Multichannel detectors attached to chromatographic columns are characteristic of the so-called hyphenated analytical techniques, such as gas chromatography-mass spectrometry (GC-MS), gas chromatography-infrared spectrometry (GC-IR), and liquid chromatography-mass spectrometry (GC-MS). A model that has been tested for this purpose employed a Gaussian peak convoluted with an exponential decay (eq. (14.2)) to account for the tailing of chromatographic peaks [7]. The model in Table 14.1 should also be appropriate but has not been tested for this purpose at the time of this writing. The main difference from the single channel problem is that, in addition to time, the multichannel detector data has a second independent variable equivalent to channel number. Therefore, the model in Table 14.2 expressed for multivariate detection would have the following form:

R(m>0 j§ hxp[-aj(t - cj)] + exp[~bj(t - cj)]j (14"9)

where m is the detector channel number, and j denotes the y'th component. It is assumed in eq. (14.9) that the peak from each detection channel would have the same shape, as governed by the chromatographic analysis, but that the Aj m will be different because of different sensitivities in the different channels.

Data obtained from multichannel detection of chromatographic peaks has been resolved by matrix regression methods similar to those described for time-resolved fluorescence (cf., Section 14.B.1). The multichannel detector data were organized in a [D] matrix of x rows and t columns, where x is the number of spectral channels and t is the number of chromatographic time channels at which the spectra are observed [8]. The time axis corresponds to the elution time of the chromatographic experiment. For resolution of overlapped chromatographic peaks, the data matrix is decomposed into its component matrices as defined by

where the component matrices are

[A]x x n = contains the normalized spectra of the n compounds in the analyte mixture as represented in the overlapped peak,

{Q\n x n = diagonal matrix representative of the amount of each compound present; and

[C\n x t = contains normalized chromatograms of each of the n compounds.

The individual components of the overlapped chromatographic peaks were modeled with the convolution of the Gaussian peak shape with a single-sided exponential to account for tailing [8]. Thus, the elements of the trail matrix [C'] for the k\h compound at time j are given conceptually by

where * represent the convolution of the two functions (cf., eq. (14.2), rk is the tailing "lifetime" for the kih compound, and At is the time interval between collection of each individual spectra. Gkj is the Gaussian peak shape, expressed in a slightly different way than in Chapters 3 and 7, as

In eq. (14.12), N is the number of theoretical plates in the chromatographic column. The tailing decay lifetimes rk were assumed to have been measured on pure standards. Therefore, only one parameter per compound, the chromatographic retention time tk, was required for the nonlinear regression analysis.

Analogous to the use of eq. (14.7), the best matrix product was found from

The error sum minimized for the regression analysis was the sum of squares of the deviations between [£)] and the model matrix [D'] weighted by the inverse of the degrees of freedom in the analysis [8], The model is expressed as

As in the time-resolved fluorescence example, the error sum included a penalty for negative elements in [>!']. The error sum was minimized using the simplex algorithm.

The preceding methodology was first evaluated with GC-MS data on two- and three-component mixtures, all of which eluted in a single, poorly resolved chromatographic peak [8]. The number of components n was found accurately by successive fits with increasing integer n values until the smallest minimum error sum was found. In most cases, the correct n was found easily by this method.

The regression method successfully resolved major components of test mixtures having a high degree of overlap in both their chromatograms and mass spectra. However, quantitative analysis of minor components was limited in this initial application because of a small spectral sampling rate; that is, At was too large. Nevertheless, the technique avoids some of the problems associated with arbitrary peak separation methods or those based on library searching or factor analysis [8].

A second application of the multichannel detection peak resolution method was made to high-pressure liquid chromotography using a multiple-wavelength photodiode array as the detector [9]. The data matrix was expressed by eq. (14.5), where A contained the spectra of the standards. Chromatographic profiles in the matrix [C] were normalized over the total elution volume. The model for each component chromatographic peak was again the Gaussian peak shape convoluted with an exponential tail, as in eq. (14.11). In this case, parameters used for each k component were retention time (tk) and peak width W = tk/Nm. The modified simplex algorithm was used to find the minimum error sum.

The procedure was tested for resolving five- to eight-component mixtures. Successful resolution of overlapped chromatographic peaks with up to seven components was achieved. Severely overlapped minor components in binary mixtures were detected at a 95% level of confidence when present in amounts >35%, depending on peak resolution factors and S/N of the data [9].

Was this article helpful?

## Post a comment