Improved reversed-phase gradient retention modeling

Performance of global retention models in the optimisation of the liquid chromatographic separation (II): Complex multi-analyte samples

BACKGROUND

Enhancing the quality control of medicinal plants is a complex challenge due to their rich variety of chemical compounds present at varying and extreme concentrations. Chromatographic fingerprints, which have become essential for characterising these complex natural materials, require achieving optimal separation conditions to effectively maximise the number of detected peaks. The challenges in optimising fingerprints and other complex multi-analyte samples include the unavailability of standards, the presence of unknown constituents and the substantial workload that would require conventional optimisation methods based on models.

RESULTS

This work introduces an interpretive optimisation approach which operates on the premise of predicting chromatograms using global models. Initially, a multi-linear gradient experimental design is sequentially executed to accommodate all peaks in the chromatogram in an adequate time window. Following this, a small set of sample peaks (reference peaks) is selected based on their consistent traceability across all chromatograms in the design. Using this reference dataset, a global model is constructed, initially focused solely on the reference peaks and later extended to encompass all detected peaks in the sample. The aim is to find gradients that maximise resolution while minimising analysis time. These optimised gradients are applied successfully to enhance the separation of medicinal plant extracts, with particular emphasis on peppermint and pennyroyal extracts.

SIGNIFICANCE

The proposed optimisation relying on global models can be applied to highly complex samples even in the absence of standards, or in cases where standards are available but their use is impractical due to workload constraints. Moreover, in discerning the most promising gradients for highly complex samples, peak purity has demonstrated superior reliability and competitiveness compared to peak capacity as chromatographic objective function.

Assisted Active Learning for Model-Based Method Development in Liquid Chromatography

In recent decades, there has been a growing interest in fully automated methods for tackling complex optimization problems across various fields. Active learning (AL) and its variant, assisted active learning (AAL), incorporating guidance or assistance from external sources into the learning process, play key roles in this automation by enabling the autonomous selection of optimal experimental conditions to efficiently explore the problem space. These approaches are particularly valuable in situations wherein experimentation is costly or time-consuming. This study explores the application of AAL in model-based method development (MD) for liquid chromatography (LC) by using Bayesian statistics to incorporate historical data and analyte information for the generation of initial retention models. The process involves updating the model parameters based on new experiments, coupled with an active data selection method to choose the most informative experiment to run in a subsequent step. This iterative process balances model exploitation and experimental exploration until a satisfactory separation is achieved. The effectiveness of this approach is demonstrated via two practical examples, resulting in optimized separations in a limited number of experiments by optimizing the gradient slope. It is shown that the ability of AAL to leverage past knowledge and compound information to improve accuracy and reduce experimental runs offers a flexible alternative approach to fixed design methods.

Analysis and classification of tea varieties using high-performance liquid chromatography and global retention models

As a result of their metabolic processes, medicinal plants produce bioactive molecules with significant implications for human health, used directly for treatment or for pharmaceutical development. Chromatographic fingerprints with solvent gradients authenticate and categorise medicinal plants by capturing chemical diversity. This work focuses on optimising tea sample analysis in HPLC, using a model-based approach without requiring standards. Predicting the gradient profile effects on full signals was the basis to identify optimal separation conditions. Global models characterised retention and bandwidth for 14 peaks in the chromatograms across varied elution conditions, facilitating resolution optimisation of 63 peaks, covering 99.95 % of total peak area. The identified optimal gradient was applied to classify 40 samples representing six tea varieties. Matrices of baseline-corrected signals, elution bands, and band ratios, were evaluated to select the best dataset. Principal Component Analysis (PCA), k-means clustering, and Partial Least Squares-Discriminant Analysis (PLS-DA) assessed classification feasibility. Classification limitations were found reasonable due to tea processing complexities, involving drying and fermentation influenced by environmental conditions.

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.

Numerical Approximation of a Nonequilibrium Model of Gradient Elution Chromatography Considering Different Functional Relationships between Model Parameters and Solvent Composition

In this paper, a rigorous theoretical study is conducted to analyze the influence of varying solvent compositions on the retention characteristics of elution profiles within a fixed-bed liquid chromatographic column. In gradient chromatography, the propagation speed of elution profiles is manipulated through a progressive variation in the mobile-phase composition. Consequently, enhanced separation of the mixture components can be achieved together with a reduction in the requisite recycling times for subsequent injections. In other words, both the efficiency and the selectivity of the column can be enhanced. The lumped kinetic model coupled with the convection-diffusion equation for the volume fraction of the solvent is applied to simulate the process. The resulting nonlinear model equations are numerically solved by applying a semidiscrete second-order finite-volume method. The numerical solutions are utilized to quantify the effects of gradient starting and ending times, solvent composition, solvent strength parameters, and gradient slope on the concentration profiles. Additionally, temporal numerical moments are plotted versus the starting and ending times of the gradient, and standard performance criteria are presented for evaluating the process performance. The outcomes of this investigation will contribute to further enhancements in gradient elution chromatography.

Deep reinforcement learning for the direct optimization of gradient separations in liquid chromatography

While Reinforcement Learning (RL) has already proven successful in performing complex tasks, such as controlling large-scale epidemics, mitigating influenza and playing computer games beyond expert level, it is currently largely unexplored in the field of separation sciences. This paper therefore aims to introduce RL, specifically proximal policy optimization (PPO), in liquid chromatography, and evaluate whether it can be trained to optimize separations directly, based solely on the outcome of a single generic separation as input, and a reward signal based on the resolution between peak pairs (taking a value between [-1,1]). More specifically, PPO algorithms or agents were trained to select linear (1-segment) or multi-segment (2-, 3-, or 16-segment) gradients in 1 experiment, based on the outcome of an initial, generic linear gradient (ϕstart=0.3, ϕend=1.0, and tg=20min), to improve separations. The size of the mixtures to be separated varied between 10 and 20 components. Furthermore, two agents, selecting 16-segment gradients, were trained to perform this optimization using either 2 or 3 experiments, in sequence, to investigate whether the agents could improve separations further, based on previous outcomes. Results showed that the PPO agent can improve separations given the outcome of one generic scouting run as input, by selecting ϕ-programs tailored to the mixture under consideration. Allowing agents more freedom in selecting multi-segment gradients increased the reward from 0.891 to 0.908 on average; and allowing the agents to perform an additional experiment increased the reward from 0.908 to 0.918 on average. Finally, the agent outperformed random experiments as well as standard experiments (ϕstart=0.0, ϕend=1.0, and tg=20min) significantly; as random experiments resulted in average rewards between 0.220 and 0.283, and standard experiments resulted in average rewards of 0.840. In conclusion, while there is room for improvement, the results demonstrate the potential of RL in chromatography and present an interesting future direction for the automated optimization of separations.

Automated method development in high-pressure liquid chromatography

Method development in liquid chromatography is a crucial step in the optimization of analytical separations for various applications. However, it is often a challenging endeavour due to its time-consuming, resource intensive and costly nature, which is further hampered by its complexity requiring highly skilled and experienced scientists. This review presents an examination of the methods that are required for a completely automated method development procedure in liquid chromatography, aimed at taking the human out of the decision loop. Some of the presented approaches have recently witnessed an important increase in interest as they offer the promise to facilitate, streamline and speed up the method development process. The review first discusses the mathematical description of the separation problem by means of multi-criteria optimization functions. Two different strategies to resolve this optimization are then presented; an experimental and a model-based approach. Additionally, methods for automated peak detection and peak tracking are reviewed, which, upon integration in an instrument, allow for a completely closed-loop method development process. For each of these approaches, various currently applied methods are presented, recent trends and approaches discussed, short-comings pointed out, and future prospects highlighted.

A perspective on the use of deep deterministic policy gradient reinforcement learning for retention time modeling in reversed-phase liquid chromatography

Artificial intelligence and machine learning techniques are increasingly used for different tasks related to method development in liquid chromatography. In this study, the possibilities of a reinforcement learning algorithm, more specifically a deep deterministic policy gradient algorithm, are evaluated for the selection of scouting runs for retention time modeling. As a theoretical exercise, it is investigated whether such an algorithm can be trained to select scouting runs for any compound of interest allowing to retrieve its correct retention parameters for the three-parameter Neue-Kuss retention model. It is observed that three scouting runs are generally sufficient to retrieve the retention parameters with an accuracy (mean relative percentage error MRPE) of 1 % or less. When given the opportunity to select additional scouting runs, this does not lead to a significantly improved accuracy. It is also observed that the agent tends to give preference to isocratic scouting runs for retention time modeling, and is only motivated towards selecting gradient scouting runs when penalized (strongly) for large analysis/gradient times. This seems to reinforce the general power and usefulness of isocratic scouting runs for retention time modeling. Finally, the best results (lowest MRPE) are obtained when the agent manages to retrieve retention time data for % ACN at elution of the compound under consideration that spread the entire relevant range of ACN (5 % ACN to 95 % ACN) as well as possible, i.e., resulting in retention data at a low, intermediate and high % ACN. Based on the obtained results, we believe reinforcement learning holds great potential to automate and rationalize method development in liquid chromatography in the future.

Experimental design and re-parameterization of the Neue-Kuss model for accurate and precise prediction of isocratic retention factors from gradient measurements in reversed phase liquid chromatography

The present work describes a re-parameterization of the Neue Kuss (NK) model for describing retention in liquid chromatography, and this re-parameterized model is used to fit a large set of isocratic retention measurements with improved convergence properties relative to the original parameterization of the model. Next, an experimental design for retention measurements using mobile phase gradient elution conditions is proposed for the purpose of obtaining accurate and precise NK parameters. Simulated retention data for mobile phase gradient elution conditions with two different levels of noise, as well as an essentially zero noise level were fit with the re-parameterized model. The results showed that the re-parameterized fits yielded average (absolute value) prediction errors for the parameters at the highest noise level of 7.2 % for S1,ref, 18 % for S2,ref and 6.2 % for kref (the re-parameterized NK model parameters). These errors were significantly smaller than those for the original parameterization of the NK model, where the errors were 23 % for S1, 25 % for S2 and 160 % for kw (the original NK model parameters). Furthermore, isocratic retention factors predicted using these model parameters were found to have an average magnitude of error of 0.51 % for the re-parameterized model, as opposed to 6800 % for the model with the original parameterization. A further test of this approach was carried out for independent experimental measurements for five solutes on a C18 column. The average magnitude of error of the isocratic retention factors predicted from parameters obtained from fits of gradient data was 1.6 %, provided that the range of organic solvent compositions that the solute sampled in the mobile phase gradient experiments was consistent with the isocratic experiments. These results indicate that the re-parameterization of the NK model allows for significant improvements in the fitting process, and that the proposed experimental design allows for NK parameters to be extracted from mobile phase gradient experiments, with prediction accuracies of isocratic retention factors on the order of 1-2 %.

Computer-driven optimization of complex gradients in comprehensive two-dimensional liquid chromatography

Method development in comprehensive two-dimensional liquid chromatography (LC × LC) is a complicated endeavor. The dependency between the two dimensions and the possibility of incorporating complex gradient profiles, such as multi-segmented gradients or shifting gradients, renders method development by "trial-and-error" time-consuming and highly dependent on user experience. In this work, an open-source algorithm for the automated and interpretive method development of complex gradients in LC × LC-mass spectrometry (MS) was developed. A workflow was designed to operate within a closed-loop that allowed direct interaction between the LC × LC-MS system and a data-processing computer which ran in an unsupervised and automated fashion. Obtaining accurate retention models in LC × LC is difficult due to the challenges associated with the exact determination of retention times, curve fitting because of the use of gradient elution, and gradient deformation. Thus, retention models were compared in terms of repeatability of determination. Additionally, the design of shifting gradients in the second dimension and the prediction of peak widths were investigated. The algorithm was tested on separations of a tryptic digest of a monoclonal antibody using an objective function that included the sum of resolutions and analysis time as quality descriptors. The algorithm was able to improve the separation relative to a generic starting method using these complex gradient profiles after only four method-development iterations (i.e., sets of chromatographic conditions). Further iterations improved retention time and peak width predictions and thus the accuracy in the separations predicted by the algorithm.

Characterization of a high throughput approach for large scale retention measurement in liquid chromatography

Many contemporary challenges in liquid chromatography-such as the need for "smarter" method development tools, and deeper understanding of chromatographic phenomena-could be addressed more efficiently and effectively with larger volumes of experimental retention data than are available. The paucity of publicly accessible, high-quality measurements needed for the development of retention models and simulation tools has largely been due to the high cost in time and resources associated with traditional retention measurement approaches. Recently we described an approach to improve the throughput of such measurements by using very short columns (typically 5 mm), while maintaining measurement accuracy. In this paper we present a perspective on the characteristics of a dataset containing about 13,000 retention measurements obtained using this approach, and describe a different sample introduction method that is better suited to this application than the approach we used in prior work. The dataset comprises results for 35 different small molecules, nine different stationary phases, and several mobile phase compositions for each analyte/phase combination. During the acquisition of these data, we have interspersed repeated measurements of a small number of compounds for quality control purposes. The data from these measurements not only enable detection of outliers but also assessment of the repeatability and reproducibility of retention measurements over time. For retention factors greater than 1, the mean relative standard deviation (RSD) of replicate (typically n=5) measurements is 0.4%, and the standard deviation of RSDs is 0.4%. Most differences between selectivity values measured six months apart for 15 non-ionogenic compounds were in the range of +/- 1%, indicating good reproducibility. A critically important observation from these analyses is that selectivity defined as retention of a given analyte relative to the retention of a reference compound (kx/kref) is a much more consistent measure of retention over a time span of months compared to the retention factor alone. While this work and dataset also highlight the importance of stationary phase stability over time for achieving reliable retention measurements, we are nevertheless optimistic that this approach will enable the compilation of large databases (>> 10,000 measurements) of retention values over long time periods (years), which can in turn be leveraged to address some of the most important contemporary challenges in liquid chromatography. All the data discussed in the manuscript are provided as Supplemental Information.

Revisiting the dependence of retention factor on the content of organic component in the mobile phase in reversed-phase HPLC

In high-performance liquid chromatography, the dependence of retention factor k on volumetric fraction ϕ of organic phase is expressed by log k = F(ϕ) with F(ϕ) obtained by measuring log k at different ϕ values. From F(ϕ), a value kw is calculated by taking ϕ = 0. The equation log k = F(ϕ) is applied for predicting k, and kw is a descriptor of hydrophobic character of solutes and stationary phases. Calculated kw should not depend on the nature of organic component of mobile phase but extrapolation procedure leads to different kw for different organic components. The present study shows that the expression of F(ϕ) changes depending on the range of ϕ and the same function F(ϕ) cannot be used for the full range of ϕ from 0 to 1. Consequently, kw obtained by extrapolation of ϕ to zero is not correct because the expression of F(ϕ) was generated by fitting the data using ϕ with higher values. The present study shows the proper way to obtain the value of kw .

High-Throughput Chromatographic Separation of Oligonucleotides: A Proof of Concept Using Ultra-Short Columns

Ion-pairing reversed-phase liquid chromatography (IP-RPLC) is the reference separation technique for characterizing oligonucleotides (ONs) and their related impurities. The aim of this study was to better understand the retention mechanism of ONs, evaluate the applicability of the linear solvent strength (LSS) retention model, and explore the potential of ultra-short columns having a length of only 5 mm for the separation of model ONs. First, the validity of the LSS model was evaluated for ONs having sizes comprised between 3 and 30 kDa, and the accuracy of retention time predictions was assessed. It was found that ONs in IP-RPLC conditions follow an "on-off" elution behavior, despite a molecular weight lower than that of proteins. For most linear gradient separation conditions, a column length between 5 and 35 mm was found to be appropriate. Ultra-short columns of only 5 mm were therefore explored to speed up separations by considering the impact of the instrumentation on the efficiency. Interestingly, the impacts of injection volume and post-column connection tubing on peak capacity were found to be negligible. Finally, it was demonstrated that longer columns would not improve selectivity or separation efficiency, but baseline separation of three model ONs mixtures was enabled in as little as 30 s on the 5 mm column. This proof-of-concept work paves the way for future investigations using more complex therapeutic ONs and their related impurities.

Performance of global retention models in the optimisation of the chromatographic separation (I): Simple multi-analyte samples

Conventional retention models lead to accurate descriptions of the elution behaviour from the fitting of data for single solutes or from a set of solutes, one by one. However, the simultaneous fitting of several solutes through a regression process that separates the contributions of column and solvent from those of each solute is also possible. The result is a global retention model constituted by a set of equations with some common parameters (those associated with column and solvent), whereas others, specific to each solute, differ for each equation. This work explores the possibilities, advantages, and limitations of global models when they are applied to the optimisation of chromatographic resolution. A set constituted by 13 drugs (diuretics and β-blockers) and a training experimental design of seven multi-linear gradients are considered. Since standards for all compounds were available, the optimisation based on global models could be compared with the conventional optimisation, which is based on individual models. In their current state, global models do not predict changes in elution order, but they do allow for incorporating additional solutes (e.g., new analytes or matrix peaks) with only one new experiment. This possibility is explored by extending the model for the 13 analytes to include 26 peaks associated with a contamination in the injector. The combination of individual and global models allows an optimisation where the effects of matrix peaks on the separation of analytes can be integrated.

Graph Neural Networks for Improved Retention Time Predictions

In this extended special feature to celebrate the 35th anniversary edition of LCGC Europe, leading figures from the separation science community explore contemporary trends in separation science and identify possible future developments.

User-Driven Strategy for In Silico Screening of Reversed-Phase Liquid Chromatography Conditions for Known Pharmaceutical-Related Small Molecules

In the pharmaceutical field, and more precisely in quality control laboratories, robust liquid chromatographic methods are needed to separate and analyze mixtures of compounds. The development of such chromatographic methods for new mixtures can result in a long and tedious process even while using the design of experiments methodology. However, developments could be accelerated with the help of in silico screening. In this work, the usefulness of a strategy combining response surface methodology (RSM) followed by multicriteria decision analysis (MCDA) applied to predictions from a quantitative structure-retention relationship (QSRR) model is demonstrated. The developed strategy shows that selecting equations for the retention time prediction models based on the pKa of the compound allows flexibility in the models. The MCDA developed is shown to help to make decisions on different criteria while being robust to the user's decision on the weights for each criterion. This strategy is proposed for the screening phase of the method lifecycle. The strategy offers the possibility to the user to select chromatographic conditions based on multiple criteria without being too sensitive to the importance given to them. The conditions with the highest desirability are defined as the starting point for further optimization steps.

Numerical Approximation of the Nonequilibrium Model of Gradient Elution Chromatography Considering Linear and Nonlinear Solvent Strength Models

In both linear and nonlinear chromatography, the lumped kinetic model is a suitable model for predicting elution bands when appropriate equilibrium functions and mass transfer coefficients are accessible. This model also works well in the case of gradient elution chromatography if variations in the equilibrium functions due to changes in the mobile phase composition are known. The rational selection of an optimum gradient is explored in this study from three different perspectives using the lumped kinetic model. Elution profiles generated by using (a) linear solvent strength, (b) quadratic solvent strength, and (c) power law are investigated. The effectiveness and reliability of the suggested numerical approach, utilizing the flux-limiting finite volume method, are demonstrated through numerical simulations. The impacts of axial dispersion, nonlinearity coefficient, Henry's constant, mass transfer coefficient, and gradient parameters are studied on single and two-component elution profiles.

Solvent strength of organic phase for two biphasic solvent systems in high speed countercurrent chromatography

In this work, relationships between solvent strength of organic phase (ψ) for two biphasic solvent systems in high speed countercurrent chromatography, hexane-ethyl-acetate-methanol-water (HEMWat) and ethyl acetate-n-butanol-water (EBuWat), and partition coefficient (K) were investigated using four retention models, including Jandera's model (ABM), Neue-Kuss model (NK), linear-solvent-strength model (LSS) and quadratic-solvent-strength model (QSS). Experimental results showed that ABM model had the best fitting results for HEMWat system while NK model and QSS model had good fitting results in EBuWat system. Thus, a mathematical relationship between partition coefficient (K) and solvent strength of organic phase (ψ) could be obtained by measurement of partition coefficients of the target compounds with three different volume ratios of organic phase. At the same time, a functional map was proposed to construct to get a maneuverable region so that an optimal two-phase solvent system for separation of a target compound could be selected easily, which saved a lot of manpower for high speed countercurrent chromatographic separation. The application of this new method was declared by successful separation of two components, apigenin-6-C-β-D-xylopyranosyl-8-C-α-L-arabinopyranoside and vicenin-3, from dried leaves of Dendrobium officinale Kimura et Migo using high speed countercurrent chromatography.

Theory of separation performance and peak width in gradient elution liquid chromatography: A tutorial

Separation performance in chromatography has been extensively studied since the dawn of the technique. Although the basic principles of band broadening and the resulting separation performance in isocratic elution are in general well known and understood, this is much less the case for gradient separations. In this tutorial, first the basic principles, concepts and parameters that determine separation performance, peak width and variance and analysis time in isocratic separations are reviewed. This is subsequently used to discuss the parameters that affect peak width in gradient elution, together with the concepts of plate count and plate height in this elution mode. In addition, the effect of peak compression in gradient elution is elaborated. Finally, the effect of extra-column dispersion on separation performance in gradient elution is discussed, and an overview of how these contributions can be experimentally evaluated is given.

Retention prediction of monoamine neurotransmitters in gradient liquid chromatography

Retention prediction of monoamine neurotransmitters has been compared for the generally applied linear solvent-strength model and quadratic polynomial three-parameter model. The design of experiments protocol has been applied to plan linear gradients within the experimental space with altered gradient time, mobile phase flowrate, and column temperature. Relative prediction errors increased at elevated temperature, which is more significant for the linear solvent-strength model when compared to the polynomial model. On the other hand, the predefined design of experiments space controls the retention time errors, as predictions for LC conditions that are outside of the plan are much less accurate and should be avoided. The final part of the work deals with the effect of extracolumn band dispersion on the peak capacity of linear gradients at various gradient times, mobile phase flowrates, and column temperature. The peak capacity determined for corrected experimental data were consistent with the published results dealing with the optimization of peak capacity in gradient elution. This article is protected by copyright. All rights reserved.

An approach to high throughput measurement of accurate retention data in liquid chromatography

Efforts to model and simulate various aspects of liquid chromatography (LC) separations (e.g., retention, selectivity, peak capacity, injection breakthrough) depend on experimental retention measurements to use as the basis for the models and simulations. Often these modeling and simulation efforts are limited by datasets that are too small because of the cost (time and money) associated with making the measurements. Other groups have demonstrated improvements in throughput of LC separations by focusing on "overhead" associated with the instrument itself - for example, between-analysis software processing time, and autosampler motions. In this paper we explore the possibility of using columns with small volumes (i.e., 5 mm x 2.1 mm i.d.) compared to conventional columns (e.g., 100 mm x 2.1 mm i.d.) that are typically used for retention measurements. We find that isocratic retention factors calculated for columns with these dimensions are different by about 20%; we attribute this difference - which we interpret as an error in measurements based on data from the 5 mm column - to extra-column volume associated with inlet and outlet frits. Since retention factor is a thermodynamic property of the mobile/stationary phase system under study, it should be independent of the dimensions of the column that is used for the measurement. We propose using ratios of retention factors (i.e., selectivities) to translate retention measurements between columns of different dimensions, so that measurements made using small columns can be used to make predictions for separations that involve conventional columns. We find that this approach reduces the difference in retention factors (5 mm compared to 100 mm columns) from an average of 18% to an average absolute difference of 1.7% (all errors less than 8%). This approach will significantly increase the rate at which high quality retention data can be collected to thousands of measurements per instrument per day, which in turn will likely have a profound impact on the quality of models and simulations that can be developed for many aspects of LC separations.

A simple mathematical treatment for predicting linear solvent strength behaviour in gradient elution: application to biomolecules

This paper describes an approach to rapidly and easily calculate the linear solvent strength parameters, namely log k₀ and S, under reversed‐phase liquid chromatography conditions. This approach, which requires two preliminary gradient experiments to determine the retention parameters, was applied to various representative compounds including small molecules, peptides, and proteins. The retention time prediction errors were compared to the ones obtained with a commercial HPLC modeling software, and a good correlation was found between the values. However, two important constraints have to be accounted for to maintain good predictions with this new approach: i) the retention factor at the initial composition of the preliminary gradient series have to be large enough (i.e., log k_i above 2.1) and ii) the retention models have to be sufficiently linear. While these two conditions are not always met with small molecules or even peptides, the situation is different with large biomolecules. This is why our simple calculation method should be preferentially applied to calculate the linear solvent strength parameters of protein samples.

An exploration into the use of Hansen solubility parameters for modelling reversed-phase chromatographic separations

The suitability of Hansen solubility parameters as descriptors for modelling analyte retention during reversed-phase chromatographic experiments was investigated. A novel theoretical model using Hansen solubility parameters as the basis for a complete mathematical derivation of the model was developed. The theoretical model also includes the cavitation volumes of the analytes, which were calculated using ab initio density functional theory methods. A set of three homologous phthalates was used for experimental data collection and subsequent model construction. The training error and the generalization error of the model were additionally evaluated using a range of chemically diverse analytes. Statistical evaluation of the results revealed that the model is suitable for analyte retention prediction but is limited to the analytes used in the model construction. Therefore, the resulting theoretical model cannot be easily generalized. A retention anomaly attributed to the column temperature and mobile phase composition was experimentally observed and mathematically investigated.

Accuracy of retention model parameters obtained from retention data in liquid chromatography

In liquid chromatography (LC), it is often very useful to have an accurate model of the retention factor, k, over a wide range of isocratic elution conditions. In principle, the parameters of a retention model can be obtained by fitting either isocratic or gradient retention factor data. However, in spite of many of our own attempts to accurately predict isocratic k values using retention models trained with gradient retention data, this has not worked in our hands. In the present study we have used synthetic isocratic and gradient retention data for small molecules under reversed-phase LC conditions. This allows us to discover challenges associated with predicting isocratic k's without the confounding influences of experimental issues that are difficult to model or eliminate. The results indicate that it is not currently possible to consistently predict isocratic retention factors for small molecules with accuracies better than 10%, even when using synthetic gradient retention data. Two distinct challenges in fitting gradient retention data were identified: 1) a lack of 'uniqueness' in the parameters; and 2) an inability to find the global optimum fit in a complex fitting landscape. Working with experimental data where measurement noise is unavoidable will only make the accuracy worse. This article is protected by copyright. All rights reserved.

Development of a retention prediction model in ion-pair reversed-phase HPLC for nucleoside triphosphates used as mRNA vaccine raw materials

The International Conference on Harmonization guidelines for quality on pharmaceutical development recommends a systematic development approach including robustness studies which assure performance of manufacturing and analytical method development of drug product. It was demonstrated that the retention prediction model for nucleoside triphosphates (NTPs) on ion-pair reversed-phase HPLC was developed by a highly accurate Kawabe's model which supports the development of robust HPLC methods. As NTPs and its derivatives are typically used for Messenger ribonucleic acid (mRNA) vaccine production, adenosine-5'-triphosphate (ATP), guanosine-5'-triphosphate (GTP), cytidine-5'-triphosphate (CTP), 5-methylcytidine-5'-triphosphate (m5-CTP), uridine-5'-triphosphate (UTP), 5-methyluridine-5'-triphosphate (m5-UTP), pseudouridine-5'-triphosphate (Ψ-UTP) and N1-methylpseudouridine-5'-triphosphate (m1Ψ-UTP) were applied for prediction model development. By a comparison of the predicted retention factor in eight studied samples with the retention factor measured under six isocratic conditions, the absolute prediction error was 0.075 and also the prediction error (%) was 2.70%. In practical examples, analytical method for residual ATP, GTP, CTP, and m1Ψ-UTP in the commercial mRNA-based drugs and purity method for UTP derivatives were optimized by QbD approach. The design space for the minimum resolution between adjacent peaks was simulated with the models developed to evaluate the robustness of peak separation, and the optimal mobile phase condition was also simulated. As a conclusion, the desired peak was successfully separated under the optimized condition, and we thought that these retention models could optimize the mobile phase condition of the NTP analysis method for applying to various quality tests, such as quantity, purity and identity test for NTPs and its derivates in the mRNA-based drugs.

Retention modeling and adsorption mechanisms in reversed-phase liquid chromatography

To interpret the dependence of solute retention behavior on modifier content in reversed-phase liquid chromatography, a theoretical framework, based on the concentration dependence of solvophobic forces imposed on solutes and the competitive adsorptions of solutes and solvent modifiers, was proposed. The generality of the developed model was demonstrated by comparing the model with conventional retention models. The linear dependence of the Gibbs energy change of solute adsorption with respect to the modifier concentration was assumed, and the model was fitted to the experimental results, with good agreement demonstrated between the experimental data and the model. Retention behaviors were inferred to be determined by two key dimensionless groups that represented the reductions in the retention factors resulting from a weakened solvophobic interaction and modifier competitive adsorption. The retention behaviors were successfully deconvoluted for each contribution as a function of the modifier concentration by using the fitted parameters. The effects of both contributions on the retention behaviors were enhanced for the solutes with aromatic groups. The standard Gibbs energy change _SL^o of benzene adsorption was found to depend linearly on the number of modifier molecules present but independent of modifier identity. For the solutes associated with hydrogen-bonding groups, the degree of reduction in the solvophobic interactions was considerably reduced. Hence, the relative contributions of both mechanisms to solute retention depend greatly on the solute structure. Perturbation method was performed to investigate the modifier adsorption mechanisms. The results show that the standard Gibbs energy change _SL^o for the first-layer adsorption of modifiers changed linearly with the carbon number of modifier molecule. These results demonstrated that the proposed model can offer a physically consistent quantitative description of retention when solvent composition is varied.

Bayesian optimization of comprehensive two-dimensional liquid chromatography separations

Comprehensive two-dimensional liquid chromatography (LC×LC), is a powerful, emerging separation technique in analytical chemistry. However, as many instrumental parameters need to be tuned, the technique is troubled by lengthy method development. To speed up this process, we applied a Bayesian optimization algorithm. The algorithm can optimize LC×LC method parameters by maximizing a novel chromatographic response function based on the concept of connected components of a graph. The algorithm was benchmarked against a grid search (11,664 experiments) and a random search algorithm on the optimization of eight gradient parameters for four different samples of 50 compounds. The worst-case performance of the algorithm was investigated by repeating the optimization loop for 100 experiments with random starting experiments and seeds. Given an optimization budget of 100 experiments, the Bayesian optimization algorithm generally outperformed the random search and often improved upon the grid search. Moreover, the Bayesian optimization algorithm offered a considerably more sample-efficient alternative to grid searches, as it found similar optima to the grid search in far fewer experiments (a factor of 16-100 times less). This could likely be further improved by a more informed choice of the initialization experiments, which could be provided by the analyst's experience or smarter selection procedures. The algorithm allows for expansion to other method parameters (e.g., temperature, flow rate, etc.) and unlocks closed-loop automated method development.

Impact of changes in physicochemical parameters of the mobile phase along the column on the retention time in gradient liquid chromatography. Part A - temperature gradient

Gradient chromatography has been widely applied in analytical and preparative chromatography since it provides better peak shapes and the ability to elute analytes in a shorter time frame. Apart from changes in the composition of a mobile phase also alteration of process temperature can be applied during separation procedures to improve efficiency. However, proper mathematical modeling of the gradient chromatography and further correct prediction of solutes' retention behavior have become a serious challenge as it involves the need to develop computational procedures that accurately account for the time and spatial gradients of crucial parameters. In this work, a computational procedure including the equilibrium-dispersive two-dimensional mass transfer model, the two-dimensional (2D) heat transfer model together with Darcy's law and the continuity equation have been proposed. Additionally, the calculation procedure was simplified by replacing the 2D model with the one-dimensional (1D) mass transfer model in order to speed up the computations. Both proposed solutions were validated employing external experimental data of temperature gradient HPLC [1] as well as with predictions based on the linear elution strength (LES) model available therein. The proposed procedures made it possible to efficiently predict the concentration profiles with average relative errors of calculated retention times not exceeding 3.22%. Moreover, the effect of the axial dispersion coefficient determination method on the obtained peak shapes was examined involving the Gunn, the Wen-Fan, and the Chung-Wen correlations, indicating that the latter produces the most accurate results. Finally, the proposed mathematical procedures were tested under UHPLC conditions, and due to a significant difference in retention times found the 2D model is strongly advised.

Testing experimental designs in liquid chromatography (II): Influence of the design geometry on the prediction performance of retention models

In liquid chromatography, the reliability of predictions carried out with retention models depends critically on the quality of the training experimental design. The search of the best design is more complex when gradient runs are used instead of isocratic experiments. In Part I of this work (JCA 1624 (2020) 461180), a general methodology based on the error propagation theory was developed and validated for assessing the quality of training designs involving gradients. The treatment relates the mathematical properties of a retention model with the geometry of the training designs and their subsequent predictions. In that work, only five usual designs were considered. Part II investigates in detail the effects on predictions when the features of the training design (number and distribution of the experiments, initial and final modifier content, gradient slope(s), and location of gradient nodes and pulses) are varied systematically. Several groups of related designs containing one or more isocratic steps, linear or multi-linear gradients, or mixed isocratic/gradient runs, among others (in total 38 designs) were evaluated. Box and whiskers and triple plots of expected relative uncertainties were used to evidence the differences in prediction performance. The purpose was to give recommendations to construct designs with good prediction performance. The best designs sample (considering all runs) concentrations as diverse as possible, at any gradient time.

Rebirth of recycling liquid chromatography with modern chromatographic columns : Extension to gradient elution

Twin column recycling semi-preparative liquid chromatography (TCRLC) is revived to prepare small amount (∼ 1 mg) of a pure targeted compound, which cannot be isolated by conventional preparative liquid chromatography. In this work, TCRLC is extended to gradient elution. The first step of this modified process consists of a gradient step, which eliminates both early and late impurities. If not discarded, some late impurities could echo during the second isocratic recycling step of the process and compromise the purity level required for the targeted compound. Additionally, the entire gradient TCRLC (GTCRLC) process is automated regarding the eluent composition programmed and the actuation times of two valves: one two-position four-port divert valve enables to shave the targeted compound from early and late impurities during the initial gradient step. The second two-position six-port recycling valve ensures the complete baseline resolution between the band of the targeted compound and those of the closest impurities, which are not fully eliminated after the initial gradient step. The automation of the whole GTCRLC process is achieved by running four preliminary scouting gradient runs (at four different relative gradient times, t_gt₀= 2, 6, 18, and 54, where t₀ is the hold-up column time) for the accurate determination of the thermodynamics (lnk versus φ plots of the retention factor as a function of the mobile phase composition) of the first impurity, the targeted compound(s), and of the last impurity. The automated GTCRLC process was successfully applied for the isolation of a polycyclic aromatic hydrocarbon (PAH), chrysene, from a complex mixture of PAHs containing two nearly co-eluting impurities (benzo[a]anthracene and triphenylene) and nine other early/late impurities (sample volume injected: 1 mL, 7.8 mm × 150 mm Sunfire-C₁₈ column, acetonitrile/water eluent mixtures, T= 55 ^∘C, 20 cycles, baseline separation in less than two hours). Additionally, the GTCRLC process is advantageously used to isolate and baseline separate the vitamins D₂ and D₃ initially present in a milk extract mixture (0.3 mL sample injection volume, 7.8 mm × 150 mm Sunfire-C₁₈ column, methanol/water eluent mixtures, T= 65 ^∘C, 14 cycles needed in 1.5 hours). These results open promising avenues toward an effective preparation of unknown targeted compounds before further physico-chemical characterization and unambiguous identification.

[Effects of peak compression in gradient elution of liquid chromatography]

谱带压缩效应是梯度洗脱区别于等度洗脱的重要特征。经典的范德姆特(van Deemter)理论塔板高度方程基于等度洗脱推导得到,因此不能对谱带压缩效应进行描述。在梯度洗脱中,保留因子(k)会随流动相组成(φ)的改变而发生变化,这就使得对梯度洗脱机理的研究要比等度洗脱复杂许多。该文对近10年来谱带压缩效应的研究进展,特别是溶剂强度模型(即描述ln k与φ关系的数学表达式)的非线性特征对谱带压缩因子(G)的影响进行了述评,指出为了更好地认识谱带压缩效应需要将这种非线性因素考虑在内。

Closed form approximations to predict retention times and peak widths in gradient elution under conditions of sample volume overload and sample solvent mismatch

Closed form expressions for the prediction of retention times and peak widths for gradient liquid chromatography are particularly useful in understanding, rationalizing and optimizing separations. These expressions are obtained by integrating differential equations, in conjunction with a model of the variation of the retention factor as a function of mobile phase composition. Two of these models, the linear solvent strength (LSS) model and the Neue-Kuss (NK) model are explored in the present work. Here, we expand on these closed form expressions to account for effects of sample volume overload and a mismatch between the sample solvent and the initial mobile phase composition for the gradient. We show that there have been errors in expressions reported in the literature, and we have evaluated the accuracy of the predictions from the closed form expressions reported here using a recently developed liquid chromatography simulator. The expressions assume a constant plate height and consider elution across four zones of the gradient profile - elution in the sample solvent, elution in the initial (isocratic) mobile phase caused by the gradient delay volume, elution during a linear gradient, and elution post-gradient at the final (isocratic) mobile phase composition. The expressions generally give reasonably accurate predictions for retention times and peak widths, except for cases where the solute elutes during transitions between the different zones. The average magnitude of the prediction errors for retention time and peak width relative to simulation were 0.093% and 0.40% for the LSS expressions for ten amphetamine solutes at 36 different separation conditions, and 0.22% and 1.8% for the NK expressions for eight alkylbenzene solutes at 36 different separation conditions, respectively.

Perspective on the Future Approaches to Predict Retention in Liquid Chromatography

The demand for rapid column screening, computer-assisted method development and method transfer, and unambiguous compound identification by LC/MS analyses has pushed analysts to adopt experimental protocols and software for the accurate prediction of the retention time in liquid chromatography (LC). This Perspective discusses the classical approaches used to predict retention times in LC over the last three decades and proposes future requirements to increase their accuracy. First, inverse methods for retention prediction are essentially applied during screening and gradient method optimization: a minimum number of experiments or design of experiments (DoE) is run to train and calibrate a model (either purely statistical or based on the principles and fundamentals of liquid chromatography) by a mere fitting process. They do not require the accurate knowledge of the true column hold-up volume V₀, system dwell volume V_dwell (in gradient elution), and the retention behavior (k versus the content of strong solvent φ, temperature T, pH, and ionic strength I) of the analytes. Their relative accuracy is often excellent below a few percent. Statistical methods are expected to be the most attractive to handle very complex retention behavior such as in mixed-mode chromatography (MMC). Fundamentally correct retention models accounting for the simultaneous impact of φ, I, pH, and T in MMC are needed for method development based on chromatography principles. Second, direct methods for retention prediction are ideally suited for accurate method transfer from one column/system configuration to another: these quality by design (QbD) methods are based on the fundamentals and principles of solid-liquid adsorption and gradient chromatography. No model calibration is necessary; however, they require universal conventions for the accurate determination of true retention factors (for 1 < k < 30) as a function of the experimental variables (φ, T, pH, and I) and of the true column/system parameters (V₀, V_dwell, dispersion volume, σ, and relaxation volume, τ, of the programmed gradient profile at the column inlet and gradient distortion at the column outlet). Finally, when the molecular structure of the analytes is either known or assumed, retention prediction has essentially been made on the basis of statistical approaches such as the linear solvation energy relationships (LSERs) and the quantitative structure retention relationships (QSRRs): their ability to accurately predict the retention remains limited within 10-30%. They have been combined with molecular similarity approaches (where the retention model is calibrated with compounds having structures similar to that of the targeted analytes) and artificial intelligence algorithms to further improve their accuracy below 10%. In this Perspective, it is proposed to adopt a more rigorous and fundamental approach by considering the very details of the solid-liquid adsorption process: Monte Carlo (MC) or molecular dynamics (MD) simulations are promising tools to explain and interpret retention data that are too complex to be described by either empirical or statistical retention models.

The development of retention time prediction model using multilinear gradient profiles of seven pharmaceuticals

The ICH guidance on pharmaceutical development recommends a systematic development approach including robustness studies which assure performance of manufacturing and analytical method development of drug product. The retention model by T. Kawabe et al have an excellent correlation between observed and predicted retention time in various kinds of pharmaceutical compounds during isocratic elution by the multiple regression modeling of solvent strength parameters. However, it cannot be successfully applied to the predictability of the retention time during multilinear gradient elution and also it does not consider the instrument dependent parameters such as dwell volume. The current study demonstrated that the solution of the fundamental gradient elution equation was applied to T. Kawabe's retention time prediction model to predict the retention time using a multilinear gradient profile with taking the delay volume of HPLC system into account. Seven pharmaceutical compounds were used for evaluation of prediction models for retention time. The predicted retention time was compared with the measured retention time obtained by several multilinear gradient using two HPLC systems with different dwell volume. The evaluated prediction error (%) was 1.10 % and 1.54 % with H-Class and Nexera XR HPLC systems, respectively. In order to evaluate the robustness of the analytical method and to set the system suitability test (SST) for proper method performance, the design space for the ACN/MeOH mixture ratio in the total organic solvent and the full width at half maximum (FWHM) relationship to the minimum resolution was simulated by the developed retention time prediction. The optimized condition of the ACN/MeOH mixture ratio, the acceptance criterion of the SST for achieving the robust separation was estimated based on the simulated design space. As a conclusion, the developed retention time prediction will be useful during analytical method transfer among different manufacturing/analytical sites of the pharmaceuticals with different HPLC systems.

Global retention models and their application to the prediction of chromatographic fingerprints

The resolution of samples containing unknown compounds of different nature, or without standards available, as is the case of chromatographic fingerprints, is still a challenge. Possibly, the most problematic aspect that prevents systematic method development is finding models that describe without bias the retention behaviour of the compounds in the samples. In this work, the use of global models (able to describe the whole sample) is proposed as an alternative to the use of individual models for each solute. Global models contain parameters that are specific for each solute, while other parameters ‒related to the column and solvent‒ are common for all solutes. A special regression procedure is presented for the construction of global models, which are applied to predict highly complex chromatograms, such as chromatographic fingerprints, for diverse experimental conditions in isocratic and gradient elution. Another interesting application is the prediction of molecular properties, such as log P_o/w, from the specific solute parameters of the global models. The examined adapted models are based on the equations proposed by Snyder, Schoenmakers, Neue and Kuss, Jandera, and Bosch Rosés to describe the retention. In all cases, the predictive capability was very satisfactory. Two cases of study were considered: chromatograms of camomile extracts analysed using acetonitrile gradients, and a set of 145 known compounds in a wide range of structures and functionalities, eluted isocratically with acetonitrile/water mobile phases.

Measuring and using scanning-gradient data for use in method optimization for liquid chromatography

The use of scanning gradients can significantly reduce method-development time in reversed-phase liquid chromatography. However, there is no consensus on how they can best be used. In the present work we set out to systematically investigate various factors and to formulate guidelines. Scanning gradients are used to establish retention models for individual analytes. Different retention models were compared by computing the Akaike information criterion and the prediction accuracy. The measurement uncertainty was found to influence the optimum choice of model. The use of a third parameter to account for non-linear relationships was consistently found not to be statistically significant. The duration (slope) of the scanning gradients was not found to influence the accuracy of prediction. The prediction error may be reduced by repeating scanning experiments or - preferably - by reducing the measurement uncertainty. It is commonly assumed that the gradient-slope factor, i.e. the ratio between slopes of the fastest and the slowest scanning gradients, should be at least three. However, in the present work we found this factor less important than the proximity of the slope of the predicted gradient to that of the scanning gradients. Also, interpolation to a slope between that of the fastest and the slowest scanning gradient is preferable to extrapolation. For comprehensive two-dimensional liquid chromatography (LC × LC) our results suggest that data obtained from fast second-dimension gradients cannot be used to predict retention in much slower first-dimension gradients.

Recent applications of retention modelling in liquid chromatography

Recent applications of retention modelling in liquid chromatography (2015–2020) are comprehensively reviewed. The fundamentals of the field, which date back much longer, are summarized. Retention modeling is used in retention‐mechanism studies, for determining physical parameters, such as lipophilicity, and for various more‐practical purposes, including method development and optimization, method transfer, and stationary‐phase characterization and comparison. The review focusses on the effects of mobile‐phase composition on retention, but other variables and novel models to describe their effects are also considered. The five most‐common models are addressed in detail, i.e. the log‐linear (linear‐solvent‐strength) model, the quadratic model, the log–log (adsorption) model, the mixed‐mode model, and the Neue–Kuss model. Isocratic and gradient‐elution methods are considered for determining model parameters and the evaluation and validation of fitted models is discussed. Strategies in which retention models are applied for developing and optimizing one‐ and two‐dimensional liquid chromatographic separations are discussed. The review culminates in some overall conclusions and several concrete recommendations.

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.

Insight into the hydrophilic interaction liquid chromatographic retention behaviors of hydrophilic compounds on different stationary phases

In this work, the correlations between retention behavior and lipophilicity of a large set of hydrophilic neutral and ionic analytes were studied based on three hydrophilic interaction liquid chromatography (HILIC) stationary phases, including zwitterionic, crosslinked diol and triazole stationary phases. It was found that HILIC, due to the diversity of retention mechanism, is a more complex chromatography separation mode than reversed-phase liquid chromatography (RPLC) which has been widely accepted for lipophilicity assessment. Because electrostatic interactions contributed to the overall retention of the charged solutes on all three stationary phases, ion-strength of the mobile phase kept the same during the whole experiment. After the correlations between retention factor log k and water volume fraction Φ were investigated, the mixed retention model was revealed to be more suitable for HILIC retention behavior than other single models including partitioning and adsorption model. Moreover, in order to bridge the relationship between HILIC log k and lipophilicity parameter log D, net charge n_e and Abraham solvation parameter were introduced in the quantitative structure-retention relationship (QSRR) model. Although the correlation coefficients between log D and log k were still moderate, the significant improvement in correlation has made HILIC a potential choice as the complement of RPLC for log D measurement.