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Pharmacokinetics

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Contents

Basic considerations
Pharmacokinetics models
Compartment modeling
One compartment model- IV bolus, IV infusion, extra
vascular
Multi compartment model
Two compartment model
Non- linear pharmacokinetics
Cause of non- linearity
Michaelis- Menten equation
Estimation of Kmax and Vmax

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Pharmacokinetics
 Refers on how the human body acts on the drug.

Definition:

• Pharmacokinetics is the study of the kinetics of drug

absorption, distribution, metabolism and elimination and

their relationship with the pharmacological, therapeutic or

toxicological response in man and animals.

➢ Now Pharmacokinetics can be better described as LADME.

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Pharmacokinetic Process

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Clinical Pharmacokinetics:

 The application of pharmacokinetic principles in the safe
and effective management of individual patient.

Population pharmacokinetics:

 The study of pharmacokinetic differences of drugs in
various population groups.

Toxicokinetics:

 the application of principle to the design, conduct and
interpretation of drug safety evaluation studies.

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Plasma Drug Concentration–Time Curve

Two categorizes of parameters can be evaluated from a plasma
concentration time profile.

Pharmacokinetic

Pharmacodynamic

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Pharmacokinetic Parameters

❖Peak plasma concentration (cmax)

❖Time of peak concentration(tmax)

❖Area under curve (AUC)

Peak Plasma Concentration(cmax):

The point at which, maximum concentration of drug in
plasma.

Units : µg/ml

Peak conc. Related to the intensity of pharmacological
response, it should be above MEC but less than MSC.

The peak level depends on administered dose and rate of
absorption and elimination.

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Time of peak concentration (tmax):

The time for the drug to reach peak concentration in plasma
(after extra vascular administration).

Units : hrs

 Useful in estimating onset of action and rate of absorption.

 Important in assessing the efficacy of single dose drugs
used to treat acute conditions (pain, insomnia ).

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Area under curve (AUC):

It represents the total integrated area under the plasma level-
time profile and expresses the total amount of the drug
that comes into systemic circulation after its
administration.

Units : µg/ml x hrs

 Represents extent of absorption – evaluating the
bioavailability of drug from its dosage form.

 Important for drugs administered repetitively for treatment
of chronic conditions (asthma or epilepsy).

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PHARMACODYNAMICS

 Pharmacodynamics is the study of the biochemical and
physiological effects of drugs on the body;

 Includes the mechanisms of drug action and the
relationship between drug concentration and effect.

 Typical example of pharmacodynamics is how a drug
interacts quantitatively with a drug receptor to produce a
response (effect).

 Receptors are the molecules that interact with specific
drugs to produce a pharmacological effect in the body.

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PHARMACODYNAMIC PARAMETERS
1.Minimum effective concentration (MEC)

Minimum concentration of drug in plasma/receptor site is
required to produce therapeutic effect.

 Concentration of drug below MEC – sub therapeutic level

2.Maximum safe concentration (MSC)

Concentration in plasma above which adverse or unwanted
effects are precipitated.

 Concentration above MSC – toxic level

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3.Onset time

Time required to start producing pharmacological response.

Time for plasma concentration to reach MEC after
administrating drug.

4.Onset of action

The beginning of pharmacologic response.

It occurs when plasma drug concentration just exceeds the
required mec.

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5.Duration of action

The time period for which the plasma concentration of
drug remains above MEC level.

6.Intensity of action

It is the minimum pharmacologic response produced by the
peak plasma conc. Of drug.

7.Therapeutic range

the drug conc. Between MEC and MSC

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Pharmacokinetic models

 Pharmacokinetic modeling is a mathematical modeling
technique.

 Predicting the absorption, distribution, metabolism and
excretion (ADME) of synthetic or natural chemical substances
in humans and other animal species.

 Why model the data ?

There are three main reasons

1. Descriptive: to describe the drug kinetics in a simple way.

2. Predictive: to predict the time course of the drug after
multiple dosing based on single dose data, to predict the
absorption profile of the drug from the iv data.

3. Explanatory: to explain unclear observations
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Applications

 Characterizing the behaviour of drugs in patients.

 Calculating the optimum dosage regimens for individual
patients.

 Evaluating the bioequivalence between different
formulations of same drug.

 Determining the influence of altered physiology or disease
state on drug ADME.

 Prediction of drug concentration in various body fluids with
any dosage regimen.

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 Estimation of the possible accumulation of drugs/
metabolites.

 Correlating plasma drug concentration with
pharmacological response.

 Explaining the drug interactions.

 Evaluating the risk of toxicity with certain dosage
regimens.

 Predicting the multiple dose concentration curves from
single dose experiments.

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Pharmacokinetic Model Approach

 Pharmacokinetic models are hypothetical structures used to
describe the fate of a drug in a biological system following
its administration.

 The two major approaches in the quantitative study of

 various kinetic processes of drug disposition in body are

 1. Model approach

 2. Model independent approach

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Pharmacokinetic Modelling

Compartment Models Non-compartment Models Physiologic Models

AUC, MRT, MAT, Cl, VSS

Caternary Mamillary Model
Model

One compt Multi compt Two compt

i v bolus
i v bolus

Single oral Dose

i v infusion
Oral drug

Intermittent i v infusion

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Multiple doses

 

Types of pharmacokinetic models

 Compartment models ; are also called empirical models.

 Physiological models ; are realistic models.

 Distributed parameter models ; are also realistic models.

A Compartment is a group of tissues with similar blood
flow and drug affinity.

A compartment is not a real physiologic or anatomic region.

 A Model is a mathematic description of a biologic system
and is used to express quantitative relationships.

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Compartment model

 Traditional and most widely used approach to
pharmacokinetic characterization of drug.

 simply interpolate the experimental data and allow on
empirical formula to estimate drug concentration with time.

 Assumptions of Compartmental Models

 The body is represented as a series of compartment
arranged in series or parallel to each other.

 Each compartment is not a real physiological or anatomical
region but fictitious or virtual one.

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 Considered as a tissue or group of tissue that have similar
drug distribution characteristics

 Within each compartments the drugs is considered to be
rapidly and uniformly distributed.

 The rate of drug movement between compartment (entry
into and exit)is described by first order kinetics.

 Rate constants are used to represent rate of entry into and
exit from compartment.

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Types of Compartment model

 Two categories

 Mammillary model

 Catenary model

❖Mammillary model

 Most common compartment model used in
pharmacokinetics.

 It consists of one or more peripheral compartments
connected to the central compartment in a manner similar
to connection of satellites to a planet .

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 They are joined parallel to the central compartment.

 The central compartment( or compartment 1) comprises of
plasma and highly perfused tissues such as lungs, liver,
kidney etc. which rapidly equilibrate with drugs.

 The peripheral compartments or tissue compartments (
denoted by numbers 2,3 etc) are those with low vascularity
and poor perfusion.

 Distribution of drugs to these compartments is through
blood.

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 Movement of drug can be defined by first-order kinetics.

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Catenary model

 The compartments are joined to one another in a series like
compartments of a train.

 It is rarely used because it is not observed that anatomicaly
or physiologically various organs are directly linked to the
blood compartment.

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Applications of Compartment Modeling

 Simple and flexible approach and is widely used

 Gives a visual representation of various rate process
involved in drug disposition.

 Shows how many rate constants are necessary to describe
these processes.

 To describe drug concentration changes in each
compartment.

 Monitoring of drug concentration change with time a
limited amount of data( plasma concentration or urinary
excretion data is sufficient.)

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 Important in the development of dosage regimens.

 Useful in predicting drug concentration time profile in both
normal and pathological conditions.

 Useful in relating plasma drug levels in therapeutic and
toxic effects in body.

 Clinically, drug data comparisons are based on
compartment models.

 Its simplicity allows for easy tabulation of volume of
distribution, half life etc.

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Disadvantages

 Several assumptions have to be facilitate data
interpretation, since the compartments and parameters bear
no relationship with the physiological function.

 Extensive efforts are required in the development of an
exact model.

 Model may vary within a study population.

 Can be applied only to a specific drug under study.

 Difficulties generally arise when using models to interpret
the differences between results from human and animal
experiments.

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Physiological Models

 Also known as physiologically – based pharmacokinetic
models (PB-PK Models)

 drawn on the basis of known anatomical and physiological
data

 present more realistic picture of drug disposition in various
organs and tissues.

 The number of compartments to be included in the model
depends upon the disposition characteristic of the drug.

 Tissues with similar perfusion properties are grouped into a
single compartment

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Types of Physiological method
 Two types

 Blood flow rate limited models

 Membrane permeation rate limited models

Blood flow rate limited models:

 More popular and commonly used

 Assumption that the drug movement within a body region is
much more rapid than its rate of delivery to that region by the
perfusing blood.

 Also called as perfusion rate limited models.

 Applicable only to the highly membrane permeable drugs.

 (low molecular weight, poorly ionised & highly lipophilic
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drugs- thiopental, lidocaine)

 

Membrane Permeation Rate Limited Models

 More complex

 Applicable to highly polar, ionised and charged drugs.

 Cell membrane act as a barrier for the drug that gradually
permeates by diffusion.

 Also called as diffusion limited models

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Advantages

 Mathematical treatment is straightforward.

 Realistic approach.

 Data filling is not required.

 Gives exact description of drug concentration –time profile
in any organ or tissue,

 The influence of altered physiology or pathology on drug
disposition can be easily predicted from changes in the
various pharmacokinetic parameters.

 Frequently used in animals.

 Mechanism of ADME of drug can be easily explained.

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Disadvantages

 Obtaining the experimental data is a very exhaustive
process.

 Prediction of individualized dosing is difficult.

 Less number of data point is to be assessed.

 Monitoring of drug concentration in body is difficult.

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Distributed Parameter Model

 Analogous to physiological model but has been designed to
take into account

 Variations in blood flow to an organ.

 Variations in drug diffusion in an organ.

 Specially useful for assessing regional difference in drug
concentration in tumours or narcotic tissues.

 Mathematical equations are more complex and collection of
drug concentration data is more difficult.

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Non Compartmental Analysis

 Also called as model independent methods

 It does not require the assumption of specific compartment
model.

 Based on the assumption that the drugs or metabolites
follow linear kinetics.

 Can be applied to any compartment model.

 Approach based on statistical moments theory.

 It involves collection of experimental data following a
single dose of drug

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 If one consider the time course of drug concentration in
plasma as a statistical distribution curve, then

MRT = AUMC/AUC
Where

 MRT= mean residence time

 AUMC= area under the first moment curve

 AUC= Area under the zero moment curve

 MRT= is defined as the average amount of time spent by
the drug in the body before being eliminated.

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Applications

 Used to estimate the important Pk parameters-
bioavailability, clearance and apparent volume of
distribution.

 Also useful in determining half life, rate of absorption and
first order absorption rate constant.

 Advantages:

 Ease of derivation of Pk parameters by simple algebraic
equations

 Same mathematical treatment can be applied almost any
drug or metabolite if they follow first order kinetics.

 Detailed description of drug disposition characteristic is not
required.

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Disadvantages

 Provides limited information regarding the plasma drug
concentration- time profile.

 Does not adequately treat non- linear cases.

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One Compartment Open Model

 Simplest model.

 The body is considered as a single , kinetically
homogenous unit.

 This model applies only to those drugs that distributes
rapidly throughout the body.

 Drugs move dynamically in an out of this compartment

 Elimination is first order (monoexponential) process with
first order rate constant

 Rate of input(absorption)> rate of output(elimination).

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• Any change in plasma concentration reflects a proportional
change in drug concentration throughout the body.

• Open model indicates that the input (availability) and
output(elimination) are unidirectional and that the drug can
be eliminated from the body.

Ka KE
Metabolism

Drug
Blood and other

Input body tissues output
(Absorption) (Elimination) Excretion

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One Compartment Open Model

 Intravenous Bolus Administration

 When drug is given in the form of rapid i.v. injection it takes
about one to three Minutes for complete circulation.

 Hence, the rate of absorption is neglected in calculations.

Blood and other body KE

tissues

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 General expression of rate of drug presentation to the body
is
dX= Rate In –Rate Out
dt

 If, Rate in absorption is absent,

 dX= Rate Out

dt

 If, the rate out or elimination follows first –order kinetics

 dX= -KEX

dt

 KE=first order elimination rate constant

 X= amount of drug in body at any time t

 remaining to be implemented.
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Estimation of Pharmacokinetic Parameters- IV
bolus Administration

 A drug that follows one compartment kinetics and
administered as rapid I.V. Injection , the decline in plasma
drug concentration is only due to elimination of drug from
the body, the phase being called as elimination phase.

 Elimination phase can be characterized by 3 parameters

❖Elimination rate constant

❖Elimination half- life

❖Clearance

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Elimination Rate Constant

 dX = – KEX
dt

 Integrating above equation yields

 ln X = ln X0 – KEt

 where Xo = amount of drug at time t=0 (initial amount of
drug injected)

 Equation can be written in exponential form as
 X= Xo e‐KEt

 The above equation shows the disposition of a drug that
follows one compartmental kinetics is monoexponential.

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 Transforming equation into logarithm form we get,

 Log X = Log X0 –‐ KEt

2.303

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 X=Vd C

 X= amount of drug in the body

 C= drug concentration in plasma

 Vd = Apparent volume of distribution

 Log C = Log C0 KEt

2.303

 Overall elimination rate Constant:

 The elimination or removal of drug from the body is the sum
of urinary excretion, metabolism, biliary excretion,
pulmonary excretion and other mechanism involved therein.

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Elimination half life

 Also called as biological half life.

 Defined as the time taken for the amount of drug in the
body as well as plasma concentration to decline by one-half
or 50% its initial value.

 t1/2 = 0.693/KE

 t1/2 = 0.693Vd/ClT

 Apparent volume of distribution:

 Two separate and independent pharmacokinetic
characteristic of drugs are

1. Apparent volume of distribution

2. clearance

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 For drugs given as i.v. bolus ,
 Vd(area) = X0

KEAUC
 For drugs administered extravascularly,
 Vd= FX0

KEAUC
 where,X0= dose administered
 F=fraction of drug absorbed in
 systemic circulation.

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Clearance

 Clearance is defined as the theoretical volume of body fluid
containing drug from which the drug is completely
removed in a given period of time.

 Cl = Rate of elimination

 Plasma drug concentration

 Cl = dX/dt

 C

 Total body clearance : elimination of drug from the body
involves processes occurring in kidney, liver, lungs and
other eliminating organs.

 Also called as total systemic clearance.

 ClT = ClR+ ClH+ Cl others

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 ClT= KEX
C

 X/C = Vd

 ClT=KEVd ; ClH=KmVd;ClR=KeVd

 ClT= 0.693Vd

t1/2

 for drugs given as IV bolus
 ClT = X 0

AUC
 for drugs given as EV
 ClT = FX 0

AUC

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Organ clearance
 Rate of Elimination = Rate of Presentation to the organ –

Rate of exit from the organ

 Rate of Presentation (input)= organ blood flow x Entering
concentration = (QCin)

 Rate of exit (output)= organ blood flow x Exiting
concentration= (QCout)

 Rate of elimination ( rate of extraction) = QCin- Qcout

 Extraction Ratio(ER)=Cin- Cout/ Cin

 Based on ER values drugs can be

classified as:

 Drugs with high ER=above0.7

 Drugs with intermediate ER=between 0.7‐0.3
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 D.DruulogMsix .cwomith low ER = below 0.3

 

Hepatic clearance

 ClH= ClT- ClR

 ClH =QHERH

 QH = hepatic blood flow(about 1.5l/min)

 ERH = hepatic extraction ratio

 Divided into 2 groups

 Drugs with hepatic blood flow rate limited clearance

 Drugs with intrinsic capacity limited clearance.

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One Compartment Open Model – Intravenous
Infusion

 Rapid i.v. injection is unsuitable when the drug has
potential to precipitate toxicity or when maintenance of a
stable concentration or amount of the drug in body desired.
In such a situation, the drug is administered at a constant
rate (zero ordered) by i.v. infusion.

 Advantages of zero order infusion of drug include

1. Ease of control of rate of infusion.

2. Prevents fluctuating maxima and minima (peak and
valley) plasma level. This is desired especially when the
drug has a narrow therapeutic index.

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 Other drugs, electrolytes and nutrients can be conveniently
administered simultaneously by the same infusion line in critically ill
patients.

 The model can be represented as follows

 During infusion, the rate of change in amt. of drug in the body, dx/dt is
the difference between the zero order rate of drug infusion Ro and first
order rate elimination, ‐KEx:

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 Integration and rearrangement of above equation yields

 Since X=Vd C the above equation can be transformed into
concentration terms as follows

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One Compartment Open Model- Extravascular
Administration

 When a drug is administered by extravascular route (e.g.
oral, i.m., rectal, etc.), absorption is a prerequisite for its
therapeutic activity.

 The rate of absorption may be described mathematically as
a zero-order or first-order process.

 A large number of plasma concentration time profiles can
be described by a one compartment model with first-order
absorption and elimination.

 However, under certain conditions, the absorption of some
drugs may be better described by assuming zero-order
(constant rate) kinetics.

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 Distinction between zero-order and first-order absorption
processes. Figure a is regular plot, and Figure b a semi log
plot of amount of drug remaining to be absorbed (ARA)
versus time t.

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 Zero-order absorption is characterized by a

constant rate of absorption

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 During absorption phase, the rate of absorption is greater than
elimination phase.

 At plasma concentration, the rate of absorption equals the rate of
elimination and the change in amount of drug in the body is zero

 During post absorption phase, the rate of elimination is greater than the
absorption rate

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Zero Order Absorption Model – EV
Administration

 This model is similar that for constant rate infusion.

 In controlled drug delivery systems, the rate of drug
absorption is constant and continues until the amount of
drug at the absorption site (GIT) is depleted.

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First Order Absorption Model – EV
Administration

 For a drug that enters the body by a first order absorption
process, gets distributed in the body according to one-
compartment kinetics and is eliminated by a first order
process

 Differential form of the equation

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 Ka = first order absorption rate constant

 Xa = amount of drug at the absorption site remaining to be
absorbed.

 integration of above equation yields

 Transforming into concentration terms,

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Assessment of Pharmacokinetic Parameters – EV
Administration

 Cmax and tmax:

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Elimination Rate Constant:

 Absorption rate is significantly greater than the elimination
rate Kat >>Ket

 Absorption rate constant:

 Calculated by method of residuals

 Te technique also known as feathering, peeling and
stripping

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Wagner- Nelson Method for Estimation of Ka

 Determination of Ka from unabsorbed amount of drug at
any time plots.

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Determination of KE from urinary excretion data

 Rate of excretion method
 Sigma-minus method
 Rate of excretion method:

 Advantage:
 Drugs that having long half- lives, urine may be collected

for only 3-4 half lives.
 No need to collect all urine samples.

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Sigma- Minus method

 (X ∞
u – Xu)=amount remaining to be excreted ARE

 Disadvantage: Total urine has to be carried out until no
unchanged drug can be detected in the urine (upto 7-half-
lives)

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Multicompartment models

 One compartment is described by mono‐exponential term i.e
.elimination.

 For large class of drugs this terms is not sufficient to descr
ibe its disposition.

 It needs a bi‐or multi‐exponential terms

 This is because the body is composed of a heterogeneous g
roup of tissues each with different degree of blood flow an
d affinity or drug and therefore different rates of eliminatio
n.

 Ideally a true pharmacokinetic model should be the one wit
h a rate constant for each tissue undergoing equilibrium.

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Multicompartment Model
( delayed distribution methods)

 Multicompartment models provide answers to such
questions as:

 (1) How much of a dose is eliminated?

 (2) How much drug remains in the plasma compartment at
any given time? And

 (3) How much drug accumulates in the tissue
compartment?

 The latter information is particularly useful for drug safety

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Multicompartment models are based on following assumptions

 Central compartment-comprising of blood/plasma and
highly perfused tissues such as brain, heart, lung, liver and
kidney.

 Peripheral compartment comprising of poorly perfused
tissues such as muscles, skin & adipose etc.

 IV administered medications are introduced directly into
the central compartment

 Irreversible drug elimination (hepatic or renal excretion )
take place only from the central compartment

 Reversible distribution occurs between central and
peripheral compartment.

 Elimination of drug follows first order kinetics

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Two Compartment Open Model
 All multicomponent models is a two compartment model.

 Body tissues are broadly classified into 2 categories.

 Central compartment or Compartment1:

 Comprises of blood and highly perfused tissues like liver
,lungs, kidneys, etc that equilibrate with the body rapidly.

 Peripheral or Tissue Compartmen or Compartment2:

 Comprises of poorly perfused and slow

equilibrating tissues such as muscles,

skin, adipose, etc.

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 Considered as a hybrid of several

functional physiologic units.

 Based on the drug elimination, two compartment model
can be categorized in to 3 types

1. With elimination from Central

compartment

1. With elimination from peripheral

compartment

1. With elimination from both the

compartments

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Two Compartment Open Model –IV bolus
Administration

 The rate constants k12 and k21 represent the first-order rate
transfer constants for the movement of drug from
compartment 1 to compartment 2

 (k12) and (k21) represents from compartment 2 to
compartment 1.

 Most two-compartment models assume that elimination
occurs from the central compartment model.

 The plasma level–time curve for a drug that follows a two-
compartment model may be divided into two parts,

 (a) a distribution phase

 b) an elimination phase

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 Distribution Phase:

 The initial, more rapid decline of drug from the central
compartment into the tissue compartment.(line a).

• Elimination phase:
 The drug concentrations in both the central and tissue

compartments decline in parallel and more slowlycompared
to the distribution phase. This decline is a first-order
process and is called the elimination phase or the beta (ß)
phase (line b) or post‐distributive phase.

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 During the distribution phase, Drug elimination and
distribution occur concurrently.

 Net transfer of drug from the central compartment to the
tissue compartment because the rate of distribution is faster
than the rate of elimination.

• The fraction of drug in the tissue compartment is
equilibrium with the fraction of drug in the central
compartment.

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 In contrast to this compartment, the
conc of drug in the peripheral compartment firs
t increases and reaches its max.

 Following peak, the drug conc declines
which corresponds to the post‐distributiv phase.

 dCc = K21Cp–K12Cc–KECc

dt
 Extending the relationship X= VdC
 dCc = K21Xp – K12Xc – KEXc

dt Vp Vc Vc

 Xc & Xp= amount of drug in the central and peripheral
compartment,

 Vc & Vp= apparent volume of drug in the central &
peripheral compartment.

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 The rate of change in drug conc in the peripheral component is given
by:

On integration equation gives conc of drug in central and peripheral
compartments at any given time t :

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 α andβ are hybrid first order constants for rapid dissolution
phase and slow elimination phase, which depend entirely
on 1st order constants K12, K21, KE

 The constants K12, and K21 that depict the reversible
transfer of drug between the compartments are called micro
or transfer constants.

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 Simplification of the equation as

 Cc = distribution exponent + elimination exponent.
 A and B are hybrid constants for two exponents and can be

resolved by graph by method of residuals.

 Co = plasma drug conc immediately after i.v. injection
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Method of residuals :
 The biexponential disposition curve obtained after i. v. bolus of a drug

that fits two compartment model can be resolved into its individual
exponents by the method of residuals

 From graph the initial decline due to distribution is more rapid than the
terminal decline due to elimination i.e. the rate constant a >> b and
hence the term e‐at approaches zero much faster than e –βt

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 In log form, the equation becomes

C = back extrapolated plasma concentration
values
A semilog plot of C vs t yields the terminal linear
phase of the curve having slope –b/2.303 and
when back extrapolated to time zero, yields
y‐intercept log B. The t1/2 for the elimination
phase can be obtained from equation

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www = 0.693/b.

 

 Residual conc values can be found as‐

 Cr = C – C = Ae‐αt

 log Cr = log A – αt

2.303
 A semilog plot

Cr vs t gives a
straight line.

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Assessment of Pharmacokinetic Parameters – IV
Bolus Administration
 Parametres of the model viz K12, K21, KE etc can be derived by

proper substitution of the values

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 For two compartment model, KE is the rate constant for
elimination of drug from the central compartment and β is
the rate constant for elimination from the entire body.
Overall elimination t1/2 can be calculated from β.

 Area under the plasma concentration –time curve can be
obtained by

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 The apparent volume of central compartment Vc is given as

 The apparent volume of peripheral compartment can be
obtained from the equation

 The apparent volume of distribution at steady state or
equilibriumcan be defined as

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It is also given as

 Total systemic clearance is given as

 The pharmacokinetic parameters can also be calculated by
using urinary excretion data

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 Rate of excretion of unchanged drug in urine

 Renal clearance is given as

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Two Compartment Open Model – IV Infusion

 The model can be depicted as shown as

 The plasma or central compartment concentration of a drug
that fits two compartment model when administered as
constant rate (zero order) IV infusion, is given by equation:

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 At steady state, the second and third term in the bracket becomes zero
and equation reduces to

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 Now VcKE = Vdß substituting this in equation

 The loading dose Xo,L to obtain Css immediately at the
start of infusion can be calculated from the equation

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Two Compartment Open Model- Extravascular
Administration – First Order Absorption

 This model can de depicted as follows

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 The rate of change in drug concentration in the central
compartment is described by 3 exponents that describe drug
disposition.

 C=absorption exponent + distribution exponent +
elimination exponent

 The 3 exponents can be resolved by stepwise application of
method of residuals assuming Ka>α>β

 Ka can also be estimated by Loo- Riegelman method.

 It requires plasma concentration-time data both after oral
and IV administration of the drug to the same subject at
different times.

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Non linear Pharmacokinetics

 It is a Dose Dependent Pharmacokinetics.

 Nonlinear pharmacokinetic models imply that some aspect
of the pharmacokinetic behaviour of the drug is saturable.

 It is also called as mixed –order and capacity limited
kinetics.

 Detection of Non linear pharmacokinetics:

 Determination of steady state plasma concentration at
different doses.-

 if the steady state concentrations are directly proportional
to the dose is not observable.

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 Determination of some important pharmacokinetic
parameters such as fraction bioavailable, elimination half
life or total systemic clearance at different doses of drug.
Any change in these parameters is indicative to non-
linearity which are usually constant.

 Causes of non-linearity:

 Drug absorption

 Distribution

 Metabolism

 Excretion

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Drug Absorption

 Three causes:-

1. Solubility / dissolution of drug is rate-limited;
Griseofulvin – at high concentration in intestine.

2. Carrier – mediated transport system; Ascorbic acid,
Riboflavin & Cyanocobalamin – saturation of transport
system at higher doses results in non linearity.

3. Presystemic gut wall / hepatic metabolism attains
saturation; Propranolol, hydralazine andverapamil-
saturation of presymetic metabolism of these drugs at
high doses leads to increased bioavailability.

 These parameters affected F, Ka, Cmax and AUC.

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Drug Distribution

 Non linearity in distribution of drugs administered at high
doses may be due to

1. Saturation of binding sites on plasma proteins. ex:
Phenylbutazone and naproxen

2. Saturation of tissue binding sites. Ex: thiopental, fentanyl.

 In both cases free plasma drug concentration increases.

Vd Increase only in (1)

 Clearance with high ER get increased due to saturation of
binding sites.

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Drug Metabolism

 Non-linearity occurs due to capacity limited metabolism,

 Small changes in dose administration can produce large
variations in plasma concentration at steady state- major
source of large intersubject variability in pharmacological
response..

 • Two important causes

1.Capacity – limited metabolism due to enzyme
&/cofactor saturation; Phenytoin, Alcohol, theophylline

 Saturation of enzymes results in – decrease in ClH –
increase in Css.

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 2.Enzyme induction–Ex. Carbamazepine.

decrease in peak plasma concentration- on repetitive
administration over a period of time.

 Common cause of both dose and time dependent kinetics.

 Enzyme induction results in increase ClH – decrease in
Css.

 Other reasons of non linearity includes saturation of
binding sites, inhibitory effects of the metabolites on
enzymes and pathological situations (hepatotoxicity and
changes in hepatic blood flow).

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Drug Excretion

 Two active processes in renal excretion of a drug which are
saturable,

1. Active tubular secretion – Penicillin G after saturation of
carrier systems – decrease in renal clearance.

2. Active tubular reabsorption – Water soluble vitamins &
Glucose.- after saturation of carrier systems – increase in
renal clearance.

 Other reasons like forced diuresis, change in urine pH,
nephrotoxicity & saturation of binding sites.

 In case of biliary excretion non – linearity due to saturation
– Tetracycline & Indomethacin.

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Michaelis Menten Equation

 The kinetics of capacity limited or saturable processes is
best described by Michaelis-Menten equation.

 -dc = Vmax C (I)
dt Km+C

 -dC/dt = rate of decline of drug conc. with time

 Vmax = theoretical maximum rate of the process

 KM = Michaelis constant

 Three situation can now be considered depending upon the
value of Km and C.

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 1.When KM = C:

under this situation , eq I reduces to,

 -dC/dt = Vmax/2 ……………….II

 The rate of process is equal to half of its maximum rate.

 2.When Km >>C

 Here Km +C = Km and the equation I reduces to

 -dc/dt = Vmax C/ Km

 above eq. is identical to the one that describe first order
elimination of drug, where Vmax/KM= KE.

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 3. When Km<< C

 under this condition, Km+C= C and the equation (I)

 -dc/dt= Vmax

 above equation is identical to the one that describe a zero
order process i.e. the rate process occurs at constant rate
Vmax and is independent of drug conc.

 E.g. Metabolism of ethanol

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Determination of Km and Vmax

 The parameters Kmand Vmax can be assessed from the
plasma concentration –time data collected after IV bolus
administration of a drug with nin linear characteristics.

 Rewriting the equation (I)

-dc = Vmax C

dt Km+C

 Integration of above equation followed by conversion to
log base 10 yields

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Km and Vmax from Steady State Concentration

 If drug is administered for constant rate IV infusion/ in a
multiple dosage regimen, the steady-state conc. is given in
terms of dosing rate (DR):

(1)

 If the steady-state is reached, then the dosing rate = the rate
of decline in plasma drug conc. & if the decline occurs due
to a

 single capacity-limited process then above equation
become as:

(2)
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Estimation of Km and Cmax

Three methods are used

1. Lineweaver-Burk/Klotz Plot

2. Direct linear plot

3. Graphical method

Lineweaver-Burk/Klotz Plot:

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Direct linear plot

A pair of Css1 and
Css2 obtained with
two different dosing
rates DR1and DR2 is
plotted.

The points Css1 and
DR1 are joined to form
a line and a second line
is obtained similarly
by joining Css2 and
DR2.
DR axis to obtain V
max and on x axis to

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Graphical Method
 In this method by rearranging eq. (2) we get

 KM & Vmax can be estimated by simultaneous eq. As

Combination of the above 2 equation yields

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