Anel. Chem. 1093, 65, 678-682
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Robust Procedure for Calibration and Calculation of the Detection Limit of Trimipramine by Adsorptive Stripping Voltammetry at a Carbon Paste Electrode M. Cruz Ortiz,’vt Julia ArcosJ Jose V. Juarros,t Jestis L6pez-Palacios,t and Luis A. Sarabiat Department of Analytical Chemistry and Department of Mathematical Analysis, Colegio Universitario, Apdo 231, E-09080Burgos, Spain
A method for the determlnatlonof trimipramlneby adsorptive stripping voltammetry using a carbon paste ehctrode has bwn developed. For routlm calibratlon and calculatlon of tho dotectlon Ikntt, a robW r e g r d o n method (least median squares) has beon proposed, providing the method wtth adoqwte reUaMlty and dotoclablllty. The methodology used In thk work overcoma the exprbnental drawbacks arldng from the need to renow the electrode surface after each measurement. The detection ibnltr thus achleved (1.16 X lo4 to 2.41 X lo4 M) take Into account the mndtlvlty of the analytlcai method, tho nature of the analyte, and the risk of fake porltlve and fake negative rewtts the analyst is willing to accept.
INTRODUCTION Trimipramine is a tricyclic compound that is widely used as an antidepressant drug. The very low therapeutic amounta required for administration make it necessary to use very sensitive techniques for ita determination in commercially availableformulationsand biological samples. In this respect, some spectxophotometricand chromatographicmethods have been developed for quantification of both the parent drug and ita metabolites.14 Electroanalyticaltechniquesin general and voltammetricmethods in particular are increasinglyused in biomedicaland pharmacologicalanalysis.73 Many tricyclic antidepressants are determined by such methods with as satisfactory resulta as those provided by chromatographic and photometric techniques, the last of which are usually more laborious and often involve preliminary complexation and extraction steps. Carbon paste electrodes are extremely useful for the determination of drugs as they typically show very low background currenta, so they allow the determination of low analyte concentrations, thereby improving analytical detectability. An additional major advantage of carbon paste is the ease of renewal of the whole electrode providing a fresh surface unaffected by electrode history.9 + Department of Analytical Chemistry. t
Department of Mathematical Analysis.
(1) Scoggins, B. A.; Maguire, K. P.; Norman, T. R.; Burrows, G. D.
Clin. Chem. 1980,26, 5-17. (2) Hattori,H.;Takaahima,E.;Yamada,T.; Suzuki, 0.J.Chromatogr. Biomed. Appl. 1990,94, 189-193. (3) Koeppel, C.; Tenczer, J. Chronatogr., Biomed. Appl. 1988, 75, 197-202. (4) Cowan, D. A.; Ozkirimli, S.; Beckett, A. H. Acta Pharm. Suec. 1988,25,141-150. ( 5 ) Hueeein, S. A.; El-Kommos, M. E.; Haasan, H. Y.; Abdel-Maboud, I.; Mohamed, A. M. I. Talanta 1989,36, 941-944. (6) Hussein, S. A.; Mohamed, A. M. I.; Haasan, H. Y. Talanta 1989, 36., 1147-1149. (7) Vir&J. C.; Kauffmann, J. M.; Patriarche, G. J. J.Pharm. Biomed. Anal. 1989, 7 (12), 1323-1335. ( 8 ) Arcos, J.; Kauffmann, J. M.; Patriarche, G. J.; Sanchez Batanero, P. Anal. Chim. Acta 1990,236, 299-305. ~~
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0003-2700/93/0365-0678$04.00/0
The presence of an amino group in trimipramine (Figure 1) is the origin of ita electrochemical activity. Anodic
electroactivity of trimipramine on glassy carbon,1° gold, and platinum electrodes11has been studied. In this context,Wang et al.10 investigated the oxidation of various tricyclic antidepressants (desipramine, imipramine, and trimipramine) at glassy carbon and carbon paste electrodes and found trimipramine to be electroinactive at the latter type of electrode. This paper reports on the occurrence of a well-defined electrochemicalresponse from trimipramine at a carbon paste electrode and ita potential use for quantitative purposes. Adsorptive stripping techniques are highly suited to surfaceactive substances and have been used for the determination of a variety of compounds of biological and pharmaceutical interest.7 In our work, an increase in the intensity of the response by adsorption of trimipramine at the electrode surface has been observed. The electrode surface must be renewed after each measurement as alternative surface cleaning methods have proved to be unsatisfactory for this purpose. As a result, the active electrode area cannot be assumed to remain constant throughout calibration, so the risk of an individual measurement being spurious must be seriously considered,which requires using a robust regression procedure in order to construct the calibration line and determine ita linear range. Under these conditions, the electrode must obviously be regarded as a random element of a theoretical electrode population; so an electrode will be reciprocallyrelated to each individualmeasurement,and each experimental data will be considered as strictly probabilistic.
THEORY The reliability of adsorptive stripping voltammetric methods based on the use of a carbon paste electrode depends heavily on careful selection of appropriate procedures for regression and estimation of the detection limit. This section deals with the theoretical foundation of the mathematical procedure used for calibrationand calculation of the detection limit. ElectrodeCalibration. Inasmuch as a fresh carbon paste surface was used for each individual measurement, any calibration point could be an outlier. As a designed experiment, the calibration should be free of leverage data. However, since the electrode behavior depends on the compositionof the paste and the active surface area, the linear range would presumably vary. Thus, we constructed a calibration line that allowed us to simultaneouslydetect both outliers and aligned pointa (linearrange) avoidingany a priori (9) McCreery, R. L. In Carbon Electrodes: Structural Effects on Electron Transfer Kinetics in Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker Inc.: New York, 1991; Vol. 17. (10) Wang, J.; Bonakdar, M.; Morgan, C. Anal. Chem. 1986,58,10241028. (11)Bishop, E.; Hussein, W. Analyst 1984, 109, 73-109. 0 1993 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 65, NO. 6, MARCH 15, 1003
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CH2 CH CH2 N(CH3)z
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estimation of their number or position. This introduced additional complexityas some effects can mask others in such a way that the usual residual analysis by least-squares regression may lead one to identify as outliers points which are not so through inclusion of others with high leverage.’* Solving both problems at once called for the development of a robust regression method. The linear calibration model used was based on the following equation
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y = a + bx e (1) where e denotes the independent normally distributed residuals. From the experimentalconcentration-current data (xi,y,) i = 1, ..., n the regression procedures provided the estimates d and 6 for the parameters in eq 1, the bias and sensitivity of the proposed chemical analysis method, respectively. The features of these estimates are determined by the optimization criterion used in the regression method and the hypotheses made on the experimental data. The least-squares solution, LS, provides the & and 6 values so that
On the assumption that residuals e in eq 1 conform to a normal distribution, the Gauss-Markoff theorem charaderizes the LS estimators as those with the smallest variance among unbiased estimators. In fact, least-squares estimation is the most precise of all exact estimations.13 The usefulness of these properties to the construction of the calibration curve are obvious. Outliers are known to result in highly asymmetric distributions and/or high cumulativeprobability falling away from central values, so the normality assumption cannot be widely fulfilled. Usually, a fairly small number of data are obtained in a routine calibration and the normality test performs poorly, so they do not detect the problem, and one may conclude that only in the absence of outliers can LS regression provide acceptably accurate and precise calibration. Robust regressions are insensitive to a greater or lesser degree to the Occurrence of outliers, so they can provide the actual linear relationship for the calibration and allow any outliers to be detected and the conditions for optimal LS estimation to be reestablished. An introduction to robust regression and ita chemical uses is provided elsewhere.’* In order to solve the problems posed by the calibration in this work, we used the method of the least median of squares (LMS),the theoreticaldevelopment and practicalapplications of which are described elsewhere.12J5 This regression method was previously used as an option in calibration techniques.”j-18 The d and 6 estimates provided by the LMS (12) Rousseeuw, P. J.; Leroy, A. M. Robust Regression & Outliers Detection; Wiley: New York, 1987. (13) Hampel, F.R.;Ronchetti, E. M.; Rousseeuw, P. J.; Stahel, W. A. Robust Statistics, The Approach Based on Influence Functions; Wiley: .~ New York, 1986. (14) Rousseeuw, P. J. J. Chemom. 1991,5, 1-20. (16)Rousseeuw, P.J. JASA, J. Am. Stat. Assoc. 1984, 79, 871-880. (16) Massart, D. L.; Kaufman, L.; Rousseeuw, P. J.; Leroy, A. Anal. Chim. Acta 1986, 187, 171-179.
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Figure 2. Residuals of different regression methods. The data palm (x& / = 1, ..., 15 generated fromthe equationy = x + 1 C, where c is 0 for the first 10 pairs and -0.1, -0.4, -0.9, -1.6, and -2.5, respectively, for the last five. (A) Least-squares method. (B) Laplace method. (C) Least median of square method.
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method are those which minimize the median of the squares of the residuals: min,, {median,,,..., {bi- (a. + bri)12W (3) The performance of robust regression methods is usually assessed through two indices, viz., the “influence function” and the “sampling breakdown”. While the influence function of the LMS method is quoted on both coordinates, that of the LS method is quoted on neither.19 The effect of an outlier on aleast-squares regression depends on ita relative location in the set of pointa;m if the experimental data include concentrations beyond the linear range of the analytical method, then they will influence the LS regression. The sample breakdown of the LMS regression is 50 % ,viz., the greatest theoretical value for this index. On the other hand, the breakdown of the LS regression is 0%. These notable advantages of the LMS regression are supplemented by one other that is of special interest when both the linear range and sensitivity of an analytical method are to be determined. This property is exact fitting:12 if at least 50% of the (x,y)data pairs conform to a linear model, then the LMS regression coincides with it. Figure 2 shows a graphic example of this property in the form of the residuals of the LS, LMS, and Laplace LI regressions. This deviation from the linear model is the calibration pattern most often obtained. Ita effect on the LS regression is highly undesirable because the method tends to offset the deviation arising from misaligned points, thereby introducing large errors in the correctpointa. Thus, the modelmay provide an overestimated linear range and fully mask the actual underlying linear relationship. The LMS regression has the disadvantage of being less efficient than the least-squaresregression,i.e., a larger sample is required to obtain the same evidence in probabilistic terms. Furthermore, the probability distribution of LMS estimations is unknown, and, therefore, no valuation about the statistical significance for slope (sensitivity)and intercept of the calibration is obtained which is essential in order to assess the quality of calibration and, hence, the performance of a given analytical method. (17) Ortiz,M. C.;L6pez-Palacios,J.;Aram,J.;Sarabia,L. A.;Piangerelli, M. G.; Cingolani,D.,111. Jornadaa Cientificas de ElectroquImicadel gmpo de la S.E.Q.A. y la R.S.E.Q. Valladolid (Espaiia), 1991. Comunicaci6n 5-3. (18) Yuzhu, H.; Smeyers-Verbeke, J.; Massart, D. L. Chem. Lab. 1990, 9, 31-44. (19) Ronchetti, E.M.In Statistical Data Analysis Based on the LINorm andRelatedMethods;Dodge, Y.,Ed.; North-Holland Amsterdam, 1987; pp 65-80. (20)Cook, R. D.; Weiberg, S.Residuals and Influence in Regression; Chapman and Hall: New York, 1982.
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Consistent with the above reasoning, the numerical calibration procedure should exploit the ability of the LMS regression to detect the group of aligned experimental data and the optimal precision and accuracy of the estimates provided by the LS regression. Consequently,we developed the following computational sequence: Step 1. Acquisition of the experimental data pairs: (xiyi), x = concentration; y = peak current; i = 1, ...,n. Step 2. Application of the LMS regression in order to obtain a robust estimation of the line y = QLMS + bLMST. Step 3. Evaluation of the standardized residuals obtained in order to detect outliers, hence determining the linear range and rejectable points on account of their experimental errors. S t e p 4 . Application of LS regression-outliers excluded-in order to obtain the calibration line y = & f 6 x . Step 5. Evaluation of the degree of coincidence between the two regressions, the quality of the regression, and hence of the calibration. If a common LS regression is made in step 2, a careful inspection of the residuals is essential in order to determine the linear range and outliers;20921 this calls for expertise in regression analysis which is not always available in routine chemical laboratories. The availability of a computerized method capable of performing this task effectively, viz., the LMS regression, is the key to the success of the proposed calibration procedure. On the other hand, the LS regression of step 4 provides an estimation of the probability distribution of the current 5 corresponding to a concentration x of trimipramine within the linear calibration range. This is known to be a Student’s t value with n - 2 degrees of freedom, the mean and variance of which are
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r being the number of replicates in the determination of the signal for a future sample. From eqs 4 and 5 it is obvious that the calibration parameters and variance must be estimated optimally. Any outliers and/or an inadequate linear range will make the estimations unsuccessful. Detection Limit. Because of the analytical purpose of this work, calculating ita detection limit was obviously compelling. A broad discussion including a wealth of references to procedures that can be used for this purpose has been reported else~here.2282~ The features of our analytical signal (the blank signal was subtracted from the recorded current) precluded evaluation of ita dispersion in the blank and, hence, the establishment of a confidence interval for the signal corresponding to a zero concentration of trimipramine. Also, we should note that most methods based on such an interval do not allow the probability of a false negative to be assessed. Moreover, those methods do not consider the sensitivity of the analytical method concerned. The above reasoning and the need to renew the electrode surface after each measurement resulted in an uncertainty (21)Draper, N.; Smith, H. Applied Regression Analysis, 2nd ed; Wiley: New York, 1981. (22)Liteanu, C.; Rica, I. Statistical Theory and Methodology of Trace Analysis; Ellis Horwood Chichester, 1980. (23)Currie, L. A. In Chemometrics: Mathematics and Statistics in Chemistry; Kowalski,B. R., Ed.; NATO AS1 Series; Reidel: Dordrecht, 1984;pp 115-146.
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 6, MARCH 15, 1993
Table I. Trimiwamine Determination by Cyclic AdsorDtive Voltammetry at Carbon Paste Electrode. LS regression LMS regression Calibration I 15 15 no. of data sensitivity X 10-6,pA mol-' dm3 1.323 2.262 intercept, pA 0.049 0.007 coeff of determination (R2) 0.989 0.998 residual standard deviation (a) 0.047 linear range, mol dm-3 2.40 x 10-9,i.wx 10-7 0.022 detection current, pA 1.16 X 10-8 detection limit, mol dm-3 Calibration I1 no. of data 15 15 sensitivity X 10-6,pA mol-' dm3 1.222 1.261 0.017 intercept, pA 0.030 0.995 coeff of determination (R2) 0.995 residual standard deviation (a) 0.286 linear range, mol dm-3 2.40 X lb9, 8.26 X 0.036 detection current, pA 2.40 X 10" detection limit, mol dm-3 Calibration I11 no. of data 15 15 sensitivity X 10-6, pA mol-' dm3 1.340 1.675 0.019 intercept, pA 0.040 coeff of determination (R2) 0.971 0.993 residual standard deviation (G) 0.801 linear range, mol dm-3 2.40 X 6.22X 10-' 0.034 detection current, pA 2.41 X 1 W detection limit, mol dm-3 Analytical parameters calculated for three independentcalibrations with contents from 2.40 X additional explanation. Confidence interval at 0.05 significance level. calculated. Clayton et aLZ4established that the probability pis a Student's t (A) distribution with n - 2 degrees of freedom, a mean y - Yd, and a noncentrality parameter A equals A = (y - 9o)lWoG
(6)
By fixing 8, the parameter A, which depends on a,through the difference y - Yd can be obtained from tables of t(A) distribution. Thus, with the aid of the following equations derived from eq 6 and the corresponding calibration equation y -yo = y - c9 = 6~ = A(c~,p)w,G
(7)
one can obtain the concentration which is the detection limit searched for = A(~,B)wO(G/~)
(8) Equation 8 for the detection limit allows us (i) to prevent obtaining false negatives and false positives through A(a,j3), (ii)to take into account the design of the calibration through wo,and (iii) to consider the calibration sensitivity, and the quality of the regression, through the parameter u. xd
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EXPERIMENTAL SECTION Apparatus. Voltammetric measurements were made on a BAS 27 potentiostat furnished with a three-electrode cell consisting of a BAS carbon paste electrode with a surface area of 0.07 cm2,a platinum wire as counter-electrode, and asaturated calomel electrode (SCE) as reference electrode. The voltammograph was connected to a Philips XY8133 recorder. Reagents. All chemicals used in this work were of analytical reagent grade. Trimipramine was kindly supplied by R h h e Poulenc Farma SAE (Alcorcbn, Madrid) and was used without further purification. Liquid paraffin oil (Nujol) was purchased from Sigma, while micronized graphite was obtained from Carbone USA Corp. A Britton-Robinson buffer of pH 5 was (24) Clayton, C. A.; Hines, J. W.; Elkins, P. D. Anal. Chem. 1987,59, 2506-2514.
681
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calibration line 9 2.383 0.1572b 0.008 O.0O6gb 0.995 0.007
*
12 1.277 i 0.0246b 0.016 0.0081" 0.999 0.103
13 1.693 0.0366" 0.014 0.0080b 0.999 0.110
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to 1.03 X 10-6 mol dm-3. See text for
used as the supporting electrolyte. All solutions were made in deionized water. Solutions of trimipramine were daily renewed. Analytical Procedure. Carbon paste containing 15% liquid paraffin was made by thoroughly mixing micronized graphite and Nujol in a mortar containing the minimum volume of chloroform. After being mixed, the solvent was allowed to evaporate overnight. The working electrode was constructed by pressing the paste inside the Teflon body of the electrode. The electrode surface was smoothed by gentle rubbing with clean paper. The paste was carefully removed after each experiment, and the electrode was submitted to ultrasonic waves before a new portion of paste was applied. After a time of accumulation of 120 s in open circuit with constant stirring, followed by a 15-speriod to achieve equilibrium, voltammograms were recorded from 0 to 1.2 V at a scan rate of 100 mV s-'. Computational Procedure. The LMS regression was implemented by using the April 1988 version of the program PROGRESS (Program for Robust reGRESSion).12 In performing step 3 of the procedure, we considered to be outliers all those points whose standardized residue with respect to the LMS line was greater than 2.5 in absolute value. This limit is quite conservative in that if errors have a normal distribution, then the probability of finding a residue of this magnitude is 0.012. Graphs were obtained by using the corresponding routines in the program STATGRAPHICS (Version 5, 1991) from STSC Inc. All computations and graphing were done on a Tandon MCS 486133 computer.
RESULTS AND DISCUSSION As can be seen in Figure 3, an anodic scan from 0 to 1.2 V recorded after stirring for 2 min provided a well-defined peak. The voltammetric response was influenced by different variables such as the pH of the medium and the accumulation time. A comprehensive study of the peak current obtained under different conditions allowed us to select pH 5 and an accumulation time of 2 min as optimal for the best possible sensitivity and precision.
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Voltammograms recorded after 0.5, 1, 3, and 5 min of stirring showed increasing peak currents, thereby suggesting the gradual accumulationof the drug at the electrodesurface. The peak shape points to a rapid adsorption/desorption process between the electrode surface and the bulk solution. The composition of the carbon paste (Nujol content) also affected the intensity of the response, the best results being obtained with 15% Nujol in the paste. The absence of cathodic peaks shows the process to be irreversible. The proposed procedure for calibration and calculation of the detection limit was applied to three independent data sets (I, 11, and 111),each of which was made up of 14 samples containing trimipramine concentrations between 2.40 X loW9 and 1.03 X 104 mol dm-3. Figure 4 shows the residuals of the LS regresion against the sequential number of the samples (increasing concentrations) for calibration I. The absence of a normal distribution and the lack of randomness in the signs-negative at the lowest concentrations, positive then again negative at the highest concentrations-are quite apparent. This trend is typical of poor line fitting despite the relatively high determination coefficientobtained (0.989). Table I lists the parameter values obtained by (a) LS regression of the first 15 data, (b) LMS regression, and (c) LS regression after exclusion of outliers. This last regression is the calibration procedure provided. Table I also gives the confidence intervals (significancelevel 0.05) for the sensitivity and intercept of each calibration. As can be seen, every intercept is significantly non-zero at this significance level. In all three calibrations, the sensitivity is markedly increased as a result of outliersbeing removed, and the residual standard deviation, G,iamuch lower, while the determination coefficient is somewhat higher. The results provided by the robust LMS regression and the LS regression in the absence of outliers are quite consistent. ABexpected, the number and position of outliers vary from calibration to calibration. The pattern of these points needs the use of a robust procedure since not all of them can be attributed to deviations caused by a curvature. We considered the linear range to be the wideat concentrationinterval outside which were only outliers. In calculating the detection limit, we assumed both the probability of false positives (a)and that of false negatives (6)to be 0.05. For a single determination on an unknown sample (r = 1in eq 51, we obtained detection limits between 1.16 X 10-8 and 2.41 X 10-8 mol dm-3. The decision rule was specific to each calibration. Thus, for calibration I, a given sample was assumed to contain trimipramine if it yielded a peak current higher than 0.022 PA, Le., if the drug content in the sample exceeded the detection limit. It is interesting to note that while an acceptable signal for quantitative
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Flgure 5. Operating characteristic curves for trimipramlne detection limit (calibration I) as a hypothesis test with probabilityof false posithre 0.05. The vertical scale is the probablittyof a false negativeassociated with each detection limit.
purposes was obtained above a concentration of 2.40 X mol dm-3, the requisites imposed on the detection limit in terms of false positives and false negatives shifted it to a much higher value. This argument can be completed by taking into account the number of determinations on the unknown sample: the larger it is, the lower the detection limit will be. Figure 5 shows the operating characteristic curves of the detection limit established from calibration I by keeping the significance level a at 0.05 in 1, 2, and 3 replicates; the detection current is 0.022,0.019, and 0.018 PA, respectively.
CONCLUSIONS The proposed methodology allows the satisfactory determination of trimipramine by adsorptive stripping voltammetry at a carbon paste electrode. The procedure used for calibration and determination of the detection limit can be routinely applied to other chemical systems and endows the analyticalmethod with adequate sensitivityand detedability. The detection limit thus obtained takes account of the sensitivity of the analytical method, the nature of the anal@, and the risk of false positives and falses negatives the analyst is willing to accept. In addition, the proposed methodology solves the problems posed by the need to renew the electrode after each measurement, so it should be equally applicable to any electrochemical method based on the use of a carbon paste electrode.
RECEIVED for review April 28, 1992. Accepted December 1, 1992.
