Optimization techniques in pharmaceutical formulation and processing

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Optimization techniques in

formulation and processing


Mpharm first year

Department of pharmaceutics

College of pharmacy
Madras Medical College



❖ Introduction
❖ Importance
❖ Concept of optimization
❖ Terms

❖ Parameters
❖ Experimental design
❖ Techniques
❖ Reference



• The term Optimize is defined as “to make perfect”. It is used in pharmacy relative to
formulation and processing. It is involved in formulating drug products in various forms.

• It is the process of finding the best way of using the existing resources while taking in to
the account of all the factors that influences decisions in any experiment.

• In development projects, one generally experiments by a series of logical steps, carefully
controlling the variables & changing one at a time, until a satisfactory system is obtained

“It is not a screening technique.”



Primary objective may not be optimize absolutely but to compromise effectively &
thereby produce the best formulation under a given set of restrictions .

Reduces the Safety and
Save Time

cost Reduces error

Innovation and




o Yield the “Best Solution” within the domain of study.

o Require fewer experiments to achieve an optimum formulation.

o Can trace and rectify problem in a remarkably easier manner.


Concept of Optimization

• The term optimize is defined as “ to make perfect”.

• In terms of sentence it is defined as choosing the best element from some set of
available alternatives.

• According to Merriam Webster dictionary, optimization means, “ An act, process or
methodology of making something (as a design, system or a decision) as a fully
perfect, functional or effective as possible; specially the mathematical procedures.

• Optimization is also defined as “The process of finding the best values for the
variables of a particular problem to minimize or maximize an objective function.”


• It is used in pharmacy relative formulation and processing

• It is involved in formulating drug products in various forms.

• Final product not only meets the requirements from the bioavailability but also from the
practical mass production criteria.

• It helps the pharmaceutical scientist to understand theoretical formulation and the target
processing parameters which ranges for each excipients & processing factors.

• In development projects, one generally experiments by a series of logical steps, carefully
controlling the variables & changing one at a time, until a satisfactory system is obtained


Optimization Parameters


Problem type Variables

Constrained Unconstrained Dependent Independent

Formulating Processing


Problem types


• In unconstrained optimization problems there are no restrictions.
• For a given pharmaceutical system one might wish to make the hardest tablet possible.
• The making of the hardest tablet is the unconstrained optimization problem.


• The constrained problem involved in it, is to make the hardest tablet possible, but it
must disintegrate in less than 15 minutes.




Independent variables/primary/input variables:

Formulation and process variables directly under the control of the formulator.
It includes level of a given ingredient, mixing time for a given process step.

Dependent variables/secondary/output variables:

Responses or the characteristics of the in-progress material or the resulting drug

delivery system.

➢These are a direct result of any change in the formulation or process.


Independent variable Dependent variable

• X1 diluent ratio • Y1 disintegration
• X2 compressional time

force • Y2 hardness
• X3 disintegrant level • Y3dissolution
• X4 binder level • Y4 friability
• X5 lubricant level • Y5 weight uniformity

• Y6 thickness
• Y7 porosity
• Y8 mean pore




• If greater the variables in a given system, then greater will be the
complicated job of optimization.

• But regardless of the number of variables, there will be relationship
between a given response and independent variables.

• Once we know this relationship for a given response, then will able to
define a response surface i.e.,


FACTOR: It is an assigned variable such as concentration ,Temperature etc..,
Quantitative: Numerical factor assigned to it

Ex; Concentration- 1%, 2%,3% etc..
Qualitative: Which are not numerical

Ex; Polymer grade, humidity condition etc

LEVELS: Levels of a factor are the values or designations assigned to the factor
Temperature 30oc
Concentration 1%, 2%

RESPONSE: It is an outcome of the experiment.
It is the effect to evaluate.
Ex: Disintegration time etc..,

EFFECT: It is the change in response caused by varying the levels
It gives the relationship between various factors & levels

INTERACTION: It gives the overall effect of two or more variables
Ex: Combined effect of lubricant and glidant on hardness of the tablet



➢ A blueprint of the procedure that enables the researcher to test his

hypothesis by reaching valid conclusions about relationships between

independent and dependent variables.

➢ It refers to the conceptual framework within which the experiment

is conducted.


Types of experimental designs:

❑Completely randomized designs
❑Randomized block designs
❑Factorial designs:

* Full
* Fractional

❑Response surface designs:
* Central composite designs
* Box-Behnken designs

❑Contour designs


❑Completely randomized Designs:

-These designs compares the values of a response variable based on different levels of

that primary factor.

-The levels of the primary factor are randomly assigned to the experimental designs.

-For example ,if there are 3 levels of the primary factor with each level to be run 2 times

then there are 6 factorial possible run sequences.

❑Randomized block designs:

-For this there is one factor or variable that is of primary interest.

-To control non-significant factors(nuisance factor), an important technique called

blocking can be used to reduce or eliminate the contribution of these factors to experimental


-General rule- block what is possible & randomise what is not possible.


❑Factorial Design(FD):

Factorial experiment is an experiment whose design consist of

two or more factor each with different possible values or “levels”.

Factorial design applied in optimization techniques.

Types of FD:


❖Full Factorial Design.

❖Fractional Factorial Design.


Factorial Design Testing:

In chromatographic condition responses can be

1. Efficiency
2. Retention factor
3. Asymmetry
4. Retention time
5. Resolution


Basics of factorial design:

▪ The experiments involve the study of the effects of two or more factors.

▪ By a factorial design, we mean that in each complete trial or replication of the

experiment all possible combination of the levels of the factors are investigated.

▪ When factors are arranged in a factorial design, they are often said to be crossed.

Software Used:

-Design expert 7.1.3 -Minitab
-SYSTAT sigma Stat 3.11 -Matrex
-CYTEL East 3.1 -Omega
-Compact 21-Apr-15 O



-It’s easier to study the combined effect of two or more factors simultaneously

and analyze their interrelationships.

-Has a wide range of factor combination are used.

-It saves time.


-Wasting of time and experimental material.

-Increase in factor size leads to increase in block size which increase the

chance of error.

-It’s complex when several factors are involved simultaneously.


Full Factorial Design:

-A design in which every setting of every factor appears with setting

of every other factor is full FD.

-Simplest design to create, but extremely inefficient.

-If there is x factor, each at y level, a full FD has yx.

Number of Runs(N)

N=y˟ Where, y= number of levels;

x= number of factors.

ex: 2³=8

3factors, 2 levels each


-It depends on INDEPENDENT VARIABLES for development of new

-It also depends LEVELS as well as CODING.
-Factors can be Quantitative (numerical number) or they are Qualitative.

Factorial design: 2², 2³, 3²,3³

2²FD=2 Factors, 2 levels=4 runs
2³FD=3 Factors, 2 levels=8 runs
3²FD=2 Factors, 3 levels=9 runs
3³FD=3 Factors, 3 levels=27 runs


Two Levels Full FD:

2 factors: X₁ and X₂
2 levels: Low and High
Coding: (-1), (+1)

Three Levels Full FD:

In three level FD,
3 factors: X₁, X₂ and X₃
3 levels are use,



Fractional Factorial Design:

-In full FD, as a number of factor or level increases , the number of

experiment required exceeds to unmanageable levels.

-In such cases, the number of experiment can be reduced systematically and

resulting design is called as fractional FD(FFD).

-Applied if number of factor are more than 5.

-Levels combinations are chosen to provide sufficient information to

determine the factor effect.


Types of fractional factorial designs:

❑Homogenous fractional

❑Mixed level fractional


❑Plackett – Burman


❑Latin square


❑Homogenous fractional:

▪Useful when large number of factors must be screened efficiently & all variables

have the same number of levels.

❑Mixed level fractional:

▪Useful when variety of factors needed to be evaluated for main effects and higher

level interactions can be assumed to be negligible.

▪Ex-objective is to generate a design for one variable, A, at 2 levels and another, X, at

three levels , mixed &evaluated.


▪Fractional designs with factors of more than two levels can be specified as

homogenous fractional or mixed level fractional



▪It is a popular class of screening design.

▪These designs are very efficient screening designs when only the main

effects are of interest.

▪These are useful for detecting large main effects economically, assuming all

interactions are negligible when compared with important main effects.

▪Used to investigate n-1 variables in n experiments, proposing experimental

designs for more than seven factors.



▪It is similar to PBDs.

▪It allows estimation of main effects while minimizing variance.

❑Latin square:

▪They are special case of fractional factorial design where there is one treatment factor

of interest and two or more blocking factors


❑Response surface designs

This model has quadratic form

γ =β0 + β1X1 + β2X2 +….β 2
11X1 + β22X2

-Designs for fitting these types of models are known as response surface designs.
If defects and yield are the outputs and the goal is to minimize defects and maximize

Two most common designs generally used in this response surface modeling are
▪Central composite designs
▪Box-Behnken designs


▪Box-Wilson Central Composite Design

-This type contains an embedded factorial or fractional factorial design with centre

points that is augmented with the group of ‘star points’.

-These always contain twice as many star points as there are factors in the design.

-The star points represent new extreme value (low & high) for each factor in the

-To picture central composite design, it must be imagined that there are several

factors that can vary between low and high values.


Central composite designs are of three types:

❑Circumscribed(CCC) designs – Cube points at the corners of the unit cube,

star points along the axes at or outside the cube and center point at origin.

❑Inscribed (CCI) designs – Star points take the value of +1 & -1 and cube

points lie in the interior of the cube.

❑Faced(CCF) designs – Star points on the faces of the cube.


Generation of a Central Composite Design for Factors


▪Box-Behnken designs:

-Box-Behnken designs use just three levels of each factor.

-In this design the treatment combinations are at the midpoints of edges of the process

space and at the center. These designs are rotatable (or near rotatable) and require 3 levels of

each factor.

-These designs for three factors with circled point appearing at the origin and possibly

repeated for several runs.

-It’s alternative to CCD.

-The design should be sufficient to fit a quadratic model , that justify equations based

on square term & products of factors.

Y=b + 2

1 b8X22+b9X3


A Box-Behnken Design


❑Contour designs:

-A Contour plot is a graphical representation of the relationships among three

numeric variables in two dimension.

-Two variables are for X and Y axes, and a third variable Z is for contour levels.

-A contour plot is a graphical technique for representing a 3 dimensional surface by

plotting constant z slice, called contours, on a 2D format. That is, given a value for z, lines

are drawn by connecting the (x, y) co-ordinates where that z value occurs.

-The contour plot is an alternative to a 3-D surface plot.


This contour plot shows that the surface 3-D representation of Contour plot
is symmetric and peaks in the centre.


The contour plots are formed by,

Vertical axis: Independent Variable 2

Horizontal axis: Independent Variable 1

Lines: Iso-response values.


Application in formulation:

➢Contour plots helps in visualizing the response surface.

➢Contour plots are useful for establishing desirable response values and operating


➢This plot shows how a response variable relates to two factors based on a model


➢Points that have the same response are connected to produce contour lines of

constant responses.

➢Such type of plots and experimental designs are used for used optimization

techniques in Pharmaceuticals formulation and processing.


The dex contour plot is a
specialized contour plot used in the
design of experiments. In particular,
it is useful for full and fractional


Other applications:

➢ Formulation and Processing

➢ Clinical Chemistry

➢ Medicinal Chemistry

➢ High Performance Liquid Chromatographic Analysis

➢ Formulation of Culture Medium in Virological Studies

➢ Study of Pharmacokinetic Parameters


Optimization Technology:

❑Classical optimization
❑Statistical optimization
❑Applied optimization



•Classical optimization is done by using the calculus to basic problem to find the
maximum and the minimum of a function.
•The curve represents the relationship between the response Y and the single independent
variable X and we can obtain the maximum and the minimum. By using the calculus the
graphical represented can be avoided. If the relationship, the equation for Y as a function of
X is,

Y = f(X)


• When the relationship for the response Y is given as the function of two independent
variables, X1 and X2 ,

Y = f(X1, X2)
•Graphically, there are contour plots on which the axes represents the two independent
variables, X1 and X2, and contours represents the response Y.


Limited applications
• Problems that are too complex.
• They do not involve more than two variables.


Techniques used divided into two types:
▪ Experimentation continues as optimization proceeds.

-It is represented by evolutionary operations(EVOP), sequential
simplex methods.

▪ Experimentation is completed before optimization takes place.
-It is represented by classic mathematical & search methods.

In later one it is necessary that the relation between any dependent variable
and one or more independent variable is known.


❖There are two possible approaches for this
Theoretical approach- If theoretical equation is known, no experimentation
is necessary.
Empirical or experimental approach – With single or more independent
variables, formulator experiments at several levels.


-Applicable only to the problems that are not too complex.
-They do not involve more than two variables.
-For more than two variables graphical representation is impossible


The effect on a real system of changing some input(some variable) is
observed directly at the output(one measures some property).

Considering the changes in input and effect on output, the
optimization techniques are categorized into five types:

1. Evolutionary operations
2. Simplex method
3. Lagrangian method
4. Search method
5. Canonical analysis


1.Evolutionary operations:

❑ It is the one of the most widely used methods of experimental optimization in fields
other than pharmaceutical technology is the evolutionary operation(EVOP)

❑ The basic idea is that the production procedure(formulation and process) is allowed to
evolve to the optimum by careful planning and constant repetition.

Where we have to select this technique?

➢ This technique is especially well suited to a production situation

➢ Applied mostly to tablets.



This process is run in a such a way that
A. It produces a product that meets all specifications.
B. Simultaneously, it generates information on product improvement.

❖ Experimenter makes a very small change in the formulation or process but makes it so
many times i.e., repeats the experiment so many times.

❖ Then he or she can be able to determine statistically whether the product has improved.
❖ And the experimenter makes further any other change in the same direction, many

times and notes the results.
❖ This continues until further changes do not improve the product or perhaps become



1. It was applied to tablets by Rubinstein.
2. It has also been applied to an inspection system for parenteral products

•Generates information on product development.
•Predict the direction of improvement.
•Help formulator to decide optimum conditions for the formulation and process.

1. It is impractical and expensive to use.
2. It is not a substitute for good laboratory scale investigation.
3.Time consuming.


2.Simplex method:

❑ It is most widely applied technique.
❑ It was proposed by Spendley et.al.
❑ This technique has even wider appeal in areas other than formulation and processing.
❑ A good example to explain its principle is the application to the development of an
analytical method i.e., a continuous flow analyser, it was predicted by Deming and king.
❑ Simplex method is a geometric figure that has one or more point than the number of
❑ If two factors or any independent variables are there, then simplex is represented
❑ Once the shape of a simplex has been determined, the method can employ a simplex of
fixed size or of variable sizes that are determined by comparing the magnitude of the
responses after each successive calculation.



❖ The two axes in the figure are nothing but two independent variables show the pump
speeds for the two reagents required in the analysis reaction.
❖ The initial simplex is represented by the lowest triangle.
❖ The vertices represent the spectrophotometric response.
❖ The strategy is moves towards a better response.
❖ The worst response is 0.25, conditions are selected at the vertex, 0.6 and indeed
improvement is obtained.
❖ Then the experiment path is followed to obtain optimum,0.721.


The simplex method is especially appropriate when:

• Process performance is changing over time.

• More than three control variables are to be changed.

• The process requires a fresh optimization with each new lot of material.

• The simplex method is based on an initial design of k+1, where k is the number of

variables. A k+1 convex geometric figure in a k-dimensional space is called a simplex.

The corners of this figure are called vertices.

It is of two types:
A. Basic Simplex Method
B. Modified Simplex Method.


A. Basic Simplex Method
• It is easy to understand and apply. Optimization begins with the initial trials.

• Number of initial trials is equal to the number of control variables plus one.

• These initial trials form the first simplex. The shapes of the simplex in a one, a two and a

three variable search space, are a line, a triangle or a tetrahedron respectively.


Rules for basic simplex:

➢ The first rule is to reject the trial with the least favorable value in the current


➢ The second rule is never return to control variable levels that have just been



B. Modified Simplex Method
• It was introduced by Nelder-Mead in 1965.

• It can adjust its shape and size depending on the response in each step. This method is

also called the variable-size simplex method.

Rules for modified simplex:
1.Contract if a move was taken in a direction of less favorable conditions.
2.Expand in a direction of more favorable conditions.

• When a set of interrelated variables is to be optimized, a systematic approach must be

used in which all variables are adjusted concurrently to obtain the maximum signal.

One such approach is called modified simplex method.



-The two independent variables(the axes) show the pump speeds for the
two reagents required in the analysis reaction is taken.

-The initial simplex is represented by the lowest triangle; the vertices
represent the spectrophotometric response.

-The strategy is to move toward a better response by moving away from the
worst response. Since the worst response is 0.25, conditions are selected at the
vortex, 0.6, and, indeed, improvement is obtained.

-One can follow the experimental path to the optimum, 0.721.


Graph representing the simplex movements to the optimum conditions


Applications of method:
1. This method was used by Shek.et.al. to search for an capsule formula.
2. This was applied to study the solubility problem involving butaconazolenitrate in a
multicomponent system.
3. Bindschaeder and Gurny published an adaptation of the simplex technique to a TI-59
calculator and applied successfully to a direct compression tablet of acetaminophen.
4. Janeczeck applied the approach to a liquid system i.e., a pharmaceutical solution and
was able to optimize physical stability.


•This method will find the true optimum of a response with fewer trials than the

non-systematic approaches or the one-variable-at-a-time method.


•There are sets of rules for the selection of the sequential vertices in the


•Require mathematical knowledge.


3. The Lagrangian method:

❑ This optimization method represents the mathematical technique, is an extension of
the classic method.
❑ Fonner.et.al gave ideas of understanding this technique by applying it to a tablet
formulation and by considering two independent variables.

Where we have to select this technique?

This technique can applied to a pharmaceutical formulation and processing.


Lagrangian method was able to handle several responses or dependent variables.


Although the Lagrangian method was able to handle several responses or

dependent variables, it was generally limited to two independent variables.


The several steps in the Lagrangian method can be written as follows:

Step-1 : Determine objective function.
Step-2 : Determine constraints.
Step-3 : Change inequality constraints to equality constraints.
Step-4 : Form the Lagrange function, F

a. One Lagrange multiplier ⅄ for each constraint.
b. One slack variable q for each inequality constraint.

Step-5 : Partially differentiate the Lagrange function for each variable and set derivatives
equal to zero.
Step-6 : Solve the set of simultaneous equations.
Step-7 : Substitute the resulting values into the objective functions.


This can be phased into four phases, this was given by Buck et.al

The phases are:

a. A preliminary planning phase
b. An experimental phase
c. An analytical phase
d. A verification phase



▪ Optimization of a tablet.
▪ Phenyl propranolol(active ingredient)-kept constant.
▪ X1 –disintegrate (corn starch).
▪ X2 –lubricant (stearic acid).
▪ X1 & X2 are independent variables.
▪ Dependent variables include tablet hardness, friability ,volume, invitro release rate

▪ It is full 32 factorial experimental design.
▪ Nine formulation were prepared.


Explanation :

❑ The active ingredient, phenylpropanolamine HCl , was kept at a constant level, and the
levels of disintegrant (corn starch) and lubricant (stearic acid) were selected as the
independent variables x1 and x2 .

❑ The dependent variables include tablet hardness, friability,volume, in vitro release rate
and urinary excretion rate in human subjects.

❑ This techniques requires that the experimentation to be completed before optimization so
that mathematical models can be generated.


The experimental data for the taken example :

Tablet formulation:


❑ The analysis is performed on a polynomial form

Y=B0+B₁X₁+B₂X₂+B₃X₁² +B4 X₂² +B5X1X2+B6X

1X +B 2 2
2 7X1 X2+B 2

8X1 X2

And these terms were retained and eliminated according to stand stepwise regression
In above equation ,
y = Any given response
Bi = The regression co-efficient for the various terms containing levels of the independent

❑ Each response have its own equation.
❑ A graphic technique may be obtained from the polynomial equations.


Figure a.
It shows the contours for tablet hardness as the levels of the independent
variables are changed.


Figure b,
It shows similar contour for the dissolution response t50%


Figure c.
The above graph shows the feasible space with the limits of
hardness be 8-10 kg and t50 % be 20-33 min .
This figure is obtained by superimposing a and b.


▪ In the mathematical technique slightly different constraints are

▪ In this example.

The constrained optimization problem is to locate levels of stearic
acid (x1) and starch (x2)

That minimizes the time of invitro release (y2)

Such that the average tablet volume (y4) did not exceed 9.422cm2 and
the average friability (y3) did not exceed 2.72%.


To apply the Lagrangian method, this problem must be expressed mathematically as


y2 = F2 (x1 , x2) ……………..(1)
such that
y3 = F3 (x1 , x2) ≤ 2.72 ………………(2)
y4 = F4 (x1 , x2) ≤ 0.422 ………………(3)
And 5 ≤ x1 ≤ 45 ………….. (4)
1 ≤ x2 ≤ 41 ………….. (5)
Equation (4) and (5) are the solution within the experimental range.


❑ The foregoing inequality constraints must be converted to equality constraints
before the operation begins and this is done by introducing a slack variable ‘q’ for each.
❑ The several equations are then combined into a Lagrange function F and this
necessitates the introduction of a Lagrange multiplier ⅄ for each constraint.
❑ The partial differentiation of Lagrange function and solving the resulting set of six
simultaneous equations, value are obtained for x1 and x2 to yield an optimum invitro time
of 17.9mm (t50 %).
❑ The solution to a constrained optimization program may depend heavily on the
constraints applied to the secondary objectives.

❑ A technique called “sensitive analysis” can provide information so that the
formulator can further trade off one property for another.
❑ For sensitivity analysis the formulator solves the constrained optimization problem
for systemic changes in the secondary objectives.


4.Search methods:

❑ It is defined by appropriate equations. It does not require continuity or differentiability of
function. It is applied to pharmaceutical system.
For optimization 2 major steps are used
• Feasibility search-used to locate set of response constraints that are just at the limit of
• Grid search – experimental range is divided in to grid of specific size & methodically

Advantages of search method:

It takes five independent variables in to account.
Persons unfamiliar with mathematics of optimization & with no previous computer

experience could carry out an optimization study.


Steps involved in search method

Select a system
Select variables
Perform experiments and test product
Submit data for statistical and regression analysis
Set specifications for feasibility program
Select constraints for grid search
Evaluate grid search printout



• Five independent variables dictates total of 32 experiments . This design is known as
five-factor, orthogonal, central ,composite, second order design.
• First 16 formulations represent a half-factorial design for five factors at two levels.
• The two levels represented by +1 & -1, analogous to high & low values in any two
level factorials.

Translation of statistical design in to physical units



Then the data is subjected to statistical analysis followed by multiple regression

The equation used in this design is second order polynomial.

y = a0+a1x1+…+a5x5+a11x1^2+…+a55x5^2+a12x1x2+a13x1x3+a45x4x5

A multivariate statistical technique called principle component analysis (PCA) is
used to select the best formulation.

PCA utilizes variance-covariance matrix for the responses involved to determine
their interrelationship.