Optimisation technique

Presented by
(Dr.) Kahnu Charan Panigrahi
Asst. Professor (PHARMACEUTICS)

Contents
Definition
Introduction
Optimization parameters
Problem type
Variables
Applied optimization method
Response surface method
Factorial Design

Definition
• It is defined as follows: choosing the best element from some set
of available alternatives.
• Optimisation techniques provide both depth of understanding and
ability to explore ranges from formulation and processing factor.
• An art, process, or methodology of making something (a design,
system, or decision) as perfect, as functional, as effective as
possible.

Introduction
In development projects pharmacist generally experiments by a
series of logical steps, carefully controlling the variables and
changing one at a time until satisfactory results are obtained.
This is how the optimization done in pharmaceutical industry.
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. Less no of experiment
required
Final product not only meets the requirements from the bio-
availability but also from the practical mass production criteria.

Optimisation
parameters
Problem
type
Constrained Unconstrained
variable
Dependent Independent
Optimisation Parameters

PROBLEM TYPES
Unconstrained
• 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.
Constrained
• In constrained optimization problems there are restrictions.
• The constrained problem involved in it, is to make the hardest tablet
possible, but it must disintegrate in less than 15 minutes.

Variables
Independent variables : The independent variables are the
formulation and process variable under the control of the formulator.
These might include the compression force or the die cavity filling or the
mixing time.
Dependent variables : The dependent variables are the responses or the
characteristics of the in-process material or the resulting drug delivery
system. These are directly result of any changes in the formulation and
process. The more the variables that are present in the system the more the
complications that are involved in the optimization.

• Once the relationship between the
variable and the response is known, it
gives the response surface as
represented in the Fig. 1.
• Surface is to be evaluated to get the
independent variables, X1 and X2,
which gave the response, Y.

ClassicalOptimization
•Classical optimization is done by using the calculus to
basic problem to find the maximum and the minimum of a
function.
•The curve in the fig 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.
•The first derivative set it to zero and solve for X to obtain the
maximum and the minimum.
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 (Fig. 3.) on which the axes
represents the two independent variables, X1 and X2, and contours
represents the response Y.
The multiple variables design use partial derivative, matrices
determinants etc.
Figure 3. Contour plot. Contour represents values of the
dependent variable Y

STATISTICAL DESIGN:
• This technique may be divided into two categories:
 Simplex method : Experimentation continues as the optimization
proceeds
 Search method: Experimentation completed before optimisation
takes place
• For the second type the relation between any dependent variable
and one or more independent variable be known.
• It can be known by either theoretical or empirical approaches.

Applied optimizationmethods
• Evolutionary operation
• Simplex method
• Lagrangian method
• Search method
• Canonical analysis
Flowchartforoptimization

Evolutionary operations
• It is a method of experimental optimization.The production procedure
(formulation and process) is allowed to evolve to the optimum by
careful planning and constant repetition.
• Small changes in the formulation or process are made (i.e. repeats the
experiment so many times) & statistically analyzed whether it is
improved.
• It continues until no further changes takes place i.e., it has reached
optimum-the peak. The result of changes are statistically analyzed.

Example
Tablet
Hardness
By changing the
concentration
of binder
How can we get hardness
Response
In this example, A formulator can changes the concentration of binder
and get the desired hardness.

Simplexmethod
than
It is an experimental method applied for pharmaceutical systems
Technique has wider appeal in analytical method other
formulation and processing
Simplex is a geometric figure that has one more point than the
number of factors. It is represented by triangle.
It is determined by comparing the magnitude of
the responses after each successive calculation.
When complete knowledge of response is not
available the simplex method is most appropriate
type.

Typesofsimplexmethod
Simplex
method
Basic simplex
method
Modified simplex
method

Basic simplexmethod
• 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.
• The first rule is to reject the trial with the least favourable value in
the current simplex. The second rule is never to return to control
variable levels that have just been rejected.

Modified simplex method
1. Contract if a move was taken in a direction of less favorable conditions.
2. Expand in a direction of more favorable conditions.
Example
The two independent variable show the pump speeds for the
two reagents required in the analysis reaction is taken.
 It can adjust its shape and size depending on the response in each
step. This method is also called the variable-size simplex method.

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 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.

Lagrangianmethod
• It represents mathematical techniques which is an extension of classic
method. It is applied to a pharmaceutical formulation and processing
problem.
• This technique require that the experimentation be completed before
optimization so that the mathematical models can be generates.
Stepsinvolved:
1. Determine the objective function.
2. Determine the constraints.
3. Change inequality constraints to equality constraints.
4. Form the Lagrange function F.
5. Partially differentiate the Lagrange function for each variable and set
derivatives equal to zero
6. Solve the set of simultaneous equations.
7. Substitute the resulting values into objective function.

Example
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, in vitro
release rate etc..,
• It is full 32 factorial experimental design. Nine formulations were prepared.
• Polynomial model relating the response variable to the independent
variable were generated by regression analysis programme.
• In this equation y represents given response and Bi represents the
regression coefficient for the various terms containing level of the
independent variable.

Formulation
no
Drug
(phenylpropan
olamine)
Dicalcium
phosphate
Starch Stearic acid
1 50 326 4(1%) 20(5%)
2 50 246 84(21%) 20
3 50 166 164(41%) 20
4 50 246 4 100(25%)
5 50 166 84 100
6 50 86 164 100
7 50 166 4 180(45%)
8 50 86 84 180
9 50 6 164 180

CONTOUR PLOT FOR TABLET HARDNESS
A
CONTOUR PLOT FOR Tablet dissolution(T50%)
B

•If the requirements on the final tablet are that hardness be 8-10 kg and t50%
be 20-33 min, the feasible solution space is indicated in above fig
• This has been obtained by superimposing Fig. A and B, and several
different combinations of X1 and X2 willsuffice.
Feasible solution space indicated
by crosshatched area

• Constrained optimization problem is to locate the levels of stearic acid(x1)
and starch (x2) to minimizes the time of in vitro release (y2), such that
average tablet volume (y4) did not exceed 9.422 cm2, average friability
(y3) did not exceed 2.72%.
• To apply the Lagrangian method, the problem must be expressed
mathematically as follows.
Y2 = f2 (X1, X2) -in vitro release
Y3 = f3(X1,X2)<2.72%-Friability
Y4 = f4 (x1, x2) <0.422 cm3 average tablet volume
Experimental range: 5< X1 < 45 , 1 < X2 < 41
• The several equation are combined into a Lagrange function F and
introducing Lagrange multiplier ƛ for each constraint.

The plots of the independent
variables, X1 and X2, can be
obtained as shown in fig. Thus
the formulator is provided
with the solution (the
formulation) as he changed
the friability restriction.
Four phases are
 Preliminary planning phase
 An experimental phase
 Analytical phase
 Verification phase
Optimizing values of stearic acid and strach
as a function of restrictions on tablet
friability: (A) percent starch; (B) percent
stearic acid

SearchMethod
 It is defined by appropriate equations and do not require
continuity or differentiability of function.
 The response surface is searched by various methods to
find the combination of independent variables yielding
an optimum.
 It can handle more than two independent variables into
account and is can be computer-aassisted.

Example
Independent Variables
X1 = Diluents ratio
X2= Compression force
X3= Disintegrate levels
X4= Binder levels
X5 = Lubricant levels
Dependent Variables
Y1 = Disintegration time
Y2= Hardness
Y3 = Dissolution
Y4 = Friability
Y5 = weight uniformity
Y6 = thickness
Y7 = porosity
Y8 = mean pore diameter
Different dependent & independent variables or
formulation factors selected for this study

• Five independent variables
dictates total of 27 experiments.
This design is known as five
factor, orthogonal, central,
composite, second order design.
• The first 16 trials are represented
by +1 and -1, analogous to the
high and low values.
• The remaining trials are
represented by a -1.547, zero or
1.547.
• The data were subjected
statistical analysis, followed by
multiple regression analysis.
The experimental design used was a modified factorial and is
shown in Table

• Again the formulations were prepared and the responses measured. The
data were subject to statistical analysis, followed by multiple regression
analysis.
• The equation used in this design is second order polynomial.
y = 1a0+a1x1+…+a5x5+a11x1
2+…+a55x2
5+a12x1x2+a13x1x3+a45 x4x5
.
• Where Y is the level of a given response, aij the regression coefficients for
second-order polynomial, and Xi the level of the independent variable.
• The full equation has 21 terms, and one such equation is generated for
each response variable .
• The usefulness of this equation is evaluated by the R2 value.or the index of
determination.

The translation of the statistical design into physical units is shown in
table.
Factor -1.54eu -1 eu Base 0 +1 eu +1.547eu
X1= ca.phos/lactose
24.5/55.5 30/50 40/40 50/30 55.5/24.5
X2= compression
pressure( 0.5 ton)
0.25 0.5 1 1.5 1.75
X3 = corn starch
disintegrant(1mg)
2.5 3 4 5 5.5
X4 = Granulating
gelatin(0.5mg)
0.2 0.5 1 1.5 1.8
X5 = mg.stearate
(0.5mg)
0.2 0.5 1 1.5 1.8

Fortheoptimizationitself,twomajorstepswereused:
The feasibility search
The grid search
1. The feasibility search :
The feasibility program is used to locate a set of response constraints
that are just at the limit of possibility. For example, the constraints in
table were fed into the computer and were relaxed one at a time
until a solution was found.

• The first possible solution was found at disintegration time 5 min,
hardness 5 kg and dissolution 100% at 50 min.
• This program is designed so that it stops after the first possibility, it
is not a full search. The formulation obtained may be one of many
possibilities satisfying the constraints.
2. The grid search or exhaustive grid search :
• It is essentially a brute force method in which the experimental range
is divided into a grid of specific size and methodically searched.
• From an input of the desired criteria, the program prints out all
points (formulations) that satisfy the constraints.
• Thus the best formulation is selected from the grid search printout to
complete the optimization.

• The multivariate stastical technique called principal component
analysis(PCA) can effectively used to identify the response to be
constrained.
• PCA utilises variance-covariance matrix for the response involved
to determine the interrelationbship
• Contour plot are also generated and specific responses is noted
on the graph.

Stepsinvolvedinsearchmethod
1. Select a system
2. Select variables
a. Independent
b. Dependent
3. Perform experiments and test product.
4. Submit data for statistical and regression analysis.
5. Set specifications for feasibility program.
6. Select constraints for grid search.
7. Evaluate grid search printout.
8. Request and evaluate.
a. “Partial derivative” plots, single or composite.
b. Contour plots.

Canonicalanalysis
• Canonical analysis, or canonical reduction, is a technique used to
reduce a second-order regression equation, to an equation consisting
of a constant and squared terms, as follows:
Y = Y0+λ1W1
2+λ2W2
2+…….
• The technique allows immediate interpretation of the regression
equation by including the linear and interaction terms in the constant
term which simplify the canonical equation.
• In canonical analysis or canonical reduction, second-order regression
equations are reduced to a simpler form by a rigid rotation and
translation of the response surface axes in multidimensional space,
as shown in fig for a two dimension system.

OtherApplications
• Formulation and processing
• Clinical chemistry Medicinal chemistry
• High performance liquid chromatographic analysis
• Formulation of culture medium in virological studies
• Study of pharmacokinetic parameters

•In statistics, response surface methodology (RSM) explores the
relationships between several explanatory variables and one or
more response variables.
•The method was introduced by George E. P. Box and K. B. Wilson
in 1951. The main idea of RSM is to use a sequence of designed
experiments to obtain an optimal response.
•Box and Wilson suggest using a second-degree polynomial model
to do this. They acknowledge that this model is only an
approximation, but they use it because such a model is easy to
estimate and apply, even when little is known about the process.
•Statistical approaches such as RSM can be employed to maximize
the production of a special substance by optimization of operational
factors. Of late, for formulation optimization, the RSM, using
proper design of experiments (DoE), has become extensively used.
Response surface methodology (RSM)

Important RSM properties and features
•ORTHOGONALITY: The property that allows individual effects of the
k-factors to be estimated independently without (or with minimal)
confounding. Also orthogonality provides minimum variance
estimates of the model coefficient so that they are uncorrelated.
•ROTATABILITY: The property of rotating points of the design about
the center of the factor space. The moments of the distribution of
the design points are constant.
•UNIFORMITY: A third property of CCD designs used to control the
number of center points is uniform precision (or Uniformity).

Basic approach of response surface methodology
•An easy way to estimate a first-degree polynomial model is to use
a factorial experiment or a fractional factorial design. This is
sufficient to determine which explanatory variables affect the
response variable(s) of interest.
•Once it is suspected that only significant explanatory variables are
left, then a more complicated design, such as a central composite
design or Box Behenken can be implemented to estimate a second-
degree polynomial model, which is still only an approximation at
best.

Central composite design
• Central composite design is an experimental
design, useful in response surface methodology,
for building a second order (quadratic) model for
the response variable without needing to use a
complete three-level factorial experiment.
• After the designed experiment is
performed, linear regression is used, sometimes
iteratively, to obtain results.

The design consists of three distinct sets of experimental runs:
•A factorial (perhaps fractional) design in the factors studied, each
having two levels; The matrix F obtained from the factorial
experiment. The factor levels are scaled so that its entries are coded
as +1 and −1.
•A set of centre points, experimental runs whose values of each
factor are the medians of the values used in the factorial portion.
This point is often replicated in order to improve the precision of the
experiment; The matrix C from the centre points, denoted in coded
variables as (0,0,0,...,0), where there are k zeros.
•A set of axial points, experimental runs identical to the centre points
except for one factor, which will take on values both below and
above the median of the two factorial levels, and typically both
outside their range. All factors are varied in this way. A matrix E from
the axial points, with 2k rows. Each factor is sequentially placed at ±α
and all other factors are at zero.

•Box–Behnken designs are experimental designs for response surface
methodology, devised by George E. P. Box and Donald Behnken in
1960, to achieve the following goals:
•Each factor, or independent variable, is placed at one of three
equally spaced values, usually coded as −1, 0, +1. (At least three
levels are needed for the following goal.
•The design should be sufficient to fit a quadratic model, that is, one
containing squared terms, products of two factors, linear terms and
an intercept.
•The ratio of the number of experimental points to the number of
coefficients in the quadratic model should be reasonable (in fact,
their designs kept in the range of 1.5 to 2.6).
•The estimation variance should more or less depend only on the
distance from the centre and should not vary too much inside the
smallest (hyper)cube containing the experimental points.
Box–Behnken designs

Factorial design
• Factorial design is a type of research
methodology that allows for the investigation of
the main and interaction effects between two or
more independent variables and on one or more
outcome variable(s).
• Factorial experiment is an experiment whose
design consist of two or more factor each with
different possible values or “levels”.
• FD technique introduced by “Fisher” in 1926 and
applied in optimization techniques.

• Factors can be “Quantitative” (numerical
number) or they are qualitative.
• Factorial design depends on independent
variables for development of new formulation
• Factorial design also depends on Levels as well
as Coding.
• Factorial Design can be either:
a. Full Factorial design
b. Fractional Factorial design

• A design in which every setting of every factor appears with
setting of every other factor is full factorial design.
• If there is k factor , each at Z level , a Full FD has ZK
EXAMPLE
• 2 2 FD = 2 Factors , 2 Levels = 4 runs
• 2 3 FD = 3 Factors , 2 Levels = 8 runs
Full Factorial design

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 experiments can be
reduced systemically and resulting design is called as
Fractional factorial design (FFD).
• Applied if no. of factor are more than 5
• Suppose a situation occurs in which three factors, each
at two levels are of interest, but the experimenter does
not want to run all eight treatment combinations
(23=8). A design with four treatment combinations can
then be performed when considering the one-half
fraction of the 23 design (23−1=4).

Gilbert S. Banker, “Modern Pharmaceutics”,4th
edi,vol.121,Marcel & Dekker publications, p.g.-
607-625.

Optimisation technique

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