optimization techniques in pharmaceutical processing

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING Facilitated by, Mr. Rajasekhar Valluru Professor Presented by, Venkata krishna Y, Dept. of pharmaceutics, EAST WEST COLLEGE OF PHARMACY

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 2 I. INTRODUCTION OPTIMIZATION It is defined as follows: choosing the best element from some set of available alternatives. In Pharmacy word “optimization” is found in the literature referring to any study of formula. 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.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 3 II . OPTIMIZATION PARAMETERS There are two optimization parameters 1.Problem Types 2.Variables PROBLEM TYPES - There are two general types of optimization problems: 1. Unconstrained 2. Constrained 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 4 VARIABLES - The development procedure of the pharmaceutical formulation involves several variables. Mathematically these variables are divided into two groups. 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 5 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 6 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 graphical represented can be avoided. If the relationship, 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 7 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 8 V. APPLIED OPTIMIZATION METHODS There are several methods used for optimization. They are

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 9 EVOLUTIONARY OPERATIONS One of the most widely used methods of experimental optimization in fields other than pharmaceutical technology is the evolutionary operation (EVOP). This technique is especially well suited to a production situation. The basic philosophy is that the production procedure (formulation and process) is allowed to evolve to the optimum by careful planning and constant repetition. The process is run in a way such that it both produces a product that meets all specifications and (at the same time) generates information on product improvement.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 10 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. So, for two factors or independent variables, the simplex is represented by a triangle. 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. THE SIMPLEX METHOD

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 11 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. Figure 5 The simplex approach to optimization. Response is spectorphotometric reading at a given wavelength .

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 12 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 13 This technique requires that the experimentation be completed before optimization so that mathematical models can be generated. The experimental design here was full 3 square factorial, and , as shown in Table- 1 nine formulations were prepared.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 14 Polynomial models relating the response variables to the independent variable were generated by a backward stepwise regression analysis program. The analyses were performed on a polynomial of the form 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 In Eq. (3), y represents any given response and B i represents the regression coefficient for the various terms containing levels of the independent variable. One equation is generated for each response or dependent variable.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 15 The active ingredient, phenyl-propanolamine 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 subject. EXAMPLE FOR THE LAGRANGIAN METHOD A graphic technique may be obtained from the polynomial equations, as follows:

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 16 Figure 6. Contour plots for the Lagrangian method: (a) tablet hardness;

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 17 Figure 6. Contour plots for the Lagrangian method: (b) dissolution (t 50% )

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 18 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 19 A technique called sensitivity 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 systematic changes in the secondary objectives. For example, the foregoing problem restricted tablet friability, y 3 , to a maximum of 2.72%. Figure 7 illustrates the in vitro release profile as this constraint is tightened or relaxed and demonstrates that substantial improvement in the t 50% can be obtained up to about 1-2%.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 20 Figure 7 illustrates the in vitro release profile as this constraint is tightened or relaxed and demonstrates that substantial improvement n the t 50% can be obtained up to about 1-2%.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 21 The plots of the independent variables, X 1 and X 2 , can be obtained as shown in Fig.8. Thus the formulator is provided with the solution (the formulation) as he changed the friability restriction. Figure 8. Optimizing values of stearic acid and strach as a function of restrictions on tablet friability: (A) percent starch; (B) percent stearic acid

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 22 Suspension design to illustrate the efficient and effective procedures that might be applied. Representation of such analysis and the available solution space is shown for the suspension in Figs. 9 and 10. Figure 9. Response surface concept and results of the second case study

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 23 Figure 10. Secondary properties of various suspensions yielding zero dose variation.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 24 Although the Lagrangian method was able to handle several responses or dependent variable, it was generally limited to two independent variables. A search method of optimization was also applied to a pharmaceutical system. It takes five independent variables into account and is computer-assisted. It was proposed that the procedure described could be set up such that persons unfamiliar with the mathematics of optimization and with no previous computer experience could carry out an optimization study. THE SEARCH METHOD

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 25 1. Select a system 2. Select variables: a. Independent b. Dependent 3. Perform experimens 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 26 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 27 The dependent variables are listed in Table 3

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 28 The experimental design used was a modified factorial and is shown in Table4. The fact that there are 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 firs 16 formulations represent a half-factorial design for five factors at two levels, resulting in ½ X 25 =16 trials. The two levels are represented by +1 and -1, analogous to the high and low values in any two level factorial design. For the remaining trials, three additional levels were selected: zero represents a base level midway between the a fore mentioned levels, and the levels noted as 1.547 represent extreme (or axial) values.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 29

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 30 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 31 The type of predictor equation usd 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 Where Y is the level of a given response, the regression coefficients for second-order polynomial, and X 1 the level of the independent variable. The full equation has 21 terms, and one such equation is generated for each response variable .

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 32 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 33 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 34 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. Graphic approaches are also available and graphic output is provided by a plotter from computer tapes.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 35 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.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 36 The output includes plots of a given responses as a function of all five variable (Fig 12).

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 37 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 38 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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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 39 . 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.14 for a two dimension system.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 40 VI. OTHER APPLICATIONS

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 41 . The graphs in Fig.15 show that for the drug hydrochlorothiazide, the time of the plasma peak and the absorption rate constant could, indeed, be controlled by the formulation and processing variables involved.

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 42 IX. REFERENCES Websters 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. Google search engine, WWW.Google.co.in 6. http://en.wikipedia.org/wiki/Optimization_(mathematics) 7. http://mat.gsia.cmu.edu/classes/QUANT/NOTES/chap4/node6.html 8. http://www.socialresearchmethods.net/kb/desexper.php 9. P .K. Shiromani and J. Clair, Drug Dev Ind Pharm., 26 (3), 357 (2000).

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICAL FORMULATION AND PROCESSING 43 THANK YOU