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 ß-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.

Aqueous liquid chromatography with mobile phases of sodium dodecyl sulphate and ionic liquid.

In conventional reversed-phase liquid chromatography (RPLC) with hydro-organic solvents, basic cationic solutes yield retained, broad, asymmetric peaks, owing to their interaction with free anionic silanols in the stationary phase. RPLC mobile phases to which the anionic surfactant sodium dodecyl sulphate (SDS), or an ionic liquid (IL) are added, have been proposed as solutions, since these additives are able to block the silanol effect thus improving the chromatographic performance. With these additives, it is however necessary to increase the elution strength by adding an organic solvent, such as an alcohol or acetonitrile. A novel aqueous liquid chromatographic mode (in the absence of organic solvent) is here proposed, where the mobile phases contain only a mixture of aqueous solutions of SDS and an IL derived from 1-alkyl-3-methylimidazolium associated to chloride, both environmentally friendly. When these reagents are added, the anionic surfactant adsorbed on the stationary phase is able to attract the cationic solutes, whereas the adsorbed IL cation repels them. The combination of both effects (attraction and repulsion) allows the modulation of retention, by varying the IL/SDS ratio. Given the character of the additives, a type of green liquid chromatography is achieved. In this work, the chromatographic behavior of six basic compounds of pharmaceutical interest, the ß-adrenoceptor antagonists acebutolol, atenolol, carteolol, metroprolol, oxprenolol and propranolol, is examined. In order to assess the chromatographic behavior of the mixed mobile phases containing SDS and IL, changes in retention, peak profile and resolution of mixtures of the analytes were explored at varying concentration of the additives.

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.

Interactions of basic compounds with ionic liquids used as oils in microemulsion liquid chromatography.

Aqueous microemulsions (MEs), where an oil coexists with water in the presence of the anionic surfactant sodium dodecyl sulphate (SDS), have been proposed as a solution to decrease the amount of organic solvent in the mobile phase needed in reversed-phase liquid chromatography (RPLC). However, the oil phase of a typical ME is volatile, toxic and flammable, and although it is added in a small amount, it would be desirable to avoid it from an environmental perspective. This is the reason for the proposal of Peng et al. (J. Chromatogr. A 1499 (2017) 132‒139) to replace the oil in microemulsion liquid chromatography (MELC) by the apolar ionic liquid (IL) 1-hexyl-3-methylimidazolium hexafluorophosphate ([C6C1IM][PF6]), to analyse neutral phenolic acids at acidic pH. Based on this report, an MELC procedure is here proposed for ß-adrenoceptor antagonists, which are basic compounds where the dominant species is cationic. To verify the formation of MEs containing SDS and IL, and elucidate the interactions between the cationic basic compounds with the SDS anion, and the cation and anion in the IL, an extensive study was carried out with several methylimidazolium ILs containing the cations [C2C1IM]+, [C4C1IM]+, or [C6C1IM]+, combined with the anions Cl-, BF4-, or PF6-, using 1-butanol as co-surfactant. The behaviour was compared with that observed in classical MELC with octane, micellar liquid chromatography with SDS and 1-propanol, and RPLC with mobile phases containing an IL and acetonitrile.

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.

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 Po/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.

Performance and modelling of retention in microemulsion liquid chromatography.

The capability of liquid chromatography with microemulsions (MEs) as mobile phases was studied for the analysis of four parabens (butylparaben, ethylparaben, methylparaben, and propylparaben) and seven ß-adrenoceptor antagonists (acebutolol, atenolol, carteolol, metoprolol, oxprenolol, propranolol, and timolol). MEs were formed by mixing aqueous solutions of the anionic surfactant sodium dodecyl sulphate, the alcohol 1-butanol that played the role of co-surfactant, and octane as oil. In order to guarantee the formation of stable MEs, a preliminary study was carried out to determine the appropriate ranges of concentrations of the three components. For this purpose, mixtures of variable composition were prepared, and the possible separation of two phases (formation of an emulsion) was visually detected. The advantage offered by the addition of octane to micellar mobile phases, inside the concentration range that allows the formation of stable MEs, was evaluated by comparing the retention behaviour, peak profile and resolution of mixtures of the probe compounds, in the presence and absence of octane. The final aim of this work was the proposal of a mathematical equation to model the retention behaviour in microemulsion liquid chromatography. The derived global model that considered the three factors (surfactant, alcohol and oil) allowed the prediction of retention times at diverse mobile phase compositions with satisfactory accuracy (in the 1.1‒2.5% range). The behaviour was compared with that found with mobile phases without octane. The model also yielded information about the retention mechanism and revealed that octane, when inserted inside the micelle, modifies the interaction between solutes and micelles.

Comparison of surfactant-mediated liquid chromatographic modes with sodium dodecyl sulphate for the analysis of basic drugs.

In reversed-phase liquid chromatography (RPLC), basic drugs are positively charged at the usual working pH range and interact with free anionic silanols present in conventional silica-based stationary phases. This translates into stronger retention and tailed and broadened peaks. This problem can be resolved by the addition of reagents to the mobile phase that are adsorbed on the stationary phase, avoiding the access of solutes to silanols. Among these additives, surfactants under micellar conditions have provided good silanol suppressing potency through the technique known as micellar liquid chromatography (MLC). The most common example of this is anionic sodium dodecyl sulphate (SDS). When SDS is at moderate concentration in the presence of high organic solvent content, micelles are not formed and the chromatographic mode is known as high submicellar liquid chromatography (HSLC). In contrast, the addition of an oil to an aqueous solution of SDS containing micelles gives rise to microemulsions in a chromatographic mode known as microemulsion liquid chromatography (MELC). A comprehensive comparison of the chromatographic behaviour of a set of basic ß-adrenoceptor antagonists analysed by MLC, HSLC and MELC is carried out in this work, in terms of retention, peak shape and organic solvent consumption. The study shows that high submicellar eluents reduce retention and enhance efficiency with respect to conventional RPLC and MLC. Meanwhile, MELC allows reduced analysis times with less organic solvent with respect to HSLC. The narrower and more symmetrical peaks in MLC, HSLC and MELC, with respect to conventional RPLC, reveal the presence of silanol masking.

Modified Gaussian models applied to the description and deconvolution of peaks in chiral liquid chromatography.

The description of the profiles of chromatographic peaks has been studied extensively, with a large number of proposed mathematical functions. Among them, the accuracy achieved with modified Gaussian models that describe the deviation of an ideal Gaussian peak as a change in the peak variance or standard deviation over time, has been highlighted. These models are, in fact, a family of functions of different complexity with great flexibility to adjust chromatographic peaks over a wide range of asymmetries and shapes. However, an uncontrolled behaviour of the signal may occur outside the region being fitted, forcing the use of different strategies to overcome this problem. In this work, the performance of the LMG (Linear Modified Gaussian), PVMG (Parabolic Variance Modified Gaussian), and PLMG (Parabolic-Lorentzian Modified Gaussian) models is compared with variants obtained by combination of the modified Gaussian models with an equation that adds an exponential tail and with other functions that limit the growth of the independent variable. The behaviour of the approaches is checked through the simultaneous fitting of enantiomeric peaks showing a wide range of characteristics, obtained in the separation of drugs with chiral activity by liquid chromatography using enantioselective columns. The study is also carried out with the purpose of performing the deconvolution of the peaks of the enantiomers, when these are not completely resolved, in order to evaluate the enantiomeric fraction.

Testing experimental designs in liquid chromatography (I): Development and validation of a method for the comprehensive inspection of experimental designs.

The basis of interpretive optimisation in liquid chromatography is the prediction of resolution, from appropriate solute retention models. The reliability of the process depends critically on the quality of the experimental design. This work develops, validates and applies a general methodology aimed to evaluate the quality of any training experimental design, which will be applied in Part II to design optimisation. The methodology is based on the systematic evaluation of the uncertainties associated to the prediction of retention times in comprehensive scans of both isocratic and gradient experimental conditions. It is able to evaluate comprehensively experimental designs of arbitrary complexity. Five common training experimental designs were used to model the retention, according to the Linear Solvent Strength (LSS) and the Neue-Kuss (NK) equations, using a set of 14 sulphonamides of different polarity. The results are presented in terms of relative uncertainties in predictions, which provide significant and interpretable results. The magnitude of such uncertainties, together with the systematic, coherent and logical changes observed at increasing solute hydrophobicity, give support to the results. The NK model gave smaller errors and unbiased predictions, whereas the LSS model gave rise to lack of fit. Isocratic training designs, which are widely accepted as the most informative, are confirmed as the best. As a general conclusion, gradients are predicted with intrinsically smaller uncertainties, independently of the training experimental design. In addition, gradients are more insensitive than isocratic predictions with regard to the type of training design used. Isocratic predictions deteriorate quickly with mobile phase composition. This explains the better performance of gradient predictions, even with biased models.

Classification of olive leaves and pulp extracts by comprehensive two-dimensional liquid chromatography of polyphenolic fingerprints.

The development of a new comprehensive two-dimensional liquid chromatographic method is described, to obtain the profiles of polyphenolic compounds present in olive (Olea europaea L.) leaves and pulps from different genetic origin. Optimisation of the stationary phase nature, particle size, column length and internal diameter, as well as other separation conditions, was performed. Along the study, three stationary phases (C18, PFP and phenyl) in the first dimension (1D), and five (C18, amide, cyano, phenyl and PFP) in the second dimension (2D) were combined to obtain the maximal number of resolved peaks. The optimised method successfully characterised the presence of 26and 29 common polyphenols in olive leaves and pulp extracts, respectively. Peak volume ratios were used to develop linear discriminant analysis models able to distinguish olive leaves and pulp extracts among seven cultivars from several Spanish regions. The results demonstrate that polyphenolic profiles were characteristic of each cultivar.

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.

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 ß-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.

Extension of the linear solvent strength retention model including a parameter that describes the elution strength changes in liquid chromatography.

Modelling the retention behaviour of solutes in liquid chromatography, based on the composition of the mobile phase is a common task in the chromatographic practice. Along the development of liquid chromatography (LC), several models have been proposed to help in understanding the retention mechanisms, and especially, allow the prediction of retention times with optimisation purposes. Particular models are used for different LC modes, such as normal phase (NPLC), reversed phase (RPLC), hydrophilic interaction (HILIC), and micellar (MLC). In this work, a general equation is proposed that includes a parameter (the elution degree, g), which characterises the way the elution strength varies with the modifier concentration. The elution degree adopts the value g = 1 when the system follows the linear solvent strength (LSS) model, where the elution strength is constant. When g > 1, the elution strength decreases, and for g < 1, it increases with the modifier concentration. The proposed equation was applied to experimental retention data obtained for several chromatographic systems in RPLC, HILIC, MLC, and microemulsion LC. It was found that values in the 1 < g < 2 range are most usual. The general behaviour of the proposed equation was studied for isocratic and gradient elution. A general expression to calculate the compression factor of chromatographic peaks in gradient elution was also developed. It is shown that an increasing g value makes retention factors close to zero more difficult, since the elution strength decreases as the modifier concentration increases. For this reason, the larger the g value, the harder it is to reach significant peak compression. In contrast, an elution mode with g < 1 would yield increased elution strength with the modifier concentration, giving rise to significant peak compression.

Peak dispersion in gradient elution: An insight based on the plate model.

Gradient elution in liquid chromatography reduces the analysis time, improves the efficiency and increases the peak capacity. The study of this chromatographic mode has been based mainly on kinetic dispersion models. The plate model has been applied to a lesser extent, despite being the basis for the concepts of plate height and chromatographic efficiency. In this work, a general equation describing peak dispersion in HPLC gradient elution is derived from the plate model. This equation is studied and validated for three types of gradients: (i) a reference gradient without ramp in which the retention factor varies with time identically throughout the column, (ii) a gradient of stationary phase in which the nature of the stationary phase varies continuously inside the column, resulting in a ramp of constant retention factor over time, and finally, (iii) a mobile phase gradient, which produces a retention factor ramp that varies over time. In the latter case, the results are similar to those derived from the mass-transport equation in linear solvent gradients when the linear solvent strength model is applied. The final equations are expressed based on the initial and final instantaneous retention factors, thus they can be applied independently of the deviation of the elution model, being fully compatible with isocratic elution. Results predicted with the proposed equations are identical to those obtained by numerical resolution of the elution differential equation system. The additional compression due to the presence of a ramp of modifier is also verified. However, useful compressions will appear only when the retention factor changes with time. Finally, the study indicates that the extra-column variance undergoes a compression process when the retention factor varies over time, whether or not there is a ramp inside the column.

Modelling retention and peak shape of small polar solutes analysed by nano-HPLC using methacrylate-based monolithic columns.

The development of methacrylate-based monolithic columns was studied for the separation of pharmaceutical hydrophilic compounds in nano-liquid chromatography. The selected polymerisation mixture consisted of 7.5% hexyl methacrylate, 4.5% methacrylic acid and 18.0% ethylene dimethacrylate (w/w), in a binary porogenic solvent (35:35 w/w 1-propanol/1,4-butanediol). The polymer synthesised with this mixture has a good permeability, not excessive back-pressure, and reasonable retention times for polar and non-polar solutes. Monolithic columns (12 cm total capillary length, 100 µm i.d.), prepared with this mixture, were able to produce hydrogen bonding and electrostatic interactions, giving rise to promising separations. To evaluate the chromatographic system, alkylbenzenes (neutral and hydrophobic compounds) and sulphonamides (hydrophilic drugs) were assayed. To optimise the chromatographic mobile phase in isocratic elution and characterise the retention mechanism for a mixture of eight sulphonamides, the performance of several mathematic models was checked in the description of retention. The behaviour of the monolithic capillary column was compared, in terms of selectivity and peak shape, to that obtained with a C18 column (9 cm × 4.6 mm i.d., 5 µm particle size) using a conventional HPLC equipment. The results revealed substantial differences in the interactions established for sulphonamides between the monolithic and C18 columns.

Effect of buffer nature and concentration on the chromatographic performance of basic compounds in the absence and presence of 1-hexyl-3-methylimidazolium chloride.

In reversed-phase liquid chromatography, the performance for basic compounds is affected by the interaction of the protonated (cationic) species with the anionic free silanols on the alkyl-bonded stationary phases. Using aqueous-organic mobile phases in the absence of additives, the retention may be too high, and the peaks be broad and asymmetric. The performance is improved by addition to the mobile phase of ionic liquids, from which 1-hexyl-3-methylimidazolium chloride ([C6MIm][Cl]) has especially good characteristics. A recent report has also revealed that the use of the phosphate system as buffer, at varying concentration and pH, may have a significant role in the chromatographic performance of basic compounds, with effects on both retention and peak shape. In this work, this study has been extended to other three buffer systems (acetate, citrate, and formate), at increasing concentrations and pH 3 and 7, in the presence and absence of [C6MIm][Cl]. The results have been compared with those obtained with the phosphate system. The retention increases by addition of larger concentration of all buffers, in both absence and presence of [C6MIm][Cl]. Without additive, peak performance is also enhanced significantly. This effect is minimal in the presence of [C6MIm][Cl], which yields highly symmetrical peaks at all buffer concentrations, due to an effective blocking of the silanol activity.

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.

Benefits of solvent concentration pulses in retention time modelling of liquid chromatography.

The advantages and disadvantages of the use of isocratic experimental designs including transient increments of organic solvent (i.e., pulses) in the mobile phase(s) of lowest elution strength are explored with modelling purposes. For retained solutes, this type of mixed design offers similar or better predictive capability than gradient designs, shorter measurement time than pure isocratic designs, and retention model parameters that agree with those derived from pure isocratic experiments, with similar uncertainties. The predicted retention times are comparable to those offered by models adjusted from pure isocratic designs, and the solvent waste is appreciably lower. Under a practical standpoint, mixed designs including pulse(s) can be easily constructed by replacing the slowest isocratic runs with runs containing a pulse of short duration at an intermediate time. This allows the elution of the fastest solutes with appreciable retention in the initial sector of the elution program, previous to the pulse, and the elution of the slow solutes after the pulse, also in acceptable times. The fitting of the retention data obtained with pulses is simpler compared to gradient elution, and involves solving the integral equation of gradient elution, simplified by the presence of isocratic sectors. Experiments involving pulses reveal the existence of discrepancies in the predictions for solutes eluting in the nearby of the pulse, offered by the fundamental equation of gradient elution when this is solved using numerical integration. The correction of such discrepancies implies the inclusion of intra-column delays, in the arrival of changes in the concentration of organic modifier in the gradient to the instantaneous position of the solute, along the whole migration.