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Anal. Chem. 1985, 57, 1552-1559
Smith, R. D.; Kallnoski, H. T.; Udseth, H. R.; Wright, B. W. Anal. Chem. 1984, 5 6 , 2476-2480. Hirata, Y.; Novotny, M. J . Chromatogr. 1979, 786, 521-528. Ishii, D.; Takeuchl, T. J . Chromatogr. S d . 1980, 78, 462-472. Guthrie, E. J.; Jorgenson, J. W.; Dluzneski, P. R. J . Chromatogr. Sci. 1984, 2 2 , 171-176. Gluckman, J.; Shelly, D.; Novotny, M. J. Chromatogr. 1984, 377, 443-453. Henion, J. I n "Microcolumn High Performance Liquid Chromatography"; Kucera, P., Ed., Elsevier: Amsterdam, 1984; pp 260-300. Talmi, Y., Ed. "Multichannel Image Detectors"; American Chemical Society: Washington, DC, 1979; ACS Symp. Ser. No. 102. Fell, A. F.; Clark, B. J.; Scott, H. P. J. fharm. Homed. Anal. 1983, 7 , 557-572. Jadamec, J.; Saner, W.; Talml, Y. Anal. Chem. 1977, 4 9 , 1316-1321, Fogarty, M.; Shelly, D.; Warner, I . HRC CC,J . High Resolut. Chromafogr. Chromatogr. Common. 1981, 4 , 581-568. Mllano, M.; Lam, S.; Grushka, E. J . Chromatogr. 1978, 125, 315-326. Jost, K. W.; Crispln. Th.; Halasz, I. Erdol Kohle, Erdgas fefrochem. 1984. 3 7 , 178. Desilets, 0.J.; Kissinger, P. T.; Lytle, F. E.; Horne, M. A.; Ludwiczak, M. S.; Jacko, R. B. Environ. Sci. Techno/. 1984, 18, 386-391. Takeuchi, T.; Ishii, D. HRC CC,J . H/gh Resolut. Chromatogr. Chromatogr. Commun. 1984, 7 , 151-152. Ryan, M. A.; Mliier, R. J.; Ingle, J. D., Jr. Anal. Chem. 1978, 50, 1772-1777. Talmi, Y. Appl. Spectrosc. 1982, 3 6 , 1-18. Fjeldsted, J. C.; Richter, B. E.; Jackson, W. P., Lee, M. L. J . Chromatogr. 1983, 279, 423-430.
(29) Novotny, M. I n "Microcolumn High Performance Liquid Chromatography"; Kucera, P., Ed.; Elsevler: Amsterdam, 1984; pp 194-259. (30) Beriman, I. "Handbook of Fluorescence Spectra of Aromatic Molecules"; Academic Press: New York, 1971. (31) Kropp, J.; Stanley, C. Chem. fhys. Left. 1971, 9, 534-538. (32) McKay, J.; Latham, D. Anal. Chem. 1972, 44, 2132-2137. (33) McKay, J.; Latham, D.Anal. Chem. 1973, 45, 1050-1055. (34) Clar, E.; Schmidt, W. Tetrahedron 1977, 3 3 , 2093-2097. (35) Ciar, E.; Schmidt, W. Tetrahedron 1978, 34, 3219-3224. (38) Clar, E.; Schmidt, W. Tetrahedron 1979, 3 5 , 1027-1032. (37) Karlsson, K.-E.; Novotny, M. HRC CC,J . H/gh Resolut. Chromatogr. Chromatogr. Commun. 1984, 7 , 411-413. (36) Novotny, M.; Konishi, M.; Hirose, A.; Gluckman, J.; Wlesler, D. Fuel, in press. (39) Hleftje, G. Anal. Chem. 1972, 44, 81A-88A. (40) Wiesler, D., Indiana University, Bloomington, Indiana, private communlcatlon, 1984.
RECEIVED for review January 17,1985. Accepted March 25, 1985. Support for this work was provided by Grant No. DOE DE AC02-81 ER 60007 from the U.S. Department of Energy, and Grant No. N14-82-K-0561 from the Office of Naval Research. J. C. Gluckman was the recipient of a full-year graduate fellowship bestowed by the American Chemical Society, Division of Analytical Chemistry, and sponsored by the Perkin-Elmer Corp.
Multiwavelength Detection and Reiterative Least Squares Resolution of Overlapped Liquid Chromatographic Peaks 5.D. Frans,l M. L. McConnell,2and J. M. Harris* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
A relteratlve least squares procedure Is used to resolve indlvldual component spectra from overlapped ilquid chromatographic peaks detected wlth a rnultlwavelength,photodlode array spectrometer. Rather than seeklng spectral or elutlon regions In the data which are free of overlap, the method makes use of the predlctabie shape of the eluted peaks, while allowing for variation In retention time and peak wldth. Successful spectral resolution of hlghiy overlapped fused peaks containlng up to seven components Is achieved. The exponentlally modHled Gausslan function used to model the elution response Is found to accurately describe of the obgerved peak shape. The effect of the peak shape model on the accuracy of the results Is evaluated; the detection of mlnor overlapped components Is also studied.
The resolution and identification of the components in complex mixtures continue to represent difficult analytical tasks in spite of improvements in instrumentation and methodology. It appears that the increasing complexity of analytical problems will always exceed the current state-ofthe-art in separation and analysis. Furthermore, Davis and Giddings (1) have shown by way of a statistical argument that a chromatogram must be more than 95% vacant in order to provide greater than 90% confidence that a component of interest will appear as an isolated peak. Since chromatograms lPresent address: Spectra-Physics, 3333 N o r t h First St., San Jose,
CA 95134.
2Present address: Tec Concepts, 985 University Ave., Suite 31,
Los Gatos, CA 95030.
of complex samples are nearly always less vacant than this criterion, the problem of overlapping peaks becomes a nearly universal concern in the practice of analytical separations. To address this concern, it is valuable to recognize that the use of multichannel detectors in chromatography can significantly increase the number of independent informational degrees of freedom in the measurement (2). Several numerical methods have been developed to utilize this additional informing power to resolve overlapped peaks. All of these data analysis methods share the assumption that the spectral pattern of a particular compound will rise and fall in unison as that compound elutes from the column. One approach to utilizing this behavior requires that one or more spectral channels in a region of chromatographic overlap belong only to one compound. The pure spectral channel is identified as the one which exhibits the sharpest peak within the elution period of interest (3-6). The single component chromatogram, thus identified, serves as a template against which the chromatograms at other spectral channels are compared by correlation to extract the spectrum of the component in question. A convenient approach to identifying the unique spectral channels attributable to single components is the factor analysis technique of Lawton and Sylvestre (7),termed self-modeling curve resolution. This method has been successfully applied to the mass spectra of mixtures (8) and presumably would work with GC/MS data as well. An alternative use of the method is to identify those elution times corresponding to the purest, ideally sisgle component spectra. These pure spectra, typically but not necessarily found at the edges of a pair of fused chromatographic peaks (9), are then used to factor individual component chromatograms out of the data matrix. This method has been applied to binary
0003-2700/85/0357-1552$01.50/00 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57,
mixtures in GC/MS (9, 10) and LC/UV (11). A major limitation of the above methods is that the accuracy of the results depends explicitly on the purity of a spectral channel (8, 10) or elution time (9, 11) for each component, which should be particularly troublesome in the determination of sample components having similar structure where spectral and chromatographic overlap are both likely. The likelihood of overlap is further aggravated in the case of LC/UV, compared to GC/MS, due to the fewer plates and correspondingly smaller peak capacity of conventional liquid chromatography as well as the broad electronic absorption bands in the UV and visible spectral region. Derivative spectra could increase the number of pure spectral channels in this region (12)due to the increased number of base line crossings but would eliminate the nonnegative intensity assumption used in their search (7, 9-11). To avoid entirely the requirement of pure spectral or elution regions, a reiterative least squares procedure has been developed (13,14)which makes use of the reproducible elution behavior of the chromatographic process to model the time response, rather than seeking this response in the chromatogram of a pure spectral channel. The first application of this method was for GC/MS, where the peak shape and its relationship to retention time were sufficiently predictable that only one parameter per component, the retention time, was required to fit the elution dimension of the data. Mass spectra of the individual Components in highly overlapped mixtures were resolved by this method in excellent agreement with spectra obtained from pure samples. The application of this method to liquid chromatography, where the kinetics of retention can change with capacity factor (15-1 7), should require a more flexible model to fit the elution profile compared with gas-liquid partition. In this work, the reiterative least squares method of resolving spectra from overlapped peaks is applied to highperformance liquid chromatography using multiwavelength, photodiode array detection. A model which allows independent variation of retention time and band broadening is used to fit the elution profiles of the components while absorption spectra of the components are obtained by a linear least squares step. The method is tested on multiwavelength chromatograms of mixtures containing up to eight overlapped components. The effect of the peak shape model on the accuracy of the results is also evaluated.
THEORY A multiwavelength chromatogram can be represented as a w by t matrix of absorbancevalues, D, where the wavelength of the measurement is the index, i, for the rows and the elution time is the index, j , for the columns. If the spectroscopic conditions are such that the Beer-Lambert law holds, then the absorbance a t any wavelength and time, d,, is the sum over the n components of each of their absorbances given as the product of their wavelength-dependentabsorptivities and time-dependent concentrations
This relationship can be expressed in matrix form
D = AC (2) where A is a w by n matrix containing the absorption spectra of the n components in ita columns and C is an n by t matrix containing the chromatographic elution curves in its rows. For ease of interpretation, the concentration profiles in the rows of C are normalized over the total elution volume (3)
NO. 8, JULY 1985
1559
where AIJ is the increment in elution volume between measurements and the elements of C, ck,, carry units of inverse volume. This constraint places all of the concentration as well spectral information into the matrix A. The corresponding units for the elements, alk, are absorbance times volume or the product of molar absorptivity, path length, and total amount in moles of the kth analyte. This formalism eliminates the chromatographic dispersion of the sample from quantitative interpretation of the results. For example, the absorbance of the kth component in the original injected sample of volume Vinj, is given simply by the ratio (aik/Vinj). Given a multiwavelength chromatogram, D, the data analysis task is to decompose the matrix into its factors A and C which can be used for subsequent identification and quantitation of the sample components. In order to carry out this decomposition, the individual elution profiles in the C matrix are modeled as Gaussian functions convoluted with a single sided exponential to account for peak tailing
where Nkis a normalization factor, At is the time interval between spectral scans indexed by j , t r k and are the retention time and bandwidth of the Gaussian component, and T~ is the time constant of the exponential tail. The use of this function to model liquid chromatographic peaks (18-20) has primarily concerned the separation of column efficiency from extracolumn band broadening, the latter of which dominates the observed exponential contribution to the peak shape. The time constant, T k , is primarily a characteristic of instrument dead volume which is not strongly compound dependent. The value of this shape parameter, therefore, may be determined and fixed for a particular instrument configuration and flow rate or “floated” as a single global parameter. The fitting of gas chromatographic peaks (13)required only the retention time of each component to be optimized, since the Gaussian bandwidth could be predicted from retention time assuming a constant number of plates. The fitting of high-performance liquid chromatographic peaks, however, requires independent optimization of both retention time and bandwidth due to compound-dependent variations in column efficiency (15-1 7). Given the model of eq 4 for the time-dependent concentration behavior of the sample components, all that is required to specify all the elements of the matrix, C, are 2n parameters: t r k and Uk, for k = 1 to n components. For a given estimate of these parameters, one can build a trial chromatographic matrix, C, which can be used to find the corresponding best spectral amplitude matrix, A, by right multiplying the data mairix by the pseudoinverse or least squares inverse (21,22) of c
A = D~T((J@)-~ (5) For a given estimate of the parameters in C. This process yields a model for the data fj = AC (6) corresponding to the leas; squared error, x2,between the actual data D and the model D
weighted by the inverse of the number of degrees of freedom in the fit. The actual value of x2,th? obtained, depends on the match between the trial matrix, C, and the elution characteristics of the sample C. The choice of retention time and bandwidth
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 8, JULY 1985
Table I. Composition of Samples Prepared for Obtaining Fused Peak Chromatograms
sample 5 component 6 component 7 component 8 component
5-indanol 2x 2X 1X 1X
10” IO” 10“ 10”
bend 2
X 1X 1X
10”
10“ 10“
2-naphthaldehyde
component Concentration,“M bis(ch1oromethy1)anthracene azulene
8 x 10“ 7 X 10“ 6 X 10” 5 X 10”
1 x 10-6 8 x 10” 7 x 10”
4
6 X 10“
4 x lo+
9-fluorenone
p-dibromobenzene
xanthen9-one 1 x 10-5
6 X 10” 5 x 10”
1 x 10-6
x 10” 4 x lo+
1 x 10-5 1 x 10-5
1 x 10-5
6
X
10”
0&20%.
parameters of C can be guided, therefore, by a minimization of x2using, for exampje, a SIMPLEX algorithm (23-25). Since negative elements in A, corresponding to negative absorbances, have no physical meaning, an additional penalty can be added to x2 to help steer the search away from such results. A somewhat more efficient error function to minimize is given by ERR = x2[1 + lo(CClajk< ol/CClaikl)] (8) k k
i k
where x2is penalized by a factor which is proportional to the fraction of negative spectral intensity. The preceding data treatment has presumed that the number of componepts in the data set is known in order to construct a model C of the proper dimension. While the reiterative least squares method can be used to determine the number of components by incrementing the value of a n and repeating the procedure (13,26,27),a factor analytic approach was found to be more convenient. Following a singular value decomposition of the data matrix, a partial autocorrelation function of the eigenvectors was calculated at a delay of one wavelength or time interval (28). At this interval, the autocorrelation function for eigenvectors describing high frequency noise is much smaller than for eigenvectors describing the slowly varying wavelength and time dependence of the sample. On the basis of the noise characteristics of the measurement, a cutoff value for the autocorrelation function can be established which identifies the number of significant eigenvectors, that is, the number of distinguishable sample components (28). The final theoretical issue concerns the criteria by which sample components are distinguishable. Since the modeling of C discriminates on the basis of elution behavior, no two compounds which coelute with the same retention time and bandwidth can be resolved. This is also apparent in eq ,5 ?here the pseudoinverse requires that the matrix product (CCT)be inverted, y)ich is impossible if two rows of C are identical making (CCT)singular. The degree to which the rows of C are linearly dependent, as two or more elution profiles begin to overlap, is reflected in the condition number of the matrix, cond (C). A convenient definition of the condition number utilizes a singular value decomposition of C (29) to obtain the singular values or eigenvalues of C cond (C)= SCrnax/SCmin (9) where SC- and Scmhare the largest and smallest eigenvalues, respectively. If the rows of C were orthogonal, cond (C)would have a minimum value of unity. As two or more rows become similar or more linearly dependent, the smallest valued eigenvectors describe only the differences between the similar rows resulting in a small denominator in eq 9 and a correspondingly large condition number. The condition number is particularly valuable in predicting an upper bound to the relative error in the spectra based on the relative error in the data (29)
Table 11. Composition of Samples Prepared for Minor Component Study
mixture 1 2
3 4 5
6
concentration, M 4,4’-dibromobiphenyl fluoranthene 8.5 x 1.0x 1.2 x 1.2 x
10-6 10-4
10-4 10-4 1.3 x 10-4 1.3 X lo4
Amax,d
6.0 X 3.7 X 10”
2.07
2.0 X
0.52 0.26 0.13
1.1 X 5.7 X 10” 2.9 X 10“
1.04
0.065
where the magnitudes of the errors are defined in terms of matrix norms, analogous to the Euclidian norm of a vector which is a root mean squared measure of its length. This equation predicts that the condition number of C can only serve to amplify error in the data matrix in determining the spectra of the components in the sample. When attempting to resolve spectra from closely spaced chromatographic peaks which generate a poorly conditioned matrix, C only data having a high signal-to-noise ratio would produce meaningful spectral results.
EXPERIMENTAL SECTION Solutions of the following polynuclear aromatic hydrocarbons were prepared in Burdick and Jackson HPLC grade methanol: azulene, benzil, bis(chloromethyl)anthracene,p-dibromobenzene, 9-fluorenone,5-indanol, 2-naphthaldehyde, and xanthen-9-one. From solutions of these eight compounds, a series of mixtures were prepared to produce fused peak chromatograms, the compositions of which are given in Table I. The concentrations were adjusted to provide approximatelyequal absorbancecontributions from each component in a mixture. A minor component study was carried out using, 4,4’-dibromobiphenyl and fluoranthene, solutions of which were also prepared in HPLC grade methanol as shown in Table 11. The ratio of the absorbance maxima of the two components is also presented for each sample. Chromatography was carried out with a Brownlee Labs 10-pm reverse-phase (RP-18)10 cm long column. The solvent was HPLC grade methanol pumped at a flow rate of 0.5 mL/min by a Beckman Model llOA chromatographic pump. A single wavelength detector at 254 nm, Beckman Model 153, was also used in series with the CMX-50 as a real time monitor. Multiwavelengthchromatograms of the isolated sample components and the above mixtures were detected with an LDCMilton Roy CMX-50 photodiode array spectrometer. The spectrometer was capable of collecting an entire UV-vis spectrum (190-700 nm) with a 1.1nm per diode resolution at a maximum scanning rate of six spectra per second. As the molecules under investigation were primarily UV absorbers, the spectra collected were restricted to a 100 nm range beginning at 200 or 220 nm. A data rate of one spectrum per second was adequate for the chromatograhic peaks under investigation;all of the peaks eluted within a 2-min data acquisition period beginning at 2 min following injection. The data matrix, measured as transmittance at 1.1nm intervals, was converted to absorbance by ratioing each spectrum collected in time to the first spectrum collected, in which there was no and* present. The CMX-50 spectrometerwas interfaced to a DEC LSI-11/23 for data storage and control through a Data Translation A/D, D/A converter. The digitized multiwavelength
ANALYTICAL CHEMISTRY, VOL. 57, NO. 8, JULY 1985
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Table IV. Condition Number of Concentration Matrix and Average Spectral Error Cond sample
5-component
(C)*
av re1 Cpnd spectral error, (Copt) %
2.6
2.6
11
4.0
3.6
15
7.0
6.7
26
35.5
7.2
60
7.8
30'
mixture
6-component mixture
7-component mixture 8-component mixture
8-component mixture, fit to 7 components
Merged peak a Determined from isolated compound data. spectrum compared to 9-fluorenone with reasonable (26%) error.
2: $ * Q,
W
chromatograms were stored on floppy disks and subsequently transferred to a DEC 20/60 for numerical analysis. Reiterative least squares was performed in FORTRAN using a modified Nelder-Mead SIMPLEX algorithm written by O'Neill (24).
rl
W
9
0
2; W
rl
RESULTS AND DISCUSSION Multicomponent Spectral Resolution. The reiterative least squares procedure for resolving overlapped, multiwavelength chromatograms was tested on five-, six-, seven-, and eight-component mixtures, with the results summarized in Tables I11 and IV. The retention times, Gaussian bandwidth, tailing parameters, and spectra were also determined for the individual, chemically isolated components to use for comparison with the numerically resolved data. All three peak parameters in the exponentially modified Gaussian model were allowed to vary in fitting the isolated compound results in order to ascertain the variation in 7. While a f12% average fluctuation in this parameter was observed, it was fixed at an average value of 7 = 8.0 s when fitting the mixture data to reduce the number of variables to be searched. Fixing the tailing parameter rather than the Gaussian bandwidth term is reasonable, since the former is largely although not entirely governed by extracolumn band broadening. To illustrate typical results of the method, the six-component mixture is presented graphically, where in Figure 1 the entire LC/UV data matrix is shown. While the spectral dimension of these data shows some potentially useful variation, the chromatographic dimension is considerably overlapped as illustrated by the total absorbance chromatogram in Figure 2, analogous to a total ion current chromatogram in GC/MS. The resolution of the peaks in the chromatogram is quantitatively described by the factors, R,, listed in Table I11
m
N
0
U
a
where the second moment, Mz = u2 + r2(30). The peaks in the six-componentmixture range in resolution from R, = 0.21 to 0.36, far from a base line criterion of R, = 1.0. The singular value decomposition, partial autocorrelation method correctly identified that n = 6 component? were in the mixture. Using this number to construct the C matrix and varying the 12 peak parameters to minimize ERR using SIMPLEX, the elution profiles and spectra of the individual components should occupy the rows of C and columns of A, respectively. The normalized elution profiles resolved from the mixture are plotted in Figure 3 along with the results for the chemically isolated components for comparison. The retention times are accurately determined, as shown in Table 111, while slight variations in peak width are observed which also appear in Figure 3 as differences in peak height due to the normalization procedure. The least squares determined absorption spectra, shown in Figure 4, are in good agreement with the chemically isolated component results. The plots
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ANALYTICAL CHEMISTRY, VOL.
57, NO. 8, JULY 1985
-
Figure 1. Six-component LC/UV data matrix.
120
240
Time (sec)
200
Figure 2. Integrated absorbance chromatogram from the six-component data matrix.
120
200
Time (sec)
e,
Figure 3. Concentratlon profile matrix, for the six-component mixture. Dashed llnes represent the chemically Isolated components while the solid lines are from numerical resolution of the mixture.
Wavelength (nm)
300
Figure 4. Absorption spectra of the chemically isolated components (dashed Ilnes) and the numerically resolved components (solid lines) for the six-component mixture plotted in order of elution from bottom to top; see Table 111. Scale represents absorbance in the injected sample. Average relative spectral error is 15%.
of these spectra include a scale to indicate their absorbance in the injected sample solution. The peak absorbances in the chromatograms were, on the average, six times smaller due to sample dispersion. The largest spectral error is observed for the third and fourth components, 2-naphthaldehyde and bis(chloromethy1)anthracene. This is not a surprising result since these components have the smallest chromatographic separation, R, = 0.21. The results obtained were found to be independent of the initial values of the SIMPLEX search. An important check on the quality of any data fitting procedure is the residual differences between the data and the best fit model given as a matrix R = D - D. This matrix, plotted in Figure 5, shows that most (95%) of the lower frequency, chromatographic, and spectral information has been extracted out of the data matrix by the model; compare absorbance scale with Figure 1. The residuals plot also shows
ANALYTICAL CHEMISTRY, VOL. 57, NO. 8, JULY 1985 0.02
1557
A
- ; :a Flgure 5. Residuals plot for the reiterative least squares flt of the six-component mixture.
the noisier absorbance data at wavelengths less than 220 nm, probably due to lower source intensity and photodiode sensitivity below this wavelength. Given the above discussion of the six-component case as a specific example, some general comments can be made about the five- through seven-componentresults, listed in Table 111. First, the singular value decomposition partial autocorrelation method (28)successfully identified the number of components in each of these cases. The spectral eigenvectors were, however, more consistent than the eigenvectors in the chromatographic dimension, presumably because of the greater spectral resolution. The retention times of each component were determined with an error of less than 1.0 s, the interval between spectral scans. Gaussian bandwidth terms, u, showed deviations as large as 1.3 s, but the errors were random having a mean of -0.1 s and a standard deviation of 0.6 s, which is not large, again considering the 1.0 s time interval between scans. The spectral errors, listed in Table IV, are proportional to the condition number of the concentration matrix as predicted by eq 10. This relationship can be used to estimate the average relative noise in the original data at 4%. This figure is in excellent agreement with the magnitude of the residuals in Figure 5. Another interesting trend an Table!!I is the tendency of the fitting procedure to optimize on a C matrix of slightly lower condition number, which means that the fit is trading off elution model accuracy for numerical stability, both of which contribute to overall error. This tendency is particularly apparent in the performance of the method on the eightcomponent mixture. This sample is beyond the resolving power of the method, particularly the azulene and 9-fluorenone components which are separated only by a factor, R, = 0.06. As a result, the SVD partial autocorrelation identifies only seven resolvable components in this sample. Proceeding with an eight-component reiterative least squares fit of these data, however, illustrates an interesting mode of failure. The optimization effectively throws out one component by allowing its retention time to increase to the edge of the data set where little- intensity exists, returning a corresponding spectrum in the A matrix with almost no absorbance; see Table 111. This behavior keeps the condition number of the C matrix small, see Table IV, reducing the numerical error associated with
Relative Peak Absorbance
1 : 0.07 1 : 0.13
1 : 0.25
E
1 : 0.52
1 : 1.04
- - _ *
--_
1 : 2.07
220
270
320
Wavelength (nm)
Figure 6. Mlnor component results. The absorption spectra of the minor component, fiuoranthene, from each of the six two-component mixtures. The solid lines are the numerically resolved spectra. The dashed lines are the chemically isolated component spectra. Spectra are scaled to equal areas for clarity.
determining the other components. Fitting this data set to n = 7 components as prescribed by partial autocorrelation yields better overall agreement with retention times and spectra, with the azulene and 9-fluorenone results being averaged together as a single unresolved peak. Since all of the peak results were obtained with components having approximately equal peak absorbances, as shown in Figure 4,a study was undertaken to determine the detectability of an unresolved minor component using this method. Using six different mixtures of 4,4-dibromobiphenyl and fluoranthene which were separated chromatographically by a factor, R, = 0.36, a two-component reiterative least squares fit was executed. The spectral vectors from the A matrices for fluoranthene are plotted in Figure 6 together with the isolated sample results. While the spectrum of fluoranthene is imperceptible when its absorbance fraction is only 7 % ,much of the spectral features are detectable when its contribution is 13% compared to the dibromobiphenyl peak. Since the condition number of this two-component C matrix is 2.0 and the relative error in D is about 4%, eq 10 predicts a relative
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 8, JULY 1985
Table V. Comparison of Triangular and Exponentially Modified Gaussian Models for Multicomponent Peak Resolution model
parameter
exponentially modified gaussian
MIa
5-indanol
bend
2naphthaldehyde
methy1)anthracene bis(ch1oro-
azulene
xanthen%one
139.4
144.9
152.2
160.9
173.0
184.5
fwhmc 12.0 11.0 14.0 12.0 13.0 14.0 Mlb 136.1 143.0 151.5 158.1 170.1 179.5 fwhm 13.0 17.0 18.0 15.0 13.0 18.0 "First moment for exponentially modified Gaussian, Ml = t, + T. bFirst moment for triangle, t, + 7/3, see eq 12 for variable definition. CFullwidth at half maximum. triangular
spectral error of 8%, corresponding to a 95% confidence detection limit or a minor component of about 16%, slightly larger than the results obtained experimentally. These results illustrate a general feature of regression methods, that the capability of extracting informationfrom data depends on two factors: the informing power of the measurement, in this case, related to peak separation quantified by the condition number of the C matrix, and the magnitude of noise in the data. An additional source of excess or determinate error for regression methods based on modeling is error in the model discussed in the following section. Model Considerations. A criticism which has been raised about chromatographic peak resolution by reiterative least squares is its dependence on a model for the elution peak shape (10,11,31). While modeling is a powerful method for extracting information from overdetermined measurements such as chromatography, one risks making determinate errors when the data fail to follow the model used. This raises a first question of how well the exponentially convoluted Gaussian model predicts the observed liquid chromatographic peak shape. The average relative error in fitting the eight individual component chromatograms was found to be 9.6% with a slightly better than average result illustrated in Figure 7 for 5-indanol with 8.5% relative error. This lack of fit is greater in magnitude than the residual error found in the reiterative least-squares analysis of mixtures, as in Figure 5, since the spectral djmension can accommodate some of the lack of fit from the C matrix. Since the single-componentresiduals for all the individual peaks showed the same form as that in Figure 7, indicating a partially mixed, dead volume affecting the elution profiles, the fit might be further improved by adding a characteristic residual vector to the model when building the rows of C. While the exponentially modified Gaussian was an adequate model for the compounds studied, a second question remains as to what effect a poor model might have on the results. To address this question, a naive, asymmetfic triangular peak shape model was used to construct the C matrix ekj
+ wk)/wk ( t r k - wk) 5 j A t 5 t r k + + -jAt)/(Wk +
= GAt - t r k = ( t r k wk
7
7)
trk
