Elsevier

Talanta

Volume 76, Issue 5, 15 September 2008, Pages 965-977
Talanta

Review
Response surface methodology (RSM) as a tool for optimization in analytical chemistry

https://doi.org/10.1016/j.talanta.2008.05.019Get rights and content

Abstract

A review about the application of response surface methodology (RSM) in the optimization of analytical methods is presented. The theoretical principles of RSM and steps for its application are described to introduce readers to this multivariate statistical technique. Symmetrical experimental designs (three-level factorial, Box–Behnken, central composite, and Doehlert designs) are compared in terms of characteristics and efficiency. Furthermore, recent references of their uses in analytical chemistry are presented. Multiple response optimization applying desirability functions in RSM and the use of artificial neural networks for modeling are also discussed.

Introduction

Optimizing refers to improving the performance of a system, a process, or a product in order to obtain the maximum benefit from it. The term optimization has been commonly used in analytical chemistry as a means of discovering conditions at which to apply a procedure that produces the best possible response [1].

Traditionally, optimization in analytical chemistry has been carried out by monitoring the influence of one factor at a time on an experimental response. While only one parameter is changed, others are kept at a constant level. This optimization technique is called one-variable-at-a-time. Its major disadvantage is that it does not include the interactive effects among the variables studied. As a consequence, this technique does not depict the complete effects of the parameter on the response [2]. Another disadvantage of the one-factor optimization is the increase in the number of experiments necessary to conduct the research, which leads to an increase of time and expenses as well as an increase in the consumption of reagents and materials.

In order to overcome this problem, the optimization of analytical procedures has been carried out by using multivariate statistic techniques. Among the most relevant multivariate techniques used in analytical optimization is response surface methodology (RSM). Response surface methodology is a collection of mathematical and statistical techniques based on the fit of a polynomial equation to the experimental data, which must describe the behavior of a data set with the objective of making statistical previsions. It can be well applied when a response or a set of responses of interest are influenced by several variables. The objective is to simultaneously optimize the levels of these variables to attain the best system performance.

Before applying the RSM methodology, it is first necessary to choose an experimental design that will define which experiments should be carried out in the experimental region being studied. There are some experimental matrices for this purpose. Experimental designs for first-order models (e.g., factorial designs) can be used when the data set does not present curvature [3]. However, to approximate a response function to experimental data that cannot be described by linear functions, experimental designs for quadratic response surfaces should be used, such as three-level factorial, Box–Behnken, central composite, and Doehlert designs.

The present paper discusses the use of RSM for optimization in analytical chemistry. First, its basic principles are presented. Then, the approach to the applications of its more frequently used second-order experimental designs is broached, as well as the optimization of procedures that generate multiple responses.

Section snippets

Definition of some terms

Before beginning the discussion on the applications of response surface in the optimization of analytical methods, it is pertinent to introduce and define some key terms. Examples are also presented to illustrate each term.

  • Experimental domain is the experimental field that must be investigated. It is defined by the minimum and maximum limits of the experimental variables studied.

  • Experimental design is a specific set of experiments defined by a matrix composed by the different level combinations

Theory and steps for RSM application

Response surface methodology was developed by Box and collaborators in the 50s [4], [10]. This term was originated from the graphical perspective generated after fitness of the mathematical model, and its use has been widely adopted in texts on chemometrics. RSM consists of a group of mathematical and statistical techniques that are based on the fit of empirical models to the experimental data obtained in relation to experimental design. Toward this objective, linear or square polynomial

Full three-level factorial designs

Full three-level factorial design is an experimental matrix that has limited application in RSM when the factor number is higher than 2 because the number of experiments required for this design (calculated by expression N = 3k, where N is experiment number and k is factor number) is very large, thereby losing its efficiency in the modeling of quadratic functions. Because a complete three-level factorial design for more than two variables requires more experimental runs than can usually be

Multiple responses optimization in analytical chemistry by using RSM

It is relatively simple to find the optimal conditions for a single response using surface response designs. However, the researcher may be interested in optimizing several responses simultaneously. The simplest strategy to adopt in this case is visual inspection. If the amount of significant factors allows the graphical visualization of adjusted models, and if the numbers of response are not very large, the surfaces can be overlapped to enable finding the experimental region that can satisfy

Use of artificial neural networks in RSM

Artificial neural networks (ANNs) offer an attractive possibility for providing non-linear modeling for response surfaces and optimization in analytical chemistry.

ANNs are inspired by the arrangement of cerebral networks and consist of groups of highly interconnected processing elements called neurons. The neurons are arranged in a series of layers: one input layer with neurons representing independent variables, one output layer with neurons representing dependent variables, and several hidden

Conclusions

Application of response surface methodology in the optimization of analytical procedures is today largely diffused and consolidated principally because of its advantages to classical one-variable-a-time optimization, such as the generation of large amounts of information from a small number of experiments and the possibility of evaluating the interaction effect between the variables on the response.

In order to employ this methodology in experimental optimization, it is necessary to choose an

Acknowledgements

The authors acknowledge grants and fellowships from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Pró-Reitoria de Pesquisa e Pós-Graduação (PPG) from the Universidade Estadual do Sudoeste da Bahia (UESB).

References (83)

  • P.W. Araújo et al.

    Trends Anal. Chem.

    (1996)
  • T. Lundstedt et al.

    Chemometr. Intell. Lab. Syst.

    (1998)
  • D. Bas et al.

    J. Food Eng.

    (2007)
  • S.L.C. Ferreira et al.

    Anal. Chim. Acta

    (2007)
  • A.M.G. Campaña et al.

    Anal. Chim. Acta

    (1997)
  • E. Sivertsen et al.

    Chemom. Intell. Lab. Syst.

    (2007)
  • F. Marini et al.

    Microchem. J.

    (2008)
  • K. Novotná et al.

    J. Chromatogr. A

    (2005)
  • G. Saccani et al.

    Food Chem.

    (2005)
  • M.P. Davies et al.

    Anal. Chim. Acta

    (2004)
  • S. Hillaert et al.

    J. Chromatogr. A

    (2002)
  • M.P. Llompart et al.

    J. Chromatogr. A

    (1996)
  • A. Stafiej et al.

    J. Biochem. Biophys. Methods

    (2006)
  • A.S. Souza et al.

    Spectrochim. Acta B

    (2005)
  • M. Gfrerer et al.

    Anal. Chim. Acta

    (2005)
  • T. Mirza et al.

    J. Pharm. Biomed.

    (2001)
  • M.E.P. Hows et al.

    J. Chromatogr. A

    (1997)
  • L. Vidal et al.

    Anal. Chim. Acta

    (2007)
  • P.A.J. Rosa et al.

    J. Chromatogr. A

    (2007)
  • R. López et al.

    J. Chromatogr. B

    (2007)
  • M.C.V. Mamani et al.

    Talanta

    (2006)
  • E.D. Guerrero et al.

    J. Chromatogr. A

    (2006)
  • V. Yusà et al.

    Anal. Chim. Acta

    (2006)
  • A.B. Baranda et al.

    Talanta

    (2005)
  • N. Jalbani et al.

    Talanta

    (2006)
  • M.A. Bezerra et al.

    Anal. Chim. Acta

    (2006)
  • R.E. Santelli et al.

    Talanta

    (2006)
  • C.R.T. Tarley et al.

    Anal. Chim. Acta

    (2005)
  • S. Furlanetto et al.

    J. Pharm. Biomed.

    (1998)
  • P. Araujo et al.

    J. Chromatogr. A

    (2006)
  • A. Navalón et al.

    J. Chromatogr. A

    (2002)
  • C. Aguilar et al.

    J. Chromatogr. A

    (1999)
  • L. Mateus et al.

    J. Chromatogr. A

    (1998)
  • Y.Y. Hu et al.

    J. Chromatogr. A

    (2005)
  • C.H. Kuo et al.

    Anal. Chim. Acta

    (2003)
  • F. Bianchi et al.

    J. Chromatogr. A

    (2002)
  • R. Rodil et al.

    J. Chromatogr. A

    (2002)
  • A. Ceccato et al.

    J. Chromatogr. A

    (1998)
  • L.V. Candioti et al.

    Talanta

    (2006)
  • B. Jancić et al.

    J. Chromatogr. A

    (2008)
  • A.T.K. Tran et al.

    Talanta

    (2007)
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