Response surface analysis

Response or effect surface analysis (RSA) uses multiple linear regressions to produce a statistically based mathematical relationship between the doses of each of the chemicals in a mixmre and the effect parameter. The equation for a mixmre containing three compounds would be 

In the case of negative values for /3, positive values for y and 8 indicate a less than additive interaction between two or three compounds. Zero values of y or 5 indicate absence of a particular interaction. The p values of these coefficients are estimated using t test. 

Cassee et al. (1998) stressed that it should be avoided to use high-effect levels in this method because samration and competition will play a major role for the combined effect and might lead to erroneous conclusions about combined action at lower (and more realistic) dose levels. They advocated using a concentration range not exceeding a 50% effect level, or even less. 

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The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

One example of the application of response surface analysis is a study of critical formulation variables for 20 mg piroxicam capsules [100]. Piroxicam is a BCS Class II drug (low solubility and high permeability). This... 

Ozcan, S. and Jackson, D. S. (2003). A response surface analysis of com starch annealing. 

Response Surface Analysis Structure-Activity Relationship Superfund Amendments and Reauthorization Act Scientific Committee on Consumer Products Sister Chromatid Exchange... 

Multiple regression and response surface analysis techniques were employed to assess the effects of cysteine and GaGl combinations on WPG gel characteristics (15). Gontour plots summarizing these data trends are presented in Figure 3. Significant interaction effects were observed between reagents with respect to hardness, cohesiveness and compressible water. At low levels of added... 

Schmidt, R. H., Illingworth, B. L., Deng, J. C. and Cornell, J. A. 1979. Multiple regression and response surface analysis of the effects of calcium chloride and crysteine on heat-induced whey protein gelation. J. Agr. Food Chem. 27, 529-532. 

Finally, the use of factorial designs, concluded by a response-surface analysis, is the latest approach used to select the most appropriate combination doses of two classes of drugs that allows complete pharmacological manipulation of the RAS, namely diuretics and ACE inhibitors (250, 251). 

Kimmel GL, Williams PL, Claggett TW, Kimmel CA (2002) Response-surface analysis of exposure-duration relationships The effects of hyperthermia on embryonic development of the rat in vitro. Toxicol Sci, 69 391-399. 

This issue is addressed, for instance, in the discussion of model coefficient aliasing in books on response surface analysis such as Myers and Montgomery (2002). It is clear that this aliasing can introduce serious problems into decision processes based upon the realized estimates of model coefficients. 

Shieh et al. (2003) indicated a biodiesel transesterification using soybean oil and methanol and commercial immobilized lipase from R. miehei (Lipozyme IM-77). The response surface analysis showed that the following variables were important reaction time, temperature, enzyme amount, molar ratio of methanol to soybean oil, and added water content on percentage weight conversion to soybean oil methyl ester by transesterification. The optimum yield based on ridge max analysis gave a 92.2% weight conversion. 

TABLE 9.5. Central composite rotatable second-order design and experimental data for 5-factor-5-level response surface analysis. 

Wang, G, Mu, Y., and Yu, H. Q. 2005. Response surface analysis to evaluate the influence of pH, temperature, and substrate concentration on the acidogenesis of sucrose-rich wastewater. Biochem. Eng J., 23,175-184. 

If the aim is to screen for such systematic deviations because a priori knowledge is lacking or interactions are expected, the complete concentration-response surface should be tested. Yet, the number of possible test combinations increases exponentially with the number of chemicals in the mixture. Full concentration-response surface analysis is therefore seldomly performed for testing more than 4 or 5 chemicals simultaneously. 

The method which yields the most accurate predictions of the optimum conditions is to use response surface modelling and to determine a model which also includes square terms to account for curvatures of the response surface. Analysis of the fitted model will then reveal how the experimental conditions should be adjusted to achieve an optimum result. 

When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a response surface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver Boyd, 1954). A quadratic of this type for the variables Xi, 2, and X3 is... 

Box, G.E.P. Draper, N.R. Empirical Model-Building and Response Surface Analysis Whey New York, 1987. 

TJ apid entrainment carbonization of powdered coal under pressure in a partial hydrogen atmosphere was investigated as a means of producing low sulfur char for use as a power plant fuel. Specific objectives of the research were to determine if an acceptable product could be made and to establish the relationship between yields and chemical properties of the char, with special emphasis on type and amount of sulfur compound in the product. The experiments were conducted with a 4-inch diameter by 18-inch high carbonizer according to a composite factorial design (1, 2). Results of the experiments are expressed by empirical mathematical models and are illustrated by the application of response surface analysis. 

Experiments in the carbonization of coal to produce low sulfur char were carried out and evaluated by means of a three-step procedure. Carbonization runs were flrst conducted to obtain data at various combinations of the three major independent variables. These data were then used to develop an empirical mathematical model that described the carbonization system. Finally, response surface analysis was used to interpret the empirical model and to predict the relationship between process variables and char yield and quality. 

The following results are based on the analyses of data by means of response surface analysis. The three-dimensional empirical models were developed at the 95% confidence level. In other words, 95% of the time the results obtained from the empirical model will match the actual data obtained. 

White, C.. and Gadd, G. M. (1996a). Mixed sulphate-reducing bacterial cultures for bioprecipitation of toxic metals factorial and response-surface analysis of the effects of dilution rate, sulphate and substrate concentration. Microbiology. 142, 2197-2205. 

Wang,X.,Yin,C.-S. and Wang, L.-S. (2002) Structure-activity relationships and response-surface analysis of nitroaromatics toxicity to the yeast Saccharomyces cerevisiae). Chemosphere, 46, 1045-1051. 

L. H. Hall and H. D. Stewart, Quant. Struct.-Act. Relat., in press (1991). Total Response Surface Analysis and Molecular Connectivity Bioconcentration Factors for Halogenated Hydrocarbons. 

The central composite design, as an examination of the literature will show, is the one most used in response surface analysis. For this reason, and also because its construction follows on naturally from that of the factorial designs covered previously, we will examine it first. However, even if the evidence of the pharmaceutical literature is overwhelmingly in favour of its employment, we do not recommend it to the reader with quite the same insistence. There are other methods, covered later in the chapter, which may be superior in quite frequent circumstances, and which certainly merit equal consideration. 

We conclude that the regression is highly significant. Using the reduced cubic model does not improve the regression and the second-order model could be used for response-surface analysis. The first-order model is seen to be inadequate. 

But here, if the domain is small, the coefficients 6, are unlikely to have a real physical meaning. They are, in actual fact, the calculated responses for the pure components, but these pure components are so far outside the experimental domain that the errors in extrapolation will probably be enormous. This is not in general a problem in the case of response surface analysis, as here we are not concerned with direct interpretation of the coefficients, but only in mapping the form of the response surface within the experimental region. 

Buzzini, P, Martini, A., Gaetani, M., Turchetti, B., Pagnoni, U.M., and Davoli, P. 2005. Optimization of carotenoid production by Rhodotorula graminis DBVPG 7021 as a function of trace element concentration by means of response surface analysis. Enzyme Microb Technol 36 687-692. 

KALIL, S. J. MAUGERY, F. and RODRIGUES, M. I. Response surface analysis and simulation as a tool for bioprocess design and optimization. Process Biochem. 35, 2000, pp. 539-550. 

Wangyang Sheng optimized the extraction conditions for a-Gal by response surface analysis [35]. And the optimal conditions are liquid ratio of 1 5, extraction time of 35 min, pH of 5.0. Under these conditions, the total enzyme activity increased 11.6%, and specific enzyme activity increased 11.2% by optimization. 

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