2. 2
I.INTRODUCTION
OPTIMIZATION
To make as perfect ,effective or functional as possible.
choosing the best element from some set of available
alternatives.
In Pharmacy word “optimization” is mainly applied to the
formulation and processing field.
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II. OPTIMIZATION PARAMETERS
There are two optimization parameters
1.Problem Types
2.Variables
• PROBLEM TYPES - two general types of optimization
problems:
1. Unconstrained
2. Constrained
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In unconstrained optimization problems there are no restrictions.
For a given pharmaceutical system one might wish to make the
hardest tablet possible.
This 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.
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• VARIABLES –
Two types
1.Independent variables
2.Dependent variables
The independent variables are under the control of the formulator.
These might include the compression force or the die cavity
filling or the mixing time.
The dependent variables are the responses or the characteristics
that are developed due to the independent variables.
The more the variables that are present in the system the more the
complications that are involved in the optimization.
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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.
Any number of variables can be
considered, it is impossible to
represent graphically, but
mathematically it can be evaluated.
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III. CLASSICAL OPTIMIZATION
•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. 2. 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 data or equations are plotted
•the equation for Y as a function of X, is available [Eq. (1)]:
Y = f(X)
Figure 2. Graphic location of optimum (maximum or minimum)
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• 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.
Figure 3. Contour plot. Contour represents values of
the dependent variable Y
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STATISTICAL DESIGN
Forms of Optimization techniques:
There are three forms of systematic optimization techniques:
1. Sequential Optimization techniques.
2. Simultaneous Optimization techniques.
3. Combination of both .
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Sequential Methods:
This method is also referred to as the “Hill climbing method”.
As first of all a small number of experiments are done
further research will be done by using the increase or decrease
of response.
In this way a maximum or minimum will be reached i.e. an
optimum solution
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Simultaneous Methods:
This method involves the use of full range of
experiments by an experimental design .
Results are then used to fit in the mathematical model.
And maximum or minimum response will then be
found through this fitted model.
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Relationship between dependent and independent variable
should be known.
One variable – simple regression analysis or by least square
method
More than one - statistical design ,multiple regression analysis.
Widely used is – factorial design.
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EVOLUTIONARY OPERATIONS (EVOP)
• One of the most widely used methods – not in pharmaceutical
technology
• This technique is especially well suited to a production situation.
• The basic philosophy is that the production procedure
(formulation and process) - optimum by careful planning and
constant repetition.
• The process is run in a way such that it both produces a
product - meets all specifications and (at the same time) generates
information on product improvement.
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(EVOP)
Experimenter makes small changes-many times and
statistically determines the improvement
But in pharm.system – insufficient latitude in formula
Subjected to regulatory constraints since validated product
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• The simplex approach to the optimum is also an experimental
method and has been applied more widely to pharmaceutical
systems.
• A simplex is a geometric figure that has one more point than the
number of factors.
• for two factors or independent variables, the simplex is represented
by a triangle.
THE SIMPLEX METHOD
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• 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 magnitudes of the responses after
each successive calculation.
•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.
Used in the development of an analytical method
- a continuous flow analyzer
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the worst response is
0.25,conditions are
selected at the vertex, 0.6,
and, indeed, improvement
is obtained.
One can follow the
experimental path to the
optimum, 0.721.
Figure 5 The simplex approach to optimization. Response is
spectorphotometric reading at a given wavelength.
The simplex algorithm begins at a starting vertex and moves
along the edges of the polytope until it reaches the vertex of
the optimum solution.
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The several steps in the Lagrangian method can be summarized as follows:
1. Determine objective function
2 .Determine constraints
3. Change inequality constraints to equality constraints.
4. Form the Lagrange function, F:
a. One Lagrange multiplier λ for each constraint
b. One slack variable q for each inequality constraint
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 the objective functions.
THE LAGRANGIAN METHOD
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The red line shows the constraint. The blue lines are contours of f(x,y).
The point where the red line tangentially touches a blue contour is our
solution.
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•The experimentation be completed before optimization - mathematical
models can be generated.
•Tablet formulation – phenylpropanolamine - constant
•Independent variable – disintegrant and lubricant(x1, x2)
•Experimental design – full 32 factorial
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• Polynomial models relating the response variables - generated by
backward stepwise regression analysis program.
•The analyses were performed on polynomial and the terms were
retained or eliminated according to standard stepwise regression
techniques.
y = B0+B1X1+B2X2+B3X1
2+B4X2
2+B5X1X2 +B6X1X2
2+B7X1
2X2+B8X1
2X2
2
y - any given response
Bi - regression coefficient for the various terms containing levels of
the independent variable.
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The dependent variables - tablet hardness, friability, volume, in
vitro release rate, and urinary excretion rate in human subject.
25
Figure 6. Contour plots for the Lagrangian method:
(a) tablet hardness;
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Figure 6. Contour plots for the Lagrangian method:
c) feasible solution space indicated by crosshatched area
•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 Fig. 6c.
•This has been obtained by superimposing Fig. 6a and b, and
several different combinations of X1 and X2 will suffice.
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Buck et al incorporated statistical design into above product design
It consist of following strategies
a. A preliminary planning phase
b. An experimental phase
c. An analytical phase
d. A verification phase
They included case studies of a tablet design and suspension
design to illustrate
• Efficient and effective procedures that might be applied
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Lagrangian method - handle several responses or dependent
variable, it limited to two independent variables.
Response surface explained by eqs. are searched by various methods
to find out the optimum combination of dependent variables.
It takes five independent variables into account and is computer-
assisted.
persons unfamiliar with the mathematics of optimization and with no
previous computer experience - carry out an optimization study.
THE SEARCH METHOD
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o The system selected here was also a tablet formulation .
The five independent variables or formulation factors selected for this
study are shown in Table 2.
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• The experimental design used was a modified factorial.
•five independent variable dictates that a total of 27 experiments or
formulations be prepared.
•This design is known as a five-factor, orthogonal, central, composite,
second-order design .
•The first 16 formulations represent a half-factorial design for five factors
at two levels, resulting in 25 − 2 design = 16 trials.
•The two levels are represented by +1 and -1, analogous to the high and
low values in any two level factorial design.
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For the remaining trials, three levels were selected: zero represents a
base level midway between the aforementioned levels, and the levels
noted as 1.547 represent extreme (or axial) values.
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• The translation of the statistical design into physical units is shown
in Table 5.
• Again the formulations were prepared and the responses measured.
The data were subject to statistical analysis, followed by multiple
regression analysis.
•This is an important step. One is not looking for the best of the 27
formulations, but the “global best.”
35. The type of predictor equation used with this type of design is
a second-order polynomial of the following form:
Y = a 0+a1X1+…..+a5X5+a11X12+…+a55X52+a12X1X2+a13X1X3+…+a45X4X5
Y = level of a given response,
X1 = level of the independent variable.
The full equation has 21 terms, and one such equation is generated
for each response variable .
The usefulness of equation is verified by R2 value
All pharma. responses will not fit 2nd order regression method
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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
THE SEARCH METHODS
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For the optimization itself, two major steps were used:
1. The feasibility search
2. The grid 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 6 were fed into the
computer and were relaxed one at a time until a solution was
found.
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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.
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•The grid search or exhaustive grid search, 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.
• From a set of feasible constraints – educated guess based on
experience with the system.
•A multivariate statistical technique called principal component
analysis (PCA) can be effectively used .
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•The output includes plots of a given responses as a function of a single variable
(fig.11).
The abscissa for both types is produced in experimental units, rather than physical
units, so that it extends from -1.547 to + 1.547.
Graphic approaches are also available and graphic output is provided by a plotter
from computer tapes.
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The output includes plots of a given responses as a function of all
five variable (Fig 12).
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Contour plots (Fig.13) are also generated in the same manner. The specific response is
noted on the graph, and again, the three fixed variables must be held at some desired
level. For the contour plots shown, both axes are in experimental unit (eu) .
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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:
CANONICAL ANALYSIS
Y = Y0+λ1W1
2+λ2W2
2+…….
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In canonical analysis or canonical
reduction, second-order regression
equations are reduced to a simpler
form by
• rigid rotation and translation
of the response surface axes in
multidimensional space, as shown in
Fig.14 for a two dimension system.
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Formulation and Processing
Clinical Chemistry
Medicinal Chemistry
High Performance Liquid Chromatographic
Analysis
Formulation of Culture Medium in Virological
Studies.
Study of Pharmacokinetic Parameters.
VI. OTHER APPLICATIONS
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IX. REFERENCES
1. Webster's Marriam Dictionary, G & C Marriam.
2. L. Cooper and N. Steinberg, Introduction to Methods of Optimization, W.B.
Sunder.
3. O.L.Davis , The Design and Analysis of the Indusrial Experimentation,
Macmillan.
4. Gilbert S. Banker, Modern Pharmaceutics, Marcel Dekker Inc.
5. http://en.wikipedia.org/wiki/Optimization_(mathematics)
6. http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4/node6.html
