Background Correction in Multicomponent Spectroscopic Analysis Using Target Transformation Factor Analysis
P A U L J. G E M P E R L I N E , * STACEY E. BOYETTE, and K I M B E R L Y T Y N D A L L t Department of Chemistry, East Carolina University, Greenville, North Carolina 27834
A principal components regression (PCR) is used to construct a cali- bration matrix for multicomponent spectroscopic analysis which cor- rects for background absorption due to variable concentrations of un- known species in a sample matrix. Mixed standards are used in the method, and spectra of the pure background components are not needed, The background correction capability of the method is demonstrated with the use of the UV spectra of aqueous standards prepared from cobalt (II) nitrate and nickel (II) nitrate. Known concentrations of chromium (III) nitrate were added to simulate a background which gave rise to relative concentration errors as high as 70% without the background correction capabilities of the method. With the background correction capabilities of the method, the relative concentration errors were re- duced to less than one percent. Preliminary results are presented from an application of PCR to determine the concentration of two active ingredients in a pharmaceutical product in the presence of weakly ab- sorbing tablet excipient materials. Index Headings: Principal components regression; Quantitative UV/ VIS spectroscopy; Background correction; Computer, applications.
INTRODUCTION
Several authors have recently described the use of principal component analysis, a form of factor analysis, followed by multiple regression to perform quantitative spectroscopic analysis of multicomponent mixtures2 4 The combination of these two techniques is frequently referred to as principal components regression (PCR). Target transformation factor analysis has also been used to perform quantitative spectroscopic analyses and is different from PCR in semantics only2 7 Until now, it has not been specifically noted in the literature that when one or more unknown background components are pres- ent in the standards and samples in variable amounts, the analyst can calculate the sample concentrations, having made exact corrections for the background com- ponents without including nonzero intercepts in the regression model to approximate the background sig- nal. s This is an advantageous feature because inclusion of nonzero intercepts in the regression requires the cal- culation of extra least-squares parameters which can in turn lead to over-fitting during calibration and poten- tially introduce bias in subsequent predictions. Since the PCR technique is general, it may be exploited to perform background correction in any spectroscopic technique which exhibits a linear additive response, such as that predicted by the Beer-Lambert model for ab- sorption spectroscopy.
Received 14 July 1986; revision received 20 August 1986. * Author to whom correspondence should be sent. t Current address: Burroughs Wellcome Co., P.O. Box 1887, Green-
ville, NC 27834.
Rao and Zerbi showed that the technique is mathe- matically similar to a least-squares multiple regression. 1 Cowe and McNicol took advantage of the background correction capabilities of PCR when they performed PCR of near-infrared reflectance spectra to determine the % moisture and % protein in wheat. 2 Fredericks et al. pre- dicted various properties of coal samples from the in- frared spectra using PCR) ,4 The predicted properties of the coal samples were assumed to be linearly correlated to spectral features in the mid-IR range of their spectra.
Several other methods have been devised to solve the problem of background correction in quantitative mul- ticomponent spectral analysis, including generalized standard addition 911 and partial least-squares t~,13 anal- yses. Recently, Kowalski and co-workers have begun ad- dressing the much more difficult task of compensating for a variable and unknown sample background which is not present in the calibration set. 14 Their method re- quires lengthy calculations which use systems of simul- taneous linear inequality constraints to estimate non- negative concentrations for the analytes. The background correction afforded by the method of PCR is simpler than these alternative methods although more limited in scope.
THEORY
Calibration. During the calibration phase of a PCR multicomponent analysis, a set of mixed standards is prepared which contains the analytes at concentration levels selected to bracket the desired working range. Pure standards may also be used. At least one standard must contain the unknown background component, although its concentration need not be known. Ideally, its spectral contribution should not contribute overwhelmingly to the analytical signal. If two or more independently vari- able background components are present in the samples, then each must be represented in the standards in in- dependently varying amounts. A minimum of at least k = i + j standards is required, where i is the number of analytes to be determined and j is the number of independently variable background components. More than k standards are preferred, for example 3(i + j). Using extra standards will tend to make the method more robust by averaging out slight errors in the pre- pared concentrations of the standards. Mixed standards which are linear combinations of other standards are to be avoided since they will contain "redundant" infor- mation.
Each standard is measured at m selected wavelengths where m, the number of wavelengths, should be greater
than k , the number of independently variable compo- nents in the standards. In order to take advantage of the "signal averaging" or "smoothing" capabilities of PCR, one should select a large number of wavelengths in regions where absorption bands are present. The re- sulting spectra are used to form rows in a raw data ma- trix, [A]r~w. The covariance matrix [Z], is calculated ac- cording to Eq. 1, where [A] T is the transpose of [A]:
[Z] = [A] [A] v (1) n x n n x m m x n
The covariance matrix is then decomposed by the meth- od of principal component analysis2 ~ The resulting col- umn and row eigenvector matrices are used to construct a linear model which can be used to approximate the original raw data according to Eq. 2:
[A],a~ = [A]p,~d (2) n x m n x m
[A]pre d = [C]abs t [E]abs t . (3) n x m n x k k x m
In Eq. 3, [C].u~t is the abstract concentration matrix whose columns are the orthonormal eigenvectors of [A]. [E],b~ t is the abstract absorptivity matrix calculated from Eq. 4:
[C]Tabst [A]~,w = [Elabst. (4) k x n n x m k x m
In the principal component analysis of [Z], k = i + j factors (eigenvectors) are expected to emerge, one factor for each analyte and an extra factor for each indepen- dently variable background component. [C]abs t therefore will have k orthonormal columns, each column corre- sponding to an abstract concentration vector of an ana- lyte or background component. [E]abs t will have k rows, orthogonal but not normalized, with each row corre- sponding to an abstract spectrum of an analyte or back- ground.
The estimated Real Error, RE, obtained during prin- cipal component analysis of [Z] is used to determine k, the number of eigenvectors included in [C]abst .16 RE is estimated according to Eq. 5:
RE = X J [ m ( n - k)l (5) =k+l
where m is the number of wavelengths (columns) in [A], n is the number of standards (rows) in [A], k is the number of principal components, and Xj are the eigen- values of [A]. In this work, the random error associated with each individual absorbance measurement is as- sumed to be fixed and is estimated from the noise spec- ification provided by the manufacturer. A sufficient number of factors are included in the principal compo- nent decomposition to give a value of RE equal to or less than the noise specification. While the assumption that the measurement error is fixed may not be strictly correct, especially at different wavelengths, variations in the random error for different wavelengths do not seem to adversely affect the selection of k.
Target Transformation. The final step in the calibra- tion procedure is to rotate the abstract concentration matrix into a least-squares approximation of the real concentrations. To this end, a transformation matrix,
[T] (k × i), is calculated with the use of the "target test" procedure according to Eq. 6.15 A test matrix, [C]test, is constructed which contains the known concentration of the standards and has the dimensions (n × i) where n is the number of standards used to generate [A] and i is the number of components to be determined:
[T] = [C]rabst [C]test (6) k x i k x n n x i
In Eq. 6, [C]T,b,t is the transpose of [C],b,t. The trans- formation matrix from Eq. 6 can be used to calculate the fitted or "predicted" concentrations according to Eq. 7:
[Cl,,od = [C],b,,[T]. (7) n x i n x k k x i
The procedure outlined so far would be identical to that described by Fredericks e t a l . if a unit vector were used to augment [C],b~t in Eq. 6, followed by a modification of Eq. 7 to include the pseudoinverse, 4 [C]pred = ( [C] abst [C] Tabst)-I [C] abst [W]. Augmenting [C],bst with a unit vector allows nonzero intercepts to be included in the regression. It is not necessary to use the augmented [C]~bst matrix when the background components are present in the standards. In this paper, we will demonstrate satis- factory background correction using PCR without the nonzero intercept.
The residuals of the fit in the concentration vector space can be used to estimate the standard deviation of the predicted concentrations from Eq. 8:
S i 2 : 2 (C(pred),J - - C(test),J ) 2 / ( n - - k ) . (8) j=1
In Eq. 8, si is the standard deviation of the predicted concentration of the ith component; c( , ,ed)z and c(te,t)j are the respective predicted and actual concentrations of the ith component of the j t h standard; n is the number of standards; and k is the number of factors included in the principal components regression.
Unknown Calculation and Background Correction. The absorption spectra of n samples to be analyzed are mea- sured at the same wavelengths as the standards. These spectra are then used to form n rows in a raw data ma- trix of unknown samples, [A],,m,. In order to calculate the unknown concentrations of the n samples, one must project the data matrix, [ALto,, into the concentration space of the calibration set according to Eq. 9:
[C]rp,o~ = ([E],b,, [E]T.b.t) -1 [E].bst [A]T.mp" (9) k x n kxm m x h hxrn m x n
Equation 9 may be simplified to give Eq. 10, where [k]-1 is the inverse of the matrix of eigenvalues from the prin- cipal component analysis:
[C] rproj = [X]-I [El,b,t [A] Tsamp. (10) k x n k x h k x m m x n
The transform matrix, [T], calculated in Eq. 6, can be used to predict the concentration of the analytes in the unknown samples according to Eq. 11.
[C]sam p ~-- [C]pro j [ T ] . (11) n x i n x h h x i
In Eq. 11, each row of [C],,mp contains the predicted
concentrations of the i components for the correspond- ing sample.
In order for Eqs. 10 and 11 to be valid, the absorption spectrum of each sample must be accurately expressed as a linear combination of the abstract spectra, [E]abst, obtained during the calibration phase. A simple test can be performed to verify this and to evaluate the suitabil- ity of the calibration model for each individual sample. The predicted absorption spectra of the samples, [A]prea, are calculated according to Eq. 12:
[ A l p r e d = [ C l p r o j [E ]abs t . ( 1 2 ) n x m n x k k x m .
The RMS spectral error of each sample is then calcu- lated from Eq. 13:
RMSn 2= ~ (a(prod).j - a(,,mp)j)2/m. (13) j= l
In Eq. 13, RMS, is the spectral error of the nth sample, and a(p,od)j and a(,amp)j are the respective predicted and actual absorbances of the nth sample at the mth wave- length of the jth sample. When the difference between
the RMS spectral error and the expected spectral error, RE, is statistically significant, it can be concluded that the linear calibration model failed to properly account for the measured sample spectrum, possibly because of the presence of additional background components or interactions between the analytes and the sample ma- trix.
EXPERIMENTAL
Two sets of experiments were conducted to test the background correction capabilities of the PCR tech- nique. In the first set of experiments, the method was tested under carefully controlled conditions using acidic solutions of the commonly available nitrates of cobalt (II), nickel (II), and chromium (III). Chromium (III) ni- trate was added in variable amounts to simulate the presence of an unknown background. The spectra of these solutions at representative concentrations are shown in Fig. 1. In the second set of experiments, a preliminary study was conducted to determine the fea- sibility of using the PCR technique to determine the active ingredients in the popular allergy and cold rem- edy, ACTIFED ® tablets. The active ingredients, pseu-
TABLE I. Calibration of cobalt (II) nitrate and nickel (II) nitrate using two-component model."
doephedrine hydrochloride and triprolidine hydrochlo- ride, are determined in this study in the presence of proprietary tablet excipient materials. Filtered solutions of the excipient materials give rise to weak background absorption in addition to light scattering from insoluble particulates. The spectra of pseudoephedrine hydro- chloride and triprolidine hydrochloride at representa- tive concentrations are shown in Fig. 2. The spectrum of the tablet excipient material is also shown in Fig. 2. Its contribution to the analytical signal is small; however subsequent calculations will demonstrate that failure to account for it can give rise to considerable bias in the results.
Apparatus. All spectra were recorded in digital form with the use of a Shimadzu Model UV-260 double-beam UV/VIS spectrophotometer. Spectra were transmitted over an IEEE-488 interface and stored for post-process- ing on an IBM-PC. All computations were performed on an IBM-PC using prototype programs written with the use of the ASYST scientific programming system.
Reagents. Stock solutions of 0.5000-M Co(NO3)2" 6H20, 0.5000-M Ni(NO3)2"6H20, and 0.02500-M Cr(N03).~. 9 H20 (ACS reagent grade) were prepared in 0.012-M nitric acid, ACS reagent grade. A total of eleven "stan- dards" and thirteen "unknowns" was prepared by di- lution of the appropriate volumes of the above stock solutions with 0.012-M nitric acid in 50-mL volumetric flasks to give the prepared concentrations listed in Table I and Table II, respectively.
Stock solutions containing 6.000 mg/mL pseudo- ephedrine hydrochloride and 0.2500 mg/mL triprolidine hydrochloride in 0.1-N hydrochloric acid (ACS reagent grade) were prepared from their USP standards, which assayed at 99.95% and 100.3% by weight, respectively. A total of nine "standards" was prepared by dilution of appropriate volumes of the above stock solutions to 100 mL with 0.1-N hydrochloric acid to give the prepared concentrations in Table III. A tenth "standard" con- taining a mixture of the excipient materials, pseudo- ephedrine hydrochloride, and triprolidine hydrochloride was prepared from the appropriate volumes of stock so- lutions dissolved with the weight equivalent to one tab- let taken from a mixture of 20 finely ground placebo tablets. The resulting solution was filtered through a 0.22-#-pore-size polyvinylidene difluoride Millipore ill-
TABLE IV. Unknown determination of pseudoephedrine hydrochlo- ride and triprolidine hydrochloride using two-component model. °
RMS Actual conc. Predicted conc. % Error
Sam- Spect. pie Trip. Pseud. Trip. Pseud. Trip. Pseud. Err.
ter. The first few milliliters of the filtrate were discarded and the rest kept for UV analysis. We prepared four "un- known" sample solutions by pipetting the appropriate volumes of pseudoephedrine hydrochloride and triprol- idine hydrochloride stock solutions into 100-mL volu- metric flasks to give 80%, 90%, 100%, and 120% of the labeled s t rength of the active ingredients in one ACTIFED ® tablet. The prepared concentrations are shown in Table IV. The weight of ground placebo equiv- alent to one ACTIFED ® tablet was added to each of the four flasks. The resulting mixtures were diluted to 100 mL with 0.1-N hydrochloric acid and filtered in the same fashion as the excipient "standard."
Procedure. The UV/visible spectra of the eleven stan- dards listed in Table I and thirteen samples listed in Table II were obtained against a 0.012-M nitric acid reference from a wavelength range of 550 nm to 350 nm with a slit width of 2 nm. Data points were recorded every 4 nm, giving a total of 41 points per spectrum. The spectra of the standards were used to form rows in a raw data matrix, [A], which was subjected to the target transformation calibration procedure described in the theory section. The spectra of the samples listed in Ta- ble II were used to form the rows of a data matrix which was subjected to the unknown determination procedure.
The spectra of the standards and samples containing pseudoephedrine hydrochloride and triprolidine hydro- chloride were obtained against a 0.1-N hydrochloric acid reference from 300 nm to 250 nm with a 2-nm slit width. Data points were recorded every 2 nm, giving a total of 26 points per spectrum. The spectra of the standards in Table III and the spectra of samples in Table IV were used in the calibration and unknown determination cal- culations, respectively.
RESULTS
We performed the calibration of cobalt (II) nitrate and nickel (II) nitrate in the presence of the chromium (III) nitrate using two and three factor models to dem- onstrate the ability of the method to correct for the chromium background. In both cases, the concentration of chromium (III) nitrate was not used in the calcula- tions. The results in Table I were obtained by the use of two factors in the target transformation calibration procedure. They are the results one would expect to ob- tain, without background correction, with the use of the traditional "K-matrix" approach to multicomponent quantitative analysis. 8 The results in Table V were ob- tained with the use of three factors in the target trans- formation calibration procedure. The estimated stan- dard deviation of the spectral data, RE, obtained from
APPLIED SPECTROSCOPY 457
TABLE V. Calibration of cobalt (II) nitrate and nickel (II) nitrate TABLE VII. Calibration of pseudoephedrine hydrochloride and tri- using three-component model." prolidine hydrochloride using three-component model2
the principal component analysis of the Co, Ni, and Cr standards was 1.4 × 10 -2 absorbance units (AU) for the two-factor model and 1.0 × 10 -3 AU for the three-factor model. The instrument manufacturer's specification for reproducibility is 1.0 x 10 -3 AU at 220 nm. The value obtained with the two-component model is clearly too large, indicating the presence of unexplained variability in the spectra. The value obtained with the three-factor model corresponds with the manufacturer's specifica- tion quite nicely, indicating that there are at least three independently variable components present in the spec- tral data matrix. Inclusion of additional factors in the calibration step always leads to slightly lower estimates for the standard deviation of spectral data. In this case the extra factor(s) reproduce the error incurred when the standards were measured. Subsequent predictions based on a regression model which includes these extra factors can potentially introduce bias into the results. Although statistical tests cannot be used to identify the "best" regression model for prediction, the criterion used here helps to select an "adequate" regression model with the minimum number of factors.
For the two-component model used in Table I, the standard deviation of the predicted concentrations cal- culated from the residuals of the fit are an order of mag- nitude higher than those obtained for the three-com-
TABLE VI. Unknown determination of cobalt (II) nitrate and nickel (II) nitrate using three-component model, a
ponent model. Inspection of the residuals for individual standards shows reproducible bias, which is consistent with the error expected because of the presence of un- compensated Cr background. In the case of the eleventh standard, which contains the highest concentration of Cr, a significant lack of fit is indicated by a comparison of its residuals with the rest of the residuals. For the three-component model used in Table V, good precision and good accuracy are observed for all the standards. The residuals are all of the same order of magnitude and do not appear to show evidence of any consistent bias. It is especially important to notice that the background due to Cr in the tenth and eleventh standards has been completely accounted for by incorporation of three fac- tors into the linear model, even though the concentra- tion and pure spectrum of chromium (III) nitrate were never used. In this case it is not necessary to include a nonzero intercept in the regression because the appro- priate linear combination of three eigenvectors com- pletely accounts for the variability introduced by the chromium background.
Table II and Table VI show the results for the deter- mination of Co and Ni in the "unknown" samples with the use of the two- and three-component models, re- spectively. The results shown in Table II are the same results expected if the traditional "K-matrix" multiple regression approach were used, which cannot correct for unknown sample background. The presence of the chro- mium background introduces unpredictable and some- times large errors into the results. Samples eleven through thirteen contain the largest amount of Cr back- ground and show correspondingly large relative concen- tration and spectral errors. For these three samples, the
TABLE VIII. Unknown determination of pseudoephedrine hydroehlo- ride and triprolidine hydrochloride using three-component model."
chromium background contributes a significantly large fraction to the total sample absorbance.
The results obtained with the three-component model in the sample determination show excellent precision and accuracy, except for sample twelve. The spectral error for this sample is much larger than the expected spectral error, 1.0 × 10 -3, which clearly indicates that the three-component calibration model does not prop- erly account for the variability in this sample's spectrum (possibly due to the presence of some contamination). Work is currently under way to establish valid statistical criteria for rejecting sample results associated with such large spectral errors. It is especially instructive to com- pare the two- and three-component results for sample thirteen, where the relative error in the determined con- centration of Ni is reduced from 71% to -0 .5 %. These results demonstrate the background correction ability of the PCR technique.
Given the success of the previous study, favorable re- sults were anticipated for the determination of pseudo- ephedrine hydrochloride and triprolidine hydrochloride in the presence of weakly absorbing and light scattering tablet excipient materials. The results for the calibra- tion of pseudoephedrine hydrochloride and triprolidine hydrochloride with the use of the two-component and the three-component models are shown in Tables III and VII, respectively. Ten standards were used. The standards numbered one through nine contained no ex- cipient background. Standard ten contained an added amount of excipient background. The two-component model failed to account for the presence of the back- ground components in the tenth standard, resulting in high residuals for that standard. The three-component model accommodates the background component in the tenth standard nicely, in this case lowering the standard deviation of the predicted concentrations by over an or- der of magnitude for both pseudoephedrine hydrochlo- ride and triprolidine hydrochloride.
The results for the unknown determinations with the use of the two-component and three-component models are shown in Tables IV and VIII, respectively. Each of the four samples contains the excipient material. The two-component model fails to correct for the sample background, while the three-component model ade- quately corrects for its spectrum. The relative errors in the determined concentrations were lower by an order of magnitude when the three-component model was used to correct for the excipient background material.
CONCLUSIONS
In this paper we have demonstrated the background correction capabilities of quantitative multicomponent spectroscopic analyses by use of the PCR technique when mixed standards are used which contain the same back- ground components in the standards and the samples. The method was tested with the UV spectra of aqueous nickel (II), cobalt (II), and chromium (III) nitrate. Ac- curate results were obtained for the concentration of Ni and Co in the presence of strongly absorbing Cr back- grounds, even though the UV spectra of nickel (II) nitrate and chromium (III) nitrate are severely overlapped. Pre- liminary results were obtained which indicate the meth- od will be useful for determining the concentration of two active ingredients in a pharmaceutical product, de- spite the presence of a complicated sample matrix. In this case the background correction capabilities of the method obviate the need for lengthy or difficult sepa- ration techniques and hopefully will circumvent the need for more time-consuming methods of analysis such as HPLC.
ACKNOWLEDGMENTS
The authors wish to acknowledge the partial support of this research by the Burroughs Wellcome Co. Thanks are given to Guy Inman and Alger Salt for writing software to control and acquire data from the Shimadzu UV-260 spectrometer.
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