Selecting initial values for iterative fitting of chromatographic peaks with exponentially modified gaussian function

Abstract

Various mathematical functions are used to describe shapes of chromatographic peaks. Some of these functions, such as exponentially modified gaussian, polynomially modified gaussian or parabolic variance gaussian are based on the normal distribution, and some are not. These functions have from 4 to 9 parameters that need to be iteratively optimized to fit the model function to the experimental data. Many of these functions are numerically unstable, therefore choosing an optimal initial guess of their parameters becomes crucial for successful fitting. The most commonly employed approaches are based on empirical equations relating the basic peak shape parameters (asymmetry value, width at 10% of peak height) and the parameters of the model function. Additionally, the algorithms for calculating the basic peak shape parameters are not thoroughly described in the literature.

Exponentially modified gaussian (EMG) was used as a model function in this work as it is a de facto standard in chromatography. Implementations of EMG in Python programming language libraries were listed. The numerical instability of SciPy implementation was investigated for symmetrical peaks and its probable causes were discussed. It was shown that Kalambet’s approach to calculating EMG (based on using several equations depending on the shape of the chromatographic peak) did not show such instability.

Approaches to calculate base peak parameters were discussed. Algorithms to find the apex coordinates, left and right halfwidths and width at selected peak height (10% to 50% of the peak maximum) were described.

It is widely known that the relation between the basic peak shape parameters and the parameters of the EMG function is not linear. The empirical equations that approximate these relations were suggested by Foley and Dorsey in the 1980s. We suggested using interpolation by splines instead. This approach significantly improved accuracy in estimating the model function parameters and allowed broadening the range of usable peak shape values. Splines can be calculated once and knots together with spline coefficients can be saved for future use.

In most of the articles and manuals width and halfwidths of the peak are calculated at 10% of the height. Alternative heights (10% to 50%) to calculate basic peaks parameters were tested. It was concluded that parameters of the EMG function can be calculated without significant difference in accuracy at different heights (from 10 to 30%) for noise-free peaks. For noisy data (S/N=100) 30-35% of the peak height can be considered as an alternative.