Polynomial retention models

It is clear from the above, that model equations for the description of retention surfaces have to meet high demands. Preferably, equations should be used that relate to reliable chromatographic theory, such as the one used to describe the retention behaviour as a function of pH in RPLC in the window diagram approach described in section 5.5.1. The use of such a chromatographic equation was clearly better in that case than a statistical approach using (for example) polynomial equations. 

Naturally, the number of initial experiments required to start the optimization procedure will increase if either the number of parameters considered or the complexity of the model equations increases. As far as the number of parameters is concerned, we have seen this to be true with any optimization procedure, and hence the number of parameters should be carefully selected. In order to avoid a large number of initial experiments, the complexity of the model equations may be increased once more data become available during the course of the procedure. For example, retention in RPLC may be assumed to vary linearly with the mixing ratio of two iso-eluotropic binary mixtures at first. When more experimental data points become available, the model may be expanded to include quadratic terms. However, complex mathematical equations, which bear no relation to chromatographic theory (e.g. higher order polynomials [537,579]) are dangerous, because they may describe a retention surface that is much more complicated than it actually is in practice. In other words, the complexity of the model may be dictated by experimental... 

The third solution to the problem may be found in the use of more efficient computers, algorithms and computational methods. For instance, if segmentation of the parameter space (linear interpolation) is used, large parts of the retention surfaces and hence of the response surface may remain unaltered when a new data point is added to the existing set. The use of simple model equations instead of linear segmentation may also be more efficient from a computational point of view. However, such simple equations may only be used for the description of the retention behaviour in a limited number of cases and if the model equations become more complex the advantage quickly disappears. For example, d Agostino et al. used up to sixth order polynomial equations [537] and their procedure also led to excessive calculation times. 

This model is a special case of the model studied by Matis and Wehrly [369] in which Ai Erl(Ai, u ) and A2 Ev A2.V2) retention-time distributions are associated with the first and second compartments, respectively. The analysis of the characteristic polynomial of this model implies that there are at least two complex eigenvalues, except for the case v = 2 with parameters satisfying the condition... 

A traditional formal approach, which applies linear, quadratic, cubic or other polynomial models to describe the relationship between the retention of solutes and the concentration of an organic solvent in a mobile phase ... 

ChromSword supports the optimization of separation for polynomial models up to a power of six. Thus, the most complex retention-concentration effects can be described and the separation optimized. AU polynomial models predict the retention of solutes rather precisely in the interpolation region of those concentrations studied. These models are less reliable in the extrapolation region. For example, if experiments were performed with 40% and 50% of organic solvent in a mobile phase, one can expect rather good prediction of retention and separation in the region between these concentrahons and less accuracy in the regions of 30-35% and 50-55%. Extrapolahon to wider hmits very often leads to substantial deviations between predicted and experimental data. 

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