Review
Mathematical functions for the representation of chromatographic peaks

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Abstract

About ninety empirical functions for the representation of chromatographic peaks have been collected and tabulated. The table, based on almost 200 references, reports for every function: (1) the most used name; (2) the most convenient equation, with the existence intervals for the adjustable parameters and for the independent variable; (3) the applications; (4) the mathematical properties, in relation to the possible applications. The list includes also equations originally proposed to represent peaks obtained in other analytical techniques (e.g. in spectroscopy), which in many instances have proved useful in representing chromatographic peaks as well; the built-in functions employed in some commercial peak-fitting software packages were included, too. Some of the most important chromatographic functions, i.e. the Exponentially Modified Gaussian, the Poisson, the Log-normal, the Edgeworth/Cramér series and the Gram/Charlier series, have been reviewed and commented in more detail.

Introduction

It is well known that there is no theoretic model for the exact description of the shape of chromatographic peaks. Several authors have proposed and/or used a number of empirical mathematical functions for the representation of these peaks [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76],[78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111], [112], [113], [114], [116], [117], [118], [119], [120], [121], [122], [123], [124],[126], [127], [128], [129], [130], [131], [132], [133], [134], [135], [136], [137], [138], [139], [140], [141], [142], [143], [144], [145], [146], [147], [148], [150], [151], [152], [153], [154], [155], [156], [157], [158], [159], [160], [161], [162], [163], [164], [165], [166], [168], [169], [170], [171], [172], [173], [174], [175], [176], [177], [178], [179], [180], [181], [182], [183], [184], [185], [186], [187], [188], [189], [190]. A number of these functions have proved useful also to represent the shape of signals obtained from other analytical methods, as e.g. flow injection analysis, various spectroscopic methods, voltammetry, mass spectrometry, thermal analysis, etc.; conversely, several functions proposed for other techniques have demonstrated suitable to represent chromatographic peaks.

Perhaps the most frequent application of mathematical functions for peak shape representation is the so-called deconvolution of partially resolved peaks [4], [6], [7], [8], [9], [10], [12], [13], [14], [15], [16], [18], [23], [24], [25], [29], [30], [39][40], [44], [45], [48], [49][54], [55], [57], [58], [61], [63], [64], [65], [66], [68], [69], [70][78], [80], [84], [85], [86], [87], [88], [95], [99][103], [104], [105], [117], [126], [127], [128], [133], [137][152], [153], [154], [156], [157], [159], [161], [163][164], [166], [168], [171], [172], [173], [174], [182], [189], [190]: the analytical signal produced by the sum of n partially overlapping peaks is reproduced by optimising (generally using the least squares criterion) the parameters of the sum of n mathematical functions. In this way the explicit expression of each of the n functions is obtained, thus achieving an artificial resolution of the n peaks. The potentialities and the limits of this method have been extensively discussed by Maddams [116], Vandeginste and De Galan [177] and, recently, by Zhang et al. [189]; in the most recent applications the least-squares fitting is combined with other mathematical methods of data treatment (see e.g. [49], [126], [127], [151], [189]; although these papers deal mainly with spectroscopic data, the methods described and the relevant conclusions are completely applicable to chromatographic data as well).

Another important chromatographic application of data fitting techniques (with or without peak “deconvolution”) is the computation of the so-called “chromatographic figures of merit” (cfom) of peaks [21], [26], [27], [28], [36], [41], [50], [52], [71], [72], [73][89], [91], [96], [101], [118], [129], [130], [134][135], [137], [138], [141], [144], [145], [150], [151], [166], [169], [175][178], [186], [187], [188]. These include the first statistical moments, which allow to calculate the peak area, mean/position, width, skew and excess (see also below), and also other less frequently computed parameters, as e.g. the number of theoretical plates [50], [89]. The computation of these parameters leads to important information on the analyte itself (for example, the mean and the dispersion of the molecular mass of a polymer, in steric exclusion chromatography) and on its interactions in the chromatographic column (for example, the mass transport properties, and the activity, virial and diffusion coefficients). If the shape of the peak is represented and fitted by an explicit function, the cfom can be calculated with exact or approximate formulae which relate the fitted parameters of the function to the cfom [21], [36], [72], [73], [89], [101], [129], [130], [134], [135], [137], [141][145], [150], [151], [169], [175], [178], [188]; if such formulae are not known, the fitting function can be analysed graphically. Both methods can be applied to overlapping peaks as well as to isolated ones, and are believed to be more accurate [89], [100], [129], [137] than methods not based on peak fitting [11], [26], [27], [50], [52], [71], [91], [96], [129][137], [138], [186], [187], [188].

A third class of applications is represented by the study of the effect of one or more experimental parameters (e.g. eluent flow-rate, response time of the detector, etc.) on the shape of the peaks [1], [19], [62], [76], [120], [121], [142], [143][155], [160], [178], [179].

Another application is the production of simulated (artificial) peaks, either for didactic or commercial purposes [108], [124], [146] or, more frequently, to validate a proposed method for the treatment of experimental data [2], [11], [32], [33], [42], [46], [47], [53], [67], [74][75], [89], [92], [93], [97], [98], [106], [107][109], [122], [139], [140], [145], [148], [162], [170], [172], [189], [190].

Recently, the fitting of experimental data with suitable functions has been employed to improve the signal-to-noise ratio [41], [43] and for the efficient and compact memorization of experimental data (see [5] and references therein).

As mentioned above, a large number of empirical functions have been proposed and used for all these applications. To our knowledge, an exhaustive list of suitable functions is not available in the literature. Several papers list their “names” without reporting further information [7], [9], [12], [18], [27], [31], [37], [55], [70], [115], [129], [134], [135][137], [146], [147], [152], [172], [174], [176], [177]. In other cases the equations are reported and sometimes commented [69], [108], [111], [113], [136], [145], [154], [169]; however only a small number of functions are considered in each case. To our knowledge, the only paper which contains a systematic list of functions is a review by Maddams [116], who quotes and comments 13 functions (intended mainly for spectroscopy).

In some cases two equations representing the same function have been presented with different denominations; on the contrary, the same denomination has been sometimes applied to different functions. Several papers include equations affected by more or less serious printing errors, which sometimes appear to have propagated (see below).

In the present paper we try to present an archive as complete as possible of functions proposed in the literature for the representation of peaks. We have also attempted to present all the functions in a form as unambiguous and homogeneous as possible. On the basis of some mathematical properties, the functions have been criticized and evaluated in view of their possible use in peak fitting. The table with the list of the functions is presented and explained in detail in the following section.

Section snippets

Description of the table of the functions (Table 1)

The first column of Table 1 contains the denomination proposed for each function, which generally coincides with the one most frequently employed in the literature; other denominations are reported in brackets. For all the functions to which a specific name (like Gaussian, EMG, Poisson, etc.) appears to have not been assigned, we propose as the denomination the names of the Authors who, to our knowledge, have first proposed or used this function for the representation of peaks. The same rule

Remarks on some functions

Five of the functions reported in Table 1, i.e. the Exponentially Modified Gaussian (EMG), Poisson, Log-normal, Gram–Charlier series and Edgeworth–Cramér series, will now be shortly reviewed and commented:

Commercial softwares

Although generic mathematical or graphic softwares could be used in principle for the purpose of peak fitting, a number of specific programmes have been described in the literature for the treatment of chromatograms and spectrograms [4], [7], [8], [16], [24], [25], [31], [41], [45], [51], [56], [57], [58], [61], [63], [82], [84], [86], [99], [101], [104], [106], [110], [124], [125], [132], [146], [149], [152], [153], [161], [163], [167], [168], [173], [188].

These programmes were mainly written

Conclusions

In this paper, we collected and tabulated 86 functions proposed for the representation of chromatographic peaks. The most used name, the most convenient equation, the applications, the mathematical properties (in relation to the possible applications) and a list of references are reported for each function (Table 1). The built-in functions employed in some commercial peak-fitting software packages have been included, too.

An important objective of this work has been the individuation and

References (190)

  • C.E. Alciaturi et al.

    Anal. Chim. Acta

    (1998)
  • G.C. Allen et al.

    Anal. Chim. Acta

    (1978)
  • B. Alsberg et al.

    Chemom. Intell. Labor. Systems

    (1997)
  • C. Baker et al.

    Spectrochim. Acta

    (1978)
  • C. Baker et al.

    Spectrochim. Acta

    (1978)
  • S.H. Brooks et al.

    Anal. Chim. Acta

    (1990)
  • S.I. Chou et al.

    J. Mol. Spectrosc.

    (2000)
  • G. Crisponi et al.

    Anal. Chim. Acta

    (1993)
  • F. Dondi et al.

    J. Chromatogr.

    (1984)
  • R.C. Eanes et al.

    Spectrochim. Acta B

    (2000)
  • B. Fahys et al.

    Spectrochim. Acta

    (1978)
  • A. Felinger et al.

    Talanta

    (1994)
  • D.J. Gerth et al.

    Comput. Chem.

    (1992)
  • K.J. Goodman et al.

    J. Chromatogr. A

    (1995)
  • T.R. Griffiths et al.

    Anal. Chim. Acta

    (1982)
  • J.O. Grimalt et al.

    Anal. Chim. Acta

    (1982)
  • J.O. Grimalt et al.

    Anal. Chim. Acta

    (1987)
  • J.O. Grimalt et al.

    Anal. Chim. Acta

    (1991)
  • E. Grushka et al.

    J. Chromatogr.

    (1979)
  • U.L. Haldna et al.

    J. Chromatogr.

    (1978)
  • R.D. Hester et al.

    J. Chromatogr.

    (1989)
  • W. Huang et al.

    Anal. Chim. Acta

    (1995)
  • W. Huang et al.

    Comput. Chem.

    (1995)
  • J. Huang et al.

    Microchem. J.

    (1999)
  • T. Jawhari et al.

    Carbon

    (1995)
  • M.S. Jeansonne et al.

    J. Chromatogr.

    (1989)
  • M.S. Jeansonne et al.

    J. Chromatogr.

    (1992)
  • D. Jin et al.

    Anal. Chim. Acta

    (2000)
  • D. Jin et al.

    Anal. Chim. Acta

    (2000)
  • S.D. Kolev

    Anal. Chim. Acta

    (1995)
  • S. Abramowitz et al.

    J. Chem. Phys.

    (1963)
  • S.H. Algie

    Anal. Chem.

    (1977)
  • A.H. Anderson et al.

    Anal. Chem.

    (1970)
  • A.H. Anderson et al.

    J. Chromatogr. Sci.

    (1970)
  • J.V. Arena et al.

    J. Chem. Educ.

    (1994)
  • W.E. Barber et al.

    Anal. Chem.

    (1981)
  • B.E. Barker et al.

    Anal. Chem.

    (1974)
  • J. Barthel et al.

    J. Sol. Chem.

    (2000)
  • J.C. Bartlet et al.

    Can. J. Chem.

    (1960)
  • R.G. Bennett et al.

    J. Chem. Phys.

    (1958)
  • D.E. Bergbreiter et al.

    J. Am. Chem. Soc.

    (1992)
  • A. Berthod

    Anal. Chem.

    (1991)
  • P.A. Boudreau et al.

    Anal. Chem.

    (1979)
  • S.H. Brooks et al.

    Anal. Chem.

    (1988)
  • T.S. Buys et al.

    Anal. Chem.

    (1972)
  • T.S. Buys et al.

    Anal. Chem.

    (1976)
  • W. Cai et al.

    Anal. Lett.

    (2000)
  • R.A. Caruana et al.

    Anal. Chem.

    (1986)
  • R.A. Caruana et al.

    Anal. Chem.

    (1988)
  • S.N. Chesler et al.

    Anal. Chem.

    (1971)
  • Cited by (0)

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