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
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
in the mixture. This paper demonstrates the application of the Kalman filter to the spectral deconvolution of overlapped Pr3+ and PrEDTA- responses, with the subsequent determination of the metal-ligand stability constant. Because the f-f transitions in Pr3+ and PrEDTA- are weak, and because the solubility of PrEDTA- species in water is limited, spectral data were obtained by using pulsed photoacoustic spectroscopy. Photoacoustic methods have previously been used to study aquo-complex ions of the lanthanides (10).
THEORY The Kalman filter is a recursive, digital algorithm for linear parameter estimation that was derived by Kalman in 1960 (11). The details of the application of the filter to chemical problems have been discussed previously (7-9). An outline of the approach is given here. The Kalman filter has four requirements for information which must be supplied to do linear parameter estimation. First, a model of the system dynamics must be specified. For the special case of determining concentrations in a multicomponent system where the concentrations are invariant, the system model is X(k) = I.X(k - 1) (1) where X(k) is the vector of concentrations of length N at wavelength k in the spectrum, N is the number of components, and I is the identity matrix. Next, a model of the measurement process is required. In this case, a Beer’s law linear relationship between the photoacoustic signal and the concentration of each individual absorber at a given wavelength is assumed. This can be expressed as
Z ( k ) = ST*X(k)+ v ( k )
(2) where Z ( k ) is a measurement of the photoacoustic signal at wavelength k , ST is a measurement function vector of proportionality factors obtained from measuring the pure component spectra, and v(k) is the noise contribution from the measurement process. In addition, initial guesses of the component concentrations and measurement variances are required. Finally, the structure of the measurement noise is assumed to be white and of zero mean, although for many chemical measurement processes, this assumption may not be necessary (12). When these requirements have been met, the filter improves the initial guesses of the concentrations and variances and yields an optimal (least-squares) fit of the component spectra to the multicomponent spectrum. The basic algorithm equations for the case described here are as follows: X(klk - 1) = X(k - llk - 1) (3)
P(klk - 1) = P(k - Ilk - 1)
(4)
K(k) = P(klk - l)*S(k)*[ST.P.S+ 0z2]-1
(5)
X(klk) = X(k - Ilk - 1) + K.[Z - ST.X]
(6)
P(klk) = [ I - K.ST].P(k - llk - 1)
(7)
A list of parameter and notation definitions is given in the Glossary.
EXPERIMENTAL SECTION Reagents. A stock solution of approximately 0.2 M Pr3+was prepared from Pr(N0J3.5Hz0 (Aldrich Chemical Co.). The EDTA stock solution was prepared from Na2EDTA-2Hz0after drying at 80 OC for 2 h and was 0.0983 M. All solutions were made up to an ionic strength of 0.10 with KNO* A series of PS+/PrEDTAmixtures were made to contain a constant total metal concentration of 1.81 X M Pr3+with total EDTA concentrations varying from 3.93 X lo4 M to 1.671 X M. The pure PrEDTAsolution was made up to contain less than 3% of the Pr3+aquo complex (calculated from an approximate stability constant), an
Figure 1. Photoacoustic detector: (A) brass cylinder, 2.5 X 8.0 cm; (6)clay filler; (C) aluminum anvil, 2.5 X 5.0 cm; (D) piezoelectric disk (PZT-SA,Vernitron, Inc.), 0.64 X 2.5 cm; (E) front surface mirror, 0.2 X 2.5 cm; (F) leads to amplifier circuit. 7 M
10 k
PZ T
I + I + -
-
Figure 2. Circuit diagram for the piezoelectric charge amplifler: (A) LF-411; (6) LA-3140-AT.
amount which was not detectable in these studies. All the PS+/PrEDTA- mixture solutions were adjusted to pH 2.81 f 0.01. Photoacoustic Spectrometer. A photoacoustic spectrometer was constructed for these studies. The light source was a flashlamp-pumped dye laser (Candela Corp. LFDL-2) utilizing Rhodamine 6G dye (Exiton Co.) at approximately lo4 M in 50% (v/v) methanol/water. The wavelength was controlled by a variable angle grating equipped with a piezoelectric positioner (Burleigh Inchworm Translator). The sample solution was placed in a standard 1-cm quartz fluorescence cuvette, and the cell was acoustically coupled to the detector with a glycerol drop. A diagram of the detector is shown in Figure 1;it is similar to that developed by Winefordner and co-workers (4),but a front-surface mirror was used in front of the PZT-5A piezoelectric detector, and acoustic reflections were minimized by terminating the PZT disk with an aluminum anvil surrounded by clay filler. A simple circuit was used to amplify the charge created on the PZT disk; a diagram is given in Figure 2. The photoacousticsignal was then sent to a 10-bit, 10-MHztransient recorder (Physical Data, Inc.) and stored in the computer memory via a DMA interface. The computer was an LSI-11/23 using an RT-11 operating system; it was used to collect and analyze the transient photoacoustic data. The computer was interfaced to the wavelength controller via two 16-bit parallel 110 ports. Programs. The photoacoustic experiment was controlled by two FORTRAN programs, TRANS and PAS. TRANS collected the first 50 ws of each photoacoustic transient for display, and/or storage, with a point averaging option. PAS collected the amplitude of the first acoustic pulse to reach the detector as a function of the laser wavelength, also with a point averaging option. Both programs utilized MACRO-11 and FORTRAN routines supplied by Physical Data, Inc., for data acquisition and transfer. In addition, MACRO-11 assembly language routines were written to drive the Inchworm translator to a desired wavelength and to reset the translator t o zero. The scan speeds varied from 300 A/min to 45 A/min, depending on the number of points averaged at each wavelength. In these studies, four points were averaged at each wavelength, and spectra of each of the above mixtures were obtained for a wavelength range of 5800 A to 6150 A. Data Analysis. All data analysis programs were written in FORTRAN. Spectra were smoothed by a Fourier smoothing
ANALYTICAL CHEMISTRY, VOL. 55, NO. 11, SEPTEMBER 1983
51 85730.a
Figure 3. Effect of EDTA complexation on the spectrum of Pr(II1): (A) 0 M total ligand; (B) 0.7'87 M total ligand; (C)excess total ligand.
1709
,
W O . B58~0.0 5408.8 j431.2 t ~ 2 . 26 w . a 6;ab.i t ~ i i . 2
N: ,..EhC-l:F: >:
Figure 4. Typical Kalman filter fit: (A) original spectrum; (B) Kalman filter fit; (C) residuals; (D) Pr3+ spectrum contribution; (E) PrEDTAspectrum contribution.
program and the contribution of the water background was subtracted out by a background subtraction routine. The Kalman filter program used in these studies has been described previously (7). The PrEDTA- constant was obtained with the nonlinear least-squares program, MINIQUAD (13,14),which was modified for the LSI-11/23.
zr r
R E S U L T S A N D DISCUSSION Performance of the Photoacoustic Spectrometer. The photoacoustic spectrometer used in this work was similar to that of Winefordner and co-workers ( 4 ) ,%withone important exception. A charge amplifier, rather than a voltage amplifier, was used to amplify the photoacoustic signal. The charge on the piezoelectric disk was amplified directly, giving an integrated signal with a larger signal to noise ratio. The reproducibility of cuvette placement was investigated for repeated removal and replacement of the cell on the glycerol drop; the relative standard deviation (RSD) was 5% for a solution of 2 x M Pr3+. This value was comparable to that of Winefordner aind co-workers (4). In addition, an analysis of variance showed that this errlor was significantly larger than the error inherent in the measurement process, where the RSD was 3%. After Fouriler-domain spectral smoothing, the R3D was reduced to approximately 1.5%. This value was used to estimate the variance in the measured data , ) : a ( which was required by the Kalman filter. The photoacoustic signal was measured as a function of concentration for Pr3+ in solution a t 5910 A and was found to have a linear relationship. The limit of detection was 3.6 X M Pr3+ (e = 1.915 L/(mol cm)) (15),defined as the concentration giving a signal size equal to two standard deviations of the blank. The limit of detection was bounded by the water background absorbance, which is approximately lV3 absorbance units at 5910 A and varies from to absorbance units throughout the visible region (3). The photoacoustic process is much more efficient in solvents such as hexane and carbon tetrachloride; however, improved limits of detection in these solvents are largely due to the much lower background absorbances. P r E D T A - Complexation. The spectral changes in the 5910 A Pr3+transition with complexation are shown in Figure 3. The peak maximuin was shifted about 30 A to longer wavelengths and the peak intensity was enhanced slightly with increased complexation. These changes were in agreement with previous observations reported in the literature (5, 16). The Kalman filter was used to fit each of the mixture spectra to a two-component mlodel, consisting of pure component spectra for Pr3+and PrEDTA-, and in this manner, the Pr3+ concentration was obtained. An example of a Kalman filter fit is shown in Figure 4. A summary of the fit results is given in Table I. The filter was able to extract the Pr3+ response from the mixtures for intensity ratios of up to 8:l (PrEDTA-/Pr3+) and concentration ratios of 6:l. Mixtures with
.
k!=
-
L L mol
9 0.00
TOTRL 8.84 LIGRND 0.08 ADDED (6.16 0.12 H ) (xlO-z) 0.20
0.24
Flgure 5. Titration curve for Pr3+ with EDTA ligand: points (0)are the values for Pr3+ obtained from the Kalman filter fits, and the curve is the theoretical titration curve obtained by using the stability constant obtained from MINIQUAD. Table I. Results from the Kalman Filter for Pr3+/PrEDTA-Mixtures fraction mix- total total free ture liganda metala metalb 1.81
6
0.393 0.787 0.983 1.180 1.376 1.573
7
1.671
1.81
1 2
3 4 5
1.81
0.745 0.547
1.81
0.491
1.81
0.351 0.261 0.204 0.152
1.81 1.81
CDC 0.9998 0.9999 0.9999 0.9999 0.9997 0.9999 0.9999
varianced 3.3 X 1.3 X 1.7 X 1.4 X 4.6 X
1.7 X 1.0 X
lom6
a Concentrations in mM. Fraction of the total metal concentration extracted by the filter as Pr3+. Coefficient of determination of the fit. Deterministic variance of the fit.
greater ratios of PrEDTA-/P$+ were found to give essentially identical spectra. The detectable ratio corresponds approximately to the expected detection limit of Pr3+in the presence of a PrEDTA- background. A total of seven mixtures were found to have detectable Pr3+ in the presence of PrEDTA-. A plot of the free metal concentration vs. ligand concentration is shown in Figure 5. The points were fit to obtain the stability constant by the nonlinear least-squares regression routine, MINIQUAD, and a conditional constant of (4.2 f 1.0) X lo4 (pH 2.81) was obtained. Figure 5 shows the fit to the experimental data. By use of published values for the EDTA acid dissociation constants in KN03 (13, a stability constant of (3.9 A 0.8) X lOI5 was obtained. At 95% confidence, this value was not significantly different from two literature values of 5.7 x obtained by using calorimetric methods (18,19).
Anal. Chem. 1983, 55, 1710-1712
1710
CONCLUSION The changes in praseodymium spectra obtained upon complexation are small, and the contributions of the PP3and PrEDTA- species are similar enough to thoroughly test the ability of the Kalman filter to deconvolute overlapped responses. In this study, good deconvolution is possible because adequate models for the component spectra are available and because the signal-to-noise ratios are high. Even so, mixtures containing less than 10% of Pr3+ could not be deconvoluted, probably because of slight fluctuations between model and overlapped spectra (model errors) combined with the similarity of the component spectra and the degree of their overlap. Where model error is small, the filter has been able to extract overlapped components from systems with peak height ratios up to 40:l (20). A potential problem in using the Kalman filter for spectral deconvolution would occur if a well-defined model was not available. In many cases, a series of two component mixture spectra may be obtained where only one of the contributing spectra can be obtained experimentally. A technique is currently being investigated which combines adaptive estimation techniques with the Kalman filter to obtain a second spectrum which cannot be obtained experimentally.
GLOSSARY I K V
P S
ST 2
2
X(kp)
identity matrix Kalman gain vector measurement noise covariance matrix measurement function vector transpose of the measurement function vector variance of the measurement process concentration vector the kth estimate of X based on the j t h measurement
ACKNOWLEDGMENT The authors wish to acknowledge John Frame for the design of the photoacoustic detector and Don Gilliland for help in-
volving design considerations and for the construction of the photoacoustic detector.
LITERATURE CITED (1) (2) (3) (4)
Oda, S.; Sawada, T.; Kamada, H. Anal. Chem. 1978, 5 0 , 865-867. Oda, S.; Sawada, T.; Kamada, H. Anal. Chem. 1979, 51, 686-688. Patel, C. K. N.; Tam, A. C. Rev. Mod. Phys. 1981, 53, 517-580. Voightman, E.; Jurgensen, A.; Wlnefordner, J. Anal. Chem. 1981, 53, 1442- 1446. (5) Moeller, T.; Martin, D. F.; Thompson, L. C.; Ferrus, R.; Felstel, G. R.; Randall, W. J. Chem. Rev. 1965, 65, 1-50. (6) Brubaker, T. A.; Tracy, R.; Pomernackl, C. L. Anal. Chem. 1978, 50, 10 17A-1024A. (7) Brown, T. F.; Brown, S. D. Anal. Chem. 1981, 53, 1410-1417. (8) Dldden, C. B. M.; Poullsse, H. N. J. Anal. Lett. 1980, 13 (A14), 1211-1234. (9) Poullsse, H. N. J. Anal. Chlm. Acta 1979, 112,361-374. (10) Sawada, T.; Oda, S.; Shimlzu, H.; Kamada, H. Anal. Chem. 1979, 51, 688-690. (11) Kalman, R. E. J. Basic Eng. 1960, 82, 35-45. (12) Brown, T. F.; Brown, S. D. "Book of Abstracts", 185th National Meeting of the American Chemical Society, Seattle, WA, March 22, 1983; American Chemical Society: Washlngton, DC, 1983; paper 111. (13) Sabatlnl, A.; Vacca, A.; Gans, P. Talanta 1974, 21, 53-77. (14) Leggett, D. J. Talanta 1977, 24, 535-542. (15) Stewart, D. C.; Kato, D. Anal. Chem. 1958, 30, 164-172. (16) Carnall, W. T.; Fields, P. R.; Rajnak, K. J. Chem. Phys. 1968, 4 9 , 4424-4442. (17) Perrln, D. D., Ed. "Stablllty Constants of Metal-Ion Complexes; Part B, Organlc Llgands"; Pergamon Press: New York, 1979; p 759. (18) Betts, R. H.; Dahllnger, 0. F. Can. J . Chem. 1959, 37,91-100. (19) Mackey, J. L.; Powell, J. E.; Spedding, F. H. J. Am. Chem. SOC. 1982, 84, 2047-2050. (20) Scolari, C. A.; Brown, S. D. "Book of Abstracts", 185th National Meeting of the American Chemical Society, Seattle, WA, March 23, 1983; American Chemical Soclety: Washington, DC; paper 130.
RECEIVED for review April 19, 1983. Accepted June 1, 1983. This work was supported, in part, by the Phillips Petroleum Foundation through a summer fellowship for S.C.R. This work was presented a t the 185th National Meeting of the American Chemical Society, Seattle, WA, March 23, 1983.
Discriminant Analysis by Double Stage Principal Component Analysis Ronald Hoogerbrugge,* Simon J. Willig, and Piet G. Kistemaker FOM-Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
Discriminant analysis Is possible by double stage principal component analysis. The mathematical justification and a flow scheme of the implementation in the ARTHUR package are glven. Also a comparison has been made between the discriminant analysis results obtained with ARTHUR and with
SPSS.
The proliferation of modern analytical techniques, which supply the chemist with large arrays of data for the objects (samples) analyzed, has forced the researcher to adapt pattern recognition techniques (1). Every chemist who has tried to interpret spectral patterns of highly complex mixtures has realized that human intelligence alone is not sufficient to unravel the composite spectra adequately. In our laboratory
highly complex pyrolysis mass spectra of organic samples like microorganisms, body fluids, biopolymers, and geological samples are interpreted routinely. The combination of this mass spectrometric analysis and automated data analysis has been successful in a wide area of applications, i.e., quantitative analysis of polymers ( 2 ) ,typing of microorganisms ( 3 ) ,and analysis of ancient and recent sediments ( 4 ) . The mathematical procedures adopted in multivariate data analysis are of a quite general nature and have been applied widely next to analytical chemistry in the social and medical sciences, biology, and image reconstruction. The output of the calculations is preferably presented as the results of a statistical evaluation of the identities of the objects analyzed. Visualization of the mutual relationship of the objects is shown in so-called scatter plots. In these two-dimensional plots the objects are represented by points of which the coordinates are
0003-2700/83/0355-1710$01.50/00 1983 Amerlcan Chemical Society
