Usefulness of Information Criteria for the Selection of Calibration

Article pubs.acs.org/ac

Usefulness of Information Criteria for the Selection of Calibration Curves E. Rozet,*,† E. Ziemons,† R.D. Marini,† and Ph. Hubert† †

Laboratory of Analytical Chemistry, Institute of Pharmacy, CIRM, University of Liege (ULg), CHU, B 36, B-4000 Liège, Belgium S Supporting Information *

ABSTRACT: The reliability of analytical results obtained with quantitative analytical methods is highly dependent on the selection of the adequate model used as the calibration curve. To select the adequate response function or model the most used and known parameter is to determine the coefficient R2. However, it is well-known that it suffers many inconveniences, such as leading to overfitting the data. A proposed solution is to use the adjusted determination coefficient Radj2 that aims at reducing this problem. However, there is another family of criteria that exists to allow the selection of an adequate model: the information criteria AIC, AICc, and BIC. These criteria have rarely been used in analytical chemistry to select the adequate calibration curve. This works aims at assessing the performance of the statistical information criteria as well as R2 and Radj2 for the selection of an adequate calibration curve. They are applied to several analytical methods covering liquid chromatographic methods, as well as electrophoretic ones involved in the analysis of active substances in biological fluids or aimed at quantifying impurities in drug substances. In addition, Monte Carlo simulations are performed to assess the efficacy of these statistical criteria to select the adequate calibration curve.

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proved their usefulness for selecting models in many scientific areas.16−21 These criteria are the Akaike’s Information Criteria (AIC),22 the small sample adjusted information criteria (AICc),23 and the Bayesian Information Criteria (BIC or Schwartz criteria).24 Nonetheless, these criteria have been rarely used to orientate the selection of the response function for the calibration curve of analytical methods. The purpose of this work is to compare the performance of the statistical information criteria, as well as R2 and Radj2, for the selection of an adequate calibration curve. First, the definition, computation, and interpretation of these statistical criteria are given. Then, these criteria are applied to several analytical methods covering liquid chromatographic methods, as well as electrophoretic ones, involved in the analysis of active substances in biological fluids or aimed at quantifying impurities in drug substances. In addition, Monte Carlo simulations are performed to assess the efficacy of these statistical criteria to select the adequate calibration curve.

ost quantitative analytical methods such as chromatographic, electrophoretic, or spectrophotometric ones require defining a calibration curve to relate the detector signal to the concentration of the target analyte.1−6 This calibration curve, sometimes called standard curve or response function, is then used to back calculate the concentration of the target analyte present in an unknown sample. Calibration curves are, hence, keystones of quantitative analytical methods. The main models used as calibration curves for analytical methods are simple linear model, weighted linear model, linear model after transformation of both the response and the concentration, where the most common transformation used are the logarithmic or the square root ones, simple quadratic model, and weighted quadratic model.7−9 The selection of the adequate calibration curve for a defined analytical procedure is, hence, essential. Several statistical parameters can be used to evaluate the suitability of a response function. The most known one is certainly the determination coefficient (R2).10 However, it is recognized that this coefficient is not adequate and generally leads to overfitting the data.8−15 Hence, another criterion adapted from this last one is the adjusted determination coefficient (Radj2) that takes into account the number of estimated parameters present in the response function.10 This parameter allows for a balance between model complexity defined by the number of parameters to estimate for a model and quality of fit. If two models have exactly the same fit to the data, the Radj2 value of the model with the least number of parameters to estimate will be higher than that of the model with more parameters to estimate. It hence favors parsimony and reduces risk of overfitting. Nonetheless, another group of statistical parameters, namely the statistical information criteria, © 2013 American Chemical Society

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THEORY Determination Coefficients R2 and Radj2. The determination coefficient R2 and the adjusted determination coefficients Radj2 are all based on the following decomposition:10 n

n

n

∑ (yi − y ̅ )2 = ∑ (yi ̂ − y ̅ )2 + ∑ (yi − yi ̂ )2 i=1

i=1

i=1

Received: February 28, 2013 Accepted: June 3, 2013 Published: June 3, 2013 6327

dx.doi.org/10.1021/ac400630k | Anal. Chem. 2013, 85, 6327−6335

Analytical Chemistry

Article

Table 1. Values of the information Criteria AIC, AICc, and BIC, as well as those of the R2 and Radj2 for the Four Analytical Methods Studied (See Text for Details) and for the Eight Response Functions.a response function

AIC

AICc

BIC

R2

R2 adjusted

Quad1_X2 Lin1_X2 Quad1_X Lin1_X Quad Lin Sqrt Log Quad1_X2 Lin1_X2 Quad1_X Lin1_X Quad Lin Sqrt Log Quad1_X2 Lin1_X2 Quad1_X Lin1_X Quad Lin Sqrt Log Quad1_X2 Lin1_X2 Quad1_X Lin1_X Quad Lin Sqrt Log

635.145 652.364 645.530 651.775 727.732 727.433 725.259 725.509 29.774 36.916 77.233 78.189 139.770 167.348 162.104 152.976 563.882 563.648 553.969 553.767 576.066 575.204 574.012 574.371 −68.010 −69.118 −73.381 −73.385 −65.025 −65.353 −65.570 −65.938

636.745 653.287 647.130 652.699 729.332 728.356 726.182 726.432 31.064 37.666 78.523 78.939 141.060 168.098 162.854 153.726 565.172 564.398 555.260 554.517 577.357 575.954 574.762 575.121 −64.933 −67.403 −70.304 −71.671 −61.948 −63.639 −63.856 −64.224

640.750 656.567 651.135 655.979 733.337 731.637 729.463 729.713 36.108 41.666 83.567 82.939 146.104 172.099 166.855 157.726 570.216 568.398 560.303 558.518 582.400 579.954 578.762 579.121 −64.448 −66.447 −69.819 −70.714 −61.463 −62.682 −62.899 −63.267

0.99404 0.98869 0.99932 0.99911 0.99907 0.99902 0.99967 0.99958 0.99587 0.99468 0.99799 0.99782 0.99905 0.99785 0.99888 0.99946 0.99927 0.99923 0.99978 0.99976 0.99980 0.99979 0.99983 0.99974 0.99810 0.99801 0.99927 0.99918 0.99906 0.99897 0.99919 0.99906

0.99360 0.98829 0.99927 0.99908 0.99901 0.99899 0.99966 0.99957 0.99562 0.99452 0.99787 0.99775 0.99900 0.99778 0.99885 0.99944 0.99922 0.99921 0.99976 0.99976 0.99978 0.99978 0.99982 0.99973 0.99785 0.99788 0.99917 0.99913 0.99894 0.99891 0.99914 0.99900

Enro plasma

HMF plasma

LC Timolol

NACE Timolol

a Quad 1_X2: weighted 1/X2 quadratic model, Lin 1_X2: weighted 1/X2 linear model, Quad 1_X: weighted 1/X quadratic model, Lin 1_X: weighted 1/X linear model, Quad: quadratic model, Lin: Linear model, Sqrt: linear model after square root transformation, Log: linear model after logarithmic transformation. The data in bold gives the model selected by each of the five statistics studied.

Information Criteria AIC, AICc, and BIC. Generally speaking, information criteria such as the Akaike information criterion (AIC), the small sample Akaike information criteria (AICc), and the Bayesian information criteria (BIC) are a measure of the relative goodness of fit of a statistical model.19 They are all based on the general information theory and offer a relative measure of the information lost when a given model is used to describe the observed phenomenon. AIC, AICc, and BIC values can then be used for model selection.16−21,25 For deeper theoretical details about the information criteria, we recommend the book written by Burnham and Anderson.19 The formulas of AIC, AICc, and BIC are

that can be symbolically written as SST = SSR + SSE. SSR is the regression sum of squares: SSR = Σin= 1(ŷi − y)̅ 2, SST is the total sum of squares: SST = Σin= 1 (yi − y)̅ 2, and SSE is the error sum of squares: SSE = Σin= 1(yi − ŷi)2. yi is the ith observed response of the analytical method, ŷi is the ith response predicted by the regression model, and y ̅ is the average value of all the observed responses obtained. n is the total sample size. The coefficient of determination is then obtained using R2 =

SSR SSE =1− SST SST

The adjusted determination coefficient is obtained by R adj

2

̂ + 2K AIC = − 2 log[S(y|θ )]

SSE (n − 1) =1− SST (n − p)

where K is the number of estimable parameters in the model and S(y|θ )̂ is the likelihood of the data.

where p is the number of parameters in the regression model. These two parameters have values included in the interval [0, 1]. The response function to retain when using R2 and Radj2 should be the one with a value of the parameters the closest to 1. As can be seen for Radj2, the correction factor [(n − 1)/(n − p)] is increasing when parameters are added (p increases), showing that Radj2 can decrease when the model adjusted is more complex.

AICc = AIC +

2K (K + 1) n−K−1

Where n is the total sample size. ̂ + K log(n) BIC = − 2 log[S(y|θ )] 6328



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Figure 1. Relative total error values obtained from the validation standards using the eight response functions tested for (a) the HPLC method with fluorescence detection quantifying enrofloxacine in pigs plasma, (b) the quantification of hydroxymethylfurfural (HMF) in human plasma using an HPLC-UV method, (c) the quantification of R-timolol in the drug substance S-timolol, using HPLC, and (d) the use of nonaqueous capillary electrophoresis. Quad 1_X2: weighted 1/X2 quadratic model, Lin 1_X2: weighted 1/X2 linear model, Quad 1_X: weighted 1/X quadratic model, Lin 1_X: weighted 1/X linear model, Quad: quadratic model, Lin: linear model, Sqrt: linear model after square root transformation, Log: linear model after logarithmic transformation.

Given a set of candidate models fitted to the data, the model to select is the one with the minimum AIC, AICc, or BIC value.

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parameters to define the adequate calibration curve. Two of them concern bioanalytical methods. The first one is aimed at quantifying enrofloxacine in the plasma of pigs, using a liquid chromatographic method with fluorescence detection.26 The second bioanalytical method aims at quantifying hydroxymethylfurfural (HMF) in human plasma using an HPLC (high-

MATERIALS AND METHODS

Analytical Methods. Four analytical methods previously published are used to assess the use of these 5 statistical 6329



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performance liquid chromatography)-UV method.27 The two last analytical methods concern the quantification of R-timolol in the drug substance S-timolol, using either HPLC28 or non aqueous capillary electrophoresis (NACE).29−31 These methods were selected as they are applied over a wide concentration range, where several models could be used as an adequate calibration curve. In addition, as they were fully validated, the comparison of the selection of the adequate calibration curves using the three information criteria and the two determination coefficients can be compared to the quantitative performances of these methods in terms of relative total error defined as |δ̂| + σ̂IP with δ̂ the relative bias and σ̂IP the intermediate precision relative standard deviation (RSD).32 Response Functions. The eight response functions tested in all cases are as follows: the simple least-squares linear model, y = β0 + β1x, the weighted least-squares linear model using weights wi = (1/xi), where xi is the ith concentration level of the analyte in the calibration standard, the weighted least-squares linear model using weights wi = (1/xi)2, the simple linear least-squares models after logarithmic transformation of both the response (y) and the concentration (x), the simple linear least-squares models after the square root transformation of both the response (y) and the concentration (x), the least square quadratic model: y = β0 + β1x + β2x2, the weighted least-squares quadratic model using weights, wi = (1/xi), and the weighted least-squares quadratic model using weights, wi = (1/xi)2. They represent the most used models for calibration in analytical chemistry. Simulations. To assess the reliability of the three information criteria as well as the R2 and Radj2, several Monte Carlo simulations were performed. Three true response functions were used during the simulations. First, the responses yi were generated from the following Gaussian distribution: yi ∼ N[(68.2 + 213.2xi, (25xi)2], which represent a linear model with variance increasing with x2i , the square of the concentration. The second case was created by generating responses yi from another Gaussian distribution: yi ∼ N(−164.2 + 6.2xi − 0.0014x2i , 52), which represents a quadratic model with constant variance. The third case is a worst case scenario where the true response function is a four parameter logistic (4PL) function and the responses yi are generated from the following Gaussian distribution: ⎛ ⎜ 19 yi ∼ N ⎜1 + 750 ⎜⎜ 1+ x ⎝ i

Article

RESULTS AND DISCUSSION

Analytical Methods Results. Table 1 gives the values of the three information criteria, as well as those of the R2 and Radj2 for the four analytical methods studied. The data in bold give the model selected by each of the five statistics studied. As seen in this table, the three information criteria always agree with each other and the two determination coefficients also. However, there is never a selection of the same regression model with the information criteria and the determination coefficients. The information criteria select the 1/X2 weighted quadratic model for the methods determining HMF in human plasma and enrofloxacin in pig plasma, while the determination coefficients select the linear model after logarithmic transformation for HMF and the one with the square root transformation for enrofloxacin. The AIC, AICc, and BIC criteria select the weighted 1/X linear model for the two methods quantifying R-timolol. The linear model after square root transformation is chosen by the determination coefficients for the HPLC method and the weighted 1/X quadratic model for the NACE one. As there is not always agreement in the regression model selected between the various statistics, the reliability of the quantitative performances of the analytical method will be highly dependent on the calibration curve selected. Figure 1 shows the quantitative performances obtained by the back-calculation of independent validation standards for the four analytical methods with the eight response functions used as the calibration curve. These quantitative performances are the relative bias, the intermediate precision RSD, and the relative total error. The three response functions that generate the smallest total error are the linear regression after logarithmic transformation, the weighted 1/X2 quadratic regression, as well as the weighted 1/X2 linear function for the quantification of enrofloxacin in human plasma. The model selected with the AIC, AICc, and BIC for enrofloxacin is also one model providing the best total error values: the weighted 1/X2 quadratic model. However, the response function selected for this analytical procedure by the determination coefficients (square root transformation) gives a total error of more than 20% for the smallest concentration. The information criteria for this model, as well as the linear regression after logarithmic transformation, the simple linear and quadratic models, are considered the worst ones. Indeed, these last two models have more than 30% total error for some of their concentration levels, as shown in Figure 1a. In the case of HMF quantification in human plasma, the information criteria selected also give one of the three models that provide the smallest total error: less than 10% for the weighted 1/X2 quadratic model (selected by AIC, AICc, and BIC), weighted 1/X2 linear model, and linear model after logarithmic transformation (Figure 1b). In this case, the determination coefficients have also allowed for the selection of a regression model with less than 10% total error. The models selected using the three information criteria for the two analytical procedures aiming at quantifying R-timolol in S-timolol are also those providing small values of total error at all concentration levels (Figure 1, panels c and d). Those selected with the use of the coefficient determinations are not the worst ones in terms of value of total error, but neither are they the best ones. Finally, there are not many differences in the quantitative performances of all the eight response functions for the R-timolol HPLC and NACE analytical methods. Simulation Results. The applications of the five statistics to the different analytical methods studied suggests that the information criteria generally provide the selection of an

⎞ ⎟ 2 ⎟ , 0.1 10 ⎟⎟ ⎠

( )

With this last scenario, none of the tested response functions corresponds to the true model. It will assess which response function will be defined as the most acceptable one using the five statistics studied. Four concentration levels of the calibration standards xi were tested: 30, 60, 300, and 1500 for the three models. Several sample sizes were also tested: 2, 4, 6, or 8 replicates per calibration standard. For each model and for each sample size, 2000 simulations were performed that involved the adjustment of each of the 8 response functions and the computations of AIC, AICc, BIC, R2, and Radj2. The distributions of these last five statistics are then assessed to find which one selects the right model. Software. All computations and simulations have been performed using the statistical language R 2.15.0 for Windows.33 6330



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Figure 2. Simulation results for the weighted linear model with two replicates per concentration level, giving the median values (red circle) of (a) AIC, (b) AICc, (c) BIC, (d) R2, and (e) R2adj and the 95% interval of their values (blue squares) obtained for the eight models tested: Quad 1_X2: weighted 1/ X2 quadratic model, Lin 1_X2: weighted 1/X2 linear model, Quad 1_X: weighted 1/X quadratic model, Lin 1_X: weighted 1/X linear model, Quad: quadratic model, Lin: linear model, Sqrt: linear model after square root transformation, Log: linear model after logarithmic transformation.

adequate response function as a calibration curve opposite to the coefficients of determination. This is corroborated by the good quantitative performances obtained by the models selected by AIC, AICc, or BIC. Nonetheless, in these case studies, the “true” response function is not known, and it is impossible to assess if the statistics studied allows selecting this true response function or at least providing a model close to it. To allow this comparison, a simulation study has to be performed. Three “true” models are selected for these simulations. The first two models selected can be very useful in analytical methods applied to a relatively large concentration range such as in bioanalyses, environmental or trace analyses, or in the case of the determination of impurities or degradation products in drug products. First, the weighted linear regression is a case where the variance of the response increases with the concentration. This is a situation not uncommon when working on large concentration ranges.34−45 The second “true” model is a quadratic regression that also reflects a highly observed behavior of analytical techniques such as atomic absorption spectroscopy, LC-MS/MS, or GC/MS/MS, among others.34−45

The third case where a 4PL model as the true model is used to understand the behavior of the five statistics in cases where none of the eight response functions studied is the correct model. Weighted Linear Regression. As can be seen in Figure 2, which concerns the simulations for the weighted linear model with two replicates per concentration level, the values of the three information criteria are the smallest for the weighted linear and weighted quadratic models with weights 1/X and 1/X2 (Figure 2, panels a−c). For AIC and BIC, the smallest median values are obtained for the weighted 1/X2 linear and quadratic models, while for AICc, the weighted 1/X and 1/X2 linear models have the smallest AICc values. When increasing the number of replicates per concentration level, a clear trend can be seen (Figures S-1−S-3 of the Supporting Information): the smallest median values of the three information criteria are obtained for the weighted 1/X2 linear and quadratic models and their dispersion is also the smallest. However for the R2 and R2adj, the weighted 1/X2 linear and quadratic models have the smallest median values when 2 6331



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Figure 3. Simulation results for the quadratic model with two replicates per concentration level, giving the median values (red circle) of (a) AIC, (b) AICc, (c) BIC, (d) R2, and (e) Radj2 and the 95% interval of their values (blue squares) obtained for the eight models tested: Quad 1_X2: weighted 1/X2 quadratic model, Lin 1_X2: weighted 1/X2 linear model, Quad 1_X: weighted 1/X quadratic model, Lin 1_X: weighted 1/X linear model, Quad: quadratic model, Lin: Linear model, Sqrt: linear model after square root transformation, and Log: linear model after logarithmic transformation.

replicates per concentration level are used (Figure 2, panels d and e). When sample size increases, this is still the case (Figures S-1− S-3 of the Supporting Information). The models showing the highest median values of R2 and Radj2 are in all cases the linear model after logarithmic transformation or with the square root transformation. These two models also have the smallest dispersion of values of R2 and Radj2. Quadratic Regression. In the case of the simulations with the quadratic model with two replicates per concentration level, the values of AIC, AICc, and BIC clearly delimit two clusters of models. The first cluster includes the unweighted quadratic response function, as well as the two weighted ones which have the smallest median values of the information criteria, as seen in Figure 3 (panels a to c) for the case of 2 replicates per concentration level. The other cluster is obtained for five others models, which are characterized by higher median values of the AIC, AICc, and BIC. The information criteria clearly indicate that the quadratic models should be selected as adequate calibration curves. In addition, the model with the smallest

median value of the information criteria is the simple quadratic model, which is the real model used for the simulations. All these observations are equally valid when the sample size is increased (Figures S-4−S-6 of the Supporting Information). The R2 and Radj2 also give preference to the three quadratic regression models (unweighted, weighted by 1/X, and 1/X2), as shown in Figure 3 (panels d and e), for the case of two replicates per concentration level. In comparison to the information criteria, the distinction between the three quadratic models and the other five ones is weaker. For example, the simple linear model also has a high R2 and Radj2 value (>0.99). Finally, it is impossible to choose between these three quadratic models using the determination coefficients. They do not allow one to properly discriminate them, as shown in Figure 3 (panels d and e). Increasing the sample size per concentration level does not improve the possibility to select the adequate regression model between the three quadratic ones (Figures S-4−S-6 of the Supporting Information). 6332



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Figure 4. Simulations results for the four parameter logistic model with two replicates per concentration level, giving the median values (red circle) of (a) AIC, (b) AICc, (c) BIC, (d) R2, and (e) Radj2 and the 95% interval of their values (blue squares) obtained for the eight models tested: Quad 1_X2: weighted 1/X2 quadratic model, Lin 1_X2: weighted 1/X2 linear model, Quad 1_X: weighted 1/X quadratic model, Lin 1_X: weighted 1/X linear model, Quad: quadratic model, Lin: Linear model, Sqrt: linear model after square root transformation, and Log: linear model after logarithmic transformation.

also has a relatively high R2 and Radj2 value (>0.96). Finally, it is impossible to choose between these three quadratic models, using the determination coefficients. They do not allow for proper discrimination, as shown in Figure 4 (panels d and e). Finally, this simulation study also highlights that from a set of candidate models; the information criteria allows for the selection of the one which is the closest to the true model, even if the true model is not included in this set.

Four Parameters Logistic Function. Figure 4 provides the results obtained with a four parameter logistic function as the true model and two replicates per concentration level. None of the candidate calibration models include this true relationship. The values of AIC, AICc, and BIC here also distinguish the three quadratic models from all others. These quadratic models have the smallest median values of the information criteria, as seen in Figure 4 (panels a to c). The other group is obtained for the five others models, which are characterized by higher median values of the AIC, AICc, and BIC. In addition, the information criteria clearly indicate that the unweighted quadratic model should be selected as an adequate calibration curve. The R2 and Radj2 also give preference to the three quadratic regression models (unweighted, weighted by 1/X, and 1/X2), as shown in Figure 3 (panels d and e), for the case of two replicates per concentration level. In comparison to the information criteria, the distinction between the three quadratic models and the other five ones is again weaker. For example, the simple linear model

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CONCLUSION In the light of the simulation studies and case studies, it can be concluded that using the determination coefficients R2 and Radj2 do not allow one to properly select an adequate response function for the calibration curve. Conversely, the information criteria AIC, AICc, and BIC have greater potential to choose among a set of models a response function that will provide acceptable quantitative performance to the analytical procedure. This is essential since the quantitative performance and reliability 6333



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of the results generated by an analytical procedure is highly dependent on its standard curve. The results of this work have not highlighted great differences between the three information criteria. The AIC and AICc are frequentist point of view, while BIC is a Bayesian one. AICc is generally recommended for small sample sizes in place of AIC. Nonetheless, for the case studies and during the simulations, these differences did not generate any practical divergences. The information criteria should then be preferred rather than the determination coefficients to orientate the selection of the calibration curve model. The role of the sample size used to build the calibration curve is also crucial, as shown by the simulations. Another point highlighted by the simulations is that if the models selected as candidate calibration curves do not include the true model, the information criteria allow one to better identify the best candidate model among those available. These criteria have a greater discriminating power than the two determination coefficients studied. Finally, it has to be noted that even if the calibration curve has been selected using one of the three information criteria, for example during a prevalidation step or during method development, a formal validation of the analytical procedure, and hence of its calibration curve, should be done with independent validation standards or quality control samples, at least in order to experimentally confirm this choice. Indeed, there is always the risk of having missed the true relationship between the concentration and signal, as highlighted by the simulations using the 4PL function. Those criteria do not provide a measure of the overall quantitative performance of the analytical procedure or of the reliability of the results obtained with it.

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ASSOCIATED CONTENT

S Supporting Information *

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +32-4-3664320. Fax: +32-43664317. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are very grateful to the anonymous reviewers for providing important comments that led to significant improvements of this article. Financial support from the Walloon Region of Belgium is gratefully acknowledged for the funding of E.R. with the CWALity convention funds N° 1217614 (iGUARANTEE).

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REFERENCES

(1) Marini, R. D.; Rozet, E.; Montes, M. L. A.; Rohrbasser, C.; Roht, S.; Rhème, D.; Bonnabry, P.; Schappler, J.; Veuthey, J.-L.; Hubert, P.; Rudaz, S. J. Pharm. Biomed. 2010, 53, 1278−1287. (2) Cavalier, E.; Rozet, E.; Dubois, N.; Charlier, C.; Hubert, P.; Chapelle, J.-P.; Krzesinski, J.-M.; Delanaye, P. Clin. Chim. Acta 2008, 396, 80−85. (3) Rozet, E.; Mertens, B.; Dewe, W.; Ceccato, A.; Govaerts, B.; Boulanger, B.; Chiap, P.; Streel, B.; Crommen, J.; Hubert, Ph. J. Pharm. Biomed. 2006, 42, 64−70. (4) Peters, F. T.; Maurer, H. H. Anal. Chem. 2007, 79, 4967−4976. (5) Almeida, A. M.; Castel-Branco, M. M.; Falcão, A. C. J. Chromatogr., B 2002, 774, 215−222. 6334



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