Appendix IV Experimental Design
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1 Experimental Design The aim of pharmaceutical formulation and development is to develop an acceptable pharmaceutical formulation in the shortest possible time, using minimum number of working hours and raw materials. The formula developed by the formulation and development center is first tried at the pilot scale and then manufacture scale. Only minor changes are to be made during scale-up. Thus, it is very ideal to study the formulation from all perspectives at laboratory levels. In addition to the art of formulation, a statistical technique is available that can aid in the pharmacist s choice of formulation components, which can optimize one or more formulation attributes. A very efficient way to enhance the value of research and to minimize the process development time is through design experiment. The need to develop this design because traditional experiments involve a good deal of efforts and time, especially where complex formulations are to be developed. The statistical problem solving approach uses a series of small carefully designed experiments. We sometimes call the statistical approach strategic experimentation or iterative problem solving strategy. We also call this the stop look and listen approach to experimentation. Analyze the results of few experiments and then plan the next experiments. Any statistical design consist of three small and efficient experiments, namely a screening experiments where from many factors affecting the process few important factors are identified, then an optimization experiment where a predictive model is build for the few factors in the region of optimum and finally a verification experiment where the results is confirmed at the predicted setting. In the present work factorial design was used for the development of effective, functional and perfect dosage form. The help of systematic formulation approach is taken to get detailed knowledge on the formulation. 275
2 In the present study, 23 and factorial design and response surface methodology like central composite designs and Box-Behnken design were used. Hence, only these designs are discussed in details. IV-A. FACTORIAL DESIGNS [1-3] Factorial designs are used in experiments when the effects of different factors or conditions, on experiment results are to be elucidated. Factorial designs are the design of choice of simultaneous determination of the effects of several factors and their interaction. Factors may be qualitative or quantitative. The levels of an each factor are the values or designations assigned to combinations, of all levels, of all factors. The effects of a factor are the change in response caused by varying the levels(s) of the factor. The important objective of a factorial experiment is to characterize the effect of changing the levels of the factor or combination of factors on the response variable. Predictions based on results of an undersigned experiment will be more variable than those, which could be obtained in a designed experiment, in particular factorial design. The optimization procedure is facilitated by construction of an equation that describes the experimental results as a function of the factor levels. A polynomial equation can be constructed, where the coefficients in the equation are related to the effects and interaction of the factors. The goal of pharmaceutical formulation and development is to develop acceptable pharmaceutical formulation in the shortest possible time using minimum time and raw materials Optimization by experimental design leads to the evolution of a statistically valid model to understand the relationship between independent and dependent variables. The equation constructed form 2n factorial experiment is in the following from. Y= B0+B,Xi+B2X2+ B3X3+B12X1X2+B13X1X3+B23X2X3+B123X1X2X3 Where, Y= the measured response Xj= level of i,h factor (independent variable) 276
3 Appendix I V Bj = the regression coefficient for the ith independent variable. B0 = intercept The magnitudes of the coefficients represent the relative importance of each factor. Once the polynomial equation has been established, an optimum formulation can be found out by grid analysis. With the use of computer a grid method can be used to identify optimum regions, and response surfaces may be depicted. A computer can calculate the response based on equation at many combinations of factor levels. The formulation whose response has optimal characteristics based on the experimenter s specification is then chosen. Advantages of Factorial Design: In absence of interaction, they have maximum efficiency in estimating main effects. Maximum use is made of the data, since all main effects and interaction are calculated from the data Since factors effects are measured over varying levels of other factors, conclusions apply to wide range of condition. Factorial designs are orthogonal; all estimated effect and interaction are independent of effect of other factors. If interaction occur; factorial designs are necessary to reveal and identity the interaction. More information is obtained with less work. The most important attribute of this design is that the effects are measured with maximum precision. 277
4 Applications of Factorial Design: It helps and interprets the mechanism of an experimental system. It is very useful in an industrial manufacturing operation because it recommend or implement, a practical procedure or a set of condition It provides guidance for further experiment It also useful for the drug-excipinents compatibility study. In most situations, where one is interested in the effect of various factors or condition on some experimental outcome, factorial designs will be optimal. Factorial design used is either full or fractional. Full factorial design is a design in which every setting of every factor appears with every setting of every other factor is called as a full factorial design. When experiments are with a large number of factors and /or a large number of levels for the factors, the number of factors needed to complete factorial design is also large. Thus, application of full factorial design becomes costly and time consuming. In this case, fractional factorial design is utilized for optimization. In fractional factorial designs, the negligible and important factors are indistinguishable, and thus, confounded. Full Factorial Design: 23 factorial designs: A common experimental design is one with all input factors set at two levels each. These levels are called 'high' and 'low' or '+!' and '-1', respectively. A design with all possible high/low combinations of all the input factors is called a full factorial design in two levels. If there are k factors, each at 2 levels, a full factorial design has 2k runs. 278
5 Table 1: Runs in 2k full Factorial Design Number of Factors Number of Runs In 23 full factorial designs three factors and two levels are used to achieve the proper result. This implies eight runs (not counting replications or center point runs). Graphically, the 2 design is represented by the cube shown in figure. The arrows show the direction of increase of the factors. The numbers 1 through 8 at the corners of the design box reference the Standard Order of runs. 6 8 Figure 1:23 two-level, full factorial design; factors Xi, X2, X3 279
6 4 The design of 2 factorial design is given in Table 2. Table 2: Design of 23' factorial design Trial X, x2 x l l i l *1 -l l +i i +i +1 The design of 32 factorial design is as given in Table 3. In 32 full factorial designs two factors and three levels are used. Total 9 trials are made if this design is employed. Table 3: 32 Factorial Design Trial X, x
7 IV-B. RESPONSE SURFACE METHODOLOGY These designs can be classified as follows: Second order designs for spherical domain Second order designs for the cubic domain Central composite designs 3k factorial designs Special designs for cubic domain Central Composite Designs [4-6] Central composite designs are response surface designs that can fit a full quadratic model. One central composite design consists of cube points at the comers of a unit cube that is the product of the intervals [-1, 1], stars points along the axes at or outside the cube, and centre points at the origin. The design has three basic parts: The design consists of three distinct sets of experimental runs: 1. A factorial (perhaps fractional) design in the factors studied, each having two levels(2k F); 2. A set of centre points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often repeated in order to improve the precision of the experiment; thus the term is central composite, and 3. A set of axial points, experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range(i.e. starpoints, 2K).. All factors are varied in this way. 281
8 The number of experimental trials (N) in a composite design is given by: N = 2k f + 2K + C Where, K = no. of variables F = fraction of the full factorial and C = number of center point replicates Design Matrix: The design matrix for a central composite design experiment involving k factors is derived from a matrix, d, containing the following three different parts corresponding to the three types of experimental runs: 1. The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and The matrix C from the centre points, denoted in coded variables as (0,0,0,...,0), where there are k zeros. 3. A matrix E from the axial points, with 2k rows. Each factor is sequentially placed at ±a and all other factors are at zero. The value of a is determined by the designer; while arbitrary, some values may give the design desirable properties. This part would look like: a 0 0 a a 0 E = 0 -a a a 282
9 Central Composite Designs are of three types: Circumscribed (CCC) designs are as described above. Inscribed (CCI) designs are as described above, but scaled so the star points take the values -1 and +1, and the cube points lie in the interior of the cube. Faced (CCF) designs have the star points on the faces of the cube. Faced designs have three levels per factor, in contrast with the other types, which have five levels per factor. It is shown in Figure 2. CCD designs start with a factorial or fractional factorial design (with center points) and add "star" points to estimate curvature Similarly, the number of center point runs the design is also depends on certain properties required for the design. Characteristic of Central Composite Designs 1. Central Composite Circumscribed (CCC): This design is the original form of the central composite design. The star points are at some distance a from the center based on the properties desired for the design and the number of factors in the design. The star points establish new extremes for the low and high settings for all factors. These designs have circular, spherical, or hyperspherical symmetry and require 5 levels for each factor. CCC designs provide high quality predictions over the entire design space, but require factor settings outside the range of the factors in the factorial part. 2. Central Composite Inscribed (CCI): For those situations in which the limits specified for factor settings are truly limits, the CCI design uses the factor settings as the star points and creates a factorial or fractional factorial design within those limits (in other words, a CCI design is a scaled down CCC design with each factor level of the CCC design divided by a to generate the CCI design). This design also requires 5 levels of each factor. CCI designs use only points within the factor ranges originally specified, but do not provide the same high quality prediction over the entire space compared to the CCC. Requires 5 levels of each factor. 283
10 3. Central Composite Face Centered (CCF): In this design the star points are at the center of each face of the factorial space, so a = ± 1. This variety requires 3 levels of each factor. Augmenting an existing factorial or resolution V design with appropriate star points can also produce this design. CCF designs provide relatively high quality predictions over the entire design space and do not require using points outside the original factor range. However, they give poor precision for estimating pure quadratic coefficients. These designs require fewer treatment combinations than a central composite design in cases involving 3 or 4 factors. Table 4: Design Matrix for Central Composite Design Runs X, x "hi Box-Behnken Designs 17-12] Screening and optimizing process for dosage form development can be simplified by the use of a statistical design that requires only a small number of experiments, thereby eliminating the need for time-consuming and detailed experimental trials. Response surface methodology (RSM) is one such approach. It is used when only a few significant factors are involved in optimization. Different types of RSM designs include 3-level factorial design, central composite design (CCD), Box-Behnken design, and D-optimal design. The comparison of number of runs required for a given number of factors for various Central Composite design and Box-Behnken design is given in table
11 Table 5: Comparison of numbers of runs for Central Composite and Box-Behnken Designs Number of Box-behnken Central Composite Design Factors Design 2 13 (5 center points) (6 center point runs) (6 center point runs) (fractional factorial) or 52 (full factorial) (fractional factorial) or 91 (full factorial) 54 The runs or trials of the composite design experiments consist of all combination of all levels of all factors. The effect of a factor is the change in response caused by varying the level(s) of the factor chosen. Box-Behnken design requires fewer runs (15 runs) in a 3-factor experimental design A3-factor, 3-level design would require a total of 27 unique runs without any repetitions and a total of 30 runs with 3 repetitions. A 3-factor, 3-level design used is suitable for exploring quadratic response surfaces and constructing second-order polynomial models. It is basically a modified central composite experimental design, which, is an independent, rotatable or nearly rotatable quadratic design (contains no embedded factorial or fractional factorial design),. It consists of 1. Center points, and 2. Points lying on one sphere, equally distant from the center point. The latter points consists of small two-level full factorials where some factors are fixed at their center values. The number of center points is chosen to establish rotatability. The nonlinear quadratic model generated by the design is expressed as follows: The optimization procedure is facilitated by construction of an equation that describes the experimental results as a function of the factor levels. A polynomial equation can be constructed, where the coefficients in the equation are related to the effects and interactions of the factors. Y = b0 + b,x, + b2x2 + b3x3 + b12x,x2- b13x,x3+ b23x2x3+ b,,x,2 + b22 X22 + b33x32 Where Y= Estimated Response associated with each factor level combination. 285
12 Xi, X2 and X3 =Independent variables bi to b33 = Regression coefficients The magnitude of the coefficients represents the relative importance of each factor. Once the polynomial equation has been established, an optimum formulation can be found out by grid analysis. With the use of computers a grid method can be used to identify optimum regions and response surfaces may be depicted. A computer can calculate the response based on equation at many combinations of factor levels. The formulation whose response has optimal characteristics based on the experimenter s specification. Table 6:Design Matrix for 3 factors Box-Behnken design Trials Variable levels in coded form X, x2 X T
13 REFERENCES: 1. Bolton S., Bon C., Pharmaceutical Statistics: Practical and clinical application, 2nd Ed., Marcel Dekker Inc., NY, 1990: ; Franz R.M., Browne J.E., and Lewis A.R.; Experimental design, modeling an optimization strategies for product and process development: In Libermann, H.A. Riger, M.M., Banker, G.S., (Eds.), Pharmaceutical dosage form: Disperse systems (Volume I), Marcel Dekker, NY, 1988: Lewis,GA, Mathieu D, Phan-Tan-Luu R. Pharmaceutical Experimental Design Marcel Dekker Inc., New York, 1999, html, Accessed on 28th August, Lewis GA, Optimization Methods in Encyclopedia of Pharmaceutical technology, Edited by Swarbrick, J., Boylan, J.C., Marcel Dekker Inc., New York, 2nd edition, 2002; Box GEP, Wilson KB. On the experimental attainment of optimum multifactorial conditions. Royal Statistics Society. 1951; 13: Singh SK, Dodge J, Durrani MJ, Khan MA. Optimization and characterization of controlled release pellets coated with experimental latex: I. Anionic drug. Int J Pharm. 1995; 125: Sanchez-Lafuente C, Furlanetto S, Femandez-Arevalo M, et al. Didanosine extendedrelease matrix tablets: optimization of formulation variables using statistical experimental design. Int J Pharm.2002; 237: Ragonese R, Macka M, Hughes J, Petocz P. The use of the Box- Behnken experimental design in the optimization and robustness testing of a capillary electrophoresis method for the analysis of ethambutol hydrochloride in a pharmaceutical formulation. J Pharm Biomed Anal. 2002: 27: Box GEP, Behnken DW. Some new three level designs for the study of quantitative variables. Technometrics. 1960;2: Wilson WI, Peng Y, Augsburger LL. Comparison of statistical analysis and Bayesian networks in the evaluation of dissolution performance of BCS class II model drugs. J. Pharma. Sci., 94(12):
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