A Geometric Approach to Robustness Testing in Analytical HPLC

Transcription

1 7 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 JANUARY 00 A Geometric Approach to Robustness Testing in Analytical HPLC The authors describe a new approach to verifying robustness testing in high performance liquid chromatography. They use the linear interpretation of the response obtained from an experimental domain close to the procedural conditions. The authors applied the procedure, based upon multiple linear regression analysis of a data set obtained from a well-balanced experimental design, to the liquid chromatography determination of biogenic amines as dabsyl derivatives. The results are in agreement with those obtained from the traditional effect definition. R. Romero, D. Gázquez, M. Sánchez-Viñas, L. Cuadros Rodríguez, and M.G. Bagur Department of Analytical Chemistry, Faculty of Sciences, University of Granada, E-18071, Granada, Spain. Address correspondence to D. Gázquez. Selecting the best set of operational conditions for an analytical method, in general, and for a chromatographic method, in particular, often is a complex task because of the large number of variables that must be optimized. Even when these factors are chosen as the optimum values, the difficulty in accurately controlling them leads to interlaboratory bias and day-to-day intralaboratory variations. Therefore, it is convenient to check the inertia of a method as another intrinsic performance characteristic of an analytical validation (1). This attribute can be checked using an interlaboratory trial ruggedness or an intralaboratory study robustness. Ruggedness should be used in an analytical chromatographic method to anticipate problems that could arise during the method s application when different reagents, chromatographic columns, environments, or instruments are used (). Robustness is defined as the capacity of a method to yield exact results in the presence of small changes of experimental conditions, such as might occur during the utilization of these procedures (3). Occasionally, analysts can conduct a robustness test using a one-factor-at-a-time procedure, but this approach is neither economical nor optimal for interpretation purposes, so screening experimental designs mainly are used, as in an optimization process (3,4). Because robustness generally is investigated when the method already has been optimized and its aim is to define the variable limitations, the experimental design approach used is different from that of optimization. According to the literature, the most appropriate tests are the two-level saturated designs (fractional factorial or Plackett Burmann designs) that focus on the procedural conditions (3,5,6). The robustness of an analytical method can be determined using six steps: identification of the factors to be tested; selection of the factor levels and level numbers; selection of the experimental design; realization of experiments and measurement of the analytical responses of the method; calculation of the variable effects; and analysis and interpretation of the results. These steps are a general procedure that outline the robustness test. Depending upon the definition of effect used and the method used to estimate its statistical significance, the interpretation of the results could be different. In this article, we propose a new definition of effect from the point of view of an analytical chemist and discuss different ways to estimate significant effects. To show the reliability of this approach, we studied the effects of several chromatographic factors on peak height, peak area, retention time, and resolution. For this purpose, we selected a chromatographic method

2 74 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 JANUARY 00 to determine eight biogenic amines as dabsyl derivatives (7). Theoretical Aspects When studying robustness, researchers introduce experimental variations to the procedural value of the operational variables in an attempt to predict the dependence of the method upon the sources that provoke their possible variation in routine practice. In this sense, two levels, at right ( 1) and left ( 1) of the nominal value, must be examined for each considered variable (factor, in this context) according to a wellbalanced experimental design pattern. It is considered well balanced because each factor appears the same number of times in the 1 and 1 levels, and all other factors also appear the same number of times in the 1 and 1 levels for each of the subsets. The nominal value of each factor is the value specified in the analytical procedure the optimized value whereas the extreme levels represent the values of the variables that deviated from the nominal value (8,9). It is convenient to add three or more replicate responses, obtained only at nominal conditions of the procedure-related variable (that is, the central points of the design), to control the precision and trueness of the experiment. General texts about design of experiments usually state that the effect of the factors on the variable response for well-balanced twolevel designs is calculated from the expression: E Xk N Y ( ) Xk Y ( ) Xk [1] where E Xk is the effect of the kth factor (X k ) on the analytical chromatographic response (Y ), N is the number of experiments defined in the design, and Y( ) Xk and Y( ) Xk indicate the sum of the responses from the experimental runs when the factor X k has the upper ( 1) and lower ( 1) values, respectively. The effect calculated in this manner, inherent of the industrial application of the design of experiments, indicates the total variation introduced in the response Y when the factor X k changes its value from lower ( 1) to upper value ( 1); that is, in an interval of two units. The significance of the effects is identified using a Student s t-test (6,9) or an analysis of variance (8,10). When evaluating the robustness of an analytical method, the effect (E Xk ) should be considered to be the variation in the response when the procedural value is modified in one interval toward the other side; for example, 0 to 1 or 0 to 1. The range of the factor levels is so small that we can assume that all effects on the response are linear (11). In this sense, the effect (E Xk ) can be expressed as: E Xk 1 N [] Furthermore, in the assayed domain, the analytical response (Y ) would be the response under nominal conditions (Y 0 ) in addition to the effect produced by the modification in the procedural value of the factors, as expressed in equation 3. [3] From the quoted definition of effect above, it also is possible to estimate effect using classic least-squares multiple linear regression, which fits the following mathematical function: [4] where b 0 and b k are the estimated intercept and regression coefficient, respectively. The equation represents the change that occurs in the response for each modified unit; that is, the sensitivity of the variable X k, which we call sensitivity coefficient. When we use coded values with the new definition of effect, we found that the effect and the regression coefficient have the same values at the extreme levels 1 (1): E Xk 1 Y ( 1) Y ( 1) 1 b 0 b ( 1) k 1 b k b k Y ( ) Xk Y ( ) Xk Y Y 0 Y b 0 E Xk b k X k b 0 b ( 1) k [5] In general, and when taking equations 3 and 4 into account: E Xk b k X k [6] Whereas the effect describes simply the observed average difference between the nominal level and an extreme level, the sensitivity coefficient enables us to evaluate the effect on the studied interval, although measurements are performed only at the extreme levels. The estimation of the significance of the effect can be done by statistical and graphical analysis. Statistical analysis: The statistical test should be a Student s t-test in which the null hypothesis (H 0 ) is E Xk 0 or b k 0 (nonsignificant effect), whereas the alternative hypothesis (H 1 ) is E Xk 0 or b k 0 (significant effect). The statistic is E Xk /SE (where SE is the standard error associated with the effect), which is compared with the critical twotailed Student s t-value, t crit (, ), where is the significance level usually equals 0.05 and is the number of degrees of freedom. SE can be estimated in different ways (5,8,13). When analysts choose randomized replicate measurements at the nominal level of the factors or replicates of any experimental run of the design, the SE values can be calculated by (1): [7] where N is the number of experiments defined in the design without counting replicates and s represents the variance obtained from these approaches calculated by: s [8] where Y ij is each replicate at the point i of the design, Y i is the mean value of these replicates, and r i is the number of replicates at each point of the design. If replicates are used only at the nominal level, equation 8 can be reduced to: s SE r i i s N Y ij Y i r i 1 Y 0j Y 0 r 0 1 [9] where Y 0j is each response obtained at the nominal level, Y 0 is the mean of these values, and r 0 is the number of replicate measurements at nominal level. In this case, the number of degrees of freedom is r 0 1. The significance test is usually performed by applying the following equations:

3 76 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 JANUARY 00 E Xk t crit SE [10] b k t crit SE [11] where t crit SE characterizes the threshold value E crit (critical effect) for significance. This test leads to a value that indicates the probability (P) that the estimated value of the effect was due to random experimental errors. P values can be calculated easily by using suitable statistics packages or programmable calculators with statistical program libraries. Thus P values higher than 0.0 corroborate the null hypothesis, whereas P values of lead to a doubtful conclusion and suggest additional experimental work such as duplicating the design or running new replicates. When P values are unavailable, analysts can apply a rule for detecting evidence of significance for estimated effects. Thus, an estimated value between E crit and 0.5E crit should be considered an indication of significance. Chromatographers also could use analysis of variance to analyze the experimental data set of a robustness study. In this case, they should test the significance of the effects (or sensitivity coefficients) using the pure error option, which is common in statistical software (). This option can be used if the experimental design is at least partially replicated, so users can estimate the variability of the experimental response from the scattering of the replicated runs. Because these measurements are taken under identical conditions at identical settings of the factor levels, the estimated variability represents pure error; that is, the variability is entirely the result of unreliabilities in the measurement of the dependent variable. The calculated (E Xk ) and critical effects (E crit ) can be normalized according to equation 1 by dividing them by the nominal response. These parameters (%E Xk or %E crit ), because of their relative values, can be used to characterize and to compare the sensitivity of the factors in the different response variables: X k crit E crit b k [13] To change to real values of factor X k, analysts must multiply X kcrit by the interval used (from 0 to 1) in the corresponding units. Graphical analysis: By accounting for the geometrical characteristics of the sensitivity coefficient and the equivalence between E Xk and b k, analysts can use a linear response plot to estimate the robustness of the method. If the procedural value is located in a plateau, the value of the coefficients will be close to zero, and, therefore, the method will be robust. Nevertheless, if the optimal value is in the closeness of a sharp-end maximum peak, then the coefficients will have a high value, and the method will not be robust. As an example, Figure 1 shows that response variable Y is affected by slight variations of X 1, and is not robust to this factor. On the other hand, Y is unaffected by small changes of X, so it is robust in relation to this variable. Experimental Apparatus and software: In our studies, we used an Agilent Technologies 1050 series liquid chromatograph equipped with a variable-wavelength UV vis detector and a model 3396-A integrator (all from Agilent Technologies, Wilmington, Delaware) and P Robustness study Y a model 715 loop injector with a 0- L sample loop (Rheodyne, Inc., Rohnert Park, California). We used a 44 mm 4.4 mm, 5- m d p Lichrospher 100 RP-18 column linked with a 10 mm 4.6 mm Lichrospher guard column (Merck KGaA, Darmstadt, Germany) for all separations. We used the Statgraphics Plus 4.1 statistical graphics system software package (Statistical Graphics Corp. Inc., Rockville, Maryland) for data treatment. Amine standard solutions: All amine standards were purchased as hydrochloride salts of the highest purity available. We obtained tryptamine, phenylethylamine, spermine, and spermidine from Fluka (Neu-Ulm, Germany). The histamine, cadaverine, putrescine, and tyramine were obtained from Sigma (St. Louis, Missouri). We purchased 1,7-diaminoheptane from Aldrich (Steinheim, Germany). We prepared.5 mm or 10 mm stock solutions of the biogenic amines by dissolving them in 0.1 M hydrochloric acid containing 0.% (w/v) 3,3 -thiodipropionic acid (Fluka) as an antioxidant. The solutions were stored at 0 C. We prepared a composite amine standard from stock solutions to yield an overall concentration of 50 M per component. Nominal chromatographic solutions: We adjusted eluent A, which comprised mol/l sodium acetate (Panreac, Barcelona, Spain), 10% (v/v) dimethylfor- X P Y X 1 % E Xk E Xk Y [1] X Bearing in mind the relationships between the effect, the sensitivity coefficient, and the X kcrit value, analysts can establish the factor values intervals (X kcrit ) in which an effect is not significant. This interval can be estimated as: X 1 Figure 1: Geometrical relationship between robustness and the response surface profile in the experimental domain assayed for the factors X 1 and X. P procedural experimental for values of X 1 and X. The procedure is not robust to X 1 and is robust to X.

4 78 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 JANUARY 00 mamide (Fluka), and 0.3% (v/v) triethylamine (Carlo Erba, Milan, Italy), to ph 5.0 with diluted acetic acid (Panreac). Eluent B comprised 87.5:10:.5 (v/v/v) acetonitrile (Panreac), tert-butylmethyl ether (Fluka), and water. We equilibrated the C18 column at 40 C with a mobile phase of 45% eluent A 55% eluent B. We injected a 0- L aliquot of the dabsyl derivative solution, and it was eluted at a flow rate of 1.0 ml/min. The gradient profile was as follows: 55% eluent B for 3 min, then 55 75% eluent B in 10 min, then % eluent B in 10 min, 100% eluent B for 10 min, then % eluent B in 5 min, and 55% eluent B for 15 min. The detection wavelength was 446 nm. We used highly purified water from a Milli-Q water-purification system (Millipore, Bedford, Massachusetts) throughout our studies for the preparation of buffers and reagents. Table I: Factor levels used in the designs Results and Discussion Experimental design: We selected seven factors for examination in the robustness test, namely gradient (understood as the effect upon the stationary phase by the first gradients when the chromatograph is started); column temperature; buffer concentration; mobile-phase buffer ph; detection wavelength; triethylamine ; and dimethylformamide. Table I summarizes these factors and their levels. The gradient levels studied were ( 1), corresponding with the first gradient made from time 0 to 55 min, (0) the second gradient made from 55 to 110 min, and ( 1) from 110 to 165 min, after the third gradient. We selected a well-balanced saturated fractional factorial 7-4 design, as shown in Table II. To estimate the experimental error, we used three replicates at the nominal level of the factors and performed all experimental runs in random order. To verify the robustness test using the effect defined in equation, we selected the following response variables: peak height and peak area for phenylethylamine; retention time and unadjusted retention time (t R ), which was defined as the ratio of retention time of the analyte to that of 1,7- diaminoheptane (internal standard) for tyramine; and resolution between cadaverine and histamine for chromatogram resolution. Nominal Lower level Upper level Variation Factor value ( 1) ( 1) (%) Gradient profile 1 0 Column temperature ( C) Buffer concentration (mm) Mobile-phase buffer ph Detection wavelength (nm) Triethylamine (%) Dimethylformamide (%) Interpretation of the robustness test: Table II summarizes the experimental values of the response variables. Table III shows the calculated effects for each factor using both definitions of effect (equations 1 and ). To detect statistically significant effects, we calculated the critical effect values according to the expression E crit t crit SE. We observed that the significant and nonsignificant factors are in agreement, regardless of which approach we used. In the case of phenylethylamine, we observed that peak area appears to be robust to the variations introduced in all factors, although we could draw no definitive conclusion with respect to gradient and detection wavelength, because their effects are greater than one-half the threshold value. On the other hand, gradient, column temperature, and detection wavelength cause significant variation of the peak height, and the other four factors are insignificant. Figure illustrates the results obtained for the peak height and peak area for phenylethylamine and shows the factors that have significant influence upon the response variables when the normalized effects are plotted. In relation to the retention time of tyramine, all the factors, except detection wavelength and triethylamine, were significant. However, when we used unadjusted relative retention as a response variable, only mobile-phase buffer ph, dimethylformamide, and column temperature were significant. Finally, we observed that none of the factors had a significant effect upon the chromatogram resolution, although the gradient showed evidence of significance. The main advantage of our new definition of effect arises from the total equiva- Table II: Experimental design for robustness test Response Variables Factor Phenylethylamine Tyramine Column Buffer Mobile-Phase Triethyl- Dimethyl- Unadjusted Temper- Concen- Buffer Detection amine formamide Peak Peak t R t R Example Gradient ature tration ph Wavelength Percentage Percentage Area* Height* (min) (min) Resolution ,365,197 1, ,375,64 6, ,304,69 6, ,39,880 5, ,378,08 1, ,383,7 4, ,448,7 34, ,384,194 34, ,356,666 1, ,311,10 18, ,311,896 19, * Arbitrary units.

5 JANUARY 00 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 79 Table III: Comparison of results using the proposed and traditional definitions of effect Column Buffer Mobile-Phase Triethyl- Dimethyl- Temper- Concen- Buffer Detection amine formamide Gradient ature tration ph Wavelength Percentage Percentage E crit Phenylethylamine Peak area E Xk 7,365* ,767,14* ,566 39,669 E Xk 54,730* ,534 44,48* 13,78 1,131 79,338 Peak height E Xk E Xk Tyramine t R (min) E Xk * E Xk * Unadjusted t R (min) E Xk * * E Xk * * * Resolution E Xk 0.040* E Xk 0.079* * Indication of significance. Significant effects ( 0.05). Factors lence between E Xk and b k, which allows us to obtain the results of the robustness test directly from the multiple linear regression analysis. It also is possible to make conclusions by examining the linear response plots obtained from this approach, because the effect and the slope (b k ) are equivalent graphically. Figure 3 illustrates the geometric approach to the robustness test proposed above and shows all possible situations. Figure 3a shows how the peak height of phenylethylamine changes as a function of buffer concentration and dimethylformamide when the rest of the factors are kept constant at their nominal levels. We observed, in accordance with Table III s results, how the peak height was insensitive to these factors. Figure 3b shows the influence of column temperature and detection wavelength (significant factors) upon this response variable. Peak height was very sensitive to both factors, as the steep slope of the linear response plot indicates. This profile would be similar to a profile obtained with the other significant factor, gradient. Finally, Figure 3c represents the influence of column temperature and triethylamine upon the peak height. In this case, the response variable is sensitive only to column temperature (significant), and the response for triethylamine (nonsignificant) is stable. As a final conclusion of the robustness study we performed, Table IV shows the intervals (equation 13) in which the factors should be maintained to eliminate the significant effects. Normalized effect (%) Gradient Peak area Peak height Column temperature Buffer concentration Buffer ph Factor Detection wavelength Triethylamine Figure : Plot of the normalized effects upon the peak area and peak height for phenylethylamine. The continuous line represents normalized E crit for peak area values, and the dotted line represents normalized E crit for peak height values. Table IV: Intervals of the factor levels in which the significant effects can be eliminated Response Variables Dimethylformamide Phenylethylamine Tyramine Factor Peak height t R (min) Unadjusted t R (min) Column temperature C C C Buffer concentration mm Mobile-phase buffer ph Detection wavelength nm Dimethylformamide % %

6 80 LCGC NORTH AMERICA VOLUME 0 NUMBER 1 JANUARY 00 (a) (b) (c) Peak height ( 10 4 ) Buffer concentration Dimethylformamide Column temperature Detection wavelength Column temperature Triethylamine Figure 3: Estimated linear response plots for phenylethylamine peak height: (a) buffer concentration and dimethylformamide ; (b) column temperature and detection wavelength; and (c) column temperature and triethylamine. Conclusion This article discusses the robustness of a chromatographic method from an analytical chemist s point of view. We have proposed a redefinition of effect, which, combined with a multiple linear regression analysis, provides an easy and powerful tool to analyze a data set obtained from a robustness study. Furthermore, this approach allows users to analyze the influence of factors upon response variables graphically by using linear response plots. Finally, analysts also can establish intervals that ensure the robustness of their chromatographic methods. Acknowledgments The authors are grateful to the Consejería de Educación y Ciencia de la Junta de Andalucía (Seville, Spain) for financial assistance (researcher groups FQM-3 and FQM-00 of the II Plan Andaluz de Investigación II P.A.I.). References (1) A.M. García Campaña, J.M. Bosque Sendra, L. Cuadros Rodríguez, and E. Almansa López, Biomed. Chromatog. 14, 7 9 (000). () Y. Vander Heyden, K. De Braekeleer, Y. Zhu, E. Roets, J. Hoogmartens, J. de Beer, and D.L. Massart, J. Pharm. Biomed. Anal. 0, (1999). (3) H. Fabre, J. Pharm. Biomed. Anal. 14, (1996). (4) J.A. Van Leeuwen, B.G.M. Vandeginste, G. Kateman, M. Mulholland, and A. Cleland, Anal. Chim. Acta 8, (1990). (5) Y. Vander Heyden, K. Luipaert, C. Hartmann, D.L. Massart, J. Hoogmartens, and J. de Beer, Anal. Chim. Acta 31, 45 6 (1995). (6) Y. Vander Heyden, F. Questier, and D.L. Massart, J. Pharm. Biomed. Anal. 18, (1998). (7) R. Romero, D. Gázquez, M.G. Bagur, and M. Sánchez-Viñas, J. Chromatogr. A 871, (000). (8) L. Cuadros Rodríguez, R. Blanc García, A.M. García Campaña, and J.M. Bosque Sendra, Chemom. Intell. Lab. Syst. 41, (1998). (9) J.M. Bosque Sendra, M. Nechar, and L. Cuadros Rodríguez, Fresenius J. Anal. Chem. 365, (1999). (10) Y. Vander Heyden, M. Jimidar, E. Hund, N. Niemeijer, R. Peeters, J. Smeyers-Verbeke, D.L. Massart, and J. Hoogmartens, J. Chromatogr. A 845, (000). (11) A. Nijhuis, H.C.M. van der Knaap, S. de Jong and B.G.M. Vandeginste, Anal. Chim. Acta 391, (1999). (1) D.L. Massart, B.G.M. Vandeginste, L.M.C. Buydens, S. de Jong, P.J. Lewi, and J. Smeyers- Verbeke, in Handbook of Chemometrics and Qualimetrics: Part A, B.G.M. Vandeginste and S.C. Rutan, Eds. (Elsevier, Amsterdam, The Netherlands, 1997), pp (13) E. Hund, Y. Vander Heyden, M. Haustein, D.L. Massart, and J. Smeyers-Verbeke, Anal. Chim. Acta 404, (000).