On the equations describing chromatographic peaks and the problem of the deconvolution of overlapped peaks

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Abstract

The problem of the appropriate choice of the function that describes a chromatographic peak is examined in combination with the deconvolution of overlapped peaks by means of the non-linear least-squares method. It is shown that the majority of the functions proposed in the literature to describe chromatographic peaks are not suitable for this purpose. Only the polynomial modified Gaussian function can describe almost every peak but it is mathematically incorrect unless it is redefined properly. Two new functions are proposed and discussed. It is also shown that the deconvolution of an overlapping peak can be done with high accuracy using a non-linear least-squares procedure, like Microsoft Solver, but this target is attained only if we use as fitted parameters the position of the peak maximum and the peak area (or height) of every component in the unresolved chromatographic peak. In case we use as fitted parameters all the parameters that describe each single peak enclosed in the multi-component peak, then Solver leads to better fits, which though do not correspond to the best deconvolution of the peak. Finally, it is found that Solver gives much better results than those of modern methods, like the immune and genetic algorithms.

Introduction

Recently, three novel methods for deconvolving overlapped chromatographic peaks have been proposed [1], [2]. All of them belong to the general case of computational methods that mimic natural processes. Thus, the immune algorithm of Ref. [1] imitates the defending process of an immune system, the genetic algorithm used also in Ref. [1] is based on Darwin’s evolution rule and finally the artificial neural network used in Ref. [2] mimics the behaviour of the neural system in the brain.

The most important conclusion of these studies is that the development of new deconvolution methods for overlapping peaks is necessary due to serious limitations of the classical non-linear least-squares method [2]. However, this conclusion is not supported by other studies [3]. For this reason we examine this issue in the present paper in more detail, i.e., whether the non-linear least-squares method is inappropriate for deconvolution of overlapping peaks or there are conditions under which it can give correct results. Note that nowadays the non-linear least-squares method is so simple that it can be used not only by scientists but also by students. For example the simplicity and the capabilities of Solver, a programme for non-linear least-squares fitting using the spreadsheets of Microsoft Excel, have been pointed out in several articles [4], [5], [6], [7], [8], [9]. On the contrary, the algorithms that mimic natural processes are still quite complicated and forbidden for the average analyst.

The deconvolution problem of overlapping peaks is closely associated with the appropriate choice of the function that describes the chromatographic peaks. In literature there is a good variety of such functions [1], [2], [3], [10], [11], [12], [13], [14], [15], [16], [17]. However, the majority of them are not general enough to describe every chromatographic peak. For example the exponentially modified Gaussian function (EMG) [10], [11], [12], [13] is considered as one of the most effective functions for this purpose. Despite this we have observed that it cannot describe peaks characterised by tailing behaviour at the end of the peaks such as those obtained by electrochemical detection (ED) under special conditions. Thus, in the present paper we first examine the problem of the appropriate choice of the function that describes a chromatographic peak and then the deconvolution of overlapped peaks by means of the non-linear least-squares method.

Section snippets

Functions for chromatographic peaks

The most important functions used up to now to describe chromatographic peaks are the following:

1. Gaussian distribution function (GD) expressed as:h(t)=hmet−tms2where t is the time, h(t) is the ordinate, i.e., the peak intensity, hm is the height of the peak, tm is the position of the peak maximum and s is a constant denoting the standard deviation of the Gaussian distribution.

2. Asymmetric Gaussian distribution function (AGD) [2]. It is expressed by Eq. (1) with s=s1 when t<tm and s=s2s1

Chromatographic system and conditions

The liquid chromatography system consisted of a Shimadzu LC-9A pump, a Model 7125 syringe loading sample injector fitted with a 50 μl loop (Rheodyne, Cotati, CA, USA), a 250×4 mm MZ- Analytical column (5 μm Inertsil ODS-3), a Shimadzu UV–visible spectrophotometric detector (Model SPD-10A) and a Gilson ED system (Model 141) equipped with a glassy carbon electrode. The detector cell volume for UV was 8 μl and for ED 7.2 μl. The UV and ED systems were connected in series so that the analytes

Choice of the appropriate function

The sum of squares of residuals (SSR) after the fitting can be used as a criterion for the choice of the function that describes satisfactorily the experimental chromatographic peaks. The values of SSR for the various solutes used in the presence of 5% isopropanol in neutral and acid mobile phases are given in Table 1. Table 2 includes the values of SSR in neutral mobile phases in the presence of 1% and 2% isopropanol.

In respect to these tables we should clarify the following: (a) the values of

Conclusions

The majority of the functions proposed to describe chromatographic peaks are not suitable for this purpose. The simple and the asymmetric Gaussian functions, the generalised exponential and the Lorentzian functions failed to describe our experimental data. The exponentially modified Gaussian (EMG) function was also problematic in describing peaks recorded by ED characterised by tailing behaviour in some cases.

From the functions of the literature only the polynomial modified Gaussian (PMG1) can

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