New approach to linear gradient elution used for optimisation in reversed-phase liquid chromatography

A new chromatographic response function with automatically adapting weight factor for automated method development

A critical factor for automated method development in chromatography is the maximization or minimization of an objective function describing the quality (and speed) of the separation. In chromatography, this function is commonly referred to as a chromatographic response function (CRF). Many CRFs have previously been introduced, but many have unfavourable properties such as featuring multiple optima, insufficient discriminatory power, and a too strong dependence on the weight factors needed to balance resolution and time penalty components. To overcome these problems, the present study introduces a new type of CRF wherein the relative weight of the time penalty term is a self-adaptive function of the separation quality. The ability to unambiguously identify the optimal gradient settings of this newly proposed CRF is compared to that of some of the most frequently used CRFs in a study covering 100 randomly composed in silico samples. Doing so, the new CRF is found to flawlessly lead to the correct solution (=linear gradient parameters providing the highest resolution in the shortest potential time) in 100 % of the cases, while the most frequently used literature CRFs were off-target for about 50 to 60 % of the samples, even when considering the availability of spectral peak shape data. Some slight alterations to the proposed CRF are introduced and discussed as well.

Application of evolutionary algorithms to optimise one- and two-dimensional gradient chromatographic separations

We report on the performance of three classes of evolutionary algorithms (genetic algorithms (GA), evolution strategies (ES) and covariance matrix adaptation evolution strategy (CMA-ES)) as a means to enhance searches in the method development spaces of 1D- and 2D-chromatography. After optimisation of the design parameters of the different algorithms, they were benchmarked against the performance of a plain grid search. It was found that all three classes significantly outperform the plain grid search, especially in terms of the number of search runs needed to achieve a given separation quality. As soon as more than 100 search runs are needed, the ES algorithm clearly outperforms the GA and CMA-ES algorithms, with the latter performing very well for short searches (<50 search runs) but being susceptible to convergence to local optima for longer searches. It was also found that the performance of the ES and GA algorithms, as well as the grid search, follow a hyperbolic law in the large search run number limit, such that the convergence rate parameter of this hyperbolic function can be used to quantify the difference in required number of search runs for these algorithms. In agreement with one's physical expectations, it was also found that the general advantage of the GA and ES algorithms over the grid search, as well as their mutual performance differences, grow with increasing difficulty of the separation problem.

Optimisation of multilinear gradient elutions in reversed-phase liquid chromatography using ternary solvent mixtures

The multilinear gradient elution theory for binary mobile phases in reversed-phase liquid chromatography presented in [P. Nikitas, A. Pappa-Louisi, A. Papageorgiou, J. Chromatogr. A 1157 (2007) 178] is extended to ternary gradients. For the evaluation of this theory and the performance of the various fitting and optimisation algorithms we used 13 o-phthalaldehyde (OPA) derivatives of amino acids with mobile phases modified by acetonitrile and methanol. It is shown that the theory can lead to high quality predictions of the retention times under gradients elutions and optimisation of ternary gradients provided that we use a six-parameter expression for the logarithm of the retention factor, lnk, and the adjustable parameters of this expression are determined from ternary isocratic data.

Simple algorithms for fitting and optimisation for multilinear gradient elution in reversed-phase liquid chromatography

The theory of the multilinear gradient elution in reversed-phase liquid chromatography (RP-HPLC) presented in [P. Nikitas, A. Pappa-Louisi, P. Agrafiotou, J. Chromatogr. A 1120 (2006) 299] is modified to increase its flexibility. In addition, it is embodied to simple algorithms for fitting gradient data and especially for resolution optimisation under multilinear gradient conditions. In particular, two new algorithms for fitting and one for optimisation are tested and compared with conventional algorithms. Their performance was examined using 13 o-phthalaldehyde (OPA) derivatives of amino acids with mobile phases modified by acetonitrile. It was found that the new proposed algorithms, a repeated application of the Levenberg-Marquardt (LM) method for fitting (R_LM) and a modified descent algorithm for optimisation (RND_D), in combination with the modified theory of the multilinear gradient elution can lead to high quality predictions of the retention times and optimisation results.

The bandwidth in gradient elution chromatography with a retained organic modifier

The variance of a chromatographic band is derived in the case of RPLC gradient elution when the organic modifier is significantly retained onto the stationary phase. This derivation is based on the extension of a model due to Poppe et al. [H. Poppe, J. Paanakker, M. Bronckhorst, J. Chromatogr., 204 (1981) 77] which assumes that the gradient front remains unchanged and propagates along the column at the same speed as the mobile phase, following piston flow. Theoretical and experimental results are compared in the case of caffeine on a C(1)-silica stationary phase eluted with an acetonitrile gradient. The actual retention behaviors of caffeine and acetonitrile were implemented in the theoretical calculations. The model predicts compression factors between 0.71 and 0.34 for relatively smooth gradient steepness, betat(0), between 0.009 and 0.054 while the corresponding experimental band compression factors vary between 1.01 and 0.43 for the very same gradient steepness. The model underestimation of these factors arises likely from the strong deviation of the actual retention behavior from the prediction of the Linear Solvent Strength Model (LSSM).

Analytical Solutions of the Ideal Model for Gradient Liquid Chromatography

The analytical solutions of the ideal model for gradient elution that ignores the influence of the solute concentration on the retention factor (k) were studied by using the method of characteristics for solving partial differential equations. It is found for any gradient profiles and solvent strength models used that the concentration of the solute will be discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is discontinuous, the product on the left side of this point is equal to that on the right side. We also discussed the basic equations to predict the retention time in gradient elution and introduced the injection time into them. For linear solvent strength stepwise and linear gradient elution, general expressions were proposed for the prediction and they can be used as the basis to derive others for specific gradient modes such as single linear, stepwise, and ladderlike gradients. For these modes, simple expressions to account for the band compression and the concentration change during the elution were also given.

Models and objective functions for the optimisation of selectivity in reversed-phase liquid chromatography

Interpretive methodologies are the most efficient tools for finding the optimal conditions in chromatography. These methodologies are supported by models or algorithms able to infer the system behaviour upon changes in the experimental factors. Once the models are built with data obtained from sets of carefully designed experiments, molecular modelling or other approaches, they can be applied to predict the performance of new conditions. The different elements involved in these methodologies, for both isocratic and gradient elution, are given. Special attention is devoted to the description of retention, owing to its major impact on the prediction of chromatographic resolution. Several models considering the main factors affecting retention (i.e. organic modifiers, pH and temperature), and procedures that enhance the predictions, are presented. Both the existence of skewed peaks and the effect of elution conditions on peak profiles are considered. Finally, the assessment of resolution, as well as other secondary aims that affect the practical suitability of the optimal conditions, is discussed.

Computer-assisted transfer of programmed elutions in reversed-phase high-performance liquid chromatography

Computer-assisted procedures were used to simulate modifications in chromatograms caused by the transfer of elution programmes between instruments with significantly different dwell volumes. Moreover, for the first time the same approach was used to modify the elution programmes to match the chromatograms produced in the different instruments. The process may consist of making minor modifications to gradient programmes or transforming the original gradient programme into a stepwise gradient profile and/or the simultaneous programming of flow and solvent composition. The combination of these approaches has been shown to have an enormous potential for producing matched chromatograms in instrumental systems with dwell volumes that differ by several millilitres. The efficiency and robustness of the proposed procedure is demonstrated with a variety of compounds (two different mixtures of 10 and 11 analytes), mobile phases (methanol and acetonitrile gradients), flow rates (0.5-1.5 mL/min range), temperatures (35-45 degrees C interval) and gradient profiles (linear, multilinear, curved and stepwise).

Can the theory of gradient liquid chromatography be useful in solving practical problems?

Advances in the theory of gradient liquid chromatography and their practical impacts are reviewed. Theoretical models describing retention in reversed-phase, normal-phase and ion-exchange modes are compared. Main attention is focused on practically useful models described by two- or three-parameter equations fitting the experimental data in the range of mobile phase composition utilized for sample migration during gradient elution. The applications of theory for gradient method development, optimization and transfer are addressed. The origins and possibilities for overcoming possible pitfalls are discussed, including the effects of the instrumental dwell volume, uptake of mobile phase components on the column and size of the sample molecules. Special attention is focused on gradient separations of large molecules.

Modelling flow rate gradient elution in reversed-phase liquid chromatography

The fundamental equation of flow programming elution was tested in several different types of flow-rate gradients (step, linear, multilinear, parabolic and more combined gradients) implemented in the separation of two multicomponent mixtures of solutes. The retention prediction obtained for all solutes under all flow-programmed conditions was excellent. In addition, although flow programming appears quite limited in its ability to provide improved performance in liquid chromatography, there are specific flow-rate profiles which provide significant improvement in the rearrangement of the peaks within a chromatogram.

Multilinear gradient elution optimisation in reversed-phase liquid chromatography using genetic algorithms

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 lnk versus phi curve, where k is the retention factor of a sample solute under isocratic conditions and phi 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 phi 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.

Levels in the interpretive optimisation of selectivity in high-performance liquid chromatography: A magical mystery tour

Interpretive approaches for selectivity optimisation, which are those supported by retention models, are able to exploit efficiently the capabilities of the chromatographic system. The resolution of a mixture is usually faced in a first trial by looking for a unique experimental condition, able to resolve all compounds in the sample. If this is not possible, the problem can be outlined with less ambitious aims, focusing on only some compounds. In an extreme case, a single analyte can be individually optimised. Current strategies that give answer to the different goals pursued in the analysis, which are classified as total, partial and specific, are reviewed. Optimisation oriented to deconvolution, useful in case of partial coelution, and robust measurements of resolution, are also outlined. The steps recognised in any chromatographic optimisation procedure, and some fundamentals and tools used in optimisation approaches for isocratic and gradient elution are commented to explain different strategies. Examples of increasing complexity are supplied to explain the problematic arose, and the convenience in applying a certain methodology. Details on the mathematical treatment for each particular optimisation strategy are also given.

Theory of Stepwise Gradient Elution in Reversed-Phase Liquid Chromatography Coupled with Flow Rate Variations: Application to Retention Prediction and Separation Optimization of a Set of Amino Acids

The coupling of stepwise mobile phase gradient elution and flow programming is proposed as an integrated approach to the general elution problem in reversed-phase liquid chromatography. A model is developed to describe the above separation process performed under simultaneous programming of two separation parameters by extending our previous work on the rigorous derivation of the fundamental equation governing the concentration gradient of organic modifier in the mobile phase, that is, a single gradient elution mode (Anal. Chem. 2005, 77, 5670-5677). The theory was tested in the retention prediction and separation optimization of 18 o-phthalaldehyde derivatives of amino acids in eluting systems modified by acetonitrile or methanol. The retention prediction obtained for all solutes under all dual-mode gradient conditions was excellent. In addition, it has been shown that the combination of mobile phase and flow rate programming modes is particularly favorable, whereas the separations among the analytes were considerably improved by using the acetonitrile eluting system, as compared to those obtained by the methanol system.

Column equilibration effects in gradient elution in reversed-phase liquid chromatography

The fundamental equations and conditions for linear and stepwise gradient elution in reversed-phase liquid chromatography are applied to a mixture of amino acids in their underivatized form in aqueous mobile phases modified by 2-propanol, acetonitrile or methanol for examining column equilibration effects. It was found in all cases systematic deviations between experimental and calculated retention times, which are prominent in 2-propanol, reduced in acetonitrile and practically negligible in methanol. These deviations appear within a chromatogram just after the first change in the composition of the mobile phase reaches the detector and last ca. 5 min, where the magnitude of errors reduces exponentially with time. Based on these observations we propose a simple way to correct the calculated from the gradient elution theory retention times of sample solutes. The origin of the discrepancies between theory and experiment as well as their impact on the resolution is also discussed.

Optimization of gradient elution conditions in multicomponent preparative liquid chromatography

Gradient elution is widely applied in analytical chromatography to reduce the separation time and/or to improve the selectivity. Increasingly the potential of modulating the solvent strength during gradient operation is exploited in preparative liquid chromatography. The purpose of this paper is to investigate theoretically the effect of optimizing free parameters available in gradient chromatography (extents and shapes of gradients) on the productivity of isolating a target component in a multicomponent mixture. An equilibrium stage model was used to quantify and compare different modes of operation (isocratic and various variants of gradient elution). By combining experimental design and artificial neural network concepts, optimal conditions were identified for the production of the second eluting component in a ternary mixture. The strong impact of the shape of gradients on process performance is elucidated.

Expressions of the Fundamental Equation of Gradient Elution and a Numerical Solution of These Equations under Any Gradient Profile

The original work carried out by Freiling and Drake in gradient liquid chromatography is rewritten in the current language of reversed-phase liquid chromatography. This allows for the rigorous derivation of the fundamental equation for gradient elution and the development of two alternative expressions of this equation, one of which is free from the constraint that the holdup time must be constant. In addition, the above derivation results in a very simple numerical solution of the various equations of gradient elution under any gradient profile. The theory was tested using eight catechol-related solutes in mobile phases modified with methanol, acetonitrile, or 2-propanol. It was found to be a satisfactory prediction of solute gradient retention behavior even if we used a simple linear description for the isocratic elution of these solutes.