Response Surface Method

The basic idea behind response surface methods (RSMs) is to develop a response surface approximation which describes the relationship between the parameters in the original model and selected target outputs. Using a sample of simulations of the full model, a model approximation or metamodel is constructed, which can then be used as a surrogate for the full model in order to perform uncertainty and sensitivity analysis. The methods have some similarities with Monte Carlo approaches in that first input parameter ranges must be selected, and then a suitable sampling approach 

Isukapalli et al. (2000) coupled the Stochastic Response Surface Method (SRSM) with ADIFOR. The ADIFOR method (see Sect. 5.2.5) is used to transform the model code into one that calculates the derivatives of the model outputs with respect to inputs or transformed inputs. The calculated model outputs and the derivatives at a set of sample points are used to approximate the unknown coefficients in the series expansions of outputs. The coupling of the SRSM and ADIFOR methods was applied for an atmospheric photochemical model. The results obtained agree closely with those of the traditional Monte Carlo and Latin hypercube sampling methods whilst reducing the required number of model simulations by about two orders of magnitude. 

A discussion of the broad literature on model approximation methods is beyond the scope of this text but some of the methods which are used in the context of 

The use of polynomial chaos expansions for the generation of response surfaces is based on the spectral uncertainty method introduced for combustion models in Reagan et al. (2003, 2004, 2005) and Najm et al. (2009) which was extended to an RSM in, e.g.. Sheen et al. (2009). Here an uncertainty factor m, is first assigned to each input variable. Note that this uncertainty factor m, is related to uncertainty parameter/to be discussed in Sect. 5.6.1 by Mj= 10. Taking the example of rate coefficients, they are then normalised into factorial variables x as follows  

The uncertainty in x may be expressed as a polynomial expansion of basis random variables 

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Kandimalla KK, Kanikkannon N, Singh M. Optimization of a vehicle mixture for the transdermal delivery of melatonin using artificial neural networks and response surface method. / Controlled Release 1999 61 71-82. 

Czapski, J., Maksymiuk, M., and Grajek, W., Analysis of biodenitrification conditions of red beet juice using the response surface method, J. Agric. Food Chem., 46, 4702, 1998. 

Carter, W.H., Jr., Jones, D.E., and Carchman, R.A. (1985), Application of Response Surface Methods for Evaluating the Interactions of Soman, Atropine, and Pralidioxime Chloride, Fundamental and Applied Tox., 5, S232-S241. 

On-line coupling of a C-nanotubes-based preconcentra-tion system with a flow injection manifold. Variables considered most relevant to control the process were evaluated and optimised using experimental designs and response surface methods... 

Hammami, C., Rene, F., Marin, M. Process-quality optimization of the vacuum freeze-drying of apple slices by the response surface method, bit. J. Food Sci. Technol. 34,145-160,1999... 

Carter, C.W.J. 1997. Response surface methods for optimizing and improving reproducibility of crystal growth. Methods Enzymol 276 74-99. 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

Isukapalli S, Roy Z, Georgopoulos PG (2000) Efficient sensitivity/uncertainty analysis using the combined stochastic response surface method (SRSM) and automatic differentiation for FORTRAN code (ADIFOR) Application to environmental and biological systems. Risk Analysis, 20 591-602. 

A mixed granulate produced from PETP and PVC postconsumer bottles has been separated using the technology of column flotation. The aqueous separation medium pH and surfactant concentration were optimised with respect to separation efficiency, using Response Surface Methods. The study has shown that column flotation could be used to give close to 100% separation of PVC from PETP in a single operation. 15 refs. 

An initial set of five para substitutes (H, Me, Cl, MeO, Me2N) acetophenones was selected. The experimental conditions which afforded the maximum yield were determined for each substrate by response surface methods. The optimum settings... 

Kittrell, J. R.. and J. Erjavec, Response surface methods in heterogeneous kinetic modelling, Jnd. Eng. Chem., Process Des. Devel, 7, 321-327 (1968). 

In a simultaneous approach the relationship between variables and results is studied as follows carry out an appropriate design, apply a mathematical model to the design, and then apply a response surface method to the data. 

Appropriate designs might be based on factorial designs (full or fractional) or a central composite design. Response surface methods frequently rely on visualization of the data for interpretation. 

MRS is the mixture response surface method (Ochsner et al., 1985). The value of MPD was taken from Table 2 of the Jouyban-Gharamaleki et al. (Jouyban-Gharamaleki et al., 1999) paper. 

With new synthetic methods, mechanistic details are still obscured. It is not likely that such details will be revealed until the preparative utility of the procedure has been demonstrated. This means that an optimization of the experimental conditions must generally precede a mechanistic understanding. Hence, the optimum conditions must be inferred from experimental observations. The common method of adjusting one-variable-at-a-time, is a poor strategy, especially in optimization studies (see below). It is necessary to use multivariate strategies also for determining the optimum experimental conditions. There are many useful, and very simple strategies for this sequential simplex search, the method of steepest ascent, response surface methods. These will be discussed in Chapters 9 - 12. 

Response surface modelling is used to locate the detailed optimum conditions. The principle is to establish a response surface model which maps the optimum region. The map can then be used for navigation in the optimum region. Close to an optimum the surface is curved and to describe the general features of the response surface quadratic models are used. The response surface methods have been and are being developed by Box and coworkers.[3]... 

Characterizing the overall uncertainties associated with the PBPK model estimates is also an important component of the PBPK model evaluation and application. This includes characterizing the uncertainties in model outputs resulting from the uncertainty in the PBPK model parameters. Traditionally, Monte Carlo has been employed for performing uncertainty analysis of PBPK models (39, 40). Some of the recent techniques that have been applied for the uncertainty analysis of PBPK models include the stochastic response surface method (SRSM) (38, 41) and the high-dimensional model reduction (HDMR) technique (42). 

S. S. IsukapalU, A. Roy, and P. G. Georgopoulos, Stochastic response surface methods (SRSMs) for uncertainty propagation application to environmental and biological systems. Risk Anal 18 351-363 (1998). 

Chen HC (1996) Optimizing the concentrations of carbon, nitrogen, and phosphorus in a citric acid fermentation with response surface method. Food Biotechnol 10 13-27... 

Strobel RJ, Nakatsukasa WM (1993) Response surface methods for optimizing Saccharopolyspora spinosa, a novel macrolide producer. J Ind Microbiol 11 121-127... 

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