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Analytica Chimica Acta 597 (2007) 179–186

Review

Box-Behnken design: An alternative for the optimizationof analytical methods

S.L.C. Ferreira a,∗, R.E. Bruns b, H.S. Ferreira a, G.D. Matos a, J.M. David a,G.C. Brandao a, E.G.P. da Silva a, L.A. Portugal a, P.S. dos Reis c,a,

A.S. Souza a, W.N.L. dos Santos c

a Universidade Federal da Bahia, Instituto de Quımica, Campus Universitario de Ondina, Salvador, Bahia 40170-290, Brazilb Universidade Estadual de Campinas, Instituto de Quımica, Campinas, Sao Paulo 13084-971, Brazil

c Universidade Do Estado da Bahia, Rua Silveira Martins, 2555, Cabula, Salvador-Bahia 41.195.001, Brazil

Received 15 May 2007; received in revised form 1 July 2007; accepted 3 July 2007Available online 23 July 2007

bstract

The present paper describes fundamentals, advantages and limitations of the Box-Behnken design (BBD) for the optimization of analyticalethods. It establishes also a comparison between this design and composite central, three-level full factorial and Doehlert designs. A detailed

tudy on factors and responses involved during the optimization of analytical systems is also presented. Functions developed for calculation ofultiple responses are discussed, including the desirability function, which was proposed by Derringer and Suich in 1980. Concept and evaluation

f robustness of analytical methods are also discussed. Finally, descriptions of applications of this technique for optimization of analytical methods

re presented.

2007 Elsevier B.V. All rights reserved.

eywords: Box-Behnken design; Multivariate optimization; Experimental design; Analytical methods; Desirability function; Robustness

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801.1. Application of multivariate techniques in analytical chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801.2. Factors and responses in multivariate optimization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801.3. The robustness of analytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.4. Box-Behnken design as a tool for multivariate optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

2. Application of Box-Behnken designs (BBD) for optimization of analytical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1822.1. Application of BBD for the optimization of the spectroanalytical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1822.2. Application of BBD for the optimization of chromatographic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832.3. Application of BBD for the optimization of capillary electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1842.4. Application of BBD for the optimization of electroanalytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1842.5. Application of BBD for the optimization of sorption process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

2.6. Other applications of BBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

∗ Corresponding author. Fax: +55 71 32355166.E-mail address: slcf@ufba.br (S.L.C. Ferreira).

003-2670/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.aca.2007.07.011

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80 S.L.C. Ferreira et al. / Analytica

. Introduction

.1. Application of multivariate techniques in analyticalhemistry

In recent years, chemometric tools have been frequentlypplied to the optimization of analytical methods, consider-ng their advantages such as a reduction in the number ofxperiments that need be executed resulting in lower reagentonsumption and considerably less laboratory work. Further-ore these methods allow the development of mathematicalodels that permit assessment of the relevance as well as sta-

istical significance of the factor effects being studied as wells evaluate the interaction effects between the factors. If thereignificant interaction effects between factors the optimal con-itions indicated by the univariate studies will be different fromhe correct results of the multivariate optimization. The larger thenteraction effects the greater the difference that will be foundsing univariate and multivariate optimization strategies. So thenivariate procedure may fail since the effect of one variablean be dependent on the level of the others involved in the opti-ization. That is why multivariate optimization schemes involve

esigns for which the levels of all the variables are changedimultaneously. The first step of multivariate optimization isccomplished screening the factors studied (full factorial or frac-ional factorial design) in order to obtain the significant effects ofhe analytical system. After determining the significant factors,he optimum operation conditions are attained by using moreomplex experimental designs such as Doehlert matrix (DM),entral composite designs (CCD) and three-level designs suchs the Box-Behnken design (BBD) [1–3].

In analytical chemistry, multivariate techniques have beenpplied to the optimization of chemical factors during theevelopment of analytical strategies involving pre-concentrationystems using solid phase extraction [4–10] cloud pointxtraction [11–13], liquid–liquid extraction [14,15] and co-recipitation [16]; procedures for sample digestion [17–20];ampling systems [21]; chromatographic methods [22–28];apillary electrophoresis [29] methods employing flow injec-ion analysis [30,31] and sequential injection analysis [32–35];lectroanalytical methods [36–39] and thermogravimetry [40].ther applications include the optimization of instrumentalarameters of equipment for analysis by graphite furnace atomicbsorption spectrometry (GF AAS) [41,42], inductively cou-led plasma optical emission spectrometry (ICP OES) [43,44]nd inductively coupled plasma mass spectrometry (ICP-MS)45]. Several review papers have been published on this subject46,47].

.2. Factors and responses in multivariate optimizationechniques

During the multivariate optimization procedure, there are two

ypes of variables: the responses and the factors. The responsesre the dependent variables. Their values depend on the levelsf the factors, which can be classified as qualitative or quanti-ative. In an optimization of a digestion process of lubricating

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ica Acta 597 (2007) 179–186

il samples using focused-microwave assistance for determina-ion of several metals employing ICP OES, the factors could be:ype of acid mixture (qualitative factor), amount of acid mix-ure, power applied and digestion time (quantitative factors).n the same study, several responses could be evaluated: (a)esidual acidity after digestion (this response is important con-idering the inconvenience of using very acidic solutions foruantification using ICP OES); (b) residual carbon (this param-ter reflects directly the efficiency of the mineralization processf the organic matrix); (c) quantification of the metals usingCP OES (this response can be evaluated by recoveries for eachetal and shows the efficiency of the digestion process, with-

ut considering the residual acidity and residual carbon). If onenows the natures of the relationships between the responsesnd the factors, i.e. the response surfaces, the optimal valuesf the factors can be determined. The optimization can be per-ormed in two ways. Response surfaces can be determined forach response and these surfaces can be analyzed simultane-usly. Or a model for a single composite function that takes intoccount all three responses can be determined to obtain a sin-le response surface. The advantages of each approach are stilleing investigated.

Another question that should be addressed is whether theifferent responses suffer similar effects on changing the fac-or levels. During the optimization of an analytical procedurenvolving a multielement technique (ICP OES, ICP-MS andhromatography) generally the composite response will be resul-ant of several single responses with similar effects. However,he digestion process described for quantification of metals inil samples clearly exemplifies a situation where the individualesponses have different effects. This aspect should be consid-red during the establishment of an appropriate optimizationtrategy.

An increasingly popular form for treating multiple responsesakes use of a desirability function D, which was proposed byerringer and Suich in 1980 [48]. Individual response surfaces

re determined for each response. Predicted values obtainedrom each response surface are transformed to a dimension-ess scale di . The scale of the desirability function rangesetween d = 0 (for an unacceptable response value) and d = 1for a completely desirable one). D is calculated combininghe individual desirability values by applying the geometric

ean: D = (d1 × d2 × . . . dm)1/m. An algorithm is then appliedo the D function in order to determine the set of variable val-es that maximize it. This function has been frequently useduring the optimization of analytical systems, which involveeveral responses. Garcia et al. optimized the chromatographiconditions for the determination of eight hormones employ-ng gas chromatography with mass spectrometry detection. Aesirability function was proposed for simultaneously optimiz-ng the resolution and the peak width of the separation process49]. Another paper used a desirability function involving theesponses, size and coefficient of variation of the analyti-

al signal, during the optimization of a flow injection systemith electrochemical detection for the determination of hydro-uinone in cosmetics [50]. Candioti et al. proposed a methodor separation and determination of four active ingredients in

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S.L.C. Ferreira et al. / Analytica

harmaceutical preparations employing capillary electrophore-is. They used a desirability function to simultaneously optimizeve responses: the three resolutions, the analysis time and theapillary current [51]. Ortiz et al. reported the use of a desir-bility function to optimize instrumental responses obtainedn analysis involving electroanalytical methods. Two applica-ions were discussed: (1) the simultaneous maximization ofhe peak current and minimization of its standard deviationor the determination of copper(II) by differential pulse anodictripping voltammetry; and (2) the simultaneous maximizationf the peak current and minimization of the blank signal forhe determinations of nickel(II) and indomethacin by adsorp-ive stripping voltammetry. In all these cases, the experimentalonditions for which the optima are found for each individualesponse are quite different so one is required to look for a com-romise solution, that can be achieved using the desirabilityunction [52]. A method was developed for the determinationf pesticide multiresidues by matrix solid-phase dispersion andas chromatography. The authors employed a desirability func-ion to optimize simultaneously pesticide recoveries and matrixleanup [53]. Concha-Herrera et al. [54] performed a chromato-raphic separation procedure for the determination of proteicrimary amino acids. They used a desirability function involv-ng resolution(s) between peaks (and) analysis time during theptimization step of method. Simultaneous optimization of theesolution and analysis time was achieved using Derringer’sesirability function for a method proposed for the determina-ion of of phenyl thiohydantoin amino acids employing micellariquid chromatography [55].

Other multi-response functions have been also proposed buthese are not based on the Derringer–Suich desirability function48]. These were established considering mainly the objectivef the analytical system being optimized. Then, during the opti-ization step of an on-line pre-concentration system for the

etermination of copper by flame atomic absorption spectrome-ry, Ferreira et al. proposed a multi-response function, involvingnalytical signal (absorbance) and pre-concentration time. Thisulti-response function was called “sensitivity efficiency” and

t has been defined as the analytical signal obtained for an on-ine enrichment system for a pre-concentration time of 1 min56]. This multi-response function was also used for the opti-ization of an on-line pre-concentration system proposed for the

etermination of lead using FAAS [57] and in another methoderformed for selective extraction and determination of cat-chol in water samples, using a polymeric sorbent based onolecular imprinting technology with subsequent determina-

ion by differential pulse voltammetry. The sensitivity efficiencyas determined considering the electrochemical signal and there-concentration time [37]. During the optimization of a pre-oncentration procedure using cloud point extraction for theetermination of six metal ions (cadmium, chromium, cop-er, manganese, nickel and lead) employing ICP OES, anotherultiple response function was found for obtaining of a simul-

aneous pre-concentration condition [58]. The behaviors of fivef these ions as a function of varying experimental condi-ions are highly correlated and can all adequately be describedy a first principal component whereas the nickel ion behav-

ica Acta 597 (2007) 179–186 181

or is quite different and is described by a second principalomponent.

.3. The robustness of analytical methods

In validation studies multivariate optimization techniques arelso used for determination of robustness, which is defined ashe capacity of an analytical method to reproduce results whenhe procedure is performed under small changes in the nominalalues of the experimental factors established in the optimizationtep [59,60,45,61–63].

.4. Box-Behnken design as a tool for multivariateptimization

Box-Behnken designs (BBD) [64] are a class of rotatabler nearly rotatable second-order designs based on three-levelncomplete factorial designs. For three factors its graphical rep-

Fig. 1. (a) the cube for BBD and three interlocking 22 factorial design (b).

182 S.L.C. Ferreira et al. / Analytica Chim

Table 1Coded factor levels for a Box-Behnken design of a three-variable system

Experiment x1 x2 x3

1 −1 −1 02 1 −1 03 −1 1 04 1 1 05 −1 0 −16 1 0 −17 −1 0 18 1 0 19 0 −1 −110 0 1 −111 0 −1 112 0 1 1C 0 0 0C 0 0 0C

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The number of experiments (N) required for the developmentf BBD is defined as N = 2k(k − 1) + C0, (where k is number ofactors and Co is the number of central points). For comparison,he number of experiments for a central composite design is

= 2k + 2k + C0. Tables 1 and 2 contain the coded values of theactor levels for BBD on three, four and five factors, respectively.

A comparison between the BBD and other response surfaceesigns (central composite, Doehlert matrix and three-level full

actorial design) has demonstrated that the BBD and Doehlertatrix are slightly more efficient than the central composite

esign but much more efficient than the three-level full fac-orial designs where the efficiency of one experimental design

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Table 2Coded factor levels for Box-Behnken designsfactors

ica Acta 597 (2007) 179–186

s defined as the number of coefficients in the estimated modelivided by the number of experiments. Table 3 establishes a com-arison among the efficiencies of the BBD and other responseurface designs for the quadratic model. This Table demonstrateslso that the three-level full factorial designs are costly when theactor number is higher than 2.

Another advantage of the BBD is that it does not containombinations for which all factors are simultaneously at theirighest or lowest levels. So these designs are useful in avoid-ng experiments performed under extreme conditions, for whichnsatisfactory results might occur. Conversely, they are not indi-ated for situations in which we would like to know the responsest the extremes, that is, at the vertices of the cube.

BBD for four and five factors can be arranged in orthogonallocks, as shown in Table 2. In this table, each (±1, ±1) com-ination within a row represents a full 22 design. Dashed lineseparate the different blocks. Because of block orthogonality, theecond-order model can be augmented to include block effectsithout affecting the parameter estimates, that is, the effects

hemselves are orthogonal to the block effects. This orthogonallocking is a desirable property when the experiments have toe arranged in blocks and the block effects are likely to be large.

. Application of Box-Behnken designs (BBD) forptimization of analytical systems

.1. Application of BBD for the optimization of thepectroanalytical method

Korn and de Oliveira used BBD during the optimization of

sequential injection analysis method proposed for the deter-ination of sulphate in ethanol automotive fuel employingolecular absorption spectrophotometry (MAS) [65]. Arau-

aria angustifolia (named pinhao) wastes were tested as solid

for optimizations involving four and five

S.L.C. Ferreira et al. / Analytica Chimica Acta 597 (2007) 179–186 183

Table 3Comparison of efficiency of central composite design (CCD), Box-Behnken design (BBD) and Doehlert design (DM)

Factors (k) Number of coefficients (p) Number of experiments (f) Efficiency (p/f)

CCD DM BBD CCD DM BBD

2 6 9 7 – 0.67 0.86 –3 10 15 13 13 0.67 0.77 0.774 15 25 21 25 0.60 0.71 0.605 21 43 31 41 0.49 0.68 0.616 28 77 43 61 0.36 0.65 0.467 578 73

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hase for extraction of chromium VI. BBD was used for opti-ization of the experimental factors [66]. Chemical factors ofmethod proposed for the quantification of amikacin in phar-aceutical formulations were optimized also using BBD [67].tero-Rey et al. used BBD for optimization of several experi-ental parameters on arsenic and selenium leaching from coaly ash samples and their determination using hydride generationoupled with atomic fluorescence spectrometry (HG AFS) [68].erreira et al. proposed two on-line pre-concentration systemsor the determination of cadmium and lead in drinking watery FAAS employing knotted reactor and 1-(2-pyridylazo)-2-aphthol (PAN) as complexing reagent. In both methods, theptimization step was performed using BBD [69,70]. Zougaght al. developed an on-line system for the pre-concentration andetermination of lead in water using ICP OES. The solid phasextraction process was optimized using BBD [71]. Bosque-endra et al. [72] described the advantages of the BBD asesponse surface methodology for obtaining second order mod-ls in full detail. They used this design for re-optimization ofhe pararosaniline classical method for the determination oformaldehyde employing MAS. A comparison with the classicalethod [73], which was optimized using a univariate strategy,

evealed that the re-optimized procedure using BBD has a sensi-ivity almost twice as large as the univariate result. BBD was usedlso for the optimization of the factors of a direct-current plasmaystem. The factors involved were horizontal position, vertical

osition, nebulizer pressure and electrode sleeve pressure. Threeesponses (precision, drift and sensitivity) were evaluated. Allhe optimization was carried out by quantification of the cop-er signal. An evaluation of the performance of the equipment

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able 4pplication of BBD for the optimization of spectroanalytical methods

nalyte Sample Anal

ulphate Ethanol automotive fuel SIA/hromium VI Water MASmikacin Pharmaceutical formulations Chemrsenic and selenium Coal fly ash HG Aadmium Drinking water KR-Fead Drinking water KR-Fead Water SPE-ormaldehyde – MAS

equential injection analysis (SIA); molecular absorption spectrophotometry (MASpectrometry (HG AFS); solid phase extraction (SPE); inductively coupled plasma op

85 0.25 0.63 0.42113 0.16 0.62 0.40

sing the optimized conditions was performed for six other ele-ents [74]. Table 4 summarizes applications of BBD for the

ptimization of chemical systems involving spectroanalyticalechniques.

.2. Application of BBD for the optimization ofhromatographic methods

Carasek and coworkers optimized a microextraction pro-ess for the determination of 2,4,6-trichloroanisole and,4,6-tribromoanisole in wine samples employing BBD [75].yrzynska and coworkers used BBD for optimization of theerivatization reaction established during the development of aethod proposed for the determination of aliphatic aldehydes byPLC [76]. BBD was used for optimization of a procedure usingicrowave-assisted extraction proposed for the determination of

ersistent organochlorine pesticides in sediment using GC–MS77]. McKenon and coworkers used BBD for optimization ofethod involving supercritical fluid extraction for the determi-

ation of fatty acid composition of castor seeds using GC-FID78]. The separation process performed for the determination ofaptopril in pharmaceutical tablets using HPLC was optimizedlso using BBD [79]. Petz and coworkers proposed a CG–MSethod for the determination of aminoglycoside antibiotics.BD was used for the optimization of the derivatization reac-

ion [80]. Walters and Qiu employed BBD for the optimization

f the separation process of hydroxamates (arginine, leucine,hreonine, histidine and Tryptophan) using paper chromatogra-hy [81]. Table 5 presents applications of BBD for optimizationf chromatographic methods.

ytical technique Optimized parameters References

MAS Instrumental factors [65]Chemical factors [66]

iluminescence Chemical factors [67]FS Chemical factors [68]AAS Chemical factors [69]AAS Chemical factors [70]ICP OES Chemical factors [71]

Chemical factors [72]

); knotted reactor (KR); hydride generation coupled with atomic fluorescencetical emission spectrometry (ICP OES).

184 S.L.C. Ferreira et al. / Analytica Chimica Acta 597 (2007) 179–186

Table 5Application of BBD for the optimization of chromatographic methods

Analyte Sample Chromatographic technique Optimized process References

2-4-6 TCA, 2-4-6 TBA Wine GC-ECD Extraction step [75]Aliphatic aldehydes – HPLC Derivatization reaction [76]Organochlorine pesticides Sediments GC–MS Extraction step [77]Fatty acid composition Castor oil GC-FID Extraction step [78]Captopril Pharmaceutical tablets HPLC Separation step [79]Aminoglycoside antibiotics – GC–MS Derivatization reaction [80]Aminoacids hydroxamates PC Separation step [81]

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,4,6-trichloroanisole (2-4-6TCA); 2,4,6-tribromoanisole (2-4-6TBA); gas chhromatography (HPLC); gas chromatography–mass spectrometry (GC–MS); gPC).

.3. Application of BBD for the optimization of capillarylectrophoresis

Gong and coworkers compared response surfaces based onomplementary three-level Box-Behnken, face-centered centralomposite and full factorial designs during the optimization ofrocedures employing capillary electrophoresis (CE) for theetermination of tamsulosin enantiomers [82] and also ascor-ic acid and isoascorbic acid [83]. These authors concludedhat the optimizations using the three designs furnished similarnd efficient results. A procedure for neuropeptide separationmploying CE was optimized using BBD [84]. Ragonese etl. performed a separation procedure for the determination ofthambutol hydrochloride in pharmaceutical formulations usingE. The optimization step was carried out using BBD [85].ows et al. proposed a method for separation and determina-

ion of sulphonamides, dihydrofolate reductase inhibitors andeta-lactam antibiotics also employing CE. The experimentalactors were optimized using BBD [86].

.4. Application of BBD for the optimization oflectroanalytical methods

BBD has not been used for the optimization of electro-nalytical methods. Only a procedure employing adsorptiontripping voltammetry for the determination of nalidxic acid and-hydroximethylanalidxic acid in urine samples was establishedsing this design [87].

.5. Application of BBD for the optimization of sorptionrocess

Kannan et al. proposed the use of straw carbon for thedsorption of the copper(II), cadmium(II) and nickel(II) metalons. The optimization step was carried out using BBD [88].

adaria et al. applied carbon aerogel for electrolytic removalf mercury from aqueous solutions. Experimental factors wereptimized employing BBD [89]. Activated carbon immobi-ized with Pseudomonas putida was evaluated as solid phase

or extraction of phenol. The optimization step was performedsing BBD [90]. The optimization of the adsorption process ofhodamine 6G from water using both chitosan and activatedarbon was performed also using BBD [91]. BBD was used for

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ography and electron-capture detection (GC-ECD); high performance liquidromatography with flame ionisation detection GC-FID; paper chromatography

he optimization of the sorption process of verofix red using aiopolymer [92].

.6. Other applications of BBD

Matthews et al. used BBD for the optimization of an enzy-atic procedure for the determination of arsenic in aqueous

olutions [93].Silva and coworkers developed a study in order to detect

he most important factors that effect the formation of the fourrihalomethanes (THM) (chloroform, bromodichloromethane,hlorodibromomethane and bromoform) in water disinfectionrocesses using chlorine. BBD was used during the optimizationtep [94].

Petz and Lamar developed a receptor protein microplatessay for the detection and determination of penicillins andephalosporins with intact beta-lactam in milk, bovine andorcine muscle juice, honey and egg samples. The optimizationtep was performed using BBD [95].

Wu and coworkers developed a photoelectrocatalytic oxida-ion system using a Ti/TiO2 electrode for the degradation ofulvic acid (FA). The optimization step was carried out usingBD [96].

BBD was employed for the optimization of an electrochem-cal process using reticulated vitreous carbon-supported-on-olyaniline cathodes for the reduction of hexavalent chromiumf industrial wastewater samples [97].

Rajkumar et al. investigated the electrochemical oxidationrocess of phenol using a Ti/TiO2–RuO2–IrO2 anode. Thexperimental factors were optimized using BBD [98].

. Conclusions

The Box-Behnken is a good design for response surfaceethodology because it permits: (i) estimation of the param-

ters of the quadratic model; (ii) building of sequential designs;iii) detection of lack of fit of the model; and (iv) use of blocks.

A comparison between the Box-Behnken design and otheresponse surface designs (central composite, Doehlert matrix

nd three-level full factorial design) has demonstrated that theox-Behnken design and Doehlert matrix are slightly more effi-ient than the central composite design but much more efficienthan the three-level full factorial designs.

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S.L.C. Ferreira et al. / Analytica

The application of Box-Behnken designs for optimization ofnalytical methods still is limited, but different applications ofhese designs in the optimization of procedures involving severalnalytical techniques are presented in this paper.

cknowledgements

The authors are grateful for Conselho Nacional de Desen-olvimento Cientıfico e Tecnologico (CNPq), Coordenacao deperfeicoamento de Pessoal do Ensino Superior (CAPES), andundacao de Amparo a Pesquisa do Estado da Bahia (FAPESB)or financial support.

eferences

[1] G.E.P. Box, J.S. Hunter, W.G. Hunter, Statistics for Experimenters, seconded., Wiley-Interscience, New York, 2005.

[2] R.E. Bruns, I.S. Scarminio, B.B. Neto, Statistical Design—Chemometrics,Elsevier, Amsterdam, 2006.

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