Multilinear Gradient Research Articles

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

BackgroundEnhancing 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. ResultsThis 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. SignificanceThe 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.

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.

Chromatographic fingerprint-based analysis of extracts of green tea, lemon balm and linden: II. Simulation of chromatograms using global models

Medicinal plants contain a large variety of chemical compounds in highly variable concentrations, so the quality control of these materials is especially complex. With this purpose, regulatory institutions have accepted chromatographic fingerprints as a valid tool to perform the analyses. In order to improve the results, separation conditions that maximise the number of detected peaks in these chromatograms are needed. This work reports the extension of a simulation strategy, based on global retention models previously developed for selected compounds, to all detected peaks in the full chromatogram. Global models contain characteristic parameters for each component in the sample, while other parameters are common to all components and describe the combined effects of column and solvent. The approach begins by detecting and measuring automatically the position of all peaks in a chromatogram, obtained preferably with the slowest gradient. Then, the retention time for each detected component is fitted to find the corresponding solute parameter in the global model, which leads to the best agreement with the measured experimental value. The process is completed by developing bandwidth models for the selected compounds used to build the global retention model based on gradient data, which are applied to all peaks in the chromatogram. The usefulness of the simulation approach is demonstrated by predicting chromatographic fingerprints for three medicinal plants with specific separation problems (green tea, lemon balm and linden), using several multi-linear gradients that lead to problematic predictions.

Chromatographic fingerprint‐based analysis of extracts of green tea, lemon balm and linden: I. Development of global retention models without the use of standards

We report here the improvement of a procedure to obtain global models, able to describe the retention behaviour of several sample components simultaneously. The reported global models include parameters that account for the general effects of column and solvent on retention and are common for all components, whereas other parameters are specific of each sample component. These models are fitted by alternate regression and offer a prediction performance comparable to individual retention models. The approach is suitable to samples of natural products including a large number of components in extremely diverse concentrations and in the absence of standards. Guidelines are given for the successful development of sample-oriented experimental designs (i.e. adapted to the elution of the components of the natural products), constituted by multi-linear gradients. These designs also facilitate peak tracking. The model proposed by Neue and Kuss to describe the retention was found to yield the best predictions. The approach is applied to the extracts of samples of green tea, lemon balm and linden, yielding excellent predictions of retention for selected components.

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.

Modelling of multilinear gradient retention time of bio-sweetener rebaudioside A in HPLC analysis

Artificial neural network (ANN), as one of the artificial intelligence methods, has been widely using in HPLC studies for modelling purposes. Stevia rebaudiana is an important industrial plant due to its bio-sweetener molecule, rebaudioside-a, in its leaves. Although rebaudioside-a is up to 300-fold sweeter than sucrose, its calorie is almost zero. In this study, HPLC optimization of rebaudioside-a was studied and the optimization data based on multilinear gradient retention times were modelled by ANN. The input parameters were selected as concentrations, column temperatures, initial acetonitrile percentage for the first step of gradient elution, initial acetonitrile percentage for the second step of gradient elution, slope of acetonitrile, wavelengths, flow rates. The retention time was the output. Also, dried S. rebaudiana leaves were extracted and the concentrations were evaluated by HPLC. According to the ANN results, the most effective parameters on the prediction of non-linear gradient retention time for rebaudioside-a were found as flow rate and initial acetonitrile percentage for the second step of gradient. The best back propagation was selected as Levenberg-Marquardt algorithm. The highest rebaudioside-a level was found as 96.53 ± 6.36 μg mL-1. ANN modelling methods can be used in preparative HPLC applications to estimate the retention time of steviol glycosides.

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.

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.

Interpretive search of optimal isocratic and gradient separations in micellar liquid chromatography in extended organic solvent domains

Micellar liquid chromatography (MLC) is a reversed-phase mode with mobile phases containing an organic solvent and a micellised surfactant. Most procedures developed in MLC are implemented in the isocratic mode, since the general elution problem in chromatography is less troublesome. However, gradient elution may be still useful in MLC to analyse mixtures of compounds within a wide range of polarities, in shorter times. MLC using gradients is attractive to determine by direct injection moderate to low polar compounds in physiological samples. In these analyses, the use of initial micellar conditions (isocratic or gradient) with a fixed amount of surfactant above the critical micellar concentration, keeping the organic solvent content low, will provide better protection of the column against the precipitation of the proteins in the physiological fluid. Once the proteins are swept away, the elution strength can be increased using a positive gradient of organic solvent to reduce the analysis time. This may give rise to the transition from the micellar to the submicellar mode, since micelles are destroyed at sufficiently high concentration of organic solvent. In this work, several retention models covering extended solvent domains in MLC are developed and tested, and applied to investigate the performance in isocratic, linear and multi-linear gradient separations. The study was applied to the screening of β-adrenoceptor antagonists in urine samples, using mobile phases prepared with sodium dodecyl sulphate and 1-propanol. Predicted chromatograms were highly accurate in all situations, although suffered of baseline problems and minor shifts for peaks eluting close to a steep gradient segment. Two columns (C18 and C8) were investigated, with the C8 column being preferable owing to the smaller amount of adsorbed surfactant.

Multi-scale optimisation vs. genetic algorithms in the gradient separation of diuretics by reversed-phase liquid chromatography

Multi-linear gradients are a convenient solution to get separation of complex samples by modulating carefully the gradient slope, in order to accomplish the local selectivity needs for each particular solute cluster. These gradients can be designed by trial-and-error according to the chromatographer experience, but this strategy becomes quickly inappropriate for complex separations. More evolved solutions imply the sequential construction of multi-segmented gradients. However, this strategy discards part of the search space in each step of the construction and, again, cannot deal properly with very complex samples. When the complexity is too large, the only valid alternative for finding the best gradient is the use of global search methods, such as genetic algorithms (GAs). Recently, a new global approach where the level of detail is increased along the search has been proposed, namely Multi-scale optimisation (MSO). In this strategy, cubic splines are applied to build intermediate curves to define any arbitrary solvent variation function. Subdivision schemes are used to generate the cubic splines and control their level of detail. The search was subjected to a number of restrictions, such as avoiding long elution and favouring a balanced peak distribution. The aim of this work is evaluating and comparing the results of GAs and MSO. Both approaches were tested with a set of 14 diuretics and probenecid, eluted with acetonitrile-water mixtures using a C18 column. Satisfactory baseline resolution was obtained with an analysis time of 15–16 min. We found that GAs optimisation offered results equivalent to those provided by MSO, when the penalisation parameters were included in the cost function.

Enhancement in the computation of gradient retention times in liquid chromatography using root-finding methods

Gradient elution may provide adequate separations within acceptably short times in a single run, by gradually increasing the elution speed. Similarly to isocratic elution, chromatograms can be predicted under any experimental condition, through strategies based on retention models. The most usual approach implies solving an integral equation (i.e., the fundamental equation of gradient elution), which has an analytical solution only for certain combinations of retention model and gradient programme. This limitation can be overcome by using numerical integration, which is a universal approach although at the cost of longer computation times. In this work, several alternatives to improve the performance in the resolution of the integral equation are explored, which can be especially useful with multi-linear gradients. For this purpose, the application of several root-finding methods that include the Newton's and bisection searches is explored in three frameworks: isolated predictions, regression modelling problems using gradient training sets, and optimisation of multi-linear gradients. Significant reductions of computation times were obtained. The substitution of non-integrable retention models by Tchebyshev polynomial approximations, which are pre-calculated before solving the integral equation in optimisation problems, is also investigated.

Estimation of peak capacity based on peak simulation

Peak capacity (PC) is a key concept in chromatographic analysis, nowadays of great importance for characterising complex separations as a criterion to find the most promising conditions. A theoretical expression for PC estimation can be easily deduced in isocratic elution, provided that the column plate count is assumed constant for all analytes. In gradient elution, the complex dependence of peak width with the gradient program implies that an integral equation has to be solved, which is only possible in a limited number of situations. In 2005, Uwe Neue developed a comprehensive theory for the calculation of PC in gradient elution, which is only valid for certain situations: single linear gradients, absence of delays and extra-column effects, Gaussian peaks and constant plate count. Going beyond these limitations implies resolving algebraic expressions that unfortunately cannot be integrated. In this work, PC is predicted for multiple situations based on peak simulation. The approach is more general and can be applied for situations out of the scope of the Neue outline, such as complex multi-linear gradients, including asymmetrical peaks. The plots of PC versus retention time of the last eluted solute give rise to Pareto fronts, and can be useful for the probabilistic enhancement of peak resolution in situations where complex multi-analyte samples are processed.

Gradient design for liquid chromatography using multi-scale optimization

In reversed phase-liquid chromatography, the usual solution to the “general elution problem” is the application of gradient elution with programmed changes of organic solvent (or other properties). A correct quantification of chromatographic peaks in liquid chromatography requires well resolved signals in a proper analysis time. When the complexity of the sample is high, the gradient program should be accommodated to the local resolution needs of each analyte. This makes the optimization of such situations rather troublesome, since enhancing the resolution for a given analyte may imply a collateral worsening of the resolution of other analytes. The aim of this work is to design multi-linear gradients that maximize the resolution, while fulfilling some restrictions: all peaks should be eluted before a given maximal time, the gradient should be flat or increasing, and sudden changes close to eluting peaks are penalized. Consequently, an equilibrated baseline resolution for all compounds is sought. This goal is achieved by splitting the optimization problem in a multi-scale framework. In each scale κ, an optimization problem is solved with Nκ ≈ 2κ variables that are used to build the gradients. The Nκ variables define cubic splines written in terms of a B-spline basis. This allows expressing gradients as polygonals of M points approximating the splines. The cubic splines are built using subdivision schemes, a technique of fast generation of smooth curves, compatible with the multi-scale framework. Owing to the nature of the problem and the presence of multiple local maxima, the algorithm used in the optimization problem of each scale κ should be “global”, such as the pattern-search algorithm. The multi-scale optimization approach is successfully applied to find the best multi-linear gradient for resolving a mixture of amino acid derivatives.

Simultaneous HPLC Quantitative Analysis of Nine Bioactive Constituents in Scirpus Yagara Ohwi. (Cyperaceae).

The tuber of Scirpus yagara Ohwi. (Cyperaceae) has long been used in traditional Chinese medicine (TCM). Several chemical constituents isolated from it possess a variety of physiologically activities such as anti-inflammatory, antitumor and antioxidant. A simultaneous high-performance liquid chromatography (HPLC) analysis was developed and validated for the determination of nine active components in tubers and aerial parts of S. yagara. The analysis was performed on a YMC-Pack ODS-A column (4.6 × 250 mm, 5 μm, 30 nm) with a multilinear gradient mobile phase of water-formic acid (100 : 0.2, v/v) and methanol. The established HPLC method was validated in terms of linearity, sensitivity, precision, accuracy, recovery and stability. All analyzed components were detected in the whole tested samples, and the contents of most components in the aerial parts were even higher than those in the tubers. Moreover, the best harvest period was discovered to be November, which is different from the traditional. The method developed was successfully applied for simultaneous qualitative and quantitative analysis of nine active components in S. yagara.

Retention prediction and separation optimization under multilinear gradient elution in liquid chromatography with Microsoft Excel macros

A package of Excel VBA macros have been developed for modeling multilinear gradient retention data obtained in single or double gradient elution mode by changing organic modifier(s) content and/or eluent pH. For this purpose, ten chromatographic models were used and four methods were adopted for their application. The methods were based on (a) the analytical expression of the retention time, provided that this expression is available, (b) the retention times estimated using the Nikitas-Pappa approach, (c) the stepwise approximation, and (d) a simple numerical approximation involving the trapezoid rule for integration of the fundamental equation for gradient elution. For all these methods, Excel VBA macros have been written and implemented using two different platforms; the fitting and the optimization platform. The fitting platform calculates not only the adjustable parameters of the chromatographic models, but also the significance of these parameters and furthermore predicts the analyte elution times. The optimization platform determines the gradient conditions that lead to the optimum separation of a mixture of analytes by using the Solver evolutionary mode, provided that proper constraints are set in order to obtain the optimum gradient profile in the minimum gradient time. The performance of the two platforms was tested using experimental and artificial data. It was found that using the proposed spreadsheets, fitting, prediction, and optimization can be performed easily and effectively under all conditions. Overall, the best performance is exhibited by the analytical and Nikitas-Pappa's methods, although the former cannot be used under all circumstances.

Optimisation of chromatographic resolution using objective functions including both time and spectral information

The optimisation of the resolution in high-performance liquid chromatography is traditionally performed attending only to the time information. However, even in the optimal conditions, some peak pairs may remain unresolved. Such incomplete resolution can be still accomplished by deconvolution, which can be carried out with more guarantees of success by including spectral information. In this work, two-way chromatographic objective functions (COFs) that incorporate both time and spectral information were tested, based on the peak purity (analyte peak fraction free of overlapping) and the multivariate selectivity (figure of merit derived from the net analyte signal) concepts. These COFs are sensitive to situations where the components that coelute in a mixture show some spectral differences. Therefore, they are useful to find out experimental conditions where the spectrochromatograms can be recovered by deconvolution. Two-way multivariate selectivity yielded the best performance and was applied to the separation using diode-array detection of a mixture of 25 phenolic compounds, which remained unresolved in the chromatographic order using linear and multi-linear gradients of acetonitrile–water. Peak deconvolution was carried out using the combination of orthogonal projection approach and alternating least squares.

Artificial neural network prediction of multilinear gradient retention in reversed-phase HPLC: comprehensive QSRR-based models combining categorical or structural solute descriptors and gradient profile parameters.

A multilayer artificial neural network (ANN) is used to model the reversed-phase liquid chromatography retention times of 16 selected compounds, including purines, pyrimidines and nucleosides. The analysed data, taken from literature, were collected in acetonitrile-water eluents under the application of 16 different multilinear gradients. The parameters describing the gradient profile together with solute descriptors are considered as the independent variables of an ANN-based model providing the retention time as response. Categorical variables or, alternatively, a selected set of molecular descriptors of computational origin are adopted to represent the solutes. Network training, validation and testing are performed preliminarily using data of 12, 2 and 4 gradients, respectively and successively, to investigate model performance under more severe calibration conditions, with data of 9, 2 and 7 gradients. The proposed approach allows a quite accurate prediction of retention times of the target analytes in external multilinear gradients. Categorical variables can successfully represent the target solutes when the model is called to transfer retention data from calibration to external gradients. In particular, using a five-dimensional bit string to represent the analytes, mean errors on retention times are 2 and 3% under the most and less favourable calibration conditions, respectively. A comparable performance is observed if the categorical variables are replaced by five molecular descriptors, selected by a genetic algorithm within a large set of structural variables of computational origin.

Optimisation of gradient elution with serially-coupled columns Part II: Multi-linear gradients

An interpretive approach is reported for the simultaneous optimisation of resolution and analysis time in the separation of complex samples by reversed-phase liquid chromatography, using serially-coupled columns of different lengths and nature, and multi-linear gradient elution. The results complement those reported in JCA 1350 (2014) 51 for single linear gradients, using combinations of five columns: three C18 columns of different characteristics, a cyano column and a phenyl column. The relationships between the logarithm of the retention factor (lnk) and the organic solvent content (φ), and between the peak half-widths and the retention times, for each analyte, were determined using isocratic retention times. From the established correlations, the gradient retention time was predicted using the fundamental equation for gradient elution (modified to be used for serially-coupled columns), and with this information, the peak profile. Two chromatographic goals (maximal peak purity and minimal analysis time) were simultaneously evaluated using the Pareto optimality concept. The rigorous predictive system can be easily applied to columns from different manufacturers and succeeds at different levels: single and combined columns, using isocratic and gradient elution. The computation time for gradient elution optimisation was significantly reduced to practical values with the help of genetic algorithms (GAs).

Enhanced selectivity and search speed for method development using one-segment-per-component optimization strategies

Linear gradient programs are very frequently used in reversed phase liquid chromatography to enhance the selectivity compared to isocratic separations. Multi-linear gradient programs on the other hand are only scarcely used, despite their intrinsically larger separation power. Because the gradient-conformity of the latest generation of instruments has greatly improved, a renewed interest in more complex multi-segment gradient liquid chromatography can be expected in the future, raising the need for better performing gradient design algorithms. We explored the possibilities of a new type of multi-segment gradient optimization algorithm, the so-called “one-segment-per-group-of-components” optimization strategy. In this gradient design strategy, the slope is adjusted after the elution of each individual component of the sample, letting the retention properties of the different analytes auto-guide the course of the gradient profile. Applying this method experimentally to four randomly selected test samples, the separation time could on average be reduced with about 40% compared to the best single linear gradient. Moreover, the newly proposed approach performed equally well or better than the multi-segment optimization mode of a commercial software package. Carrying out an extensive in silico study, the experimentally observed advantage could also be generalized over a statistically significant amount of different 10 and 20 component samples. In addition, the newly proposed gradient optimization approach enables much faster searches than the traditional multi-step gradient design methods.

Analytical separation of enantiomers by gas chromatography on chiral stationary phases.

Analytical separation of enantiomers by gas chromatography on chiral stationary phases.