1703
Anal. Chem. 1083, 55, 1703-1707 (2) Demming, S. N.; Morgaln, S. L. Clin. Cbem. (Winston-Salem, N.C.) 1979, 25,840-855. (3) Sternberg, J. C.; Stillo, 6 . L.; Schwendeman, FI. H. Anal. Chem. 1960, 32,84-90. (4) Barnett, H. A.; Bartoli, A. Anal. Chem. 1980, 32, 1153-1156. (5) Davis, R. B.; Thompson, J. F.; Pardue, H. L. Clin. Chem. (Winston&/em, N.C.)1978, 24,(111-620.
(6) Wesolowsky, G. 0. “Multiple Regression and Analysis of Variance”; Wiley: New York, 1976; pp 43-45. (7) Margoshes, M.; Rasberry, S. D. Anal. Chem. 1969, 4 1 , 1163-1172.
for review January
317
lgg3* Accepted June 13,
1983.
Multiple Analytical Frequencies and Standards for the Least-Squares Spectrometric Analysis of Serum Lipids Harold J. Kisner’ a n d C h r i s W. Brown*
Department of Chemistry, University of Rhode Island, Kingston, Rhode Island 02881 George J. Kavarnos
Cyto Medical Laboratory, Inc., 12 Case Street, Norwich, Connecticut 06360
Thls paper descrlbes an appllcatlon of multlple least-squares regresslon analysis to the slmultaneous determlnatlon of the predominant serum Ilplds. Severe band overlap of the ester carbonyl absorbance peaks used for analysls Is offset by selectlng a large number of evenly spaced analytlcal wavelengths extending over the carbonyl absorptlon reglon. Callbratlon by standard mlxtures Incorporates molecular Interactions Into the analysis matrix. Over-determlnatlon wlth respect to the number of standards extends the workable concentration range of the Components wlthout loss of accuracy wlthln that range. Llnear curve flttlng models contalnlrrg dlfferent comblnatlons of zero-, flrst-, and second-order terms have been applied to the data. The optlmum least-squares model for all three components Is a llneer relatlonshlp In the form y = a bx. P matrlx notatlon for the Beer-Lambert law, In the form C = PA Po, was used. Analysls tlme, depending on frequency selectlon and the number of slqnals averaged, Is less than 5 mln.
+
+
Often a limiting factor in multicomponent quantitative spectrometry is the lack of mutually independent absorbance bands. Recently, we experienced this problem while investigating the simultaneous determination of triglycerides, phospholipids, and cholesteryl esters by infrared spectrometry based on the difference in their carbonyl ester absorbance peaks (1). Severe overlap of the carbonyl bands (Figure 1, ref 1)limited the effectiveness of the two-point calibration method used for analy!sis. Difficulties encountered when using strongly overlapping bands for quantitative analysis can be overcome by applying a least-squares curve fitting technique to absorbance data (2-4). Calibration by mixtures incorporates molecular interactions into the analysis matrix and, depending on the number of standards used, can extend the workable concentration range of the components. Selection of analytical wavelengths is not restricted by the number of components; rather, it is limited by the computer capacity for matrix inversion (4). Sternberg et al. (2) recommended using a large number of evenly spaced wavelengths, over the absorption region of interest. Sustek ( 5 ) suggested that a 3- to 4-foPd ‘Present address: Postdoctoral Resident in Clinical Chemistry, Department of Pathology, University of Maryland School of Medicine, Baltimore, MD 21201.
Table I. Matrix Representations of Beer-Lambert Law for Least-Squares Quantitative Analysis K matrix
Beer-Lambert lawa A = KC + K , calibration A = KC A c t = KCCt K = ACt(CCt)-’ unknowns
P matrix C = P A + P, C = PA CAt = PAAt P = CAt(AAt)-’
A = KC C=PA K~= AK~KC c = ( K ~ K ) -K‘ ~ A
a Zero-order terms are included in the appropriate matrix (K or P) for computations.
overdetermination in analytical positions offers optimum results, even in the case of overlapping absorption bands. In our previous study, the use of zero-order terms allowed for the linear aproximation of nonlinear data over a limited coneentration range. Second-order terms should be more appropriate for a least-squares regression equation representing nonlinear data, as suggested by Barnett and Bartoli (3). When deviation from linearity was due to intermolecular interactions, tFey introduced “product terms” that corrected for molecular association between two compounds. Recently, we ( 4 ) reported on matrix representations for spectroscopic multicomponent analyses, and these methods are summarized in Table I. Two methods of least-squares analysis are presented: traditional K-matrix notation where A is a function of C, and P-matrix notation, where C is a function of A . The use of P-matrix notation for the BeerLambert law facilitates matrix computations and is more compatible for the inclusion of zero or higher-order terms (8). (For a more detailed description of the advantages of the P-matrix approach, see ref 4 and 8.) We have examined the utilization of a multiple linear least-squares regression analysis of overdetermined absorbance data for the simultaneous determination of triglycerides, phospholipids, and cholesteryl esters. Results comparing wavelength selection and curve fitting using linear, power series, and quadratic models in P-matrix notation are discussed herein. EXPERIMENTAL SECTION Apparatus. Uncompensated infrared spectra were measured with the solution contained in a 13-mm cell, which was constructed in-house. The cell consisted of 25 X 25 X 4 mm AgCl windows
0003-2700/83/0355-1703$01.50/0 0 1983 American Chemical Society
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SEPTEMBER 1983
Table 11. Concentrations of 1 8 Stock Solutions (6 per Component) Used to Generate 85 Calibration Standards' row no. TG PC CE 1 2 3 4 5 6 a
3.0 5.9 8.9 12.6 16.9 20.9
5.4 8.9 12.8 17.9 22.8 29.0
5.9 9.9 14.0 19.3 23.9 29.8
Concentrations are expressed in mg/dL of CHC1,.
~~
~~
~
TabIe 111. Concentrations of 21 Stock Solutions (7 per Component) Used to Make Simulated nown Mixtures' row no. 1 2 3 4 5 6 7
TG 0 4.3 7.6 10.5 14.6 19.3 25.8
PC
CE
0
0 8.3 11.9 17.0 21.7 26.8 36.0
7.0 10.8 15.4 20.4 26.1 35.2
' Concentrations are expressed in mg/dL of CHCl,. (Harshaw Chemical Co.) and a Teflon spacer with a total volume of 0.50 mL. All spectrometric procedures were performed with a Beckman MICROLAB 620MX microprocessor-centrolled, compQbinginfrared spectrophotometer. All computer calculations were Mrformed on a Data General NOVA-3 computer. Reagents. Solutions were prepared in reagent grade chloroform 99%), DL-(U(Fisher Scientific C-574). Tripalmitin (T-8002, phosphatidylcholine, dipalmitoyl (P-6138,99%),and cholesteryl palmitate (CH-SP, 99%) were purchased from Sigma Chemical Co. The palmitate esters were selected since they closely reflect the awerage chain length of these serum lipids. F m the 18 stock solutions (6 per component) listed in Table 11, calibration standards were prepared as follows: mixtures from the three lipids were made for all combinations where the difference in row number between the three Components for any one standard was always 12. For instance, a standard containing TG = 5.9 mg/dL, PC = 17.9 mg/dL, and CE = 19.3 mg/dL was acceptable while a standard of TG = 5.9 mg/dL, PC = 17.9 mg/dL, and CE = 23.9 mg/dL was not. In the latter case, the difference in row number between TG and CE is 3. In this fashion 85 standards (including TG = 0.0, PC = 0.0, CE = 0.0) were generated. Two simulated unknown sets (A and B) were generated from the stock solutions listed in Table 111. Set A (18 unknowns) was created to test accuracy within calibration range, using the same guidelines set forth while preparing calibration standards. Set B (16 unknowns) was created to test accuracy outside of the calibration range, and solutions were prepared accordingly. The concentrations listed in Tables I1 and I11 were chosen to confine absorbance values between 0.15 and 1.0 A (70% to 10% T ) , where maximum instrument accuracy occurs. These concentrations correspond to approximately 10-15% of normal serum levels of these lipids (6) and could be easily adapted to the study of serum lipids by appropriate dilution. Procedure. All spectra (calibration and unknown) were measured on the MICROLAB 620MX, using the Advanced Quantitative Analysis Compuset 111. Sixteen consecutive readings were averaged at each frequency, increasing the S / N ratio by a factor of 4. For the determination of Iothe empirical ratio method was employed (7), where Io is measured at a transmission maxima (1828 cm-l) adjacent to the absorption band. Absorbance readings were taken from 1783 cm-l to 1713 cm-I a t 6-cm-' increments, as shown in Figure 1. Extending the wavelength range to 1783 cm-I incorporates the effect of the = 1784 cm-l) into the analysis. The choice of a 5-cm-' solvent (v, increment allowed for the measurement of absorbance values within 1 cm-l of the analytical frequencies of the individual
MIXTURE TG PC Clr 8
IO
12
( MG/DL)
0
1850
Id00 IS50 I?OO FREQUENCY, CM-I
1L10
Figure 1. Carbonyl ester peak of three-component mixture of tripalmitin (20mg/dL), phosphatldylchotine dipalmltoyl (25 mg/dL), and
cholesteryl palmltate (30 mg/dL) In chloroform. The hash marks corres ond to 15 wavelength settings from 1783 to 1713 cm-' via a 5-cm-P increment. components (triglycerides vmax = 1742 cm-l, phospholipids vmax = 1737 cm-l, and cholesteryl esters vmax = 1723 cm-l). Three curve fitting models were applied to the absorbance data: linear, LN (zero- and first-order terms); second-order power series, PN2 (zero-, first-, and second-order terms); and quadratic, QN (zero-, first-, and second-order terms and first-order cross terms) (8). The inclusion of zero-order terms frees the intercept to be fixed by regression. By using 100% CHC13as one of the standards, absorbance values at zero concentration where TG = PC = CE = 0 are incorporated into the regression analysis. Statistical Analysis. The slopes and intercept which define the line of best fit comprise the coefficients of the calibration or analysis matrix. Supporting statistical tests that evaluate least-squares results (9-11) are summarized in Table IV. Since the mean percent error is overly sensitive to the low end of the concentration range, the mean deviation (mean absolute difference) better serves as an indicator of the predictive value of the analysis matrix. The standard error of the estimate (Se) is analogous to the standard deviation about the mean and thus provides a degree of scatter about the regression line. The coefficient of multiple determination (R2)characterizes the fit of the data to the entire function, as R2approaches one, the greater the degree of statistical relation in the observation. A comparison of R2 for different models is useful in optimizing the theoretical functional form. Some statisticians (11) suggest adjusting the coefficient of multiple determination and standard error of estimate to recognize variations in the number of independent variables (degrees of freedom). Adjusted statistical parameters (indicated by a subscript a) are included when applicable. Since we are simultaneously analyzing for three different lipids, a combined statistic (summation of the three lipid classes) is introduced to facilitate comparisons between different models. R E S U L T S AND DISCUSSION Overdetermination w i t h Respect t o t h e N u m b e r of Standards. Successful analysis of multicomponent systems using mutually interfering absorption bands often depends upon the number of standards used for calibration. Since all three components in the present study contribute to the ab-
ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
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Table IV. Statistical Parameters Used Tlo Evaluate Least-Squares Results equation“ individual
statistic
com bined n
2 individual n
mean % error m
$ individual
A
z Yj - Yj
i=l
mean deviation
n
m i/a
standard error of estimate (Se)
n ( m - 1) substitute m - t for m - 1
adjusted standard error of estimate (Sea)
m
A
2 ( Y ,- Y)’
i=l
coefficient of multiple determination ( R ’)
m
adjusted coefficient of multiple det,ermination ( R
1-
-
n m
z: (Yi- Er)’
2.2 (Yi- Y)’
i=l
1=1
[(iS)(l-. R2J
n
Key: m is number of standards, n is number of components, Yj is observed value, Y , is estimated value, t is total number of coefficients, 7 is mean of observed values. a
Table V. Analysis of Calibration Standards Using Combined Statistics mean % mean deviation, std error of CM-1,increment model“ error w/dL estimate,c mg/dL 1743 1738 1723 1743 to 5 1723 1783 to 10 1713 1783 to 5 1713
LN-3 PN2-3 QN-3 LN-5 PN2-5 QN- 5 LN-8 PN2-8 QN-8 LN-15 PN2-15 QN-15
10.56 9.97 9.76 10.14 9.48 9.41 7.40 6.18 4.20 5.82 5.08
1.34 1.29 1.23 1.15 1.09 0.97 0.94 0.80 0.50 0.71 0.60
1.83 (1.87) 1.79 (1.86) 1.72 (1.82) 1.57 (1.61) 1.53 (1.63) 1.28 (1.47) 1 . 2 8 (1.35) 1.06 (1.18) 0.69 (1.00) 0.97 (1.07) 0.80 (0.99)
coeff of multiple determination 0.926 (0.917) 0.930 (0.910) 0.935 (0.903) 0.946 (0.934) 0.949 (0.919) 0.964 (0.865) 0.964 (0.949) 0.975 (0.940) 0.990 ( b ) 0.979 (0.954) 0.986 ( b )
a Model LN-X = linear, nonzero intercept; PN2-X = power-series, nonzero intercept, second order; QN-X = quadratic, nonzero intercept; X = number of analytical wavelengths. Requires inversion of a (147 X 147) matrix; we are limited to a ( 7 0 X 70) matrix by the size of the computer memory. (’ Adjusted statistic in parentheses.
sorbance at each of the analytical wavelen,gths,changes in the absorbance due to any one component will significantly influence the estimated concentrations of ithe other two components. It is laborious tx, generate enough standards to statistically cover all possible variations over a large concentration gradient in a three-component mixture. On the other hand, calibration with too small a set of standards will result in inaccurate analysis of simulated unknowns. Errors in the determination of unknowns should reflect the accuracy achieved when standards represented by the calibration coefficients are reestimated. In this fashion, the least-squares coefficients could be routinely applied in the analysis of any sample in which the concentrations fall within the confines of the calibration standards. Initial attempts on our part to calibrate with 16 and then 32 standards proved unsatisfactory, eventually giving way to thle use of 85 calibration standards. Conceivably, the use of calibration sets still larger would
further minimize deviations in the analysis of unknown mixtures. Frequency Selection a n d Least-Squares Models. We investigated the use of several different wavelength combinations extending over the resultant carbonyl absorbance peak of the three-component mixture. After the calibration coefficients were determined for a particular set of parameters (wavelengths, curve fitting), the resultant analysis matrix was for every Xi used used to determine the estimated response Yi, for calibration. Results summarizing the “goodness of fit” for the calibration standards appear i t T a b l e V. Of greater interest than the “goodness of fit” parameters is the predictability of the calibration matrices in the determination of unknowns. Results obtained from the analysis of both simulated unknowns sets, using the appropriate calibration coefficients, are summarized in Table VI. Two interesting patterns warrant discussion: (A) Increasing the number of analytical wavelengths improved both the fit
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ANALYTICAL CHEMISTRY, VOL. 55, NO.
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Table VI. Analysis of Simulated Unknowns Using Combined Statistics
CM-1, increment
modela
1743 1738 1723 1743 to 5 1723 1783 to 10 1713 1783 to 5 1713
LN-3 PN2-3 QN-3 LN-5 PN 2-5 QN-5 LN-8 PN2-8 QN-8 LN-15 PN2-15 QN-15
set A set B mean deviation, std error of mean deviation, std error of mg/dL estimate, mg/dL mg/dL estimate, mg/dL 1.74 1.81 2.22 1.55 1.63 2.22 1.05 1.09 1.57 0.86 0.91
2.38 2.39 3.06 2.06 2.12 3.02 1.44 1.43 2.06 1.18 1.27
2.12 2.10 2.69 1.74 1.82 2.71 1.27 1.23 1.90 1.21 1.11
2.80 2.86 3.94 2.12 2.26 3.93 1.66 1.83 2.46 1.70 1.60
a Model LN-X = linear, nonzero intercept; PN2-X = power-series, nonzero intercept, second order; QN-X = quadratic, nonzero intercept; X = number of analytical wavelengths. Computer limitations inhibit QN-15; see footnote b in Table V.
-
Table VII. Analysis of Mixture Number 16 of Simulated Unknown Set B concentration, mg/dL lipid
observed calculated LN-15 PN2-15 25.8 35.2 36.0
TG PC CE
26.3 29.7 39.2
25.9 33.2 37.7
7% error LN-15 PN2-15 1.9 15.6 8.9
0.4 5.7 4.7
of the data and the accuracy in the analysis of unknowns. (B) The use of cross terms offers no statistical improvement in this analysis. Although it appears that the “goodness of fit” indicators favor a quadratic model (QN), the adjusted coefficient of multiple determination reveals that this improvement is due to an increase in the degrees of freedom while fitting the data. Examination of Table VI reinforces this. Comparison of curve fitting models when the number of analytical wavelengths is held constant shows that the anaylsis of simulated unknowns is least accurate when a quadratic model is used. Inclusion of second-order terms (PN2) for calibration shows comparable results with a model using only zero- and firstorder terms (LN). A slight improvement is evident in the analysis of unknowns outside the calibration limits (set B) using Model PN2-15. This improvement can be traced to a mixture (no. 16) in this unknown set where the concentrations of all three components are greater than the highest concentrations used for calibration. The carbonyl ester peak for this mixture has transmission values below 10% T. Results from
the analysis of this mixture, summarized in Table VII, suggest that deviations from linearity in the Beer-Lambert plots are not significant until absorbance values exeed 1 A . If the unknown concentrations can be adjusted (by dilution) to fall within calibration limits, the optimum least-squares model appears to be a straight line relationship of the form y = a bx. If concentrations remain above the high calibration limit, the optimum least-squares model appears to be a linear relationship of the form y = a bx cx2. Calibration using 15 evenly spaced wavelengths shows approximately a 2-fold improvement over calibration using three wavelengths corresponding to the peak maxima of the individual lipids. Results comparing models LN-3 and LN-15 for the individual componentP are presented in Table VIII. The degree of spectral interference varies for the three components, the middle peak (phospholipid) having the greatest interference. One might, therefore, predict that the phospholipid determination is less accurate than the determination of the other two components. This appears true for the analysis using three wavelengths (LN-3), as seen in Table
+
+ +
VIII. Multiwavelength calibration apparently negates this effect. The improved analysis of triglycerides with respect to the other two components in model LN-15 is probably due to a greater number of analytical wavelengths spaced equally around its peak maxima (1742 cm-I). Conceivably, recalibrating with an increase in analytical wavelengths below 1713 cm-l would produce a similar effect on the other two components. In the present study, the results using model LN-15 are acceptable for all three components, with coefficients of
Table VIII. Comparison of Models LN-3 and LN-15 for Individual Lipids std error of estimate,a mean deviation, mg/dL mg/dL lipid standard set LN-3 LN-15 LN-3 LN-15 TG
calibration A B
PC CE
calibration A B calibration A B
a
1.05 1.17 1.68 1.78 2.59 2.97 1.17 1.46 1.73
Adjusted statistic in parentheses.
0.47 0.52 0.68 0.86 1.10 1.71 0.80
0.98 1.24
1.40 (1.42) 1.57 2.04 2.39 (2.44) 3.32 3.92 1.56 (1.58) 1.86 1.98
0.61 (0.68) 0.73
coeff of multiple determinationa
LN-3
LN-15
0.935 (0.932)
0.987 (0.985)
0.891 (0.887)
0.975 (0.970)
0.955 (0.953)
0.979 (0.974)
0.88
1.14 (1.25) 1.46 2.28 1.07 (1.18) 1.24 1.63
Anal. Chem. 1983, 55. 1707-1710
multiple determination (R2)exceeding 0.97. For the analysis of simulated unknown mixtures within the calibration range (set A), the mean deviation was less than 1.10 mg/dL for any of the components. Folr unknown mixtures outside of the calibration range (set EN),errors are still tolerable, with the mean deviation not exceeding 1.71 mg/dL for the least accurate component, phospholipids. The mean deviation for triglycerides (0.68 mg/dIJ and cholesteryl esters (1.24 mg/dL) was much better. CONCLUSION The first known simultaneous determination of triglycerides, phospholipids, and cholesteryl esters has been made possible by using a multiple least-squares regression method in P-matrix form. The analysis matrix was prepared from calibration of 85 standards, using absorbance data measured at 15 evenly spaced wavelengths over the absorption region of interest. A linear model in the form C = P A Po gave optimum results for the analysis of mixtures within calibration limits. This method of analyzing the predominant serum lipids by class is presently being applied to the analysis of native serum and results will be reported in a future paper. The methods developed are applicable to the analysife of other complex mixtures, especially systems involving components with highly overlapping spectra.
+
1707
ACKNOWLEDGMENT We wish to thank R. J. Obremski for many helpful suggestions and M. A. Maris for computer assistance. Registry No. Tripalmitin, 555-44-2; dipalmitoyl-DL-a-phosphatidylcholine, 2797-68-4; cholesteryl palmitate, 601-34-3. LITERATURE CITED Kisner, H. J.; Brown, C. W.; Kavarnos, G. J. Anal. Chem. 1982, 5 4 , 1479- 1485. Sternberg, J. C.; Stillo, H. S.;Schwendeman, R. H. Anal. Chem. 1980, 32, 84-90. Barnett, H . A.; Bartoll, A. Anal. Chem. 1960, 32, 1153-1156. Brown, C. W.; Lynch, P. F.; Obremskl, R. J.; Lavery, D. S. Anal. Chem. 1982, 5 4 , 1472-1479. Sustek, J. Anal. Chem. 1974, 46, 1676-1679. Tietz, N. W. "Fundamentals of Clinical Chemistry", 2nd ed.; W. B. Saunders Co.: Philadelphia, PA, 1976. Freeman, N. K. I n "Blood Lipids and Lipoproteins: Quantitation, CornDosition, and Metabolism"; Nelson, G. J., Ed.; Why-Interscience: New York, 1972; p 159. Mark, M. A,; Brown, C. W.; Lavery, D. S. Anal. Chem. 1983, 5 5 , 1694-1703. Davls, R. 6.; Thompson, J. E.; Pardue, H. L. Clln. Chem. (Winston-Sa/em, N.C.) 1976, 24, 611-620. Demlng, S. N.; Morgan, S. L. Clln. Chern. (Winston-Salem, N . C . ) 1979, 2 5 , 840-855. Neter, J.; Wasserman, W.; Whitmore, G. A. "Applled Statlstics", 2nd ed.; Allyn and Bacon: Boston, MA, 1978.
RECEIVED for review January 31, 1983. Accepted June 13, 1983.
Pulsed Photoacoustic Spectroscopy and Spectral Deconvolution with the Kalman Filter for Determination of Metal Complexation Parameters Sarah C. Rutari and Steven D. Brown*
Department of Chemistry, Washington State University, Pullman, Washington 99164-4630
A pulsed photoacoustic spectrometer has been constructed for the purpose of monltorlng metal complexatlon equlllbrla In aqueous solutlon. The spectra of several mixtures of praseodymium and sthylenedlamlnetetraacetlcacld (EDTA) were measured and were deoonvoluted with the Kalman fliter. The amount of free praseodymlum was monltored as a functlon of ligand added, and a stablllty constant of 3.9 X 1015 was obtalned for the PrEDTA- complex.
Photoacoustic spectrloscopy has been shown to be an excellent method for studying weak absorptions in liquids. Both chopped continuous wave (CW) and pulsed laser excitation have been utilized (1-4); however, pullsed photoacoustic spectroscopy ha5 been shown to be the more sensitive technique, since the pulsed photoacoustic signal is directly dependent upon the laser pulse energy (3,4). This sensitivity allows pulsed photoacousitic spectroscopy to be used as a probe of trace metal species in aqueous environments. The lanthanide ions have unique spectroscopic behavior in aqueous solution. The spectral transitions occurring in the visible region are relatively sharp, due to the fact that the transitions are f-P transitions and are relatively well shielded from ligand and solvent effects by the 5s and 5p electrons. In particular, the praseodymium transition at 590 nm shows
only a slight shift and broadening when a complexing agent such as ethylenediaminetetraaceticacid (EDTA) is added. In addition, the lanthanide spectroscopic transitions in the visible region are weak (molar absorptivities are on the order of 1 to 10 L/(mol cm)), so that a sensitive technique, such as pulsed photoacoustic spectroscopy, is required to measure concentrations of these ions a t low levels. The spectroscopic and complexation characteristics of the lanthanides are of interest as they closely resemble those of the tripositive actinides, such as americium (6). In the past, the minute shifts with complexation in the visible spectra of the lanthanides have made it nearly impossible to determine complexation parameters from spectroscopic data, since solubilities are fairly low and since reliable deconvolution of the overlapping spectral responses is necessary to determine the amounts of free and complexed species. Recently, however, linear parameter estimation techniques have been used for spectral deconvolution of severely overlapped responses (6). The Kalman filter is a recursive, digital filtering algorithm for linear parameter estimation which has been used to resolve overlapped responses in both electrochemistry and spectroscopy (7-9). The filter requires a well-defined model consisting of a spectrum for each of the components suspected to contribute to the overall spectral response. When this model is available, the Kalman filter allows the estimation of all the component concentrations
0003-2700/83/0355-1707$01.50/0 0 1983 American Chemical Society
