# Optimization Techniques

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describes various optimization parameters, optimization techniques and its applications in pharmaceuticals

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### Optimization Techniques in Pharmaceutical Formulation and Processing:

Optimization Techniques in Pharmaceutical Formulation and Processing S.R. College of Pharmacy P. Raja Abhilash , M.pharm (Ph.D.) Assistant professor, S.R. college of pharmacy.

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Contents Introduction Optimization Parameters Classic Optimization Statistical Design Applied Optimization Methods Use of Computers for Optimization Applications Conclusion References

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3 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 . OPTIMIZATION is an act, process, or methodology of making design, system or decision as fully perfect, functional or as effective as possible . Optimization of a product or process is the determination of the experimental conditions resulting in its optimal performance.

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Optimization Parameters Optimization parameters Variable types Problem types Independent variables Unconstrained Dependent variables Constrained

### Problem types in optimization :

Problem types in optimization Unconstrained Constrained no restrictions are restrictions are placed placed on the system on the system eg: preparation of hardest eg: preparation of hardest tablet without any disintegration tablet which has the ability of or dissolution parameters. disintegrate in less than 15min

### variables in optimization:

variables in optimization Independent Dependent variables variables directly under the control responses that are developed of formulator due to the independent variables eg: eg: disintegrant level disintegration time compression force hardness binder level weight uniformity lubricant level thickness mixing time etc

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response surface curve 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. Fig I; response surface curve

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Classic 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 ) ---eqn (1) Figure 2. Graphic location of optimum (maximum or minimum)

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9 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 . Here the contours are showing the response. (contour represents the connecting point showing the peak level of response) Figure 3. Contour plot. Contour represents values of the dependent variable Y Classic Optimization

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Optimization Techniques The techniques for optimization are broadly divided into two categories: (A) simultaneous method: Experimentation continues as optimization study proceeds. E.g.: a. Evolutionary Operations Method b. Simplex Method (B) sequential method: Experimentation is completed before optimization takes place. E.g.: a. Mathematical Method b. Search Method In case (B), the formulator has to obtain the relationship between the response and one or more independent variables. This includes two approaches: Theoretical Approach & Empirical Approach .

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Optimization Strategy : Problem definition Selection of factors and levels Design of experimental protocol Formulating and evaluating the dosage form Prediction of optimum formula Validation of optimization

### Factorial Designs :

Full factorial designs: Involve study of the effect of all factors(n) at various levels(x) including the interactions among them with total number of experiments as X n . SYMMETRIC ASYMMETRIC Fractional factorial designs: It is a fraction ( 1/ x p ) of a complete or full factorial design, where ‘p’ is the degree of fractionation and the total number of experiments required is given as x n -p . Factorial Designs

### Factorial Designs :

Pictorial representation , where each point represents the individual experiment Factorial Designs

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Applied optimization methods Evolutionary Operations (EVOP) Simplex Method Lagrangian Method Search Method canonical analysis

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A. Evolutionary operations (EVOP) Most widely used method of experimental optimization in fields other than pharmaceutical technology.. Experimenter makes very small changes in formulation repeatedly. The result of changes are statistically analyzed. If there is improvement, the same step is repeated until further change doesn’t improve the product. Can be used only in industries and not on lab scale.

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B. Simplex Method It was introduced by Spendley et.al, which has been applied more widely to pharmaceutical systems. A simplex is a geometric figure, that has one more point than the no. of factors. so, for two factors ,the simplex is a triangle. It is of two types: A. Basic Simplex Method B. Modified Simplex Method Simplex methods are governed by certain rules. 1 3 2

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Basic Simplex Method 2 1 5 4 3 6 7 8 9 10 12 11 (W) (N) (B) (R) s10 s1 s2 s9 s8 s6 s7 s5 s4 s3 Rule 1 : The new simplex is formed by keeping the two vertices from preceding simplex with best results, and replacing the rejected vertex (W) with its mirror image across the line defined by remaining two vertices.

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Basic Simplex Method 2 1 5 4 3 6 7 8 9 10 12 11 (W) (N) (B) (R) s10 s1 s2 s9 s8 s6 s7 s5 s4 s3 (W) (W) (W) Rule 2 : When the new vertex in a simplex is the worst response, the second lowest response in the simplex is eliminated and its mirror image across the line; is defined as new vertices to form the new simplex.

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Basic Simplex Method 2 1 5 4 3 6 7 8 9 10 12 11 (W) (N) (B) (R) s10 s1 s2 s9 s8 s6 s7 s5 s4 s3 (W) (W) (W) Rule 3 : When a certain point is retained in three successive simplexes, the response at this point or vertex is re determined and if same results are obtained, the point is considered to be the best optimum that can be obtained.

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Basic Simplex Method 2 1 5 4 3 6 7 8 9 10 12 11 (W) (N) (B) (R) s10 s1 s2 s9 s8 s6 s7 s5 s4 s3 (W) (W) (W) Rule 4 : If a point falls outside the boundaries of the chosen range of factors, an artificially worse response should be assigned to it and one proceeds further with rules 1 to 3. This will force the simplex back into the boundaries.

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Modified Simplex Method It was introduced by Nelder -Mead in 1965. This method should not be confused with the simplex algorithm of Dantzig for linear programming. Nelder -Mead method is popular in chemistry, chemical engg ., pharmacy etc. This method involves the expansion or contraction of the simplex formed in order to determine the optimum value more effectively.

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Modified Simplex Method W N B C1 R1 E1 If response at R1 > B, expansion of simplex to E1. If response at N<= R1<=B, no expansion or contraction is done. If response at R1<N, contraction of the simplex is done.

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C. Lagrangian Method It represents mathematical method of optimization. Steps involved: 1.Determine the objective function. 2. Determine the constraints. 3. Introduce the Lagrange Multiplier ( λ) for each constraint. 4. Partially differentiate Lagrange Function (F). 5. Solve the set of simultaneous equations. 6. Substitute the resulting values into objective function.

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Lagrangian Method (polynomial model) Total Cost = 3x 2 + 6y 2 – xy ------ objective function determined! Subject to: x+y = 20 ------------- constraints determined! We can rewrite the condition as, 0 = 20-x-y ------- This has to be embedded in objective function L TC = 3x 2 + 6y 2 – xy + λ ( 20 -x - y) ---------- Lagrange multiplier (λ) introduced L TC = 3x 2 + 6y 2 – xy + 20 λ - x λ - y λ --------- Lagrange function (F) Partial differentiation done! Now solve the simultaneous equations

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Lagrangian Method 6x – y - λ = 0 x – 12y + λ = 0 7x - 13y = 0 i.e. 7x = 13y so Insert in any of the simultaneous equations

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Lagrangian Method Total Cost = 3x 2 + 6y 2 – xy ------ objective function We have determined using Lagrange function, x= 13 and y= 7 Substituting these values in the objective function, Total Cost = 3x 2 + 6y 2 – xy Total Cost = 3(13) 2 + 6(7) 2 – (13)(7) Total Cost = 507 + 294 – 91 Hence the total cost to produce 20 units is \$ 710

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The active ingredient , phenyl- propanolamine HCl, was kept at a constant level, and the level of the levels of disintegrant (corn starch) and lubricant (stearic acid) were selected as the independent variables. X 1 and X 2 . the dependent variables include tablet hardness, friability,invitro release rate, and urinary excretion rate in human subject. A graphic technique may be obtained from the polynomial equations, as follows: Example for the Lagrangian Method

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Lagrangian Method (contour plots) (a) Tablet Hardness (b) Dissolution (c) Feasible solution indicated by crosshatched area.

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D. Search methods Unlike the Lagrangian method, do not require differentiability of the objective function. It can be used for more than two independent variables. The response surface is searched by various methods to find the combination of independent variables yielding an optimum. select a system select variables: independent and dependent Perform experiments and test product Submit data for statististical and regressional analysis Set specifications for feasibility program Select constraints for grid research Evaluate grid search printout as contour plots

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Independent Variables Dependent Variables X1 = Diluent ratio Y1 = Disintegration time X2= Compressional force Y2= Hardness X3= Disintegrant levels Y3 = Dissolution X4= Binder levels Y4 = Friability X5 = Lubricant levels Y5 = Porosity Example for the Search methods

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Search methods The first 16 trials are represented by +1 and -1. The remaining trials are represented by a -1.547, zero or 1.547 The type of predictor equation used in this example is :

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32 The output includes plots of a given responses as a function of all five variables. Search methods

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33 Contour plots for (a) disintegration time (b) tablet hardness (c) dissolution response (d) tablet friability. Search methods

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E. Canonical Ana lysis 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 +…….

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Canonical Analysis 35 . 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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Use of Computers for optimization Statistical Analysis Systems (SAS) RS/Discover eCHIP Xstat JMP Design Expert FICO Xpress Optimization Suite Multisimplex

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Applications Formulation and Processing Clinical Chemistry HPLC Analysis Medicinal Chemistry Studying pharmacokinetic parameters Formulation of culture medium in microbiology studies.

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Conclusion Optimization techniques are a part of development process. The levels of variables for getting optimum response is evaluated. Different optimization methods are used for different optimization problems. Optimization helps in getting optimum product with desired bioavailability criteria as well as mass production. More optimum the product = More \$\$ the company earns in profits!!!

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References Joseph B. Schwartz. Optimization techniques in product formulation. Journal of the Society of Cosmetic Chemists. (1981) Vol 32; p: 287-301. Gilbert S. Banker, Christopher T. Rhodes. Modern Pharmaceutics. 4 th edition. CRC Press. (2002); p: 900-928. Optimization. 2012. In Merriam-Webster Online Dictionary. Retrieved March 07, 2012, from http://www.merriam-webster.com/dictionary/optimization N. Arulsudar, N. Subramanian & R.S.R. Murthy. Comparison of artificial neural network and multiple linear regressions in the optimization of formulation parameters of leuprolide acetate loaded liposomes. Journal of Pharmacy & Pharmaceutical Sciences. (2005) Vol. 8(2); p: 243-258. Roma Tauler, Steven D. Brown, Beata Walczak. Comprehensive Chemometrics: Chemical and Biochemical data analysis. Elsevier. (2009); p: 555-560. Khaled S. Al-Sultan, M.A. Rahim. Optimization in Quality Control. Springer. (1997); p: 6-8. Donald H.Mc Burney, Theresa L.White. Research Methods. 7 th edition. Thomson Wadsworth. (2007); p: 119. Rosilene L. Dutra, Heloisa F. Maltez, Eduardo Carasek, Development of an on-line preconcentration system for zinc determination in biological samples, Talanta, (2006) Vol 69(2), p:488-493.