Anal. Chem. 1997, 69, 1722-1725
Quantitative Determination of the Components in Overlapping Chromatographic Peaks Using Wavelet Transform Xueguang Shao,*,† Wensheng Cai,‡ Peiyan Sun,‡ Maosen Zhang,‡ and Guiwen Zhao†
Departments of Applied Chemistry and Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
A novel application of the wavelet transform in retrieving the separate signals from overlapping chromatographic peaks and quantitative determination of the components in the overlapping chromatograms is described. The signals can be very easily separated by decomposing an overlapping chromatographic peak into localized contributions according to their frequency, and quantitative calculation can be done by studying the contributions of higher frequency. Overlapping peaks of two- and threecomponent mixtures were investigated by the method, and the results show excellent correlations between peak areas of the retrieved signals and the concentrations for all of the components. Mixture analysis represents one of the most difficult and challenging areas within analytical chemistry. Resolution of overlapping chromatographic peaks is a problem of universal concern in the practice of analytical separations. For unresolved chromatograms, several curve resolution techniques have been reported, including the Fourier self-deconvolution (FSD) method,1,2 the Fourier derivation (FD) method,3 and factor analysis methods (EFA,4 WAF,5 HELP6,7). The essence of the FSD method and the FD method is to multiply the Fourier transform of the original signal by a weighting function, which decays more slowly, and then to transform the multiplied signal back to the time domain. This will increase the higher frequency part of the time domain signal and make the broad peak narrower. But sometimes it is not easy to choose the appropriate apodization function for the FSD method or smoothing function for the FD method and the appropriate value of the parameter xτ (the point at which the apodization or smoothing function goes to zero).8 Furthermore, the sidelobes, especially the positive sidelobes, will cause extra component peaks. The factor analysis methods have been proved to be very useful and powerful techniques for the resolution of overlapping chromatograms, but they can only cope with the data matrices obtained with diode array detection (DAD). * Corresponding author. E-mail:
[email protected]. † Department of Applied Chemistry. ‡ Department of Chemical Physics. (1) Jackson, R. S.; Griffiths, P. R. Anal. Chem. 1991, 63, 2557. (2) Kauppinen, J. K.; Moffatt, D. J.; Mantsch, H. H.; et al. Anal. Chem. 1981, 53, 1454. (3) Cameron, D. G.; Moffatt, D. J. Appl. Spectrosc. 1987, 41, 539. (4) Maeder, M. Anal. Chem. 1987, 59, 527. (5) Schostack, K. J.; Malinowski, E. R. Chemom. Intell. Lab. Syst. 1993, 20, 173. (6) Grung, B.; Kvalheim, O. M. Chemom. Intell. Lab. Syst. 1995, 29, 75. (7) Liang, Y.-Z.; Kvalheim, O. M.; Rahmani, A.; et al. J. Chemom. 1993, 7, 15. (8) Mantsch, H. H.; Moffatt, D. J.; Casal, H. L. J. Mol. Struct. 1988, 173, 285.
1722 Analytical Chemistry, Vol. 69, No. 9, May 1, 1997
Wavelet transform is a high-performance signal processing technique which has been found to be a useful approach in several fields.9-12 Applications of the technique in analytical chemistry have been also reported in recent years.13-15 The main characteristic of the wavelet transform is that it decomposes a signal into localized contributions labeled by a scale and a position parameter. Each of the contributions represents the information of different frequency contained in the original signal. Furthermore, the wavelet transform is a linear operation, which is very important to keep the linearity of the decomposed signals. Therefore, for a multicomponent overlapping chromatogram, the separated peaks, i.e., the higher frequency part, can be retrieved by wavelet transform decomposition, and quantitative determination can be done by the separated peak of each component, because it keeps its linearity. In this paper, two-components (benzene and methylbenzene) and three-components (benzene, methylbenzene, and ethylbenzene) overlapping chromatograms are analyzed with the wavelet transform technique. Signals of each component obtained from wavelet transform decomposition maintain their linearity, and satisfactory quantitative determination results are obtained. THEORY AND CALCULATION PROCEDURE The wavelet transform, in some respects, resembles the Fourier transform (FT), in which the sine and cosine are the basic analyzing functions. The analyzing function of wavelet transform is defined in two-dimensional space of scale and time and is derived from a mother wavelet denoted by Ψ(t). A number of “self-similar” wavelets can be obtained from this mother wavelet by two processes, (1) shifts in the time variable and (2) dilations, which both act on the time and the scale variables. If the variable b is used to control the shift in time and the variable a is used to control the dilation, then the series of wavelets can be described by
Ψa,b(t) )
1 t-b , a,b ∈ R,a * 0 Ψ a x|a|
( )
(1)
In this equation, 1/|a|1/2 is a normalizing constant that ensures (9) Morelet, J.; Arens, G.; Fourgeau, I.; Giand, D. Geophysics 1982, 47, 203. (10) Kronland-Martinet, R.; Morelet, J.; Grossmann, A. Int. J. Pattern Recognit. Artif. Intell. 1987, 1, 273. (11) Liandrat, J.; Moret-Bailly, F. Eur. J. Mechanics B: Fluids 1990, 9, 1. (12) Li, Z.; Borrmann, A.; Martens, C. C. Chem. Phys. Lett. 1993, 214, 362. (13) Bos, M.; Hoogendam, E. Anal. Chim. Acta 1992, 267, 73. (14) Bos, M.; Vrielink, J. A. M. Chemom. Intell. Lab. Syst. 1994, 23, 115. (15) Pan, Z.; Shao, X.; Zhong, H.; et al. Chinese J. Anal. Chem. 1996, 24 (2), 149. S0003-2700(96)00867-0 CCC: $14.00
© 1997 American Chemical Society
that all analyzing functions have unit energy. If f(t) is a signal, then the continuous wavelet transform is defined as
Wf(a,b) )
1 x|a|
∫
+∞
(t -a b) dt
f(t)Ψ
-∞
(2)
In practice, the signals to be analyzed are generally discrete signals. The discrete form of above two equations can be described by
Ψm,n(t) ) a0-m/2Ψ(a0-mt - nb0)
∫
Cm,n(f) )
+∞
Ψm,n(t)f(t) dt
Table 1. Concentrations of the Samples (µL/mL) no.
benzene
methylbenzene
I-1 I-2 I-3 I-4 I-5
2.0 4.0 6.0 8.0 10.0
10.0 8.0 6.0 4.0 2.0
II-1 II-2 II-3 II-4 II-5
2.0 4.0 6.0 7.0 8.0
2.0 4.0 6.0 8.0 10.0
ethylbenzene
9.0 8.0 7.0 5.0 3.0
(3) (4)
finite discrete dyadic wavelet transform will be accomplished by the following procedure:
-∞
j)0 generally, a0 ) 2 and b0 ) 1. Ψm,n(t) will be used to generate the series of wavelets. Because a0 ) 2, the wavelet is called dyadic. But sometimes it is still difficult to calculate the wavelet +∞ transform of the signals directly by Cm,n(f) ) ∫-∞ Ψm,n(t)f(t) dt. 16 The discrete dyadic wavelet transform is generally used, which denotes the Ψm,n(t) as discrete filters: H ) {hl}, l ∈ Z and G ) {gm}, m ∈ Z. H and G are given by hl ) 〈φ2-1(u),φ(u - l)〉 and gm ) 〈φ2-1(u),φ(u - m)〉, where φ(x) and φ(x) are the wavelet function and the corresponding scaling function.16 If S20f(n),n ∈ Z, denotes the discrete signal of f(t), then the decomposition can be described by
S20f(n) f S21f(n) f ... S2J-1f(n) f S2Jf(n) V V V W21f(n) W22f(n) W2Jf(n)
(5)
where J is the number of decompositions and
S21f(n) )
∑h S
l 20f(n
- 20l)
(6)
l∈Z
W21f(n) )
∑
gmS20f(n - 20m)
(7)
m∈Z
while j < J W2j+1 f ) S2j fGj S2j+1 f ) S2j fHj j)j+1 end of while In fact, an overlapping chromatogram, which is a signal of relatively low frequency, is the sum of the chromatograms of each component, which are the signals of relatively high frequency. Therefore, by means of the above calculation, the separated chromatographic peaks can be easily retrieved from the overlapping chromatogram. The program was compiled by C language in DOS and UNIX OS. The analyzing mother wavelet used in this paper is the Haar wavelet function:
{
1 0 e x < 1/2 Ψ(x) ) -1 1/2 e x < 1 0 otherwise
(10)
The corresponding discrete filters are H ) {1/21/2, 1/21/2} and G ) {-1/21/2, 1/21/2}.16,17
Generally,
S2j f(n) )
∑h S
- 2j-1l)
(8)
∑g
- 2j-1m)
(9)
l 2j-1f(n
l∈Z
W2j f(n) )
mS2j-1f(n
m∈Z
As for S20f(n), S2jf(n) and W2jf(n) are called the discrete approximation and the discrete detail at the jth level or at the resolution of 2j. S2jf(n) is the low-frequency part of signal S20f(n) with the frequency lower than 2-j, and W2jf(n) is the highfrequency part of the signal with the frequency between 2-j and 2-j+1. Therefore, if Hj and Gj are used to denote the filters generated by inserting 2j-1 zeros to the adjacent H and G, calculation of the (16) Mallat, S. G. IEEE Trans. Pattern Anal. Machine Intell. 1989, 11 (7), 674.
EXPERIMENTAL SECTION A Shimadzu LC-6A HPLC system with two LC-6A solvent delivery pumps (Shimadzu, Kyoto, Japan), a 7125 sampling valve (Rheodyne, Cotati, CA) with a 20-µL sample loop, and an SPD-6A UV-visible detector (Shimadzu) was used for separation. The column was packed with 10-µm ODS silica (250 mm × 5 mm, Shimadzu). The mobile solvent was redistilled H2O and methanol (61:39), and the flow rate was 1 mL/min. The operation temperature was controlled at 30 °C, and the detection wavelength was set to 254 nm. The signal from the detector was sampled by a C-R3A data processing unit (Shimadzu) at 1.0-s intervals. The sample solution was prepared by adding benzene, methylbenzene, and ethylbenzene (chromatography standard, First Chemical Reagent Factory, Shanghai, China) into a volumetric (17) Liu, G.; Di, S. Wavelet Analysis and Its Application; Press of University of Electronic Science and Technology of Xi’an: Xi’an, PRC, 1992.
Analytical Chemistry, Vol. 69, No. 9, May 1, 1997
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Table 2. Calculated Results of the Five Mixed Samples benzene
methylbenzene
ethylbenzene
added µL/mL calcd (µL/mL) recovery (%) added (µL/mL) calcd (µL/mL) recovery (%) added (µL/mL) calcd (µL/mL) recovery (%) 3.0 9.0
3.05 8.33
101.7 92.6
7.0 7.0
7.31 7.30
104.4 104.3
3.0 7.0 2.5
2.76 7.09 2.32
92.0 101.3 92.8
2.5 9.0 8.0
2.48 9.41 7.69
99.2 104.6 96.1
Figure 1. Chromatograms of group I samples.
5.0 5.0 9.0
5.24 5.01 8.65
104.8 100.2 96.1
Figure 3. Discrete details obtained from four decompositions of sample II-3.
Figure 2. Chromatograms of group II samples.
Figure 4. D3 signals of group I samples.
flask and diluting with methanol. The concentration of each sample solution is tabulated in Table 1. Both of the mobile solvents and the samples are filtered through a 0.20-µm membrane filter before use.
from the chromatogram of sample II-3 by four decompositions using the Haar analyzing wavelet. Theoretically, any of the Dj signals could be used for quantitative calculation, because the wavelet transform is a linear operation. But, from the figure, it can be seen that the resolution of D3 is greatest among these discrete details. Therefore, D3 is chosen for the further discussion. Figures 4 and 5 show the signals of D3 corresponding to the chromatograms in Figures 1 and 2. Linearities of the D3 Signals. To calculate the areas of every peak in Figures 4 and 5, baseline correction is needed. Figure 6 shows the baseline-corrected D3 signals of group II samples
RESULTS AND DISCUSSION Retrieval of the Resolved Information. Figures 1 and 2 are the chromatograms of the samples of groups I and II in Table 1. It is impossible to determine quantitative information from such chromatograms because there is little overlapping. Figure 3 shows the discrete details Dj [W2jf(n), j ) 1, 2, 3, 4] retrieved 1724 Analytical Chemistry, Vol. 69, No. 9, May 1, 1997
Figure 5. D3 signals of group II samples.
Figure 7. Calibration curves of group II samples: (a) benzene, (b) methylbenzene, and (c) ethybenzene.
Figure 6. Baseline-corrected D3 signals of group II samples. Figure 8. Calibration curves of group I samples: (a) benzene and (b) methylbenzene.
(corresponding to Figure 5), obtained by linking the minimum points of every peak as baseline, and Figure 7 shows the relationship between the areas, which are calculated by integral from Figure 6, and the concentrations. Figure 8 is obtained in the same way for the samples of group I. The correlation coefficients are 0.9919 for benzene and 0.9918 for methylbenzene in group I, and 0.9982 for benzene, 0.9924 for methylbenzene, and 0.9914 for ethylbenzene in group II. Quantitative Determination of Mixed Samples. The chromatograms of five mixed samples (two for group I and three for group II) were measured at the same experimental conditions as used for the samples of groups I and II. Table 2 gives the calculated results obtained by decomposing and calculating the areas with the same procedures as above. The recoveries are between 92.0 and 104.8%. CONCLUSION The application of wavelet transform to resolution of overlapping chromatograms and quantitative multicomponent determination was proved to be a convenient and efficiency method by
this study. The resolved chromatographic peaks can be easily retrieved by decomposing the overlapping signal into its local contributions, and each of these contributions keeps its linearity, which ensures that quantitative results can be obtained from the retrieved signals. Correlation coefficients above 0.99 were obtained for both the two- and three-component samples, and the recoveries of five mixed samples lay between 92.0 and 104.8%. ACKNOWLEDGMENT This study was supported by the National Natural Science Foundation of China.
Received for review August 26, 1996. Accepted February 10, 1997.X AC9608679 X
Abstract published in Advance ACS Abstracts, March 15, 1997.
Analytical Chemistry, Vol. 69, No. 9, May 1, 1997
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