Statistical tests for the selection of the optimum parameters set in models describing response surfaces in reversed-phase liquid chromatography

Computer aided optimization of multilinear gradient elution in liquid chromatography

Analytical expressions for retention time and peak compression factor are deduced by assuming quadratic solvent strength model and multilinear gradient elution. Based on these expressions, a program for the optimization of multilinear gradient profile is written with Visual Basic for Applications in Excel using genetic algorithm. The program is applied to search for a gradient profile for the separation of twelve compounds that are degraded from lignin. It is shown that the predicted and experimental chromatograms are well consistent. A better separation of the compounds is achieved under an S-shaped multilinear gradient profile than that obtained under linear gradient profile.

A simple approach for retention prediction in the pH-gradient reversed-phase liquid chromatography

A simple approach for retention modeling of solutes under pH-gradient conditions at various organic contents in the mobile phase is proposed. This approach is based on a retention model arising from the evaluation of the retention data of a set of 17 OPA derivatives of amino acids obtained in two series of 22 pH-gradient runs performed between a given initial and final pH value (between 2.8 and 10.7 or 3.2 and 9.0) with different gradient duration and with different organic modifier content in the eluent. The derived model is a fifth-parameter equation easily manageable through a linear least-squares fitting. It requires only 6 initial pH-gradient experiments, allows a very satisfactory prediction for various pH-changes of the same kind with those used in the fitting procedure and seems to be very promising in separation optimization under pH-gradient conditions. The pH-gradient method appears to be especially suitable and effective for separation of amino acid derivatives whereas the application of pH-gradients from 3.2 to 9.0 proved to be beneficial.

Separation optimization in reversed-phase liquid chromatography by using alkanol additives in the mobile phase: application to amino acids

In an effort to enhance complex mixture separations by using small amounts of a homologous series of alkanols as additives in the mobile phases, it was proposed an optimization algorithm based on a sixth-parameter retention model. This model considers simultaneously the contents of the main organic modifier and of the alkanol additive in the mobile phase as well as of the number of alkyl chain of the additive. This model is in fact a modification of a previously one derived in a recently published paper for the retention description of a mixture of purely hydrophobic alkylbenzenes under isocratic conditions with mobile phases containing alkanol additives. The effectiveness of the new retention model as well as the optimization algorithm was successfully applied to the separation of ten o-phthalaldehyde (OPA) derivatives of amino acids. Indeed, the new retention model exhibited an excellent prediction performance since the obtained overall predictive error between calculated and experimental times was only 2.8% for all isocratic runs by using a variety of mobile phase compositions containing any alkanol homologue even different than those used in the starting/fitting experiments. Moreover, a perfect resolution of the above amino acid mixture was achieved within only 7.4 min in the chromatogram recorded using the optimal mobile phase determined by means of the simple optimization algorithm proposed in this study.

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.

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.

Heteroscedasticity of retention factor and adequate modeling in micellar liquid chromatography

The two concepts of micelle formation (pseudo-phase and mass-action) could be the basis of retention models in micellar liquid chromatography (MLC). The separation of 4-hydroxybenzoic acid esters and seven polyaromatic hydrocarbons were performed to study the repeatability of retention factor in MLC. The full two factor experimental design was used for studying the dependence of retention factor variance on mobile phase composition (sodium dodecylsulfate, 1-butanol). The experimentally observed heteroscedasticity and perturbations after linearization were taken into account by using statistical weights obtained on the basis of errors propagation law and the modeling of retention by non-weighted and weighted least squares method was performed. The mechanistical retention models based on pseudo-phase and mass-action concepts of micelle formation were compared by fitting quality and prediction capability and high robustness of bilogarithmic dependence was observed. The significance of retention factor heteroscedasticity for retention hydrophobicity relationships was shown.

Micellar liquid chromatography retention model based on mass-action concept of micelle formation

Mass-action model of surfactant micelle formation has been used to develop a conceptual retention model in micellar liquid chromatography (MLC). The retention model bases on the consideration of the changes of the sorbate microenvironment at its transferring from the mobile phase (hybrid micellar eluent) to the stationary phase (a modified surface of alkyl-bounded sorbent). Principal retention equation contains the characteristics of hybrid micelles (critical micelle concentration, degree of counterion binding, partition coefficient of modifier between aqueous solution and micellar pseudo-phase) as well as three fitting parameters. The fitting parameters are an absolute term and coefficients that are equal to the number of molecules of surfactant and modifier, which are attached/detached by sorbate transferring from a hybrid micellar eluent to a modified surface of the stationary phase. On the MLC separation of five antibiotics of rubomicin derivatives and four esters of 4-hydroxybenzoic acid the model of the change of sorbate microenvironment has been tested. The adequateness of model to experimental data has been shown. A simple three-parameter function connecting log k with log cS and log cR that provides a high goodness-of-fit follows from principal retention equation (cS and cR are the molar concentrations of surfactant and organic modifier in the micellar eluent, respectively).

Robust interpretive optimisation in high-performance liquid chromatography considering uncertainties in peak position

In the context of interpretive chromatographic optimisation, robustness is usually calculated by introducing deliberated shifts in the nominal optimal conditions and evaluating their effects on the monitored objective function, mimicking thus the experimental procedures used in method validation. However, such strategy ignores a major source of error: the uncertainties associated to the modelling step, that may give rise to deceiving results when conditions that were expected to yield baseline separation are reproduced in the chromatograph. Two approaches, based on the peak purity concept, are here proposed to evaluate the robustness of the objective function under the perspective of measurement errors and modelling. The first approach implements these uncertainties as an extra band broadening for each chromatographic peak. The second one implements them as peak fluctuations in simulated replicated assays, which gives rise to a distribution of peak purities, easily computed through Monte-Carlo simulations. Both approaches predict satisfactorily a decreased separation capability, with respect to the conventional approach, for those situations where the uncertainties in peak position make the objective function critical. The first approach is less optimistic and formally less rigorous than the second one, but its computation is simpler. It can be used to map the critical resolution regions, to be comprehensively appraised further by the slower, although more rigorous, Monte-Carlo approach.

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

A new mathematical treatment concerning the gradient elution in reversed-phase liquid chromatography when the volume fraction psi of an organic modifier in the water-organic mobile phase varies linearly with time is presented. The experimental ln k versus psi 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 tR is calculated by means of an analytical expression arising from the fundamental equation of gradient elution. The validity of the proposed analytical expression and the methodology followed for the calculation of tR was tested using eight catechol-related solutes with mobile phases modified by methanol or acetonitrile. It was found that in all cases the accuracy of the predicted gradient retention times is very satisfactory because it is the same with the accuracy of the retention times predicted under isocratic conditions. Finally, the above method for estimating gradient retention times was used in an optimisation algorithm, which determines the best variation pattern of psi that leads to the optimum separation of a mixture of solutes at different values of the total elution time.