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.     

Multilinear gradient elution optimization in reversed-phase liquid chromatography based on logarithmic retention models: application to separation of a set of purines, pyrimidines and nucleosides

The analytical solutions of the fundamental equation of the multilinear gradient elution are derived in two cases, when the dependence of the logarithm of the solute retention (lnk) upon the volume fraction of organic modifier (φ) is a three-parameter logarithmic expression, and when a simple linear relationship between lnk and lnφ is adopted. The derived theoretical expressions for retention times under multilinear gradient conditions are embodied to simple algorithms for fitting gradient data and especially for resolution optimization. Their performance was examined by using a mixture of 16 model compounds chosen among purines, pyrimidine and nucleosides in eluting systems modified by acetonitrile. It was found that the accuracy of the predicted gradient retention times is very satisfactory even if the simple logarithmic expression for the retention behavior of solutes, i.e. the linear dependence of lnk upon lnφ, is used.     

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.     

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.     

Retention prediction of a set of amino acids under gradient elution conditions in hydrophilic interaction liquid chromatography

The analysis of amino acids presents significant challenges to contemporary analytical separations. The present paper investigates the possibility of retention prediction in hydrophilic interaction chromatography (HILIC) gradient elution based on the analytical solution of the fundamental equation of the multilinear gradient elution derived for reversed‐phase systems. A simple linear dependence of the logarithm of the solute retention (ln k) upon the volume fraction of organic modifier (φ) in a binary aqueous‐organic mobile is adopted. Utility of the developed methodology was tested on the separation of a mixture of 21 amino acids carried out with 14 different gradient elution programs (from simple linear to multilinear and curved shaped) using ternary eluents in which a mixture of methanol and water (1:1, v/v) was the strong eluting member and acetonitrile was the weak solvent. Starting from at least two gradient runs, the prediction of solute retention obtained under all the rest gradients was excellent, even when curved gradient profiles were used. Development of such methodologies can be of great interest for a wide range of applications.     

Optimisation technique for stepwise gradient elution in reversed-phase liquid chromatography

An optimisation technique of reversed-phase liquid chromatographic separations based on gradient elution with a stepwise variation pattern of the volume fraction phi of the organic modifier in the water-organic mobile phase is presented. It uses a non-linear least-squares programme with a Monte-Carlo search for initial estimates in order to determine the best variation pattern that leads to the optimum separation of a mixture of solutes. The validity of the above methodology was tested by separating eight catechol-related solutes with mobile phases modified by methanol or acetonitrile and variation patterns of two, three or four steps in the psi values. It was found in all cases a very satisfactory accuracy of the predicted gradient elution times, which is of the same order with the accuracy of the retention times predicted under isocratic or linear gradient conditions. In addition, it was shown that the proposed optimisation technique is both effective and flexible but well-shaped chromatograms are obtained under electrochemical detection only if steps with increasing psi are used and the change in psi is programmed to occur at the intermediate of the predicted peaks.     

Improved chemical composition separation of ethylene–propylene random copolymers by high-temperature solvent gradient interaction chromatography

High-temperature solvent gradient interaction chromatography (HT-SGIC) is a fast and efficient fractionation technique for the chemical composition analysis of olefin copolymers. The separation of ethylene–propylene random copolymers (EPRs) was achieved on a graphitic stationary phase, Hypercarb, at 160 °C by using linear solvent gradient elution from 1-decanol to 1,2,4-trichlorobenzene (TCB). In the present work, the solvent gradient profile was modified to improve the chromatographic separation of EPRs. With the aim to obtain a better resolution in separation, a slow increase in the volume fraction of TCB was applied. This allowed for a relatively large retention region for linear polyethylene (PE) chains on the column; thereby, a broader elution volume zone between the start of the gradient and the PE elution was achieved. The efficiency of this new gradient profile was demonstrated by analysing two fully amorphous EPR samples. Clear differences in the chemical composition of these EPR samples with similar ethylene contents have been proven by using this modified solvent gradient. The comprehensive chemical composition and microstructure analysis of the SGIC-separated fractions by FTIR revealed that ethylene/propylene (EP) copolymer chains were eluted according to their ethylene/propylene contents and E or P sequence lengths, even though they are distributed in a random manner. These results showed that the solvent composition is an important factor to affect the interactive adsorption or desorption behaviour of EP chains on Hypercarb. In this way, for the first time, the determination of the complex composition and chain structure of EPR samples was achieved within short analysis time, which is not possible till now using other fractionation techniques reported.

The Varigrad and similar apparatus in gradient liquid chromatography: A systematic computational approach. Part 1: Theoretical development

Different methods for producing a pre-assigned gradient in liquid chromatography are presented. A systematic approach to the calculations involved is given. The derived equations are mostly Poisson and Poisson summation distributions which are tabulated in the literature. The incremental method of gradient elution developed by Scott has been modified. In the modified apparatus two mixing chambers are used instead of one. This leads to an appreciable decrease in the number of reservoirs needed for the same precision in fitting a desired gradient. The application of the derived equations, together with other Varigrad modifications, will be given in Part 2 of this paper (to be published shortly in this Journal).     

Computer-assisted modelling and optimisation of reversed-phase high-temperature liquid chromatographic (RP-HTLC) separations

The use of high temperatures (above 100 °C) in reversed-phase liquid chromatography (RP-HTLC) has opened up novel and enhanced applications for this essential separation technique. Although the favourable effects of temperature on LC have been extensively studied both theoretically and practically, its potential application to method development has barely been investigated. These favourable effects include enhanced speed, efficiency, resolution and detectability, as well as changes in selectivity, especially for polar and ionisable compounds, and the emergence of new options such as temperature programming and the concomitant use of solvent and temperature gradients, green separations, and so on. The recent availability of silica-based columns that routinely support high temperatures in addition to more conventional temperature-resistant columns (based on graphitised carbon, polymers and zirconium oxide) and dedicated column ovens that allow accurate temperature control up to 200 °C makes it possible to conceive of RP-HTLC as a routine separation technique in the modern analytical laboratory. On the other hand, the addition of temperature as a new optimisable parameter to RPLC adds further complexity to method development. Thus, new computer-assisted optimisation tools that extend the capabilities of current computer-assisted tools are being specifically developed for this type of separation. A new specially developed computer-assisted method development (CAMD) tool is presented herein, and its efficiency is demonstrated. This CAMD is based on the development of a rugged retention model for peaks, allowing the simulation of any kind of RP-HTLC separation, including isocratic, linear, curved, multilinear and stepwise gradients of solvent composition concomitant with constant, linear and multilinear temperature gradients. Both the retention models and the unattended optimisation of separations are driven by evolutionary algorithms, thus providing negligible-cost, rapid, highly efficient, and user-friendly optimisation processes.     

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).     

Retention models for isocratic and gradient elution in reversed-phase liquid chromatography

One- and multi-variable retention models proposed for isocratic and/or gradient elution in reversed-phase liquid chromatography are critically reviewed. The thermodynamic, exo-thermodynamic or empirical arguments adopted for their derivation are presented and discussed. Their connection to the retention mechanism is also indicated and the assumptions and approximations involved in their derivation are stressed. Special attention is devoted to the fitting performance of the various models and its impact on the final predicted error between experimental and calculated retention times. The possibility of using exo-thermodynamic retention models for prediction under gradient elution is considered from a practical point of view. Finally, the use of statistical weights in the fitting procedure of a retention model and its effect on the calculated elution times as well as the transferability of retention data among isocratic and gradient elution modes are also examined and discussed.     

利用HCI程序确定儿茶酚化合物的反相高效液相色谱最佳分离条件

有效地确定了反相高效液相色谱分离儿茶酚化合物的最佳条件。在水和甲醇的二元流动相里分别加入乙酸缓冲液,利用基于ln k=ln kw +SF, k=A+B/F, ln k=L+MF+NF2 (F是流动相中有机物甲醇的体积分数)等保留因子的一次或二次方程式的塔板理论得到色谱分离结果;利用保留原理得到等度和梯度洗脱的最佳条件。得出最佳初始流动相是含0.1%乙酸的水和含0.1%乙酸的甲醇(体积比为75∶25)的混合溶液;梯度洗脱条件：初始流动相保持15 min,然后用10 min的时间将上述二元流动相的体积比线性变换成50∶50,直到完成全部分离。通过实验证实该计算结果与实验值相近。     

Relationship between isocratic and gradient retention times in the high-performance ion-exchange chromatography of proteins. Theory and experiment

Using isocratic retention parameters, the gradient elution retention time for several proteins has been calculated. The gradient retention time calculation is based on fitting the isocratic retention data to an equation of the form: log k' = m log (1/[Ca2+]) + log K and on applying well-established principles of gradient elution. A good correlation between the observed and calculated retention times for several test proteins was obtained at various total gradient times and column flow-rates. Conversely, isocratic retention parameters characterizing protein retention can be calculated from gradient elution retention data. However, even with retention data of high quality, small errors are amplified by the log-log nature of the ion-exchange isocratic retention model employed. Based on the close correlation between predicted and observed gradient retention times, no evidence for protein denaturation resulting from immobilization of the protein at high initial k' values at or near the column inlet was observed.     

Accelerated extraction for determination of polycyclic aromatic hydrocarbons in marine biota

A rapid and simple method is proposed for determination of polycyclic aromatic hydrocarbons (PAH) in complex matrices such as marine biota. The method uses sonication, by means of an ultrasonic probe, as a new tool for assisted extraction, coupled with reversed-phase liquid chromatography (RP-LC) with fluorescence detection (FL) for determination of 16 US EPA priority PAH. Separation and detection of the 16 PAH were complete in 45 min by RP-LC with a C₁₈ column and acetonitrile–water gradient elution. Multivariate optimisation of the variables affecting extraction (ultrasound radiation amplitude, sonication time, and temperature of the water-bath in which the extraction cell was placed) was conducted. The accuracy of the method was determined by analysis of a certified reference material and comparison of the results obtained with those from another method (microwave-assisted extraction and GC–MS). The new technique avoids the main problems encountered in the determination of PAH in complex matrices such as marine biota, and no clean-up step is necessary. The method was applied to determination of PAH in estuarine biota samples from the Urdaibai estuary (Biscay, Spain).     

Ancestral population genomics using coalescence hidden Markov models and heuristic optimisation algorithms

With full genome data from several closely related species now readily available, we have the ultimate data for demographic inference. Exploiting these full genomes, however, requires models that can explicitly model recombination along alignments of full chromosomal length. Over the last decade a class of models, based on the sequential Markov coalescence model combined with hidden Markov models, has been developed and used to make inference in simple demographic scenarios. To move forward to more complex demographic modelling we need better and more automated ways of specifying these models and efficient optimisation algorithms for inferring the parameters in complex and often high-dimensional models.In this paper we present a framework for building such coalescence hidden Markov models for pairwise alignments and present results for using heuristic optimisation algorithms for parameter estimation. We show that we can build more complex demographic models than our previous frameworks and that we obtain more accurate parameter estimates using heuristic optimisation algorithms than when using our previous gradient based approaches.Our new framework provides a flexible way of constructing coalescence hidden Markov models almost automatically. While estimating parameters in more complex models is still challenging we show that using heuristic optimisation algorithms we still get a fairly good accuracy.     

Automatic program for peak detection and deconvolution of multi-overlapped chromatographic signals part II: peak model and deconvolution algorithms

Several interlinked algorithms for peak deconvolution by non-linear regression are presented. These procedures, together with the peak detection methods outlined in Part I, have allowed the implementation of an automatic method able to process multi-overlapped signals, requiring little user interaction. A criterion based on the evaluation of the multivariate selectivity of the chromatographic signal is used to auto-select the most efficient deconvolution procedure for each chromatographic situation. In this way, non-optimal local solutions are avoided in cases of high overlap, and short computation times are obtained in situations of high resolution. A new algorithm, fitting both the original signal and the second derivatives is proved to avoid local optima in intermediate coelution situations. This allows achieving the global optimum without the need of background knowledge by the user. A previously reported peak model, a Gaussian with a polynomial standard deviation whose complexity can be modulated to enhance the fitting quality, was applied. However, the original formulation was modified to account baseline outside the peak region. Also, the optimal model complexity was auto-selected via error propagation theory. The method is able to process simultaneously several related chromatograms. The software was tested with both simulated and experimental chromatograms obtained with monolithic silica columns.     

A parallel pore and surface diffusion model for predicting the adsorption and elution profiles of lispro insulin and two impurities in gradient-elution reversed phase chromatography

Lispro insulin (LPI), a widely used insulin analog, is produced on tons per year scale. Linear gradient reversed phase chromatography (RPC) is used in the production to separate LPI from two impurities, which differ from LPI by a single amino acid residue. A chromatography model for the ternary separation in this RPC process is unavailable from the literature. In this study, a parallel pore and surface diffusion model is developed and verified for LPI and the two impurities. The LPI can be recovered with high yield (≥95%) and high purity (>99.5%). A new method, which requires a small amount of materials and an order of magnitude fewer experiments, has been developed to estimate the solvent-modulated isotherm parameters. A modified reversed phase modulator model is developed to correlate the adsorption isotherms of LPI and impurities. A strategy has been developed for estimating the intrinsic pore diffusivity and surface diffusivity. Since the adsorption affinities decrease by more than three orders of magnitude as organic fraction (φ) increases from 0.19 to 0.40, the apparent diffusivities based on a pore diffusion model or a surface diffusion model can also vary by several orders of magnitude. For this reason, a pore diffusion model or a surface diffusion model with a constant apparent diffusivity cannot predict closely the chromatograms over the same range of organic fractions, concentrations, and loadings. The parallel pore and surface diffusion model with constant diffusivities can predict closely the frontal and elution profiles over a wide range of organic fractions (0.19-0.40), LPI concentrations (0.05-18 g/L), linear velocities (<10 cm/min), and loading volume (0.0004-13 CV). For large loading stepwise and linear gradient elution, the peaks of LPI and the impurities are strongly focused by self-sharpening and gradient focusing effects as a result of the steep decrease of adsorption affinity from the loading φ (0.19) to elution φ (≥0.27). When the ratio of diffusion rate to convection rate is greater than 10, spreading due to diffusion is largely compensated by the focusing effects. As a result, a pore diffusion model with a constant pore diffusivity can predict closely the elution profiles in stepwise and linear gradient elution. The experimental yield values (≥95%) can be predicted to within ±1% by the model.     

反相高效液相三元流动相台阶梯度分离条件的快速优化方法

 根据溶质在柱内的迁移规律 ,建立了一种利用线性梯度实验快速获得溶质保留值方程系数 ,然后以串行响应函数为优化指标进行多台阶梯度分离条件优化的方法。与利用等度实验获得保留值方程的方法相比 ,该法可以大大缩短优化时间。通过该方法对芳香胺和衍生化氨基酸样品进行了分离 ,获得了满意的分离度 ,表明该方法的预测精度很好。     

Development of a perfusion reversed-phase high performance liquid chromatography method for the characterisation of maize products using multivariate analysis

A perfusion reversed-phase high performance liquid chromatography (RP-HPLC) method has been designed to allow rapid (3.4 min) separations of maize proteins with high resolution. Several factors, such as extraction conditions, temperature, detection wavelength and type and concentration of ion-pairing agent were optimised. A fine optimisation of the gradient elution was also performed by applying experimental design. Commercial maize products for human consumption (flours, precocked flours, fried snacks and extruded snacks) were characterised for the first time by perfusion RP-HPLC and their chromatographic profiles allowed a differentiation among products relating the different technological process used for their preparation. Furthermore, applying discriminant analysis makes it possible to group the samples according with the technological process suffered by maize products, obtaining a good prediction in 92% of the samples.     

Hybrid genetic algorithm–tabu search approach for optimising multilayer optical coatings

Constructing multilayer optical coatings (MOCs) is a difficult large-scale optimisation problem due to the enormous size of the search space. In the present paper, a new approach for designing MOCs is presented using genetic algorithms (GAs) and tabu search (TS). In this approach, it is not necessary to specify how many layers will be present in a design, only a maximum needs to be defined. As it is generally recognised that the existence of specific repeating blocks is beneficial for a design, a specific GA representation of a design is used which promotes the occurrence of repeating blocks. Solutions found by GAs are improved by a new refinement method, based on TS, a global optimisation method which is loosely based on artificial intelligence. The improvements are demonstrated by creating a visible transmitting/infrared reflecting filter with a wide variety of materials.