, the time for the 400-mg dose

time t: to decline to 20 mg in the body is 2.46 hours, whereas
when the Vmax is decreased to 100 mg/h, the time for
the 400-mg dose to decrease to 20 mg is increased to

1  D 
= ln 0 1
t D −D + (

K
Vma  0 t M  0.6)

4.93 hours (see Fig. 10-3). Thus, there is an inverse
x Dt 

relationship between the time for the dose to decline
to a certain amount of drug in the body and the Vmax

The relationship of KM and Vmax to the time for an IV as shown in Equation 10.6.
bolus injection of drug to decline to a given amount of Using a similar example, the effect of KM on
drug in the body is illustrated in Figs. 10-3 and 10-4. the time for a single 400-mg dose given by IV bolus
Using Equation 10.6, the time for a single 400-mg injection to decline to 20 mg in the body is
dose given by IV bolus injection to decline to 20 mg described in Table 10-5 and Fig. 10-4. Assuming

Vmax is constant at 200 mg/h, the time for the drug
to decline from 400 to 20 mg is 2.46 hours when KM
is 38 mg/L, whereas when KM is 76 mg/L, the time for

1000 the drug dose to decline to 20 mg is 3.03 hours. Thus,
an increase in KM (with no change in Vmax) will

Vmax = increase the time for the drug to be eliminated from
100 mg/h the body.

100 The one-compartment open model with capacity-
limited elimination pharmacokinetics adequately

Vmax = describes the plasma drug concentration–time pro-
200 mg/h files for some drugs. The mathematics needed to

describe nonlinear pharmacokinetic behavior of
10

0 1 2 3 4 5 drugs that follow two-compartment models and/or
Time (hours) have both combined capacity-limited and first-order

FIGURE 103 kinetic profiles are very complex and have little
Amount of drug in the body versus time for

a capacity-limited drug following an IV dose. Data generated practical application for dosage calculations and
using Vmax of 100 (O) and 200 mg/h (). KM is kept constant. therapeutic drug monitoring.

Amount of drug (mg)

Amount of drug (mg)

 

Nonlinear Pharmacokinetics 235

TABLE 104 Capacity-Limited Pharmacokinetics: TABLE 105 Capacity-Limited Pharmacokinetics:
Effect of Vmax on the Elimination of Druga Effects of KM on the Elimination of Druga

Time for Drug Elimination (h) Amount of Time for Drug Elimination (h)
Amount of Drug in Body
Drug in Body Vmax = Vmax = (mg) K = 38 mg/L K = 76 mg/L

M M

(mg) 200 mg/h 100 mg/h
400 0 0

400 0 0
380 0.109 0.119

380 0.109 0.219
360 0.220 0.240

360 0.220 0.440
340 0.330 0.361

340 0.330 0.661
320 0.442 0.484

320 0.442 0.884
300 0.554 0.609

300 0.554 1.10
280 0.667 0.735

280 0.667 1.33
260 0.781 0.863

260 0.781 1.56
240 0.897 0.994

240 0.897 1.79
220 1.01 1.12

220 1.01 2.02
200 1.13 1.26

200 1.13 2.26
180 1.25 1.40

180 1.25 2.50
160 1.37 1.54

160 1.37 2.74
140 1.49 1.69

140 1.49 2.99
120 1.62 1.85

120 1.62 3.25
100 1.76 2.02

100 1.76 3.52
80 1.90 2.21

80 1.90 3.81
60 2.06 2.42

60 2.06 4.12
40 2.23 2.67

40 2.23 4.47
20 2.46 3.03

20 2.46 4.93
aA single 400-mg dose is given by IV bolus injection. The drug is

aA single 400-mg dose is given by IV bolus injection. The drug is distributed into a single compartment and is eliminated by capacity-
distributed into a single compartment and is eliminated by capacity- limited pharmacokinetics. Vmax is 200 mg/h. The time for drug to
limited pharmacokinetics. KM is 38 mg/L. The time for drug to decline decline from 400 to 20 mg is calculated from Equation 9.6 assuming
from 400 to 20 mg is calculated from Equation 9.6 assuming the drug the drug has KM = 38 mg/L or KM = 76 mg/L.
has Vmax = 200 mg/h or Vmax = 100 mg/h.

the time for 50% of the dose to be elimi-

PRACTICE PROBLEMS nated. Explain why there is a difference in
the time for 50% elimination of a 400-mg

1. A drug eliminated from the body by capacity- dose compared to a 320-mg dose.

limited pharmacokinetics has a KM of
100 mg/L and a Vmax of 50 mg/h. If 400 mg Solution

of the drug is given to a patient by IV bolus Use Equation 10.6 to calculate the time for the
injection, calculate the time for the drug to dose to decline to a given amount of drug in
be 50% eliminated. If 320 mg of the drug is the body. For this problem, Dt is equal to 50%
to be given by IV bolus injection, calculate of the dose D0.

 

236 Chapter 10

If the dose is 400 mg, the rate of enzymatic reaction of a drug in vitro
(Equation 10.7). When an experiment is performed

1  400
t = 400 − 200+100 ln  = 5.39 h with solutions of various concentration of drug C, a

50  200
series of reaction rates (n) may be measured for

If the dose is 320 mg, each concentration. Special plots may then be used

1  320 to determine KM and Vmax (see also Chapter 12).
t = 320 −160+100 ln  = 4.59 h

50  160 Equation 10.7 may be rearranged into
Equation 10.8.

For capacity-limited elimination, the elimina-
V C

max
tion half-life is dose dependent, because the ν = (10.7)

K +C
drug elimination process is partially saturated. M

Therefore, small changes in the dose will pro- 1 KM 1 1
= + (10.8)

duce large differences in the time for 50% drug ν Vmax C Vmax

elimination. The parameters KM and Vmax deter-
mine when the dose is saturated. Equation 10.8 is a linear equation when 1/n is plotted

2. Using the same drug as in Problem 1, calculate against 1/C. The y intercept for the line is 1/Vmax, and

the time for 50% elimination of the dose when the slope is KM/Vmax. An example of a drug reacting

the doses are 10 and 5 mg. Explain why the times enzymatically with rate (n) at various concentrations

for 50% drug elimination are similar even though C is shown in Table 10-6 and Fig. 10-5. A plot of 1/n

the dose is reduced by one-half. versus 1/C is shown in Fig. 10-6. A plot of 1/n versus
1/C is linear with an intercept of 0.33 mmol. Therefore,

Solution 1
= 0.33 min ⋅mL/µmol

As in Practice Problem 1, use Equation 10.6 to V
max

calculate the time for the amount of drug in the
Vmax = 3 µmol/mL ⋅min

body at zero time (D0) to decline 50%.
If the dose is 10 mg, because the slope = 1.65 = KM/Vmax = KM/3 or KM =

1  10 3 × 1.65 mmol/mL = 5 mmol/mL. Alternatively,
t = 10 − 5+100 ln  = 1.49 h

50  5  KM may be found from the x intercept, where −1/KM
is equal to the x intercept. (This may be seen by

If the dose is 5 mg, extending the graph to intercept the x axis in the

1  5  negative region.)
t = 5− 2.5+100 ln  = 1.44 h

50  2.5 With this plot (Fig. 10-6), the points are clus-
tered. Other methods are available that may spread

Whether the patient is given a 10-mg or a 5-mg the points more evenly. These methods are derived
dose by IV bolus injection, the times for the from rearranging Equation 10.8 into Equations 10.9
amount of drug to decline 50% are approximately and 10.10.
the same. For 10- and 5-mg doses, the amount
of drug in the body is much less than the KM of C 1 KM (10.9)

= C +
100 mg. Therefore, the amount of drug in the ν Vmax Vmax

body is well below saturation of the elimination
process and the drug declines at a first-order rate. ν

ν = −K +V (10.10)
M C max

Determination of KM and Vmax A plot of C/n versus C would yield a straight line
Equation 10.1 relates the rate of drug biotransfor- with 1/Vmax as slope and KM/Vmax as intercept
mation to the concentration of the drug in the body. (Equation 10.9). A plot of n versus n/C would yield a
The same equation may be applied to determine slope of −KM and an intercept of Vmax (Equation 10.10).

 

Nonlinear Pharmacokinetics 237

TABLE 106 Information Necessary for Graphic Determination of Vmax and K
M

Observation C V 1/V 1/C

Number ( lM/mL) ( lM/mL/min) (mL/min/lM) (mL/lM)

1 1 0.500 2.000 1.000

2 6 1.636 0.611 0.166

3 11 2.062 0.484 0.090

4 16 2.285 0.437 0.062

5 21 2.423 0.412 0.047

6 26 2.516 0.397 0.038

7 31 2.583 0.337 0.032

8 36 2.63 0.379 0.027

9 41 2.673 0.373 0.024

10 46 2.705 0.369 0.021

3.0 The necessary calculations for making the above
plots are shown in Table 10-7. The plots are shown

2.5 in Figs. 10-7 and 10-8. It should be noted that the
data are spread out better by the two latter plots.

2.0
Calculations from the slope show that the same KM

1.5 and Vmax are obtained as in Fig. 10-6. When the data
are more scattered, one method may be more accu-

1.0 rate than the other. A simple approach is to graph the

0.5
0 12 24 36 48

Drug concentration (C )
TABLE 107 Calculations Necessary for

FIGURE 105 Plot of rate of drug metabolism at various Graphic Determination of KM and Vmax
drug concentrations. (KM = 0.5 mmol/mL, Vmax = 3 mmol/mL/min.)

C V C/V V/C
( lM/mL) ( lM/mL/min) (min) (1/min)

2.0
1 0.500 2.000 0.500

1.8

1.6 6 1.636 3.666 0.272

1.4 11 2.062 5.333 0.187

1.2 16 2.285 7.000 0.142
1.0

21 2.423 8.666 0.115
0.8

0.6 26 2.516 10.333 0.096

0.4 31 2.583 12.000 0.083
0.2

0 0.2 0.4 0.6 0.8 1.0 36 2.634 13.666 0.073
1/C

41 2.673 15.333 0.065
FIGURE 106 Plot of 1/n versus 1/C for determining KM

46 2.705 17.000 0.058
and Vmax.

1/u Rate of drug metabolism (u)

 

238 Chapter 10

18 An example for the determination of KM and
16 Vmax is given for the drug phenytoin. Phenytoin
14 undergoes capacity-limited kinetics at therapeutic
12 drug concentrations in the body. To determine KM
10 and Vmax, two different dose regimens are given at
8 different times, until steady state is reached. The
6 steady-state drug concentrations are then measured
4 by assay. At steady state, the rate of drug metabolism
2 (n) is assumed to be the same as the rate of drug input
0 R (dose/day). Therefore, Equation 10.11 may be writ-
0 12 24 36 48

C ten for drug metabolism in the body similar to the
way drugs are metabolized in vitro (Equation 10.7).

FIGURE 107 Plot of C/n versus C for determining KM and
However, steady state will not be reached if the drug

Vmax.
input rate, R, is greater than the Vmax; instead, drug
accumulation will continue to occur without reaching
a steady-state plateau.

data and examine the linearity of the graphs. The
same basic type of plot is used in the clinical litera-

VmaxCss (10.11)
ture to determine KM and Vmax for individual patients R =

KM +Css
for drugs that undergo capacity-limited kinetics.

where R = dose/day or dosing rate, Css = steady-state

Determination of KM and Vmax in Patients plasma drug concentration, Vmax = maximum meta-
bolic rate constant in the body, and KM = Michaelis–

Equation 10.7 shows that the rate of drug metabo-
Menten constant of the drug in the body.

lism (n) is dependent on the concentration of the
drug (C). This same basic concept may be applied to
the rate of drug metabolism of a capacity-limited EXAMPLE »» »
drug in the body (see Chapter 12). The body may be
regarded as a single compartment in which the drug Phenytoin was administered to a patient at dos-
is dissolved. The rate of drug metabolism will vary ing rates of 150 and 300 mg/d, respectively. The
depending on the concentration of drug Cp as well as steady-state plasma drug concentrations were 8.6
on the metabolic rate constants KM and Vmax of the and 25.1 mg/L, respectively. Find the KM and Vmax
drug in each individual. of this patient. What dose is needed to achieve a

steady-state concentration of 11.3 mg/L?

3.0 Solution

There are three methods for solving this problem, all
2.5

based on the same basic equation (Equation 10.11).
2.0

Method A

1.5 Inverting Equation 10.11 on both sides yields

1 K 1 1
1.0 M

= +
R V C V (10.12)

max ss max

0.5
0.06 0.18 0.30 0.42 0.54 Multiply both sides by CssVmax,

u/C
VmaxCss

FIGURE 108 Plot of n versus n/C for determining KM and = KM + Css
R

Vmax.

u C/u

 

Nonlinear Pharmacokinetics 239

30
800

Vmax = 630 mg/d
20 600

Vmax = 630 mg/d

Slope KM = 27.5 mg/L
400

10

200

0
0.02 0.04 0.06 0.08 0.10 0

0 5 10 15 20
Css/dose rate (R)

(L/d) Clearance (dose/day/Css) (L/d)

–10
FIGURE 1010 Plot of R versus R/Css or clearance
(method B). (From Witmer and Ritschel, 1984, with
permission.)

–20

KM = 27.6 mg/L 2. Mark points for R of 150 mg/d and Css of 8.6 mg/L
as shown. Connect with a straight line.

–30
3. The point where lines from the first two steps

FIGURE 109 Plot of Css versus Css/R (method A). cross is called point A.
(From Witmer and Ritschel, 1984, with permission.)

4. From point A, read Vmax on the y axis and KM on

Rearranging the x axis. (Again, Vmax of 630 mg/d and KM of
27 mg/L are found.)

V C
Css = max ss − KM (10.13)

R

A plot of Css versus Css/R is shown in Fig. 10-9. Vmax is
700

equal to the slope, 630 mg/d, and KM is found from the Vmax = 630 mg/d
y intercept, 27.6 mg/L (note the negative intercept). A

600
Method B
From Equation 10.11, 500

RKM + RCss =VmaxCss

400
Dividing both sides by Css yields

K R
R =V M

max − 300
(10.14)

Css

A plot of R versus R/Css is shown in Fig. 10-10. The 200

KM and Vmax found are similar to those calculated
by the previous method (Fig. 10-9). 100

Method C
A plot of R versus Css is shown in Fig. 10-11. To 30 20 10 0 10 20 30
determine K Phenytoin Css (mg/L)

M and Vmax:
1. Mark points for R of 300 mg/d and Css of 25.1 mg/L

FIGURE 1011 Plot of R versus Css (method C).
as shown. Connect with a straight line. (From Witmer and Ritschel, 1984, with permission.)

Phenytoin Css (mg/L)

Dose rate (mg/d)

Dose/day (mg/d)

KM = 27 mg/L

 

240 Chapter 10

is the second dosing rate. To calculate KM and Vmax,
This Vmax and KM can be used in Equation 10.11 to use Equation 10.15 with the values C1 = 8.6 mg/L,
find an R to produce the desired Css of 11.3 mg/L. C2 = 25.1 mg/L, R1 = 150 mg/d, and R2 = 300 mg/d.
Alternatively, join point A on the graph to meet The results are
11.3 mg/L on the x axis; R can be read where this
line meets the y axis (190 mg/d). 300 −150

KM = − = 27.3 mg/L
To calculate the dose needed to keep steady- (150/8.6) (300/25.1)

state phenytoin concentration of 11.3 mg/L in this
patient, use Equation 10.7. Substitute KM into either of the two simultaneous

equations to solve for Vmax.
(630 mg/d)(11.3 mg/L)

R =
27 mg/L+11.3 mg/L V (8.6)

150 max
=

27.3+8.6
7119

= =186 mg/d
38.3 Vmax = 626 mg/d

This answer compares very closely with the value
obtained by the graphic method. All three meth- Interpretation of KM and Vmax
ods have been used clinically. Vozeh et al (1981) An understanding of Michaelis–Menten kinetics
introduced a method that allows for an estimation provides insight into the nonlinear kinetics and helps
of phenytoin dose based on steady-state concentra- avoid dosing a drug at a concentration near enzyme
tion resulting from one dose. This method is based saturation. For example, in the above phenytoin
on a statistically compiled nomogram that makes it dosing example, since KM occurs at 0.5Vmax, KM =
possible to project a most likely dose for the patient. 27.3 mg/L, the implication is that at a plasma con-

centration of 27.3 mg/L, enzymes responsible for
phenytoin metabolism are eliminating the drug at

Determination of KM and Vmax 50% Vmax, that is, 0.5 × 626 mg/d or 313 mg/d. When

by Direct Method the subject is receiving 300 mg of phenytoin per day, the
plasma drug concentration of phenytoin is 8.6 mg/L,

When steady-state concentrations of phenytoin are
which is considerably below the KM of 27.3 mg/L.

known at only two dose levels, there is no advantage
In practice, the KM in patients can range from 1 to

in using the graphic method. KM and Vmax may be
15 mg/L, and V

calculated by solving two simultaneous equations max can range from 100 to 1000 mg/d.
Patients with a low KM tend to have greater changes

formed by substituting Css and R (Equation 10.11)
in plasma concentrations during dosing adjustments.

with C1, R1, C2, and R2. The equations contain two
Patients with a smaller KM (same Vmax) will show a

unknowns, KM and Vmax, and may be solved easily.
greater change in the rate of elimination when plasma

VmaxC1 drug concentration changes compared to subjects
R1 =

K with a higher KM. A subject with the same Vmax, but
M +C1

V different KM, is shown in Fig. 10-12. (For another
maxC2

R2 = example, see the slopes of the two curves generated
KM +C2

in Fig. 10-4.)
Combining the two equations yields Equation 10.15.

Dependence of Elimination Half-Life on Dose
R

2 − R1
KM = (10.

(R For drugs that follow linear kinetics, the elimination
1 /C1)− 15)

(R2 /C2 )
half-life is constant and does not change with dose or

where C1 is steady-state plasma drug concentration drug concentration. For a drug that follows nonlinear
after dose 1, C2 is steady-state plasma drug concen- kinetics, the elimination half-life and drug clearance
tration after dose 2, R1 is the first dosing rate, and R2 both change with dose or drug concentration. Generally,

 

Nonlinear Pharmacokinetics 241

8.00 Within a certain drug concentration range, an average
or mean clearance (Clav) may be determined. Because

7.00 the drug follows Michaelis–Menten kinetics, Clav
is dose dependent. Clav may be estimated from the

6.00
area under the curve and the dose given (Wagner

5.00 et al, 1985).
According to the Michaelis–Menten equation,

4.00
dC

p VmaxCp
= (10.17)

3.00 dt KM +Cp

2.00
KM = 4 Inverting Equation 10.17 and rearranging yields

1.00
KM = 2

K C
M p

0.00 Cpdt = dC dC
V p − p (10.18)

0 10 20 30 40 50 60 70 80 m′ ax Vm′ ax
Cp

FIGURE 1012 Diagram showing the rate of metabolism The area under the curve, [AUC]∞0 , is obtained by
when Vmax is constant (8 mg/mL/h) and KM is changed (KM = integration of Equation 10.18 (ie, [AUC]∞


0 = ∫ C d

0 p t).
2 mg/mL for top curve and KM = 4 mg/mL for bottom curve).
Note the rate of metabolism is faster for the lower KM, but

∞ ∞ K ∞ C
saturation starts at lower concentration. ∫ M p

C dt dC dC
0 p = ∫ p +

C0 ∫
V 0 p (10.19)

p C
m′ ax p Vm′ ax

the elimination half-life becomes longer, clearance where V ′ is the maximum velocity for metabolism.
max

becomes smaller, and the area under the curve Units for V ′ are mass/compartment volume per
max

becomes disproportionately larger with increasing unit time. Vm′ ax =Vmax /VD; Wagner et al (1985) used

dose. The relationship between elimination half-life Vmax in Equation 10.20 as mass/time to be consistent

and drug concentration is shown in Equation 10.16. with biochemistry literature, which considers the

The elimination half-life is dependent on the initial mass of the substrate reacting with the enzyme.

Michaelis–Menten parameters and concentration. Integration of Equation 10.18 from time 0 to
infinity gives Equation 10.20.

0.693
t1/2 = (K C )

V M + p (10.16) C0 
p C0 

max [AUC]∞ =  p
+K 

0 V /V  2 M (10.20)
max D

Some pharmacokineticists prefer not to calculate
the elimination half-life of a nonlinear drug because where VD is the apparent volume of distribution.
the elimination half-life is not constant. Clinically, Because the dose D0 =C0

pVD, Equation 10.20 may
if the half-life is increasing as plasma concentration be expressed as
increases, and there is no apparent change in meta- 0

D C 
bolic or renal function, then there is a good possibil- [AUC]∞ = 0  p 

0  +K (10.21)
V M

ity that the drug may be metabolized by nonlinear max 2

kinetics. To obtain mean body clearance, Clav is then calcu-
lated from the dose and the AUC.

Dependence of Clearance on Dose D0 V
max

Clav = ∞ = (10.22)
The total body clearance of a drug given by IV bolus [AUC] 0

0 (Cp /2)+KM
injection that follows a one-compartment model
with Michaelis–Menten elimination kinetics changes V

max
Clav = (10.23)

with respect to time and plasma drug concentration. (D0 /2VD )+KM

Rate of metabolism

 

242 Chapter 10

A. Metoprolol 200 mg B. Timolol 20 mg

600 120

400 80

200 40

0 0
0 4 8 12 16 18 24 0 4 8 12 16 20 24

Time (hours) Time (hours)

FIGURE 1013 Mean plasma drug concentration-versus-time profiles following administration of single oral doses
of (A) metoprolol tartrate 200 mg to 6 extensive metabolizers (EMs) and 6 poor metabolizers (PMs) and (B) timolol maleate 20 mg
to six EMs (O) and four PMs (•). (Data from Lennard MS, et al: Oxidation phenotype—A major determinant of metoprolol metabolism
and response. NEJM 307:1558–1560, 1982; Lennard MS, et al: The relationship between debrisoquine oxidation phenotype and the
pharmacokinetics and pharmacodynamics of propranolol. Br J Clin Pharmac 17(6):679–685, 1984; Lewis RV: Timolol and atenolol:
Relationships between oxidation phenotype, pharmacokinetics and pharmacodynamics. Br J Clin Pharmac 19(3):329–333, 1985.)

Alternatively, dividing Equation 10.17 by Cp gives available supporting variable metabolism due to
Equation 10.24, which shows that the clearance of a genetic polymorphism (Chapter 12). The clearance
drug that follows nonlinear pharmacokinetics is (apparent) of many of these drugs in patients who
dependent on the plasma drug concentration Cp, KM, are slow metabolizers changes with dose, although
and Vmax. these drugs may exhibit linear kinetics in subjects

with the “normal” phenotype. Metoprolol and many
VD (dCp /dt) V

max
Cl = = (10.24) b-adrenergic antagonists are extensively metabolized.

Cp KM +Cp The plasma levels of metoprolol in slow metabolizers

Equation 10.22 or 10.23 calculates the average clear- (Lennard et al, 1986) were much greater than other

ance Cl patients, and the AUC, after equal doses, is several
av for the drug after a single IV bolus dose

over the entire time course of the drug in the body. times greater among slow metabolizers of metoprolol

For any time period, clearance may be calculated (Fig. 10-13). A similar picture is observed with another

(see Chapters 7 and 12) as b-adrenergic antagonist, timolol. These drugs have
smaller clearance than normal.

dD
E /dt

ClT = (10.25)
Cp

CLINICAL FOCUS
In Chapter 12, the physiologic model based on blood
flow and intrinsic clearance is used to describe drug The dose-dependent pharmacokinetics of sodium
metabolism. The extraction ratios of many drugs are valproate (VPA) was studied in guinea pigs at 20,
listed in the literature. Actually, extraction ratios are 200, and 600 mg/kg by rapid intravenous infusion.
dependent on dose, enzymatic system, and blood The area under the plasma concentration–time curve
flow, and for practical purposes, they are often increased out of proportion at the 600-mg/kg dose
assumed to be constant at normal doses. level in all groups (Yu et al, 1987). The total clear-

Except for phenytoin, there is a paucity of KM and ance (ClT) was significantly decreased and the beta
Vmax data defining the nature of nonlinear drug elimi- elimination half-life (t1/2) was significantly increased
nation in patients. However, abundant information is at the 600-mg/kg dose level. The dose-dependent

mg/L

mg/L

 

Nonlinear Pharmacokinetics 243

kinetics of VPA were due to saturation of metabolism. case of overdose, high liver drug concentrations and
Metabolic capacity was greatly reduced in young an extensive tissue distribution (large VD) made the
guinea pigs. drug difficult to remove. Vermeulen (1998) reported

Clinically, similar enzymatic saturation may be that saturation of CYP2D6 could result in a dispro-
observed in infants and in special patient populations, portionally higher plasma level than could be
whereas drug metabolism may be linear with dose in expected from an increase in dosage. These high
normal subjects. These patients have lower Vmax and plasma drug concentrations may be outside the range
longer elimination half-life. Variability in drug metab- of 20–50 mg normally recommended. Since publica-
olism is described in Chapters 12 and 13. tion of this article, more is known about genotype

CYP2D6*10 (Yoon et al, 2000), which may contrib-
ute to intersubject variability in metabolism of this

Frequently Asked Questions drug (see also Chapter 13).

»»What is the Michaelis–Menten equation? How are
Vmax and KM obtained? What are the units for Vmax
and KM? What is the relevance of V Frequently Asked Questions

max and KM?
»»What does autoinhibition mean? Would you expect

»»What are the main differences in pharmacokinetic paroxetine (Paxil) plasma drug concentrations, Cp, to
parameters between a drug that follows linear be higher or lower after multiple doses? Would the Cp
pharmacokinetics and a drug that follows nonlinear change be predictable among different subjects?
pharmacokinetics?

»»Name an example of SSRI and MAOI drug. Read
Chapter 13 to learn how another CYP2D6 drug may
greatly change the Cp of a drug such as Paxil.

CLINICAL FOCUS
Paroxetine hydrochloride (Paxil) is an orally admin-
istered psychotropic drug. Paroxetine is extensively
metabolized and the metabolites are considered to be DRUGS DISTRIBUTED AS
inactive. Nonlinearity in pharmacokinetics is ONE-COMPARTMENT MODEL
observed with increasing doses. Paroxetine exhibits AND ELIMINATED BY NONLINEAR
autoinhibition. The major pathway for paroxetine
metabolism is by CYP2D6. The elimination half-life PHARMACOKINETICS
is about 21 hours. Saturation of this enzyme at clini- The equations presented thus far in this chapter
cal doses appears to account for the nonlinearity of have been for drugs given by IV bolus, distributed
paroxetine kinetics with increasing dose and increas- as a one-compartment model, and eliminated only
ing duration of treatment. The role of this enzyme in by nonlinear pharmacokinetics. The following are
paroxetine metabolism also suggests potential drug– useful equations describing other possible routes of
drug interactions. Clinical drug interaction studies drug administration and including mixed drug
have been performed with substrates of CYP2D6 elimination, by which the drug may be eliminated
and show that paroxetine can inhibit the metabolism by both nonlinear (Michaelis–Menten) and linear
of drugs metabolized by CYP2D6 including itself, (first-order) processes.
desipramine, risperidone, and atomoxetine.

Paroxetine hydrochloride is known to inhibit
metabolism of selective serotonin reuptake inhibitors Mixed Drug Elimination

(SSRIs) and monoamine oxidase inhibitors (MAOIs) Drugs may be metabolized to several different metab-
producing “serotonin syndrome” (hyperthermia, olites by parallel pathways. At low drug doses corre-
muscle rigidity, and rapid changes in vital signs). sponding to low drug concentrations at the site of the
Three cases of accidental overdosing with paroxetine biotransformation enzymes, the rates of formation
hydrochloride were reported (Vermeulen, 1998). In the of metabolites are first order. However, with higher

 

244 Chapter 10

doses of drug, more drug is absorbed and higher drug the nonlinear relationship between niacin dose and
concentrations are presented to the biotransformation plasma drug concentrations following multiple doses
enzymes. At higher drug concentrations, the enzyme of Niaspan (niacin) extended-release tablets (Niaspan,
involved in metabolite formation may become satu- FDA-approved label, 2009).
rated, and the rate of metabolite formation becomes
nonlinear and approaches zero order. For example, Zero-Order Input and Nonlinear Elimination
sodium salicylate is metabolized to both a glucuro-

The usual example of zero-order input is constant IV
nide and a glycine conjugate (hippurate). The rate of

infusion. If the drug is given by constant IV infusion
formation of the glycine conjugate is limited by the

and is eliminated only by nonlinear pharmacokinetics,
amount of glycine available. Thus, the rate of forma-

then the following equation describes the rate of
tion of the glucuronide continues as a first-order

change of the plasma drug concentration:
process, whereas the rate of conjugation with glycine
is capacity limited. dCp k V ′

0 maxC

The equation that describes a drug that is elimi- = − p
(10.27)

dt VD KM +Cp
nated by both first-order and Michaelis–Menten kinet-
ics after IV bolus injection is given by where k0 is the infusion rate and VD is the apparent

volume of distribution.
dCp Vm′ axCp

− = kC 2
dt p + (10. 6)

KM +Cp First-Order Absorption and
Nonlinear Elimination

where k is the first-order rate constant representing The relationship that describes the rate of change in
the sum of all first-order elimination processes, while the plasma drug concentration for a drug that is
the second term of Equation 10.26 represents the given extravascularly (eg, orally), absorbed by first-
saturable process. V ′ is simply Vmax expressed as

max order absorption, and eliminated only by nonlinear
concentration by dividing by VD. pharmacokinetics, is given by the following equation.

CGI is concentration in the GI tract.

dCp V ′
CLINICAL FOCUS = −k t maxCp

k C a

dt a GIe − (10.28)
KM +Cp

The pharmacokinetic profile of niacin is complicated
due to extensive first-pass metabolism that is dosing- where ka is the first-order absorption rate constant.
rate specific. In humans, one metabolic pathway is If the drug is eliminated by parallel pathways
through a conjugation step with glycine to form nico- consisting of both linear and nonlinear pharmaco-
tinuric acid (NUA). NUA is excreted in the urine, kinetics, Equation 10.28 may be extended to
although there may be a small amount of reversible Equation 10.29.
metabolism back to niacin. The other metabolic path-

dCp V ′
way results in the formation of nicotinamide adenine = −k maxCp

k C e at

dt a GI − − kC
K p (10.29)

dinucleotide (NAD). It is unclear whether nicotinamide M +Cp

is formed as a precursor to, or following the synthesis
where k is the first-order elimination rate constant.

of, NAD. Nicotinamide is further metabolized to at
least N-methylnicotinamide (MNA) and nicotinamide-
N-oxide (NNO). MNA is further metabolized to two Two-Compartment Model with

other compounds, N-methyl-2-pyridone-5-carboxamide Nonlinear Elimination

(2PY) and N-methyl-4-pyridone-5-carboxamide (4PY). RhG-CSF is a glycoprotein hormone (recombinant
The formation of 2PY appears to predominate over human granulocyte-colony stimulating factors, rhG-
4PY in humans. At doses used to treat hyperlipidemia, CSF, MW about 20,000) that stimulates the growth of
these metabolic pathways are saturable, which explains neutropoietic cells and activates mature neutrophils.

 

Nonlinear Pharmacokinetics 245

The drug is used in neutropenia occurring during CHRONOPHARMACOKINETICS
chemotherapy or radiotherapy. Similar to many bio-

AND TIME-DEPENDENT
technological drugs, RhG-CSF is administered by
injection. The drug is administered subcutaneously PHARMACOKINETICS
and absorbed into the blood from the dermis site.

Chronopharmacokinetics broadly refers to a tempo-
This drug follows a two-compartment model with

ral change in the rate process (such as absorption or
two elimination processes: (1) a saturable process of

elimination) of a drug. The temporal changes in drug
receptor-mediated elimination in the bone marrow

absorption or elimination can be cyclical over a
and (2) a nonsaturable process of elimination. The

constant period (eg, 24-hour interval), or they may
model is described by two differential equations as

be noncyclical, in which drug absorption or elimi-
shown below:

nation changes over a longer period of time. Chrono-
dC  V pharmacokinetics is an important consideration during

= − + + max 
1 k X

k12 k +  1 2
C1 + 2 (10.29a)

dt  V drug therapy.
1(C1 KM ) V1

Time-dependent pharmacokinetics generally
dX

2 = refers to a noncyclical change in the drug absorp-
k C V − k X (10.29b)

dt 12 1 1 21 2
tion or drug elimination rate process over a period
of time. Time-dependent pharmacokinetics leads to

where k12 and k21 are first-order transfer constants
nonlinear pharmacokinetics. Unlike dose-dependent

between the central and peripheral comparments; k
pharmacokinetics, which involves a change in the rate

is the first-order elimination constant from the cen-
process when the dose is changed, time-dependent

tral compartment; V1 is the volume of the central
pharmacokinetics may be the result of alteration in

compartment and the steady-state volume of distri-
the physiology or biochemistry in an organ or a

bution is Vss; X2 is the amount in the peripheral com-
region in the body that influences drug disposition

partment; C1 is the drug concentration in the central
(Levy, 1983).

compartments at time t; and Vmax and KM are
Michaelis–Menten parameters that describe the satu- Time-dependent pharmacokinetics may be due

rable elimination. to autoinduction or autoinhibition of biotransforma-

The pharmacokinetics of this drug was tion enzymes. For example, Pitlick and Levy (1977)

described by Hayashi et al (2001). Here, a is a func- have shown that repeated doses of carbamazepine

tion of dose with no dimensions, and granulocyte induce the enzymes responsible for its elimination

colony-stimulating factor (G-CSF) takes a value from (ie, auto-induction), thereby increasing the clearance

0 to 1. When the dose approaches 0, a of the drug. Auto-inhibition may occur during the
= 1; when the

dose approaches ∞, a course of metabolism of certain drugs (Perrier et al,
= 0.

According to Hayashi et al (2001), the drug 1973). In this case, the metabolites formed increase

clearance may be considered as two parts as shown in concentration and further inhibit metabolism of

below: the parent drug. In biochemistry, this phenomenon
is known as product inhibition. Drugs undergoing

Dose/AUC = αCl c)
int +Cln =Cl (10.29 time-dependent pharmacokinetics have variable

clearance and elimination half-lives. The steady-state

CK
∫ M

dt concentration of a drug that causes auto-induction
C +K may be due to increased clearance over time. Some

Dose/AUC = 0 M (10.29d)
Clint +Cl

∞ n anticancer drugs are better tolerated at certain times
∫ Cdt of the day; for example, the antimetabolite drug fluo-
0

rouracil (FU) was least toxic when given in the
where Clint is intrinsic clearance for the saturable path- morning to rodents (Von Roemeling, 1991). A list of
way; Cln is nonsaturable clearance; and C is serum drugs that demonstrate time dependence is shown
concentration. in Table 10-8.

 

246 Chapter 10

TABLE 108 Drugs Showing Circadian or and the incidence of nephrotoxicity were studied in
Time-Dependent Disposition 221 patients (Prins et al, 1997). Each patient received

an IV injection of 2–4 mg/kg gentamicin or tobramy-
Cefodizime Fluorouracil Ketoprofen Theophylline

cin once daily: (1) between midnight and 7:30 am,
Cisplatin Heparin Mequitazine (2) between 8 am and 3:30 pm, or (3) between 4 pm

Data from Reinberg (1991). and 11:30 pm. In this study, no statistically significant
differences in drug trough levels (0–4.2 mg/L) or
peak drug levels (3.6–26.8 mg/L) were found for the

In pharmacokinetics, it is important to recognize
three time periods of drug administration. However,

that many isozymes (CYPs) are involved in drug
nephrotoxicity occurred significantly more frequently

metabolisms. A drug may competitively influence
when the aminoglycosides were given during the rest

the metabolism of another drug within the same
period (midnight–7:30 am). Many factors contribut-

CYP subfamily. Sometimes, an unrecognized effect
ing to nephrotoxicity were discussed; the time of

from the presence of another drug may be misjudged
administration was considered to be an independent

as a time-dependent pharmacokinetics. Drug metab-
risk factor in the multivariate statistical analysis.

olism and pharmacogenetics are discussed more
Time-dependent pharmacokinetics/pharmacodynam-

extensively in Chapter 13.
ics is important, but it may be difficult to detect the
clinical difference in drug concentrations due to

Circadian Rhythms and Influence multivariates.
on Drug Response Another example of circadian changes on drug
Circadian rhythms are rhythmic or cyclical changes response involves observations with chronic obstruc-
in plasma drug concentrations that may occur daily, tive pulmonary disease (COPD) patients. Symptoms
due to normal changes in body functions. Some of hypoxemia may be aggravated in some COPD
rhythmic changes that influence body functions and patients due to changes in respiration during the
drug response are controlled by genes and subject to sleep cycle. Circadian variations have been reported
modification by environmental factors. The mam- involving the incidence of acute myocardial infarc-
malian circadian clock is a self-sustaining oscillator, tion, sudden cardiac death, and stroke. Platelet
usually within a period of ~24 hours, that cyclically aggregation favoring coagulation is increased after
controls many physiological and behavioral systems. arising in the early morning hours, coincident with
The biological clock attempts to synchronize and the peak incidence of these cardiovascular events,
respond to changes in length of the daylight cycle although much remains to be elucidated.
and optimize body functions. Time-dependent pharmacokinetics and physio-

Circadian rhythms are regulated through peri- logic functions are important considerations in the
odic activation of transcription by a set of clock treatment of certain hypertensive subjects, in whom
genes. For example, melatonin onset is associated early-morning rise in blood pressure may increase the
with onset of the quiescent period of cortisol secre- risk of stroke or hypertensive crisis. Verapamil is a
tion that regulates many functions. Some well- commonly used antihypertensive. The diurnal pattern
known circadian physiologic parameters are core of forearm vascular resistance (FVR) between hyper-
body temperature (CBT), heart rate (HR), and other tensive and normotensive volunteers was studied at
cardiovascular parameters. These fundamental phys- 9 pm on 24-hour ambulatory blood pressure monitor-
iologic factors can affect disease states, as well as ing, and the early-morning blood pressure rise was
toxicity and therapeutic response to drug therapy. studied in 23 untreated hypertensives and 10 matched,
The toxic dose of a drug may vary as much as sev- normotensive controls. The diurnal pattern of FVR
eral-fold, depending on the time of drug administra- differed between hypertensives and normotensives,
tion—during either sleep or wake cycle. with normotensives exhibiting an FVR decline

For example, the effects of timing of aminoglyco- between 2 pm and 9 pm, while FVR rose at 9 pm in
side administration on serum aminoglycoside levels hypertensives. Verapamil appeared to minimize the

 

Nonlinear Pharmacokinetics 247

diurnal variation in FVR in hypertensives, although 2200 versus 1000 hours, indicating that the rate of
there were no significant differences at any single excretion during the night time was slower (Sarveshwer
time point. Verapamil effectively reduced ambulatory Rao et al, 1997).
blood pressure throughout the 24-hour period, but it
did not blunt the early-morning rate of blood pressure Clinical and Adverse Toxicity Due to
rise despite peak S-verapamil concentrations in the Nonlinear Pharmacokinetics
early morning (Nguyen et al, 2000).

The presence of nonlinear or dose-dependent phar-
macokinetics, whether due to saturation of a process

CLINICAL FOCUS involving absorption, first-pass metabolism, binding,
or renal excretion, can have significant clinical con-

Hypertensive patients are sometimes characterized
sequences. However, nonlinear pharmacokinetics

as “dippers” if their nocturnal blood pressure drops
may not be noticed in drug studies that use a narrow

below their daytime pressure. Non-dipping patients
dose range in patients. In this case, dose estimation

appear to be at an increased risk of cardiovascular
may result in disproportionate increases in adverse

morbidity. Blood pressure and cardiovascular events
reactions but insufficient therapeutic benefits.

have a diurnal rhythm, with a peak of both in the
Nonlinear pharmacokinetics can occur anywhere

morning hours, and a decrease during the night. The
above, within, or below the therapeutic window.

circadian variation of blood pressure provides assis-
The problem of a nonlinear dose relationship in

tance in predicting cardiovascular outcome (de la
population pharmacokinetics analysis has been inves-

Sierra et al, 2011).
tigated using simulations (Hashimoto et al, 1994,

The pharmacokinetics of many cardiovascular
1995; Jonsson et al, 2000). For example, nonlinear

acting drugs have a circadian phase dependency
fluvoxamine pharmacokinetics was reported (Jonsson

(Lemmer, 2006). Examples include b-blockers, cal-
et al, 2000) to be present even at subtherapeutic doses.

cium channel blockers, oral nitrates, and ACE inhib-
By using simulated data and applying nonlinear

itors. There is clinical evidence that antihypertensive
mixed-effect models using NONMEM, the authors

drugs should be dosed in the early morning in
also demonstrated that use of nonlinear mixed-effect

patients who are hypertensive “dippers,” whereas for
models in population pharmacokinetics had an impor-

patients who are non-dippers, it may be necessary to
tant application in the detection and characterization

add an evening dose or even to use a single evening
of nonlinear processes (pharmacokinetic and pharma-

dose not only to reduce high blood pressure (BP) but
codynamic). Both first-order (FO) and FO conditional

also to normalize a disturbed non-dipping 24-hour
estimation (FOCE) algorithms were used for the

BP profile. However, for practical purposes, some
population analyses. Population pharmacokinetics is

investigators found diurnal BP monitoring in many
discussed further in Chapter 25.

individuals too variable to distinguish between dip-
pers and non-dippers (Lemmer, 2006).

The issue of time-dependent pharmacokinetics/
pharmacodynamics (PK/PD) may be an important BIOAVAILABILITY OF DRUGS
issue in some antihypertensive care. Pharmacists THAT FOLLOW NONLINEAR
should recognize drugs that exhibit this type of time- PHARMACOKINETICS
dependant PK/PD.

Another example of time-dependent pharmaco- The bioavailability of drugs that follow nonlinear
kinetics involves ciprofloxacin. Circadian variation pharmacokinetics is difficult to estimate accurately.
in the urinary excretion of ciprofloxacin was inves- As shown in Table 10-1, each process of drug absorp-
tigated in a crossover study in 12 healthy male vol- tion, distribution, and elimination is potentially satu-
unteers, ages 19–32 years. A significant decrease in rable. Drugs that follow linear pharmacokinetics
the rate and extent of the urinary excretion of cipro- follow the principle of superposition (Chapter 9). The
floxacin was observed following administrations at assumption in applying the rule of superposition is

 

248 Chapter 10

that each dose of drug superimposes on the previous 50

dose. Consequently, the bioavailability of subsequent
doses is predictable and not affected by the previous
dose. In the presence of a saturable pathway for drug 10

absorption, distribution, or elimination, drug bio-
5 B A

availability will change within a single dose or with
subsequent (multiple) doses. An example of a drug
with dose-dependent absorption is chlorothiazide
(Hsu et al, 1987). 1

Time

The extent of bioavailability is generally esti-
mated using [AUC]∞0 . If drug absorption is saturation FIGURE 1014 Plasma curve comparing the elimination

of two drugs given in equal IV doses. Curve A represents a drug
limited in the gastrointestinal tract, then a smaller

90% bound to plasma protein. Curve B represents a drug not
fraction of drug is absorbed systemically when the bound to plasma protein.
gastrointestinal drug concentration is high. A drug
with a saturable elimination pathway may also have the protein-bound drug is eliminated at a slower,
a concentration-dependent AUC affected by the nonlinear rate. Because the two drugs are eliminated
magnitude of KM and Vmax of the enzymes involved by identical mechanisms, the characteristically slower
in drug elimination (Equation 10.21). At low Cp, the elimination rate for the protein-bound drug is due to
rate of elimination is first order, even at the begin- the fact that less free drug is available for glomerular
ning of drug absorption from the gastrointestinal filtration in the course of renal excretion.
tract. As more drug is absorbed, either from a single The concentration of free drug, Cf, can be calcu-
dose or after multiple doses, systemic drug concen- lated at any time, as follows:
trations increase to levels that saturate the enzymes

Cf =Cp (1− fraction bound) (10.30)
involved in drug elimination. The body drug clear-
ance changes and the AUC increases disproportion- For any protein-bound drug, the free drug concentra-
ately to the increase in dose (see Fig. 10-2). tion (Cf) will always be less than the total drug con-

centration (Cp).
A careful examination of Fig. 10-14 shows that

NONLINEAR PHARMACOKINETICS the slope of the bound drug decreases gradually as the

DUE TO DRUG–PROTEIN BINDING drug concentration decreases. This indicates that the
percent of drug bound is not constant. In vivo, the per-

Protein binding may prolong the elimination half-life cent of drug bound usually increases as the plasma
of a drug. Drugs that are protein bound must first dis- drug concentration decreases (see Chapter 11). Since
sociate into the free or nonbound form to be elimi- protein binding of drug can cause nonlinear elimina-
nated by glomerular filtration. The nature and extent tion rates, pharmacokinetic fitting of protein-bound
of drug–protein binding affects the magnitude of the drug data to a simple one-compartment model with-
deviation from normal linear or first-order elimina- out accounting for binding results in erroneous esti-
tion rate process. mates of the volume of distribution and elimination

For example, consider the plasma level–time half-life. Sometimes plasma drug data for drugs that
curves of two hypothetical drugs given intravenously are highly protein bound have been inappropriately
in equal doses (Fig. 10-14). One drug is 90% protein fitted to two-compartment models.
bound, whereas the other drug does not bind plasma Valproic acid (Depakene) shows nonlinear phar-
protein. Both drugs are eliminated solely by glo- macokinetics that may be due partially to nonlinear
merular filtration through the kidney. protein binding. The free fraction of valproic acid is

The plasma curves in Fig. 10-14 demonstrate 10% at a plasma drug concentration of 40 mg/mL and
that the protein-bound drug is more concentrated in 18.5% at a plasma drug level of 130 mg/mL. In addi-
the plasma than a drug that is not protein bound, and tion, higher-than-expected plasma drug concentrations

Plasma level

 

Nonlinear Pharmacokinetics 249

occur in the elderly patients, and in patients with
5000

hepatic or renal disease.

One-Compartment Model Drug 1000
with Protein Binding 500

The process of elimination of a drug distributed in a 100 mg/kg
single compartment with protein binding is illus- 100
trated in Fig. 10-15. The one compartment contains

50
both free drug and bound drug, which are dynami-
cally interconverted with rate constants k 5 mg/kg 20 mg/kg

1 and k2.
Elimination of drug occurs only with the free drug, at 10
a first-order rate. The bound drug is not eliminated. Time

Assuming a saturable and instantly reversible drug- FIGURE 1016 Plasma drug concentrations for various
binding process, where P = protein concentration in doses of a one-compartment model drug with protein binding.

plasma, Cf = plasma concentration of free drug, kd = (Adapted from Coffey et al, 1971, with permission.)

k2/k1 = dissociation constant of the protein drug com-
plex, Cp = total plasma drug concentration, and Cb = This differential equation describes the relationship
plasma concentration of bound drug, of changing plasma drug concentrations during elim-

ination. The equation is not easily integrated but can
Cb (1/kd )C f

= (10.31) be solved using a numerical method. Figure 10-16
P 1+ (1/kd )Cf shows the plasma drug concentration curves for a one-

This equation can be rearranged as follows: compartment protein-bound drug having a volume of
distribution of 50 mL/kg and an elimination half-life

PC
f

Cb = =C −C 1 . 2 of 30 minutes. The protein concentration is 4.4% and
k p f ( 0 3 )
d +Cf the molecular weight of the protein is 67,000 Da. At

Solving for Cf ,
various doses, the pharmacokinetics of elimination of
the drug, as shown by the plasma curves, ranges from

1 linear to nonlinear, depending on the total plasma
C  ( ) ( 2

f = − P + kd −Cp + P + kd −Cp ) + 4k
2 dC


p  drug concentration.

(10.33) Nonlinear drug elimination pharmacokinetics
occurs at higher doses. Because more free drug is avail-

Because the rate of drug elimination is dCp/dt, able at higher doses, initial drug elimination occurs
more rapidly. For drugs demonstrating nonlinear phar-

dC
p = −kC macokinetics, the free drug concentration may increase
dt f

slowly at first, but when the dose of drug is raised
beyond the protein-bound saturation point, free plasma

dC
p −k

= 
−(P + kd −Cp )+ (P + kd −C )2+ 4 

k C
dt 2 p d p  drug concentrations may rise abruptly. Therefore, the

concentration of free drug should always be calculated
(10.34) to make sure the patient receives a proper dose.

Bound Determination of Linearity in Data Analysis
k1

k During new drug development, the pharmacokinetics
k2
Free of the drug is examined for linear or nonlinear phar-

macokinetics. A common approach is to give several
FIGURE 1015 One-compartment model with drug– graded doses to human volunteers and obtain plasma
protein binding. drug concentration curves for each dose. From these

Plasma level

 

250 Chapter 10

data, a graph of AUC versus dose is generated as shown When the third AUC point is above the trend line, it
in Fig. 10-2. The drug is considered to follow linear is risky to draw a conclusion. One should verify that
kinetics if AUC versus dose for various doses is propor- the high AUC is not due to a lower elimination or
tional (ie, linear relationship). In practice, the experi- clearance due to saturation.
mental data presented may not be very clear, especially In Fig. 10-18, a regression line was obtained by
when oral drug administration data are presented and forcing the same data through point (0,0). The linear
there is considerable variability in the data. For exam- regression analysis and estimated R2 appears to show
ple, the AUC versus three-graded doses of a new drug is that the drug followed nonlinear pharmacokinetics.
shown in Fig. 10-17. A linear regression line was drawn The line appears to have a curvature upward and the
through the three data points. The conclusion is that the possibility of some saturation at higher doses. This
drug follows dose-independent (linear) kinetics based pharmacokineticist recommends additional study by
upon a linear regression line through the data and a cor- adding a higher dose to more clearly check for dose
relation coefficient, R2 = 0.97. dependency.

• Do you agree with this conclusion after inspecting • What is your conclusion?
the graph?

Considerations
The conclusion for linear pharmacokinetics in

• The experimental data are composed of three dif-
Fig. 10-17 seems reasonable based on the estimated

ferent drug doses.
regression line drawn through the data points.

• The regression line shows that the drug follows
However, another pharmacokineticist noticed that

linear pharmacokinetics from the low dose to the
the regression line in Fig. 10-17 does not pass through

high dose.
the origin point (0,0). This pharmacokineticist consid-

• The use of a (0.0) value may provide additional
ered the following questions:

information concerning the linearity of the
• Are the patients in the study receiving the drug pharmacokinetics. However, extrapolation of

doses well separated by a washout period during curves beyond the actual experimental data can
the trial such that no residual drug remained in the be misleading.
body and carried to the present dose when plasma • The conclusion in using the (0.0) time point shows
samples are collected? that the pharmacokinetics is nonlinear below the

• Is the method for assaying the samples validated? lowest drug dose. This may occur after oral dos-
Could a high sample blank or interfering mate- ing because at very low drug doses some of the
rial be artificially adding to elevate 0 time drug drug is decomposed in the gastrointestinal tract
concentrations? or metabolized prior to systemic absorption. With

• How does the trend line look if the point (0,0) is higher doses, the small amount of drug loss is not
included? observed systemically.

AUC AUC

Dose (mg/kg) Dose (mg/kg)

FIGURE 1017 Plot of AUC versus dose to determine lin- FIGURE 1018 Plot of AUC versus dose to determine
earity. The regression line is based on the three doses of the drug. linearity.

 

Nonlinear Pharmacokinetics 251

TABLE 109 Some Common Issues during Data Analysis for Linearity

Oral Data Issues during Data Analysis Comments

Last data point may be below the LOD or limit of Last sample point scheduled too late in the study
detection. What should the AUC tailpiece be? protocol.

Last data point still very high, much above the LOD. Last sample point scheduled too early.
What should be the AUC tailpiece? A substantial number of data points may be

incorrectly estimated by the tailpiece method.

Incomplete sample spacing around peak. Total AUC estimated may be quite variable or
unreliable.

Oral AUC data are influenced by F, D, and Cl. When examining D0/Cl vs D0, F must be held
constant. Any factor causing change in F during
the trial will introduce uncertainty to AUC.

F may be affected by efflux, transporters (see Chapter 13), Nonlinearity of AUC vs D0 may not be
and GI CYP enzymes. An increase in F and decrease in Cl evident and one may incorrectly conclude a drug
or vice versa over doses may mask each other. follows linear kinetics when it does not.

IV data AUC data by IV are influenced by D0 and Cl only. When examining D0/Cl vs D0, F is always constant.
Therefore, it is easier to see changes in AUC when
Cl changes by IV route.

LOD, limit of detection.

Note if VD of the drug is known, determining k from POTENTIAL REASONS FOR
the terminal slope of the oral data provides another

UNSUSPECTED NONLINEARITY1
way of calculating Cl (Cl = VD k) to check whether
clearance has changed at higher doses due to satura-

1. Nonlinearity caused by membrane resident
tion. Some common issues during data analysis for

transporters
linearity are listed in Table 10-9.

2. Nonlinearity caused by membrane CYPs
Note: In some cases, with certain drugs, the oral

3. Nonlinearity caused by cellular proteins
absorption mechanism is quite unique and drug

4. Nonlinearity caused by transporter proteins at
clearance by the oral route may involve absorption

the GI tract
site-specific enzymes or transporters located on the

5. Nonlinearity caused by bile acid transport
brush border. Extrapolating pharmacokinetic infor-

(apical/bile canaliculus)
mation from IV dose data should be done cautiously
only after a careful consideration of these factors. It
is helpful to know whether nonlinearity is caused by
distribution, or absorption factors. Frequently Asked Questions

Unsuspected nonlinear drug disposition is one »»What is the cause of nonlinear pharmacokinetics
of the biggest issues concerning drug safety. that is not dose related?
Although pharmacokinetic tools are useful, nonlin-

»»For drugs that have several metabolic pathways,
earity can be easily missed during data analysis

must all the metabolic pathways be saturated for the
when there are outliners or extreme data scattering

drug to exhibit nonlinear pharmacokinetics?
due to individual patient factors such as genetics,
age, sex, and other unknown factors in special popu-
lations. While statistical analysis can help minimize
this, it is extremely helpful to survey for problems
(eg, epidemiological surveillance) and have a good 1Source: Evaluation of hepatotoxic potential of drugs using
understanding of how drugs are disposed in various transporter-based assays. Jasminder Sahi AAPS Transporter Meeting,
parts of the body in the target populations. 2005 at Parsippanny, New Jersey.

 

252 Chapter 10

DOSE-DEPENDENT transporters may critically enhance or reduce local
cell drug concentrations, allowing influx of drugs into

PHARMACOKINETICS
the cell or removing drug from the cell by efflux trans-

Role of Transporters porters, a defensive mechanism of the body. Many of

Classical pharmacokinetics studied linear pharmaco- the cells express transporters genetically, which may

kinetics of a drug by examining the area under plasma also be triggered on or turned off in disease state.

drug concentration curve at various doses given intra- Whether the overall pharmacokinetic process is linear

venously. The method is simple and definitive. The or nonlinear must be determined locally. The knowl-

method is useful revealing the kinetics in the body as edge of the local effects of transporters on pharmaco-

a whole. However, more useful information must now kinetics can improve safe and effective drug dosing.

be obtained through studies based on regional phar- The impact of transporters are discussed by various

macokinetics by studying the roles of transporters in authors in a review book edited by You and Morris

individual organs. Over the last few decades, trans- (2007). Table 10-10 summarizes some of the trans-

porters have been characterized in individual cells or porters that play an important role in drug distribution

in various types of cells (Chapters 11 and 13). These and how they may impact drug linearity.

TABLE 1010 Drug Transports and Comments on Roles in Altering Linearity of Absorption
or Elimination

Transporters Comments

Xenobiotic transporter Transporters may be age and gender related. These differences may change the linearity
expression of a drug through saturation.

Polymorphisms of drug Polymorphisms may have a clinical relevance affecting toxicity and efficacy in a similar
transporters way through change in pharmacokinetics.

Interplay of drug transporters The role of transporters on hepatic drug is profound and may greatly change the overall
and enzymes in liver linearity of a drug systemically.

The concept of drug clearance, Cl, and intrinsic clearance has to be reexamined as a
result of the translocation of transporters, at cellular membranes as suggested in a recent
review.

Drug–drug interaction change Clinical relevance, pharmacokinetics, pharmacodynamics, and toxicity may decrease
due to transporters or increase if a drug is a transporter substrate or inhibitor. Less clear is the change from

linear to nonlinear kinetics due to drug–drug interaction.

Drug transporters in the ABC transporters are very common and this can alter the absorption nature of a drug
intestine product, for example, the bioavailability and linearity of drug absorption. Bile acid

transporters affect drug movement and elimination by biliary excretion. The nature of the
process must be studied.

Drug transport in the kidney Various organic anion and cation drug transporters have been described. These trans-
porters may alter the linearity of systemic drug elimination if present in large quantity.

Multidrug resistance protein: These proteins may affect drug concentration in a cell or group of cells. Hence, they are
P-glycoprotein important elements in determining PK linearity.

Mammalian oligopeptide These transporters play a role in drug absorption and distribution.
transporters

Breast cancer resistance These transporters play a role in drug linearity and dosing in cancer therapy.
protein

 

Nonlinear Pharmacokinetics 253

CLINICAL EXAMPLE increase in AUC0–72) compared to the fasted state.
A standard meal also increased the rate of absorption

Zmax® (Pfizer) is an extended-release microsphere for- (119% increase in Cmax), with less effect on the extent
mulation of the antibiotic azithromycin in an oral sus- of absorption (12% increase in AUC0–72) compared to
pension. According to the approved label,2 based on administration of a 2-g Zmax dose in the fasted state.
data obtained from studies evaluating the pharmacoki-
netics of azithromycin in healthy adult subjects, a higher Distribution
peak serum concentration (Cmax) and greater systemic The serum protein binding of azithromycin is concen-
exposure (AUC 0–24) of azithromycin are achieved tration dependent, decreasing from 51% at 0.02 mg/mL
on the day of dosing following a single 2-g dose of to 7% at 2 mg/mL. Following oral administration,
Zmax versus 1.5 g of azithromycin tablets admin- azithromycin is widely distributed throughout the
istered over 3 days (500 mg/d) or 5 days (500 mg body with an apparent steady-state volume of distri-
on day 1, 250 mg/d on days 2–5) (Table 10-11). bution of 31.1 L/kg.
Consequently, due to these different pharmacokinetic Azithromycin concentrates in fibroblasts, epithe-
profiles, Zmax is not interchangeable with azithro- lial cells, macrophages, and circulating neutrophils
mycin tablet 3-day and 5-day dosing regimens. and monocytes. Higher azithromycin concentrations

in tissues than in plasma or serum have been observed.
Absorption Following a 2-g single dose of Zmax, azithromycin
The bioavailability of Zmax relative to azithromycin achieved higher exposure (AUC0–120) in mononuclear
immediate release (IR) (powder for oral suspension) leukocytes (MNL) and polymorphonuclear leuko-
was 83%. On average, peak serum concentrations cytes (PMNL) than in serum. The azithromycin
were achieved approximately 2.5 hours later following exposure (AUC0–72) in lung tissue and alveolar cells
Zmax administration and were lower by 57%, com- (AC) was approximately 100 times than in serum,
pared to 2 g azithromycin IR. Thus, single 2-g doses of and the exposure in epithelial lining fluid (ELF) was
Zmax and azithromycin IR are not bioequivalent and also higher (approximately 2–3 times) than in serum.
are not interchangeable. The clinical significance of this distribution data is

Effect of food on absorption: A high-fat meal unknown.
increased the rate and extent of absorption of a 2-g
dose of Zmax (115% increase in Cmax, and 23% Metabolism

In vitro and in vivo studies to assess the metabolism
2http://labeling.pzer.com/ShowLabeling.aspx?id=650#section-12.3. of azithromycin have not been performed.

TABLE 1011 Mean (SD) Pharmacokinetic Parameters for Azithromycin on Day 1 Following the
Administration of a Single Dose of 2 g Zmax or 1.5 g of Azithromycin Tablets Given over 3 Days
(500 mg/d) or 5 Days (500 mg on Day 1 and 250 mg on Days 2–5) to Healthy Adult Subjects

Azithromycin Regimen
Pharmacokinetic
Parameter * Zmax (N = 41) 3-Day (N = 12) 5-Day (N = 12)

Cmax (mg/mL) 0.821 (0.281) 0.441 (0.223) 0.434 (0.202)

T §
max (h) 5.0 (2.0–8.0) 2.5 (1.0–4.0) 2.5 (1.0–6.0)

AUC0–24 (mg·h/mL) 8.62 (2.34) 2.58 (0.84) 2.60 (0.71)

AUC0–∞¶ (mg·h/mL) 20.0 (6.66) 17.4 (6.2) 14.9 (3.1)

t1/2 (h) 58.8 (6.91) 71.8 (14.7) 68.9 (13.8)

∗Zmax, 3-day and 5-day regimen parameters obtained from separate pharmacokinetic studies

Adapted from Zmax approved label, October 2013.

 

254 Chapter 10

Excretion Based on the information,

Serum azithromycin concentrations following a single 1. The bioavailability of this drug may be quite
2-g dose of Zmax declined in a polyphasic pattern different for different dosage forms due to
with a terminal elimination half-life of 59 hours. The absorption profile.
prolonged terminal half-life is thought to be due to a 2. Absorption is likely to be affected by GI
large apparent volume of distribution. residence time of the product and the type of

Biliary excretion of azithromycin, predomi- dosage form.
nantly as unchanged drug, is a major route of elimi- 3. The drug is widely distributed.
nation. Over the course of a week, approximately 4. Drug binding may be nonlinear resulting in
6% of the administered dose appears as unchanged different free drug concentrations at different
drug in urine. serum drug concentrations.

CHAPTER SUMMARY
Nonlinear pharmacokinetics refers to kinetic pro- nonlinearity curving. A common cause of overdosing
cesses that result in disproportional changes in in clinical practice is undetected saturation of a meta-
plasma drug concentrations when the dose is bolic enzyme due to genotype difference in a subject,
changed. This is also referred to as dose-dependent for example, CYP2D6. A second common cause of
pharmacokinetics or saturation pharmacokinetics. overdosing in clinical practice is undetected satura-
Clearance and half-life are usually not constant with tion of a metabolic enzyme due to coadministration
dose-dependent pharmacokinetics. Carrier-mediated of a second drug/agent that alters the original linear
processes and processes that depend on the binding elimination process. Drug transporters play an impor-
of the drug to a macromolecule resulting in drug tant role in the body. Membrane-located transporters
metabolism, protein binding, active absorption, and may cause uneven drug distribution at cellular level,
some transporter-mediated processes can potentially and hiding concentration-dependent kinetics may
exhibit dose-dependent kinetics, especially at higher occur at the local level within body organs. These
doses. The Michaelis–Menten kinetic equation may processes include absorption and elimination and are
be applied in vitro or in vivo to describe drug dispo- important in drug therapy. Some transporters are trig-
sition, for example, phenytoin. gered by disease or expressed differently in individu-

An approach to determine nonlinear pharmaco- als and should be recognized by pharmacists during
kinetics is to plot AUC versus doses and observe for dosing regimen recommendation.

LEARNING QUESTIONS
1. Define nonlinear pharmacokinetics. How do 2. What processes of drug absorption, distribution,

drugs that follow nonlinear pharmacokinetics and elimination may be considered “capacity
differ from drugs that follow linear pharmaco- limited,” “saturated,” or “dose dependent”?
kinetics? 3. Drugs, such as phenytoin and salicylates, have
a. What is the rate of change in the plasma been reported to follow dose-dependent elimi-

drug concentration with respect to time, nation kinetics. What changes in pharmacoki-
dCp/dt, when Cp << KM? netic parameters, including t1/2, VD, AUC, and

b. What is the rate of change in the plasma Cp, could be predicted if the amounts of these
drug concentration with respect to time, drugs administered were increased from low
dCp/dt, when Cp >> KM? pharmacologic doses to high therapeutic doses?

 

Nonlinear Pharmacokinetics 255

4. A given drug is metabolized by capacity-limited 10. Which of the following statements is/are true
pharmacokinetics. Assume KM is 50 mg/mL, regarding the pharmacokinetics of diazepam
Vmax is 20 mg/mL per hour, and the apparent VD (98% protein bound) and propranolol
is 20 L/kg. (87% protein bound)?
a. What is the reaction order for the metabo- a. Diazepam has a long elimination half-life

lism of this drug when given in a single because it is difficult to be metabolized due
intravenous dose of 10 mg/kg? to extensive plasma–protein binding.

b. How much time is necessary for the drug to b. Propranolol is an example of a drug with high
be 50% metabolized? protein binding but unrestricted (unaffected)

5. How would induction or inhibition of the metabolic clearance.
hepatic enzymes involved in drug biotransfor- c. Diazepam is an example of a drug with low
mation theoretically affect the pharmacokinet- hepatic extraction.
ics of a drug that demonstrates nonlinear phar- d. All of the above.
macokinetics due to saturation of its hepatic e. a and c.
elimination pathway? f. b and c.

6. Assume that both the active parent drug and 11. Which of the following statements describe(s)
its inactive metabolites are excreted by active correctly the properties of a drug that follows
tubular secretion. What might be the conse- nonlinear or capacity-limited pharmacokinetics?
quences of increasing the dosage of the drug a. The elimination half-life will remain con-
on its elimination half-life? stant when the dose changes.

7. The drug isoniazid was reported to interfere with b. The area under the plasma curve (AUC) will
the metabolism of phenytoin. Patients taking both increase proportionally as dose increases.
drugs together show higher phenytoin levels in c. The rate of drug elimination = Cp × KM.
the body. Using the basic principles in this chap- d. All of the above.
ter, do you expect KM to increase or decrease in e. a and b.

patients taking both drugs? (Hint: see Fig. 10-4.) f. None of the above.
8. Explain why KM sometimes has units of mM/mL 12. The hepatic intrinsic clearances of two

and sometimes mg/L. drugs are
9. The Vmax for metabolizing a drug is 10 mmol/h. drug A: 1300 mL/min

The rate of metabolism (n) is 5 mmol/h when drug B: 26 mL/min
drug concentration is 4 mmol. Which of the fol- Which drug is likely to show the greatest increase
lowing statements is/are true? in hepatic clearance when hepatic blood ow is
a. KM is 5 mmol for this drug. increased from 1 L/min to 1.5 L/min?
b. KM cannot be determined from the informa- a. Drug A

tion given. b. Drug B
c. KM is 4 mmol for this drug. c. No change for both drugs

ANSWERS

Frequently Asked Questions hepatic enzyme systems. Alcoholics may have liver

Why is it important to monitor drug levels carefully cirrhosis and lack certain coenzymes. Other patients

for dose dependency? may experience enzyme saturation at normal doses
due to genetic polymorphism. Pharmacokinetics

• A patient with concomitant hepatic disease may provides a simple way to identify nonlinear kinet-
have decreased biotransformation enzyme activ- ics in these patients and to estimate an appropriate
ity. Infants and young subjects may have immature dose. Finally, concomitant use of other drugs may

 

256 Chapter 10

cause nonlinear pharmacokinetics at lower drug the metabolic profile based on Vmax and KM. The
doses due to enzyme inhibition. Michaelis–Menten model was applied mostly to

describe in vitro enzymatic reactions. When Vmax What are the main differences in pharmacokinetic
and KM are estimated in patients, blood flow is not

parameters between a drug that follows linear phar-
explicitly considered. This semiempirical method

macokinetics and a drug that follows nonlinear
was found by many clinicians to be useful in dos-

pharmacokinetics?
ing phenytoin. The organ clearance model was

• A drug that follows linear pharmacokinetics gen- more useful in explaining clearance change due to
erally has a constant elimination half-life and a impaired blood flow. In practice, the physiologic
constant clearance with an increase in the dose. model has limited use in dosing patients because
The steady-state drug concentrations and AUC blood flow data for patients are not available.
are proportional to the size of the dose. Nonlinear
pharmacokinetics results in dose-dependent Cl, t1/2, Learning Questions
and AUC. Nonlinear pharmacokinetics are often

2. Capacity-limited processes for drugs include:
described in terms of Vmax and KM. • Absorption

What is the cause of nonlinear pharmacokinetics Active transport
that is not dose related? Intestinal metabolism by microflora

• Distribution
• Chronopharmacokinetics is the main cause of non-

Protein binding
linear pharmacokinetics that is not dose related. • Elimination
The time-dependent or temporal process of drug

Hepatic elimination
elimination can be the result of rhythmic changes

Biotransformation
in the body. For example, nortriptyline and theoph-

Active biliary secretion
ylline levels are higher when administered between • Renal excretion
7 and 9 am compared to between 7 and 9 pm after

Active tubular secretion
the same dose. Biological rhythmic differences in

Active tubular reabsorption
clearance cause a lower elimination rate in the
morning compared to the evening. Other factors dose 10,000 µg

4. C0
p = = = 0.5 µg/mL

that cause nonlinear pharmacokinetics may result VD 20,000 mL
from enzyme induction (eg, carbamazepine) or

From Equation 10.1,
enzyme inhibition after multiple doses of the drug.
Furthermore, the drug or a metabolite may accu- dCp V C

Elimination rate = − = max p
mulate following multiple dosing and affect the dt KM +Cp
metabolism or renal elimination of the drug.

Because KM = 50 mg/mL, Cp << KM and the reac-
What are the main differences between a model based tion rate is first order. Thus, the above equation
on Michaelis–Menten kinetic (Vmax and KM) and the reduces to Equation 10.3.
physiologic model that describes hepatic metabolism

dCp V
based on clearance? maxCp

− = = k ′C
dt K p

M
• The physiologic model based on organ drug clear-

ance describes nonlinear drug metabolism in Vmax 20 µg/h
k ′ = = = 0.4 h−1

terms of blood flow and intrinsic hepatic clear- KM 50 µg
ance (Chapter 12). Drugs are extracted by the For first-order reactions,
liver as they are presented by blood flow. The

0.693 0.693
physiologic model accounts for the sigmoid pro- t1/2 = = =1.73 h

k ′ 0.4
file with changing blood flow and extraction,
whereas the Michaelis–Menten model simulates The drug will be 50% metabolized in 1.73 hours.

 

Nonlinear Pharmacokinetics 257

7. When INH is coadminstered, plasma phenytoin per liter, or micromoles per milliliter, because
concentration is increased due to a reduction reactions are expressed in moles and not milli-
in metabolic rate n. Equation 10.1 shows that n grams. In dosing, drugs are given in milligrams
and KM are inversely related (KM in denomi- and plasma drug concentrations are expressed
nator). An increase in KM will be accompanied as milligrams per liter or micrograms per
by an increase in plasma drug concentration. milliliter. The units of KM for pharmacoki-
Figure 10-4 shows that an increase in KM is netic models are estimated from in vivo data.
accompanied by an increase in the amount of They are therefore commonly expressed as
drug in the body at any time t. Equation 10.4 milligrams per liter, which is preferred over
relates drug concentration to KM, and it can be micrograms per milliliter because dose is usu-
seen that the two are proportionally related, ally expressed in milligrams. The two terms
although they are not linearly proportional to may be shown to be equivalent and convert-
each other due to the complexity of the equa- ible. Occasionally, when simulating amount of
tion. An actual study in the literature shows drug metabolized in the body as a function of
that k is increased severalfold in the presence time, the amount of drug in the body has been
of INH in the body. assumed to follow Michaelis–Menten kinetics,

8. The KM has the units of concentration. In and KM assumes the unit of D0 (eg, mg). In this
laboratory studies, KM is expressed in moles case, KM takes on a very different meaning.

REFERENCES
Coffey J, Bullock FJ, Schoenemann PT: Numerical solution of Lemmer B: The importance of circadian rhythms on drug response

nonlinear pharmacokinetic equations: Effect of plasma pro- in hypertension and coronary heart disease—From mice and
tein binding on drug distribution and elimination. J Pharm Sci man. Pharmacol Ther 111(3):629–651, 2006.
60:1623, 1971. Lennard MS, Tucker GT, Woods HF: The polymorphic oxidation

de la Sierra A, Segura J, Banegas JR, Gorostidi M, de la Cruz of beta-adrenoceptor antagonists—Clinical pharmacokinetic
JJ, Armario P, Oliveras A, Ruilope LM. Clinical features of considerations. Clin Pharmacol 11:1–17, 1986.
8295 patients with resistant hypertension classified on the Levy RH: Time-dependent pharmacokinetics. Pharmacol Ther
basis of ambulatory blood pressure monitoring. Hypertension 17:383–392, 1983.
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Evans WE, Schentag JJ, Jusko WJ. 1992. Applied Pharmacokinetics, turnal blood pressure reduction in resistant hypertension. Arch
3rd ed., Applied Therapeutics, Vancouver, WA. Intern Med 169(9):874–880, 2009.

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netics of zonisamide in epileptic patients. Biol Pharm Bull vascular resistance and blood pressure. J Clin Pharmacol
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Hashimoto Y, Koue T, Otsuki Y, Yasuhara M, Hori R, Inui K: Perrier D, Ashley JJ, Levy G: Effect of product inhibition in
Simulation for population analysis of Michaelis–Menten kinetics of drug elimination. J Pharmacokinet Biopharmacol
kinetics. J Pharmacokinet Biopharmacol 23:205–216, 1:231, 1973.
1995. Pitlick WH, Levy RH: Time-dependent kinetics, I. Exponential

Hayashi N, Aso H, Higashida M, et al: Estimation of rhG-CSF autoinduction of carbamazepine in monkeys. J Pharm Sci
absorption kinetics after subcutaneous administration using a 66:647, 1977.
modified Wagner–Nelson method with a nonlinear elimination Prins JM, Weverling GJ, Van Ketel RJ, Speelman P: Circadian
model. European J Pharm Sci 13:151–158, 2001. variations in serum levels and the renal toxicity of aminogly-

Hsu F, Prueksaritanont T, Lee MG, Chiou WL: The phenomenon cosides in patients. Clin Pharmacol Ther 62:106–111, 1997.
and cause of the dose-dependent oral absorption of chloro- Reinberg AE: Concepts of circadian chronopharmacology. In
thiazide in rats: Extrapolation to human data based on the Hrushesky WJM, Langer R, Theeuwes F (eds). Temporal
body surface area concept. J Pharmacokinet Biopharmacol Control of Drug Delivery. New York, Annals of the Academy
15:369–386, 1987. of Science, 1991, vol 618, p 102.

Jonsson EN, Wade JR, Karlsson MO: Nonlinearity detection: Sarveshwer Rao VV, Rambhau D, Ramesh Rao B, Srinivasu P:
Advantages of nonlinear mixed-effects modeling. AAPS Circadian variation in urinary excretion of ciprofloxacin after
Pharmsci 2(3), article 32, 2000. a single dose oral administration at 1000 and 2200 hours in

 

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human subjects. India Antimicrob Agents Chemother (USA) Witmer DR, Ritschel WA: Phenytoin isoniazid interaction: A kinetic
41:1802–1804, 1997, Kakatiya Univ., Warangal-506 009. approach to management. Drug Intell Clin Pharm 18:483–486,

Vermeulen T: Distribution of paroxetine in three postmortem 1984.
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Von Roemeling R: The therapeutic index of cytotoxic chemo- tion to metoprolol metabolic ratio and CYP2D6*10 genotype of
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Drug Delivery. New York, Annals of the Academy of Science, tion and Role in Drug Disposition. John Wiley and Sons Ltd,
1991, vol 618, pp 292–311. Sep 2007.

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kinetics: Areas under curves, steady-state concentrations, and
clearances for compartment models with different types of
input. Biopharm Drug Disp 6:177–200, 1985.

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Levy G: Pharmacokinetics of salicylate in man. Drug Metab Rev mination. In Welling PG, Tse FLS (eds). Pharmacokinetics:
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Physiologic Drug

11 Distribution and
Protein Binding
He Sun and Hong Zhao

Chapter Objectives PHYSIOLOGIC FACTORS OF DISTRIBUTION
»» Describe the physiology of drug After a drug is absorbed systemically from the site of administra-

distribution in the body. tion, the drug molecules are distributed throughout the body by the
»» Explain how drug distribution is systemic circulation. The location, extent, and distribution are

affected by blood flow, protein, dependent on the drug’s physicochemical properties and individual
and tissue binding. patient characteristics such as organ perfusion and blood flow. The

drug molecules are carried by the blood to the target site (receptor)
»» Describe how drug distribution

for drug action and to other (nonreceptor) tissues as well, where side
can affect the apparent volume

effects or adverse reactions may occur. These sites may be intra-
of distribution.

and/or extracellular. Drug molecules are distributed to eliminating
»» Explain how volume of organs, such as the liver and kidney, and to noneliminating tissues,

distribution, drug clearance, such as the brain, skin, and muscle. In pregnancy, drugs cross the
and half-life can be affected by placenta and may affect the developing fetus. Drugs can also be
protein binding. secreted in milk via the mammillary glands, into the saliva and into

other secretory pathways. A substantial portion of the drug may be
»» Determine drug–protein binding

constants using in vitro methods. bound to proteins in the plasma and/or in the tissues. Lipophilic
drugs deposit in fat, from which the drug may be slowly released.

»» Evaluate the impact of change Drug distribution throughout the body occurs primarily via
in drug–protein binding or the circulatory system, which consists of a series of blood vessels
displacement on free drug that carry the drug in the blood; these include the arteries that carry
concentration. blood to tissues, and the veins that return the blood back to the

heart. An average subject (70 kg) has about 5 L of blood, which is
equivalent to about 3 L of plasma (Fig. 11-1). About 50% of the
blood is in the large veins or venous sinuses. The volume of blood
pumped by the heart per minute—the cardiac output—is the product
of the stroke volume of the heart and the number of heartbeats per
minute. An average cardiac output is 0.08 L/69 left ventricular
contractions (heart beats)/min, or approximately 5.5 L/min in sub-
jects at rest. The cardiac output may be five to six times higher
during exercise. Left ventricular contraction may produce a sys-
tolic blood pressure of 120 mm Hg, and moves blood at a linear
speed of 300 mm/s through the aorta. Mixing of a drug solution in
the blood occurs rapidly at this flow rate. Drug molecules rapidly
diffuse through a network of fine capillaries to the tissue spaces

259

 

260 Chapter 11

Blood (4.5–5 L) both the drug and the cell membrane. Cell membranes
are composed of protein and a bilayer of phospho-

Plasma Blood cells lipid, which act as a lipid barrier to drug uptake.
(3 L) (2 L)

Thus, lipid-soluble drugs generally diffuse across
cell membranes more easily than highly polar or

Plasma
Intra- water-soluble drugs. Small drug molecules generally

(3 L) Extracellular
cellular water diffuse more rapidly across cell membranes than
water Interstitial

(15 L)
(27 L) water large drug molecules. If the drug is bound to a

(12 L) plasma protein such as albumin, the drug–protein
complex becomes too large for easy diffusion across

FIGURE 111 Major water volumes (L) in a 70-kg human.
the cell or even capillary membranes. A comparison
of diffusion rates for water-soluble molecules is
given in Table 11-1.

filled with interstitial fluid (Fig. 11-2). The intersti-
tial fluid plus the plasma water is termed extracel-
lular water, because these fluids reside outside the Diffusion and Hydrostatic Pressure
cells. Drug molecules may further diffuse from the The processes by which drugs transverse capillary
interstitial fluid across the cell membrane into the membranes into the tissue include passive diffu-
cell cytoplasm. sion and hydrostatic pressure. Passive diffusion is

Drug distribution is generally rapid, and most the main process by which most drugs cross cell
small drug molecules permeate capillary membranes membranes. Passive diffusion (see Chapter 14) is
easily. The passage of drug molecules across a cell the process by which drug molecules move from
membrane depends on the physicochemical nature of an area of high concentration to an area of low

Arteriole
(from artery)

Intracellular

uid

a
b

Tissue cell

Blood
capillary

Interstitial

uid

Venule
A (to vein)

Plasma Interstitial and Tissues and
lymph uids other body water

Bound Bound Bound

Free Free Free
B

FIGURE 112 Diffusion of drug from capillaries to interstitial spaces.

 

Physiologic Drug Distribution and Protein Binding 261

TABLE 111 Permeability of Molecules of Various Sizes to Capillaries

Diffusion Coefficient

Radius of Equivalent In Water Across Capillary
Molecular Weight Sphere A (0.1 mm) (cm2/s) × 105 (cm2/s × 100 g)

Water 18 3.20 3.7

Urea 60 1.6 1.95 1.83

Glucose 180 3.6 0.91 0.64

Sucrose 342 4.4 0.74 0.35

Raffinose 594 5.6 0.56 0.24

Inulin 5,500 15.2 0.21 0.036

Myoglobin 17,000 19 0.15 0.005

Hemoglobin 68,000 31 0.094 0.001

Serum albumin 69,000 0.085 <0.001

Data from Pappenheimer, JR: Passage of molecules through capillary walls, Physiol Rev 33(3):387–423, July 1953; Renkin EM: Transport of large molecules
across capillary walls, Physiologist 60:13–28, February 1964.

concentration. Passive diffusion is described by Fick’s pressure that allows small drug molecules to be
law of diffusion: filtered in the glomerulus of the renal nephron

(see Chapter 7).
Blood flow–facilitated drug distribution is rapid

dQ −DKA(Cp −Ct )
Rate of drug diffusion = and efficient, but requires pressure. As blood pres-

dt h sure gradually decreases when arteries branch into
(11.1) the small arterioles, the speed of flow slows and dif-

fusion into the interstitial space becomes diffusion or
where Cp − Ct is the difference between the drug concentration driven and facilitated by the large
concentration in the plasma (Cp) and in the tissue surface area of the capillary network. The average
(Ct); A is the surface area of the membrane; h is the pressure of the blood capillary is higher (+18 mm Hg)
thickness of the membrane; K is the lipid–water par- than the mean tissue pressure (−6 mm Hg), resulting
tition coefficient; and D is the diffusion constant. in a net total pressure of 24 mm Hg higher in the
The negative sign denotes net transfer of drug from capillary over the tissue. This pressure difference is
inside the capillary lumen into the tissue and extra- offset by an average osmotic pressure in the blood of
cellular spaces. Diffusion is spontaneous and tem- 24 mm Hg, pulling the plasma fluid back into the
perature dependent. Diffusion is distinguished from capillary. Thus, on average, the pressures in the tissue
blood flow–initiated mixing, which involves hydro- and most parts of the capillary are equal, with no net
static pressure. flow of water.

Hydrostatic pressure represents the pressure gra- At the arterial end, as the blood newly enters the
dient between the arterial end of the capillaries enter- capillary (Fig. 11-2A), the pressure of the capillary
ing the tissue and the venous capillaries leaving the blood is slightly higher (about 8 mm Hg) than that of
tissue. Hydrostatic pressure is responsible for pene- the tissue, causing fluid to leave the capillary and
tration of water-soluble drugs into spaces between enter the tissues. This pressure is called hydrostatic or
endothelial cells and possibly into lymph. In the filtration pressure. This filtered fluid (filtrate) is later
kidneys, high arterial pressure creates a filtration returned to the venous capillary (Fig. 11-2B) due to a

 

262 Chapter 11

lower venous pressure of about the same magnitude. electrolyte levels in renal and hepatic diseases,
The lower pressure of the venous blood compared resulting in net flow of plasma water into the inter-
with the tissue fluid is termed as absorptive pressure. stitial space (edema). This change in fluid distribu-
A small amount of fluid returns to the circulation tion may partially explain the increased extravascular
through the lymphatic system. drug distribution during some disease states.

Blood flow, tissue size, and tissue storage (par-

Distribution Half-Life, Blood Flow, titioning and binding) are also important in deter-

and Drug Uptake by Organs mining the time it takes the drug to become
completely distributed. Table 11-2 lists the blood

Because the process of drug transfer from the capil-
flow and tissue mass for many tissues in the human

lary into the tissue fluid is mainly diffusional,
body. Drug affinity for a tissue or organ refers to the

according to Fick’s law, the membrane thickness,
partitioning and accumulation of the drug in the tis-

diffusion coefficient of the drug, and concentration
sue. The time for drug distribution is generally mea-

gradient across the capillary membrane are impor-
sured by the distribution half-life or the time for 50%

tant factors in determining the rate of drug diffusion.
drug distribution. The factors that determine the distri-

Kinetically, if a drug diffuses rapidly across the
bution constant of a drug into an organ are the blood

membrane in such a way that blood flow is the rate-
flow to the organ, the volume of the organ, and the

limiting step in the distribution of drug, then the
process is perfusion or flow limited. A person with
congestive heart failure has a decreased cardiac out- TABLE 112 Blood Flow to Human Tissues

put, resulting in impaired blood flow, which may Percent Percent Blood Flow
reduce renal clearance through reduced filtration Body Cardiac (mL/100 g

pressure and blood flow. In contrast, if drug distribu- Tissue Weight Output tissue/min)

tion is limited by the slow diffusion of drug across Adrenals 0.02 1 550
the membrane in the tissue, then the process is
termed diffusion or permeability limited (Fig. 11-3). Kidneys 0.4 24 450

Drugs that are permeability limited may have an Thyroid 0.04 2 400
increased distribution volume in disease conditions
that cause inflammation and increased capillary Liver

membrane permeability. The delicate osmotic pres- Hepatic 2.0 5 20
sure balance may be altered due to changes in
albumin level and/or blood loss or due to changes in Portal 20 75

Portal-drained 2.0 20 75
Organ Organ viscera

Ct R Ct R
Heart (basal) 0.4 4 70

Ca Cv Ca Cv

Brain 2.0 15 55

Blood Blood Skin 7.0 5 5

V V
b b Muscle (basal) 40.0 15 3

pool pool

Connective 7.0 1 1
Diffusion-limited Perfusion-limited tissue

model model
(Slow diffusion (Rapid diffusion Fat 15.0 2 1

into tissue) into tissue)

Data from Spector WS: Handbook of Biological Data, Saunders,
FIGURE 113 Drug distribution to body organs by blood Philadelphia, 1956; Glaser O: Medical Physics, Vol 11, Year Book
flow (perfusion). Right panel for tissue with rapid permeability; Publishers, Chicago, 1950; Butler TC: Proc First International
Left panel for tissue with slow permeability. Pharmacological Meeting, Vol 6, Pergamon Press, 1962.

 

Physiologic Drug Distribution and Protein Binding 263

partitioning of the drug into the organ tissue, as
100 3 4 5

shown in Equation 11.2. 1+2

Q 80
k = (11.2)

d VR

where kd is first-order distribution constant, Q is 60

blood flow to the organ, V is volume of the organ, R
is ratio of drug concentration in the organ tissue to 40
drug concentration in the blood (venous). The distri- 1 = adrenal
bution half-life of the drug to the tissue, td1/2, may 2 = kidney

20 3 = skin
easily be determined from the distribution constant 4 = muscle [basal]
in the equation of td1/2 = 0.693/kd.

5 = fat

The ratio R is determined experimentally from 0
0 50 100 150 200 250 300 350

tissue samples. With many drugs, however, only Time (minutes)
animal tissue data are available. The ratio R is usu-
ally estimated based on knowledge of the partition FIGURE 115 Drug distribution in five groups of tissues

at various rates of equilibration.
coefficient of the drug. The partition coefficient is a
physical property that measures the ratio of the solu-
bility of the drug in the oil phase to solubility in
aqueous phase. The partition coefficient (Po/w) is longer time is needed to fill a large organ volume
defined as a ratio of the drug concentration in the oil with drug. Figure 11-5 illustrates the distribution
phase (usually represented by octanol) to the drug time (for 0%, 50%, 90%, and 95% distribution) for
concentration in the aqueous phase measured at the adrenal gland, kidney, muscle (basal), skin, and
equilibrium under specified temperature in vitro in fat tissue in an average human subject (ideal body
an oil/water two-layer system (Fig. 11-4). The parti- weight, IBW = 70 kg). In this illustration, the blood
tion coefficient is one of the most important factors drug concentration is equally maintained at 100 mg/
that determine the tissue distribution of a drug. mL, and the drug is assumed to have equal distribu-

If each tissue has the same ability to store the tion between all the tissues and blood, i.e., when
drug, then the distribution half-life is governed by fully equilibrated, the partition or drug concentration
the blood flow, Q, and volume (size), V, of the organ. ratio (R) between the tissue and the plasma will
A large blood flow, Q, to the organ decreases the equal 1. Vascular tissues such as the kidneys and
distribution time, whereas a large organ size or vol- adrenal glands achieve 95% distribution in less than
ume, V, increases the distribution time because a 2 minutes. In contrast, drug distribution time in fat

tissues takes 4 hours, while less in vascular tissues,
such as the skin and muscles, take between 2 and

Oil Diffusion into oil = k 4 hours (Fig. 11-5). When drug partition of the tissues
12 Cwater

(octanol) is the same, the distribution time is dependent only
on the tissue volume and its blood flow.

Coil
k Blood flow is an important factor in determin-

21
Diffusion into water = k21 Coil ing how rapid and how much drug reaches the

k receptor site. Under normal conditions, limited
12

Cwater blood flow reaches the muscles. During exercise,
the increase in blood flow may change the fraction

Water At steady state, k12 Cwater = k21 Coil of drug reaching the muscle tissues. Diabetic
patients receiving intramuscular injection of insulin

FIGURE 114 Diagram showing equilibration of drug may experience the effects of changing onset of drug
between oil and water layer in vitro. action during exercise. Normally, the blood reserve

mg/mL

 

264 Chapter 11

of the body stays mostly in the large veins and of the plasma drug concentration; thus, the anti-
sinuses in the abdomen. During injury or when androgen effect of the drug may not be fully
blood is lost, constriction of the large veins redirects achieved until distribution to this receptor site is
more blood to needed areas, and therefore, affects complete. Digoxin is highly bound to myocardial
drug distribution. Accumulation of carbon dioxide membranes. Digoxin has a high tissue/plasma con-
may lower the pH of certain tissues and may affect centration ratio (R = 60 − 130) in the myocardium.
the level of drugs reaching those tissues. This high R ratio for digoxin leads to a long distribu-

Figure 11-6 illustrates the distribution of a drug tional phase (see Chapter 5) despite abundant blood
to three different tissues when the partition of the flow to the heart. It is important to note that if a tis-
drug for each tissue varies. For example, the drug sue has a long distribution half-life, a long time is
partition shows that the drug concentration in the needed for the drug to leave the tissue as the blood
adrenal glands is five times of the drug concentration level decreases. Understanding drug distribution is
in the plasma, while the drug partition for the kidney important because the activities of many drugs are
is R = 3, and for basal muscle, R = 1. In this illustra- not well correlated with plasma drug levels.
tion, the adrenal gland and kidney take 5 and 3 times Kinetically, both drug–protein binding and drug
as long to be equilibrated with drug in the plasma. lipid solubility in the tissue site lead to longer distri-
Thus, it can be seen that, even for vascular tissues, bution times.
high drug partition can take much more time for the Chemical knowledge in molecular structure
tissue to become fully equilibrated. In the example in often helps estimate the lipid solubility of a drug. A
Fig. 11-6, drug administration is continuous (as in drug with large oil/water partition coefficient tends
IV infusion), since tissue drug levels remain constant to have high R values in vivo. A reduction in the
after equilibrium. partition coefficient of a drug often reduces the rate

Some tissues have great ability to store and of drug uptake into the brain. This may decrease
accumulate drug, as shown by large R values. For drug distribution into the central nervous system and
example, the anti-androgen drug, flutamide and its decrease undesirable central nervous system side
active metabolite are highly concentrated in the pros- effects. Extensive tissue distribution is kinetically
tate. The prostate drug concentration is 20 times that evidenced by a large volume of distribution. A sec-

ondary effect is a prolonged drug elimination half-
life, since the drug is distributed within a larger
volume (thus, the drug is more diluted) and there-

600 fore, less efficiently removed by the kidney or the
Top = adrenal liver. For example, etretinate (a retinoate derivative)

R = 5
500 for acne treatment has an unusual long elimination

half-life of about 100 days (Chien et al, 1992), due
400 to its extensive distribution to body fats. Newly syn-

R = 3 Middle = kidney thesized agents have been designed to reduce the
300 lipophilicity and drug distribution. These new agents

have less accumulation in the tissue and less poten-
200 tial for teratogenicity.

R = 1 Bottom = muscle
100

Drug Accumulation

0 The deposition or uptake of the drug into the tissue
0 50 100 150 200 250 300 is generally controlled by the diffusional barrier of

Time (minutes) the capillary membrane and other cell membranes.

FIGURE 116 Drug distribution in three groups of tissues For example, the brain is well perfused with blood,
with various abilities to store drug (R). but many drugs with good aqueous solubility have

mg/mL

 

Physiologic Drug Distribution and Protein Binding 265

high drug concentrations in the kidney, liver, and lung high doses of phenothiazine to chronic schizo-
and yet little or negligible drug concentration in the phrenic patients. The antibiotic tetracycline forms
brain. The brain capillaries are surrounded by a layer an insoluble chelate with calcium. In growing teeth
of tightly joined glial cells that act as a lipid barrier and bones, tetracycline complexes with the calcium
to impede the diffusion of polar or highly ionized and remain in these tissues.
drugs. A diffusion-limited model can be used to Some tissues have enzyme systems that actively
describe the pharmacokinetics of these drugs that are transport natural biochemical substances into the tis-
not adequately described by perfusion models. sues. For example, various adrenergic tissues have a

Tissues receiving high blood flow equilibrate specific uptake system for catecholamines, such as
quickly with the drug in the plasma. However, at norepinephrine. Thus, amphetamine, which has a
steady state, the drug may or may not accumulate phenylethylamine structure similar to norepineph-
(concentrate) within the tissue. The accumulation of rine, is actively transported into adrenergic tissue.
drug into tissues is dependent on both the blood flow Other examples of drug accumulation are well docu-
and the affinity of the drug for the tissue. Drug affin- mented. For some drugs, the actual mechanism for
ity for the tissue depends on partitioning and also drug accumulation may not be clearly understood.
binding to tissue components, such as receptors. In a few cases, the drug is irreversibly bound
Drug uptake into a tissue is generally reversible. The into a particular tissue. Irreversible binding of drug
drug concentration in a tissue with low capacity may occur when the drug or a reactive intermediate
equilibrates rapidly with the plasma drug concentra- metabolite becomes covalently bound to a macro-
tion and then declines rapidly as the drug is elimi- molecule within the cell, such as to a tissue protein.
nated from the body. Many purine and pyrimidine drugs used in cancer

In contrast, drugs with high tissue affinity tend chemotherapy are incorporated into nucleic acids,
to accumulate or concentrate in the tissue. Drugs causing destruction of the cell.
with a high lipid/water partition coefficient are very
lipid soluble and tend to accumulate in lipid or adi-
pose (fat) tissue. In this case, the lipid-soluble drug Permeability of Cells and

partitions from the aqueous environment of the Capillary Membranes

plasma into the fat. This process is reversible, but Cellular and plasma membranes vary in their perme-
the extraction of drug out of the tissue is so slow ability characteristics, depending on the tissue. For
that the drug may remain for days or even longer in example, capillary membranes in the liver and kid-
adipose tissues, long after the drug is depleted from neys are more permeable to transmembrane drug
the blood. Because the adipose tissue is poorly per- movement than capillaries in the brain. The sinusoi-
fused with blood, drug accumulation is slow. dal capillaries of the liver are very permeable and
However, once the drug is concentrated in fat tissue, allow the passage of large-size molecules. In the
drug removal from fat may also be slow. For exam- brain and spinal cord, the capillary endothelial cells
ple, the insecticide, chlorinated hydrocarbon DDT are surrounded by a layer of glial cells, which have
(dichlorodiphenyltrichloroethane) is highly lipid tight intercellular junctions. This added layer of cells
soluble and remains in fat tissue for years. around the capillary membranes acts effectively to

In addition to partitioning, drugs may accumu- slow the rate of drug diffusion into the brain by act-
late in tissues by other processes. For example, ing as a thicker lipid barrier. This lipid barrier, which
drugs may accumulate by binding to proteins or slows the diffusion and penetration of water-soluble
other macromolecules in a tissue. Digoxin is highly and polar drugs into the brain and spinal cord, is
bound to proteins in cardiac tissue, leading in a called the blood–brain barrier.
large volume of distribution (440 L/70 kg) and long Under certain pathophysiologic conditions, the
elimination t1/2 (approximately 40 hours). Some permeability of cell membranes, including capil-
drugs may complex with melanin in the skin and lary cell membranes, may be altered. For example,
eye, as observed after long-term administration of burns will alter the permeability of skin and allow

 

266 Chapter 11

drugs and larger molecules to permeate inward or for any of the transporters or enzyme systems. It is
outward. In meningitis, which involves inflamma- also important to determine whether the pharmaco-
tion of the membranes of the spinal cord and/or kinetic models have adequately taken transporter
brain, drug uptake into the brain will be enhanced. information into consideration.

The diameters of the capillaries are very small
and the capillary membranes are very thin. The
high blood flow within a capillary allows for inti- Drug Distribution to Cerebral Spinal Fluid,
mate contact of the drug molecules with the plasma CSF, and Brain: Blood–Brain Barrier
membrane, providing for rapid drug diffusion. For

The blood–brain barrier permits selective entry of
capillaries that perfuse the brain and spinal cord,

drugs into the brain and spinal cord due to (1) ana-
the layer of glial cells functions effectively to

tomical features (as mentioned above) and (2) the
increase the thickness (term h in Equation 11.1),

presence of cellular transporters. Anatomically, the
thereby slowing the diffusion and penetration of

layer of cells around the capillary membranes of
water-soluble and polar drugs into the brain and

the brain acts effectively as a thicker lipid barrier
spinal cord.

that slows the diffusion and penetration of water-
soluble and polar drugs into the brain and spinal

Drug Distribution within Cells and Tissues cord. However, some small hydrophilic molecules
Pharmacokinetic models generally provide a good may cross the blood–brain barrier by simple diffu-
estimation of plasma drug concentrations in the body sion. Efflux transporter is often found at the entry
based on dose, volume of distribution, and clearance. point into vital organs in the body. P-glycoprotein
However, drug concentrations within the cell or expression in the endothelial cells of human capil-
within a special region in the body are also governed lary blood vessels at the blood–brain was detected
by special efflux and metabolizing enzyme systems by special antibodies against the human multidrug-
that prevent and detoxify foreign agents entering the resistance gene product. P-gp may have a physiolog-
body. Some proteins are receptors on cell surfaces ical role in regulating the entry of certain molecules
that react specifically with a drug. The transporters into the central nervous system and other organs
are specialized proteins in the body that can associ- (Cordoncardo et al, 1989). P-gp substrate examples
ate transiently with a substrate drug through the include doxorubicin, inmervectin, and others.
hydrophobic region in the molecule, for example, Knocking out P-gp expression can increase brain
P-glycoprotein, P-gp. Drug-specific transporters are toxicity with inmervectin in probe studies. Kim et al
very important in preventing drug accumulation in (1998) studied transport characteristics of protease
cells and may cause drug tolerance or drug resis- inhibitor drugs, indinavir, nelfinavir, and saquinavir
tance. Transporters can modulate drug absorption in vitro using the model P-gp expressing cell lines
and disposition (see Chapters 13 and 14). Special and in vivo administration in the mouse model. After
families of transporters are important and well docu- IV administration, plasma concentrations of the drug
mented (You and Morris, 2007). For example, mono- in mdr1a (−/−) mice, the brain concentrations were
carboxylate transporters, organic cation transporters, elevated 7 to 36-fold. These data demonstrate that
organic anion transporters, oligopeptide transporters, P-gp can limit the penetration of these drugs into the
nucleoside transporters, bile acid transporters, and brain. Efflux transporters (ie, P-gp) effectively pre-
multidrug resistance protein (eg, P-gp) that modulate vent certain small drug substances from entering into
distribution of many types of drugs. Drug transport- the brain, whereas influx transporters enable small
ers in the liver, kidney, brain, and gastrointestinal are nutrient molecules such as glucose to be actively
discussed by You and Morris (2007) (see also taken into the brain. There is now much interest in
Chapter 13 and Fig. 14-1 in Chapter 14). When con- understanding the mechanisms for drug uptake into
sidering drug utilization and drug–drug interactions, brain in order to deliver therapeutic and diagnostic
it is helpful to know whether the drug is a substrate agents to specific regions of the brain.

 

Physiologic Drug Distribution and Protein Binding 267

CLINICAL FOCUS that system. The volume of the system may be esti-
mated if the amount of drug added to the system and

Jaundice is a condition marked by high levels of bili- the drug concentration after equilibrium in the sys-
rubin in the blood. New born infants with jaundice tem are known.
are particularly sensitive to the effects of bilirubin
since their blood–brain barrier is not well formed at Volume (L)
birth. The increased bilirubin, if untreated, may
cause jaundice, and damage the brain centers of infants amount (mg) of drug added to system

=
drug concentration (mg/L) in system after equilibrium

caused by increased levels of unconjugated, indirect
bilirubin which is free (not bound to albumin). This (11.3)
syndrome is also known as kernicterus. Depending
on the level of exposure to bilirubin, the effects range Equation 11.3 describes the relationship of concentra-
from unnoticeable to severe brain damage. Treatment tion, volume, and mass, as shown in Equation 11.4.
in some cases may require phototherapy that requires
special blue lights that work by helping to break down Concentration (mg/L) × volume (L) = mass (mg)
bilirubin in the skin. (11.4)

Frequently Asked Questions Considerations in the Calculation of Volume
»»How does a physical property, such as partition coef- of Distribution: A Simulated Example

ficient, affect drug distribution?
The objective of this exercise is to calculate the fluid

»»Why do some tissues rapidly take up drugs, whereas volume in each beaker and to compare the calculated
for other tissues, drug uptake is slower? volume to the real volume of water in the beaker.

»»Does rapid drug uptake into a tissue mean that the Assume that three beakers are each filled with 100 mL

drug will accumulate into that tissue? of aqueous fluid. Beaker 1 contains water only; bea-
kers 2 and 3 each contain aqueous fluid and a small

»»What physical and chemical characteristics of a drug compartment filled with cultured cells. The cells in
that would increase or decrease the uptake of the

beaker 2 can bind the drug, while the cells in beaker 3
drug into the brain or cerebral spinal fluid?

can metabolize the drug. The three beakers represent
the following, respectively:

APPARENT VOLUME DISTRIBUTION Beaker 1. Drug distribution in a fluid (water)
compartment only, without drug binding and

The concentration of drug in the plasma or tissues
metabolism

depends on the amount of drug systemically
Beaker 2. Drug distribution in a fluid compartment

absorbed and the volume in which the drug is dis-
containing cell clusters that reversibly bind

tributed. The apparent volume of distribution, VD in
drugs

a pharmacokinetic model, is used to estimate the
Beaker 3. Drug distribution in a fluid compart-

extent of drug distribution in the body (see Chapters 3
ment containing cell clusters (similar to tissues

and 5). Although the apparent volume of distribu-
in vivo) in which the drug may be metabolized

tion does not represent a true anatomical or physical
and the metabolites bound to cells

volume, the VD represents the result of dynamic
drug distribution between the plasma and the tissues Suppose 100 mg of drug is then added to each
and accounts for the mass balance of the drug in the beaker (Fig. 11-7). After the fluid concentration of
body. To illustrate the use of VD, consider a drug drug in each beaker is at equilibration, and the con-
dissolved in a simple solution. A volume term is centration of drug in the water (fluid) compartment
needed to relate drug concentration in the system has been sampled and assayed, the volume of water
(or human body) to the amount of drug present in may be computed.

 

268 Chapter 11

Assume that the above measurements were made
Fluid (water) and that the following information was obtained:
compartment

• Drug concentration in fluid compartment =
Cell
compartment 0.5 mg/mL

Beaker 1 Beaker 2 Beaker 3 • Drug concentration in cell cluster = 10 mg/mL
• Volume of cell cluster = 5 mL

FIGURE 117 Experiment simulating drug distribution in • Amount of drug added = 100 mg
the body. Three beakers, each contains 100 mL of water (fluid • Amount of drug taken up by the cell cluster =
compartment) and 100 mg of a water-soluble drug. Beakers 2
and 3 also contain 5 mL of cultured cell clusters. 10 mg/mL × 5 mL = 50 mg

• Amount of drug dissolved in uid (water) com-
partment = 100 mg (total) − 50 mg (in cells) =

Case 1 50 mg (in water)
The volume of water in beaker 1 is calculated from Using the above information, the true volume of the
the amount of drug added (100 mg) and the equili- fluid (water) compartment is calculated using
brated drug concentration using Equation 11.3. Equation 11.3.
After equilibration, the drug concentration was

50 mg
measured to be 1 mg/mL. Volume of fluid compartment = = 100 mL

0.5 mg/mL

Volume = 100 mg/1 mg/mL = 100 mL The value of 100 mL agrees with the volume of fluid
we put into the beaker.

The calculated volume in beaker 1 confirms that the If the tissue cells were not accessible for sam-
system is a simple, homogeneous system and, in this pling as in the case of in vivo drug administration,
case, represents the “true” fluid volume of the beaker. the volume of the fluid (water) compartment is cal-

culated using Equation 11.3, assuming the system is
Case 2 homogenous and that 100 mg drug was added to the
Beaker 2 contains cell clusters stuck to the bottom of system.
the beaker. Binding of drug to the proteins of the cells
occurs on the surface and within the cytoplasmic 100 mg

Apparent volume = = 200 mL
interior. This case represents a heterogeneous system 0.5 mg/mL

consisting of a well-stirred fluid compartment and a
The value of 200 mL is a substantial overestimation

tissue (cell). To determine the volume of this system,
of the true volume (100 mL) of the system.

more information is needed than in Case 1:
When a heterogeneous system is involved, the

1. The amount of drug dissolved in the fluid com- real or true volume of the system may not be accu-
partment must be determined. Because some of rately calculated by monitoring only one compart-
the drug will be bound within the cell compart- ment. Therefore, an apparent volume of distribution
ment, the amount of drug in the fluid compart- is calculated and the infrastructure of the system is
ment will be less than the 100 mg placed in the ignored. The term apparent volume of distribution
beaker. refers to the lack of true volume characteristics. The

2. The amount of drug taken up by the cell cluster apparent volume of distribution is used in pharmaco-
must be known to account for the entire amount kinetics because the tissue (cellular) compartments
of drug in the beaker. Therefore, both the cell are not easily sampled and the true volume is not
and the fluid compartments must be sampled known. When the experiment in beaker 2 is per-
and assayed to determine the drug concentra- formed with an equal volume of cultured cells that
tion in each compartment. have different binding affinity for the drug, then the

3. The volume of the cell cluster must be apparent volume of distribution is very much affected
determined. by the extent of cellular drug binding (Table 11-3).

 

Physiologic Drug Distribution and Protein Binding 269

TABLE 113 Relationship of Volume of Distribution and Amount of Drug in Tissue (Cellular)
Compartmenta

Total Drug Volume of Cells Drug in Cells Drug in Water Drug Concentration
(mg) (mL) (mg) (mg) in Water (mg/mL) VD in Water (mL)

100 15 75 25 0.25 400

100 10 50 50 0.50 200

100 5 25 75 0.75 133

100 1 5 95 0.95 105

aFor each condition, the true water (fluid) compartment is 100 mL. Apparent volume of distribution (VD) is calculated according to Equation 11.3.

As shown in Table 11-3, as the amount of drug volume of distribution. A true VD that exceeds the
in the cell compartment increases (column 3), the volume of the body is physically impossible. Only if
apparent VD of the fluid compartment increases (col- the drug concentrations in both the tissue and plasma
umn 6). Extensive cellular drug binding effectively compartments are sampled, and the volumes of each
pulls drug molecules out of the fluid compartment, compartment are clearly defined, can a true physical
decreases the drug concentration in the fluid com- volume be calculated.
partment, and increases VD. In biological systems,
the quantity of cells, cell compartment volume, and Case 3
extent of drug binding within the cells affect VD.

The drug in the cell compartment in beaker 3
A large cell volume and/or extensive drug binding in

decreases due to undetected metabolism because the
the cells reduce the drug concentration in the fluid

metabolite formed is bound to be inside the cells.
compartment and increase the apparent volume of

Thus, the apparent volume of distribution is also
distribution.

greater than 100 mL. Any unknown source that
In this example, the fluid compartment is com-

decreases the drug concentration in the fluid com-
parable to the central compartment and the cell

partment will increase the VD, resulting in an overes-
compartment is analogous to the peripheral or tissue

timated apparent volume of distribution. This is
compartment. If the drug is distributed widely into

illustrated with the experiment in beaker 3. In beaker 3,
the tissues or concentrated unevenly in the tissues,

the cell cluster metabolizes the drug and binds the
the VD for a drug may exceed the physical volume of

metabolite to the cells. Therefore, the drug is effec-
the body (about 70 L of total volume or 42 L of body

tively removed from the fluid. The data for this
water for a 70-kg subject). Besides cellular protein

experiment (note that metabolite is expressed as
binding, partitioning of drug into lipid cellular com-

equivalent intact drug) are as follows:
ponents may greatly inflate VD. Many drugs have
oil/water partition coefficients above 10,000. These • Total drug placed in beaker = 100 mg
lipophilic drugs are mostly concentrated in the lipid • Cell compartment:
phase of adipose tissue, resulting in a very low drug Drug concentration = 0.2 mg/mL
concentration in the extracellular water. Generally, Metabolite-bound concentration = 9.71 mg/mL
drugs with very large VD values have very low drug Metabolite-free concentration = 0.29 mg/mL
concentrations in plasma. Cell volume = 5 mL

A large VD is often interpreted as broad drug • Fluid (water) compartment:
distribution for a drug, even though many other fac- Drug concentration = 0.2 mg/mL
tors also lead to the calculation of a large apparent Metabolite concentration = 0.29 mg/mL

 

270 Chapter 11

To calculate the total amount of drug and metab- drug concentration to the amount of drug in the
olite in the cell compartment, Equation 11.3 is rear- body (Equation 11.3). Equation 11.3 relating
ranged as shown: the total mass of drug to drug concentration

and volume of distribution is important in
Total drug and metabolite in cells = 5 mL pharmacokinetics.
× (0.2 + 9.96 + 0.29 mg/mL) = 52.45 mg

Therefore, the total drug and metabolite in the fluid PRACTICE PROBLEM
compartment is 100 − 52.45 mg = 47.55 mg. The amount of drug in the system calculated from VD

If only the intact drug is considered, VD is calcu- and the drug concentration in the fluid compartment
lated using Equation 11.3. is shown in Table 11-3. Calculate the amount of drug

100 mg in the system using the true volume and the drug
VD = = 500 mL

0.2 mg/mL concentration in the fluid compartment.

Considering that only 100 mL of water was Solution
placed into beaker 3, the calculated apparent volume In each case, the product of the drug concentration
of distribution of 500 mL is an overestimate of the (column 5) and the apparent volume of distribution
true fluid volume of the system. (column 6) yields 100 mg of drug, accurately

The following conclusions can be drawn from accounting for the total amount of drug present in
this beaker exercise: the system. For example, 0.25 mg/mL × 400 mL =

1. Drug must be at equilibrium in the system 100 mg. Notice that the total amount of drug present

before any drug concentration is measured. In cannot be determined using the true volume and the

nonequilibrium conditions, the sample removed drug concentration (column 5).

from the system for drug assay does not repre- The physiologic approach requires detailed

sent all parts of the system. information, including (1) cell drug concentration,

2. Drug binding distorts the true physical volume (2) cell compartment volume, and (3) fluid compart-

of distribution when all components in the ment volume. Using the physiologic approach, the

system are not properly sampled and assayed. total amount of drug is equal to the amount of drug

Extravascular drug binding increases the in the cell compartment and the amount of drug in

apparent V the fluid compartment.
D.

3. Both intravascular and extravascular drug bind-
(15 mg/mL × 5 mL) + (100 mL × 0.25 mg/mL)

ing must be determined to calculate meaningful
volumes of distribution. = 100 mg

4. The apparent VD is essentially a measure of
The two approaches shown above each account

the relative extent of drug distribution outside
correctly for the amount of drug present in the sys-

the plasma compartment. Greater tissue drug
tem. However, the second approach requires more

binding and drug accumulation increases VD,
information than is commonly available. The second

whereas greater plasma protein drug binding
approach does, however, make more physiologic

decreases the VD distribution.
sense. Most physiologic compartment spaces are not

5. Undetected cellular drug metabolism
clearly defined for measuring drug concentrations.

increases VD.
6. An apparent VD larger than the combined vol-

ume of plasma and body water is indicative of Complex Biological Systems and VD

(4) and (5), or both, above. The above example illustrates how the VD repre-
7. Although the VD is not a true physiologic sents the apparent volume into which a drug

volume, the VD is useful to relate the plasma appears to distribute, whether into a beaker of fluid

 

Physiologic Drug Distribution and Protein Binding 271

or the human body. The human body is a much the drug concentration in the plasma compartment
more complex system than a beaker of water con- (Fig. 11-9A). In a physiological system involving a
taining drug metabolizing cells. Many components drug distributed to a given tissue from the plasma
within cells, tissues, or organs can bind to or fluid (Fig. 11-9B), the two-compartment model is
metabolize drug, thereby influencing the apparent not assumed, and drug distribution from the plasma
VD. Only free, unbound drug diffuses between the to a tissue is equilibrated by perfusion with arterial
plasma and tissue fluids. The tissue fluid, in turn, blood and returned by venous blood. The model
equilibrates with the intracellular water inside the tis- parameter Vapp is used to represent the apparent dis-
sue cells. The tissue drug concentration is influenced tribution volume in this model, which is different
by the partition coefficient (lipid/water affinity) of from VDSS used in the compartment model. Similar
the drug and tissue protein drug binding. The distri- to the apparent volume simulated in the beaker
bution of drug in a biological system is illustrated experiment in Equation 11.3, Vapp is defined by
by Fig. 11-8. Equation 11.5, and the amount of drug in the body

is given by Equation 10.6.

Apparent Volume of Distribution
D

The apparent volume of distribution, in general, Vap = B
p (11.5)

Cp
relates the plasma drug concentration to the amount
of drug present in the body. In classical compart- DB = VpCp + VtCt (11.6)
ment models, VDSS is the volume of distribution
determined at steady state when the drug concentra- where DB is the amount of drug in the body, Vp is the
tion in the tissue compartment is at equilibrium with plasma fluid volume, Vt is the tissue volume, Cp is

TISSUES

Clinical response Drug – Receptor
PLASMA

Receptor Protein Drug – Protein
+ +

Drug Drug

KIDNEY LIVER

Carrier + Drug Drug + Enzymes

Drug – Carrier Metabolites

Excretion Active renal Excretion Excretion
in urine secretion in urine in bile

FIGURE 118 Effect of reversible drug–protein binding on drug distribution and elimination. Drugs may bind reversibly with
proteins. Free (nonbound) drugs penetrate cell membranes, distributing into various tissues including those tissues involved in drug
elimination, such as kidney and liver. Active renal secretion, which is a carrier-mediated system, may have a greater affinity for free
drug molecules compared to plasma proteins. In this case, active renal drug excretion allows for rapid drug excretion despite drug–
protein binding. If a drug is displaced from the plasma proteins, more free drug is available for distribution into tissues and interac-
tion with the receptors responsible for the pharmacologic response. Moreover, more free drug is available for drug elimination.

 

272 Chapter 11

Plasma Tissue calculations of steady-state VD involve some assump-
tions on how and where the drug distributes in the

k12
Binding? Partition? body; it could involve a physiologic or a compartmen-

k Compartment model
21 Binding? tal approach.

Partition? Adsorption?
For a drug that involves protein binding, some

models assume that the drug distributes from the
plasma water into extracellular tissue fluids, where
the drug binds to extravascular proteins, resulting in

Blood Arterial Tissue a larger VD due to extravascular protein binding.
blood However, drug binding and distribution to lipoid tis-

Albumin and ow
Tissue and sues are generally not distinguishable. If the pharma-

AAG albumin Physiologic model
binding binding (Only one tissue shown) cokineticist suspects distribution to body lipids

Venous because the drug involved is very lipophilic, he or she
blood
ow may want to compare results simulated with different

models before making a final conclusion.
FIGURE 119 A diagram showing (upper panel) a two- Figure 11-10 lists the steady-state volume of dis-
compartment model approach to drug distribution; (lower tribution of 10 common drugs in ascending order.
panel) a physiologic approach to drug distribution.

Most of these drugs follow multicompartment kinetics
with various tissue distribution phases. The physio-

the plasma drug concentration, and Ct is the tissue logic volumes of an ideal 70-kg subject are also plotted
drug concentration. for comparison: (1) the plasma (3 L), (2) the extracel-

For many protein-bound drugs, the ratio of lular fluid (15 L), and (3) the intracellular fluid (27 L).
D Drugs such as penicillin, cephalosporin, valproic acid,

B/Cp is not constant over time, and this ratio
depends on the nature of dissociation of the protein– and furosemide are polar compounds that stay mostly
drug complex and how the free drug is distributed; within the plasma and extracellular fluids and there-
the ratio is best determined at steady state. Protein fore have a relatively low VD.
binding to tissue has an apparent effect of increasing In contrast, drugs with low distribution to the
the apparent volume of distribution. Several V extracellular water are more extensively di