OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING Presented by Muhamed Haris V A M.Pharm(2011-13) Calicut medical college Kerala 1

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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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3 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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4 f

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5 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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6 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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8 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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9 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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10 When the relationship for the response Y is given as the function of two independent variables , X 1 and X 2 , Y = f (X 1, X 2 ) Graphically , there are contour plots (Fig. 3.) on which the axes represents the two independent variables, X 1 and X 2 , and contours represents the response Y. Figure 3. Contour plot. Contour represents values of the dependent variable Y

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11 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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12 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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13 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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14 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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15 Applied optimization methods Evolutionary Operations The Simplex Method The Lagrangian Method Search Method Canonical Analysis

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16 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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17 (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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18 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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19 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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20 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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21 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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22 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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23 The experimentation be completed before optimization - mathematical models can be generated. Tablet formulation – phenylpropanolamine - constant Independent variable – disintegrant and lubricant(x 1 , x 2 ) Experimental design – full 3 2 factorial

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24 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 = B 0 +B 1 X 1 +B 2 X 2 +B 3 X 1 2 +B 4 X 2 2 +B 5 X 1 X 2 + B 6 X 1 X 2 2 +B 7 X 1 2 X 2 +B 8 X 1 2 X 2 2 y - any given response B i - regression coefficient for the various terms containing levels of the independent variable.

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25 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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26 Figure 6. Contour plots for the Lagrangian method: (b) dissolution (t 50% )

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27 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 t 50% 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 X 1 and X 2 will suffice.

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28 Buck et al incorporated statistical design into above product design It consist of following strategies A preliminary planning phase An experimental phase An analytical phase 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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29 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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30 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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31 The dependent variables are listed in Table 3

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32 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 2 5 − 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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33 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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34 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.”

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The type of predictor equation used with this type of design is a second-order polynomial of the following form: Y = a 0 +a 1 X 1 +…..+a 5 X 5 +a 11 X 12 +…+ a 55 X 52 +a 12 X 1 X 2 +a 13 X 1 X 3 +…+a 45 X 4 X 5 Y = level of a given response, X 1 = 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 R 2 value All pharma . responses will not fit 2 nd order regression method

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36 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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37 For the optimization itself, two major steps were used: The feasibility search 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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38 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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39 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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40 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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41 The output includes plots of a given responses as a function of all five variable ( Fig 12).

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42 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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43 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 = Y 0 +λ 1 W 1 2 +λ 2 W 2 2 +…….

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44 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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45 VI. OTHER APPLICATIONS

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46 IX. REFERENCES 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

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47 THANK YOU