166 Fermentation and Biochemical Engineering Handbook The successful use ofEVoP depends heavily upon the choice for the initial experimental runs. If the initial points are far from the optimum and elatively close to one another, many iterations will be required. Reasonable step sizes must be chosen to insure that a significant effect of the variable is observed between the points, however, the step size should not be so great to encompass the optimum. a second factor to consider is magnitude effects If one variable is measured over a range of 0. 1 to 1.0 while another is measured over a range of l to 100 the magnitude difference between th variables can effect the simplex. Scaling factors should be used to keep all variables within the same order of magnitude 4.0 RESPONSE SURFACE METHODOLOGY The best method for process optimization is response surface method- ology(RSm). This process will not only determine optimum conditions, but also give the information necessary to design a process Response surface methodology(rSm) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkin(] and Box and Wilson. 51 It is a scientific approach to determining optimum conditions which combines special experimental de- signs with Taylor first and second order equations. The rsm process determines the surface of the Taylor expansion curve which describes the response(yield, impurity level, etc. )The Taylor equation, which is the heart of the rsm method, has the form Response=A+B XI+C: X2+.H X12+I X22+ MX1X2+NX1X3+ where A, B, C,.are the coefficients of the terms of the equation, and XI= linear term for variable 1 X2= linear term for variable 2 x1= nonlinear squared term for variable X22= nonlinear squared term for variable 2
166 Fermentation and Biochemical Engineering Handbook The successfid use of EVOP depends heavily upon the choice for the initial experimental runs. If the initial points are far from the optimum and relatively close to one another, many iterations will be required. Reasonable step sizes must be chosen to insure that a significant effect of the variable is observed between the points, however, the step size should not be so great as to encompass the optimum. A second factor to consider is magnitude effects. If one variable is measured over a range of 0.1 to 1.0 while another is measured over a range of 1 to 100 the magnitude difference between the variables can effect the simplex. Scaling factors should be used to keep all variables within the same order of magnitude. 4.0 RESPONSE SURFACE METHODOLOGY The best method for process optimization is response surface methodology (RSM). This process will not only determine optimum conditions, but also give the information necessary to design a process. Response surface methodology (RSM) is a method of optimization using statistical techniques based upon the special factorial designs of Box and Behenkir~[~~] andBox and Wilson.[ls] It is a scientific approach to determining optimum conditions which combines special experimental designs with Taylor first and second order equations. The RSM process determines the surface of the Taylor expansion curve which describes the response (yield, impurity level, etc.) The Taylor equation, which is the heart of the RSM method, has the form: Response = A + B.X1 + CaX2 + . . . H-X12 + I.X22 + ... M*Xl*X2 +N*Xl*X3 + .,. where A,B,C,. . . are the coefficients of the terms of the equation, and X1 = linear term for variable 1 X2 = linear term for variable 2 Xl2 = nonlinear squared term for variable 1 X22 = nonlinear squared term for variable 2
Statistical Methods for Fermentation Optimization 167 X1X2 interaction term for variable l and variable Xl X3= interaction term for variable 1 and variable 3 The Taylor equation is named after the English mathematician Brook Taylor who proposed that any continuous function can be approximated by a power series. It is used in mathematics for approximating a wide variety of continuous functions. The RSM protocol, therefore, uses the Taylor equation to approximate the function which describes the response in nature, coupled with the special experimental designs for determining the coefficients of the Taylor equation The use of RSM requires that certain criteria must be met. These are 1. The factors which are critical for the process are known RSM programs are limited in the number of variables that they are designed to handle. As the number of variables increases the number of experiments required by the designs increases exponentially. Therefore, most RSM programs are limited to 4 to 5 variables. Fortunately for the scale up of most fermentations thenumber of variables to be optimized are limited. Some of the more important variables are listed in Table 1 Table 1. Typical Variables in a Fermentation Aeration rate Agitation rate Carbon/Nitrogen ratio Phosphate level Magnesium level Back Carbon Source Nitrogen source Dissolved oxygen level Power input
Statistical Methods for Fermentation Optimization I67 X1-X2 = interaction term for variable 1 and variable 2 XleX3 = interaction term for variable 1 and variable 3 The Taylor equation is named after the English mathematician Brook Taylor who proposed that any continuous function can be approximated by a power series. It is used in mathematics for approximating a wide variety of continuous functions. The RSM protocol, therefore, uses the Taylor equation to approximate the function which describes the response in nature, coupled with the special experimental designs for determining the coefficients of the Taylor equation. The use of RSM requires that certain criteria must be met. These are: 1. The factors which are critical for the process are known. RSM programs are limited in the number ofvariables that they are designed to handle. As the number of variables increases the number of experiments required by the designs increases exponentially. Therefore, most RSM programs are limited to 4 to 5 variables. Fortunately for the scale up of most fermentations the number of variables to be optimized are limited. Some of the more important variables are listed in Table 1. Table 1. Typical Variables in a Fermentation Aeration rate Agitation rate Temperature CarbodNitrogen ratio Phosphate level Magnesium level Back pressure Sulhr level Carbon Source Nitrogen source PH Dissolved oxygen level Power input