Enhancement in the computation of gradient retention times in liquid chromatography using root-finding methods

Gradient elution may provide adequate separations within acceptably short times in a single run, by gradually increasing the elution speed. Similarly to isocratic elution, chromatograms can be predicted under any experimental condition, through strategies based on retention models. The most usual approach implies solving an integral equation (i.e., the fundamental equation of gradient elution), which has an analytical solution only for certain combinations of retention model and gradient programme. This limitation can be overcome by using numerical integration, which is a universal approach although at the cost of longer computation times. In this work, several alternatives to improve the performance in the resolution of the integral equation are explored, which can be especially useful with multi-linear gradients. For this purpose, the application of several root-finding methods that include the Newton's and bisection searches is explored in three frameworks: isolated predictions, regression modelling problems using gradient training sets, and optimisation of multi-linear gradients. Significant reductions of computation times were obtained. The substitution of non-integrable retention models by Tchebyshev polynomial approximations, which are pre-calculated before solving the integral equation in optimisation problems, is also investigated.