optimization techniques

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OPTIMIZATION TECHNIQUES IN PHARMACEUTICS, FORMULATION AND PROCESSING: 

OPTIMIZATION TECHNIQUES IN PHARMACEUTICS, FORMULATION AND PROCESSING AJU S. SAM 1 st yr. M.PHARM DEPARTMENT OF PHARMACEUTICS NEHRU COLLEGE OF PHARMACY

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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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II . 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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VARIABLES 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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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 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 : Y = f (X)

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Evolutionary Operations The Simplex Method The Lagrangian Method Search Method Canonical Analysis APPLIED OPTIMIZATION METHODS Plackett and Burman method

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EVOLUTIONARY OPERATIONS (EVOP) This technique is well suited to a production situation. 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 produces a product that meets all specifications and (at the same time) generates information on product improvement.

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THE SIMPLEX METHOD 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. 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.

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The worst response is 0.25,conditions are selected at the vortex, 0.68, 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 spectrophotometric reading at a given wavelength .

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THE LAGRANGIAN METHOD Determine objective function Determine constraints Change inequality constraints to equality constraints. Form the Lagrange function, F: One Lagrange multiplier λ for each constraint One slack variable q for each inequality constraint Partially differentiate the Lagrange function for each variable and Set derivatives equal to zero. Solve the set of simultaneous equations. Substitute the resulting values into the objective functions.

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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. The active ingredient, phenyl-propanolamine HCl, was kept at a constant level, and the levels of disintegrant (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.

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Polynomial models relating the response variables to the independent variable were generated by a backward stepwise regression analysis program. 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., 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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EXAMPLE FOR THE LAGRANGIAN METHOD A graphic technique may be obtained from the polynominal equations, as follows: Figure 6. Contour plots for the Lagrangian method: (a) tablet hardness; c Figure 6. Contour plots for the Lagrangian method: (b) dissolution (t 50% )

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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 Figure 6. Contour plots for the Lagrangian method: c) feasible solution space indicated by crosshatched area

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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 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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The plots of the independent variables, X 1 and X 2 , can be obtained as shown in Fig.8. Figure 8. Optimizing values of stearic acid and strach as a function of restrictions on tablet friability: (A) %starch; (B) %stearic acid

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THE SEARCH METHOD It takes five independent variables into account and is computer-assisted. The five independent variables or formulation factors selected for this study are shown in Table 2. The dependent variables are listed in Table 3.

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STEPS INVOLVED: 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

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The experimental design used was a modified factorial and is shown in Table4. 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 * 2 5 =16 trials. The two levels are represented by +1 and -1. For the remaining trials, three additional levels were selected: zero represents a base level midway between the mentioned levels, and the levels noted as 1.547 represent extreme values.

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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 1 2 +…+a 12 X 5 2 +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, a ij 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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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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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. Graphic approaches are also available and graphic output is provided by a plotter from computer tapes.

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CANONICAL ANALYSIS 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 = Y 0 +λ 1 + W 1 2 +λ 2 W 2 2 +……. In canonical analysis second-order regression equations are reduced to a simpler form by a rigid rotation and translation of the response surface axes in multidimensional space, for a two dimension system.

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PLACKETT AND BURMAN METHOD The problems due to interactions among several excipients that affect a particular character or stability can be overcome. Optimized formulation can be formulated. The experimental design is 12 factorial. X1, X2 …. X12 are variables under investigation. + variable at high level. - variable at low level. avg = (sum+ - sum-) / no. of times variable oocurs. EMS = SFE* t.

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VI. OTHER APPLICATIONS

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REFERENCES Shobha Rani R Hiremath. Text book of industrial pharmacy, drug delivery systems and cosmetic and drug technology; 2008: 158 – 168 Shyamala Bhaskar. Text book of industrial pharmacy, first edition; 2005:96-113