Journal of Chromatography A, 1120 (2006) 299–307 Multilinear gradient elution optimisation in reversed-phase liquid chromatography using genetic algorithms P. Nikitas 1 , A. Pappa-Louisi ∗ , P. Agrafiotou Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Available online 19 January 2006 Abstract The treatment presented in a recent paper [P. Nikitas, A. Pappa-Louisi, J. Chromatogr. A, 1068 (2005) 279] is extended to multilinear gradients, i.e. continuous gradients consisting of a certain number of linear portions. Thus, the experimental ln k versus ϕ curve, where k is the retention factor of a sample solute under isocratic conditions and ϕ is the volume fraction of the organic modifier in the water–organic mobile phase, is subdivided into a finite number of linear portions resulting in simple analytical expressions for the solute gradient retention time. These expressions of the retention time are directly used in an optimisation technique based on genetic algorithms. This technique involves first the determination of the theoretical dependence of k upon ϕ by means of gradient measurements, which in turn is used by the genetic algorithm for the prediction of the best gradient profile. The validity of the analytical expressions and the effectiveness of the optimisation technique were tested using fifteen underivatized amino acids and related compounds with mobile phases modified by acetonitrile. It was found that the adopted methodology exhibits significant advantages and it can lead to high quality predictions of the gradient retention times and optimisation results. © 2006 Elsevier B.V. All rights reserved. Keywords: Gradient elution; Optimisation techniques; Liquid chromatography 1. Introduction In a recent paper [1], we have presented a new mathematical treatment concerning the gradient elution [2–4] in reversed- phase liquid chromatography (RP-HPLC) when the volume fraction ϕ of an organic modifier in the water–organic mobile phase varies linearly with time. According to this treatment, the experimental ln k versus ϕ curve, where k is the retention factor under isocratic conditions in a binary mobile phase, is subdivided into a finite number of linear portions and the solute gradient retention time is calculated by means of an analytical expression arising from the fundamental equation of gradient elution [2–8]. This expression was further used in an optimisa- tion algorithm, which determines the best variation pattern of ϕ that leads to the optimum separation of a mixture of solutes at different values of the total elution time. We should point out that when gradient elution is used in HPLC, linear gradients do not necessarily result in a simple and explicit expression of the retention time in terms of the gradient mode characteristics. ∗ Corresponding author. Tel.: +30 2310 997765; fax: +30 2310 997709. E-mail addresses: nikitas@chem.auth.gr (P. Nikitas), apappa@chem.auth.gr (A. Pappa-Louisi). 1 Tel.: +30 2310 997773; fax: +30 2310 997709. This is possible only if ln k varies linearly with ϕ, a feature that is not met in the majority of the experimental systems unless the ϕ range is very narrow. The combination of linear gradient with a linear dependence of ln k upon ϕ is called linear solvent strength gradient [9–16]. This approach constitutes the base of DryLab, the most widely published HPLC simulation package to date [17,18]. It is evident that the approach presented in [1] lifts the con- straint that ln k should vary linearly with ϕ. However, a notice- able weakness of this approach is that a gradient profile with a single linear portion may be inadequate for the complete separa- tion of a mixture of solutes. For this reason in the present paper we attempt to extend it to multilinear gradients, i.e. continu- ous gradients consisting of a certain number of linear portions. The analytical expressions of the solute gradient retention time derived by this approach are used in a genetic algorithm (GA) for optimization in RP-HPLC. 2. Analytical expressions of the gradient retention time The retention time t R of a solute under gradient conditions may be calculated from the fundamental equation [1]  t R -t 0 -t in -t D 0 dt t 0 k ϕ = 1 - t D + t in t 0 k ϕ in (1) 0021-9673/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2006.01.005