Theoretical study of the accuracy of the elution by characteristic points method for bi-langmuir isotherms

The bi-Langmuir equation has recently been proven essential to describe chiral chromatographic surfaces and we therefore investigated the accuracy of the elution by characteristic points method (ECP) for estimation of bi-Langmuir isotherm parameters. The ECP calculations was done on elution profiles generated by the equilibrium-dispersive model of chromatography for five different sets of bi-Langmuir parameters. The ECP method generates two different errors; (i) the error of the ECP calculated isotherm and (ii) the model error of the fitting to the ECP isotherm. Both errors decreased with increasing column efficiency. Moreover, the model error was strongly affected by the weight of the bi-Langmuir function fitted. For some bi-Langmuir compositions the error of the ECP calculated isotherm is too large even at high column efficiencies. Guidelines will be given on surface types to be avoided and on column efficiencies and loading factors required for adequate parameter estimations with ECP.     

Improvement in the generation of adsorption isotherm data in the elution by characteristic points method--the ECP-slope approach

The elution by characteristic points (ECP) method is a very rapid and precise method for determination of the phase system equilibrium of phase systems in broad solute concentration ranges. Thus, the method is especially suitable for rapid characterization of high efficient separation systems. One important source of error, the effects by the post-loop dispersion, was eliminated in a recent investigation. In this study, the systematic error caused by the selection of the integration starting point at concentration equal to 0 is eliminated. This is done by developing and validating a new procedure for isotherm data generation; the ECP-slope method. The method generates raw slope data of the adsorption isotherm instead of raw adsorption data by integrations as the classical ECP does. Both numerical and experimental data were used for the comparison of the classical ECP approach with the slope-ECP method.     

An estimation of the column efficiency made by analyzing tailing peak profiles

It has been shown previously that most columns are not radially homogeneous but exhibit radial distributions of the mobile phase flow velocity and the local efficiency. Both distributions are best approximated by fourth-order polynomial, with the velocity in the column center being maximum for most packed columns and minimum for monolithic columns. These distributions may be an important source of tailing of elution peaks. The numerical calculation of elution peaks shows how peak tailing is related to the characteristics of these two distributions. An approach is proposed that permits estimations of the true efficiency and of the degree of column radial heterogeneity by inversing this calculation and using the tailing profiles of the elution peaks that are experimentally measured. This method was applied in two concrete cases of tailing peak profiles that had been previously reported and were analyzed by applying this new inverse approach. The results obtained prove its validity and demonstrate that this numerical method is effective for deriving the true column efficiency from experimental tailing profiles.     

Influence of column radial heterogeneity on peak fronting in linear chromatography.

Using numerical calculations of elution peak profiles, an explanation of the fronting behavior of elution peaks in linear chromatography was found in certain radial distributions of the mobile phase flow velocity and local bed efficiency. Fronting peaks are observed only if the flow velocity is higher in the wall region than in the center part of the column and the local efficiency is lower near the wall than in the center. By contrast, tailing or symmetrical peaks are observed if only the flow velocity or the local efficiency are radially heterogeneous. The degree of peak fronting increases with increasing amplitude of the radial distributions. The influence of the radial heterogeneity of the flow velocity on the degree of peak fronting is more severe for high than for low efficiency columns. An equation is suggested to correlate peak fronting behavior for columns of different efficiencies and a procedure proposed for the estimation of the radial distributions of the flow velocity and the local efficiency by analyzing some characteristics of asymmetric peaks.     

Peak tailing and column radial heterogeneity in linear chromatography

The correlation between the radial heterogeneity of a column and the tailing of the elution profiles of chromatographic peaks was studied using a numerical method. A parabolic distribution of the linear flow velocity of the mobile phase and of the column efficiency in the radial direction were assumed. Moment analysis showed that peak tailing takes place under such experimental conditions and that it increases with increasing range of radial variations of the flow velocity and the column efficiency. It was also found that the higher the column efficiency, the larger the effect of a given degree of radial heterogeneity on the extent of peak tailing. Peak tailing behavior of columns having different efficiencies could be correlated with each other by an equation. Some characteristic features of tailing peaks were analyzed in connection with the column radial heterogeneity.     

Influence of the column hold-up time measurement accuracy on the prediction of chromatographic band profiles

The influence of the column hold-up time measurement accuracy on the determination of equilibrium isotherms by classical frontal analysis and the prediction of overloaded elution band profiles were investigated. The ideal model of chromatography in combination with a Langmuir isotherm was used. Breakthrough curves and overloaded elution profiles were computer generated with a known hold-up time value (true hold-up time). Then these data were evaluated the same way as it is done with experimental chromatographic data where the true hold-up time is unknown, i.e. to determine the equilibrium isotherm by the frontal analysis procedure, to fit the isotherm data to the Langmuir model and then to predict chromatographic band profiles using, e.g. the ideal model of chromatography. A comparison of overloaded elution profiles obtained with different deviations of the hold-up time from its true value shows that the effect of its measurement error is significant in preparative liquid chromatography because the isotherm is usually strongly nonlinear in this case.     

Influence of the particle porosity on chromatographic band profiles

The mass transfer kinetics of butyl benzoate, eluted on a monolithic RPLC column with methanol-water (65:35, v/v) as the mobile phase was investigated, using the perturbation method to acquire isotherm data and the mobile phase velocity dependence of the height equivalent to a theoretical plate of perturbation peaks to acquire kinetics data. The equilibrium isotherm of butyl benzoate is accounted for by the liquid-solid extended multilayer BET isotherm model. The total porosity of the column varies much with the butyl benzoate concentration, influencing strongly the parameters of its mass transfer kinetics and the profiles of the breakthrough curves. Using all these parameters, the general rate model of chromatography predicts band profiles and Van Deemter curves that are in excellent agreement with experimental results provided the influence of concentration on the porosity is properly taken into account. This agreement confirms the validity of the models selected for the isotherm and for the mass transfer kinetics.     

Determination of chromatographic separation parameters of tryptophan enantiomers on a Chirosil-SCA chiral stationary phase by using the inverse method based on the initial guesses estimated from elution by characteristic point method

An effective chiral stationary phase (CSP) for enantioseparation of amino acids was established previously by bonding (18-crown-6)-2, 3, 11, 12-tetracarboxylic acid to silica gel. This CSP has recently been commercialized under the name of Chirosil-SCA. As a first step for developing a Chirosil-SCA simulated moving bed chromatographic process for separation of tryptophan enantiomers, the adsorption isotherm and mass-transfer parameters of each tryptophan enantiomer on the Chirosil-SCA CSP were determined in this study while using only water as a mobile phase. For this task, inverse method (IM) was applied on the basis of the initial guesses estimated from elution by characteristic point (ECP) method, which was found to be more advantageous in the aspects of both accuracy and computational efficiency than the case of utilizing individually only IM or ECP method. The results revealed that the adsorption behavior of each tryptophan enantiomer on the Chirosil-SCA could be well described by the Langmuir-Freundlich isotherm. The model predictions based on the determined parameter values were in close agreement with the experimental chromatograms from a series of single-component or mixture pulse tests that were performed under various feed concentrations and flow rates. It was also found that the Langmuir-Freundlich isotherm parameters of each enantiomer were largely affected by temperature. Such a marked dependence of the parameters on temperature was investigated quantitatively. The results of such an investigation indicated that as the temperature decreases, the adsorption affinities of both enantiomers become higher and the heterogeneity of the Chirosil-SCA becomes more pronounced.     

Comparison of the binary equilibrium isotherms of the 1-indanol enantiomers on three high-performance liquid chromatography columns of different sizes

The competitive isotherm data for the enantiomers of 1-indanol were measured on three columns, a microbore column (15 cm x 0.1 cm), a conventional analytical column (15 cm x 0.46 cm), and a semi-preparative column (20 cm x 1.0 cm), packed with Chiralcel OB. The sets of isotherm data measured on each one of these three columns could be fitted well by a bi-Langmuir isotherm model. The experimental elution band profiles of mixtures of the 1-indanol isomers were recorded on the three columns. The isotherm model, combined with the equilibrium dispersive model of chromatography, gave calculated profiles that are in excellent agreement with the experimental profiles in all cases investigated. It was found that the value of the inner diameter of the column is an important parameter in the calculation of the isotherm parameters from the measured isotherm data. In order to use isotherm data obtained on one column to account for the phase equilibrium on another one, the inner diameters of these columns must be measured accurately. The diameters of the three columns were all slightly off their nominal value. Without correction, an important systematic error was made on the isotherm data obtained with the microbore column while only negligible errors were made on the data obtained with the other two columns. After due correction for this effect, the relative difference between the isotherm data for the microbore and the semi-preparative column is still, on the average, about 10%, a difference that might be explained by the limited precision of the measurement of the microbore column diameter. The relative difference between the isotherm data for the analytical and the semi-preparative columns was about 1%, a reasonable value, since the two columns came from different batches of the same packing material.     

Structural radial heterogeneity of a silica-based wide-bore monolithic column

The radial distribution of the main characteristics (elution time and standard deviation) of the elution profiles of a flat injected band recorded at the exit of a monolithic column were determined. These distributions provide the radial distributions of the average mobile phase velocity, the elution time and the maximum height of the peak of an analyte, the column efficiency and the analyte concentration. The band profiles were measured at the exit of a 10-mm i.d., 100-mm long silica-based monolithic column. An on-column local electrochemical amperometric detector allowed the recording of the elution profiles at different spatial positions throughout the column cross-section. The local spatial distribution of the mobile phase velocity does not follow a piston-flow behavior but exhibits radial heterogeneity. The local efficiency near the wall is lower than that near the column center. The radial distribution of the maximum concentration of the peaks varies throughout the column exit section, partially due to the radial variations of the column efficiency. These results might explain the rather large value of the A term of the Van Deemter or the Knox equations reported previously for monolithic columns.     

Effect of the flow rate on the measurement of adsorption data by dynamic frontal analysis

The adsorption data of propyl benzoate were acquired by frontal analysis (FA) on a Symmetry-C18 column, using a mixture of methanol (65%, v/v) and water as the mobile phase, at three different flow rates, 0.5, 1.0 and 2.0 mL/min. The exact flow rates Fv were measured by collecting the mobile phase in volumetric glasses (deltaFv / Fv < or = 0.2%). The extra-column volumes and the column hold-up volume were accurately measured at each flow rate by tracer injections. The detailed effect of the flow rate on the value of the amount adsorbed was investigated. The best isotherm model accounting for the adsorption data was the same BET isotherm model at all three flow rates. Only slight differences (always less than 5%) were found between the three different sets of isotherm parameters (saturation capacity, q(s), equilibrium constant on the adsorbent, b(s) and equilibrium constant on successive layers of propyl benzoate, bL). The reproducibility of the same isotherm parameters measured by the inverse method (IM) is less satisfactory, leading to R.S.D.s of up to 10%. A flow rate increase is systematically accompanied by a slight increase of the amount adsorbed. This phenomenon is consistent with the influence of the pressure on the equilibrium constant of adsorption due to the difference between the partial molar volumes of the solute and the adsorbate. The larger average pressure along the column that is required to achieve a larger flow rate causes a larger amount of solute to be adsorbed on the column at equilibrium. This result comforts the high sensitivity and versatility of the FA method for isotherm determination under any kind of situation.     

A random forest of combined features in the classification of cut tobacco based on gas chromatography fingerprinting

We applied the random forest method to discriminate among different kinds of cut tobacco. To overcome the influence of the descending resolution caused by column pollution and the subsequent deterioration of column efficacy at different testing times, we constructed combined peaks by summing the peaks over a specific elution time interval Δt. On constructing tree classifiers, both the original peaks and the combined peaks were considered. A data set of 75 samples from three grades of the same tobacco brand was used to evaluate our method. Two parameters of the random forest were optimized using out-of-bag error, and the relationship between Δt and classification rate was investigated. Experiments show that partial least squares discriminant analysis was not suitable because of the overfitting, and the random forest with the combined features performed more accurately than Naïve Bayes, support vector machines, bootstrap aggregating and the random forest using only its original features.     

A comparison of frontal and nonfrontal methods for determining solid-liquid adsorption isotherms using inverse liquid chromatography

Inverse liquid chromatography (ILC) has been used to determine experimental isotherms for the equilibrium adsorption of cyclohexanone onto a silica (61.8 m(2)/g) from hexane using the peak maximum (PM), elution by characteristic point (ECP), frontal analysis (FA), and frontal analysis by characteristic point (FACP) methods. Isotherms obtained using these four approaches gave good internal agreement, as well as being in good agreement with classically determined isotherms. Columns were successfully packed using a dry powder packing method with 9 microm diameter silica particles, and excellent intercolumn and instrument to instrument reproducibility was obtained for PM isotherms. The theoretical background to the PM, ECP, FA, and FACP methods, as well as the practical facets of isotherm determination using these methods, is outlined in this work.     

Impact of an error in the column hold-up time for correct adsorption isotherm determination in chromatography II. Can a wrong column porosity lead to a correct prediction of overloaded elution profiles?

The adsorption isotherm was determined for phenol in methanol/water on a C-8 stationary phase using frontal analysis in staircase mode, assuming different total column porosities, from 1 to 87%. Each set of adsorption isotherm data, with a certain column porosity, was fitted to various adsorption models and the generated parameters were used to calculate overloaded elution band profiles that were compared with experiments. It was found that the bi-Langmuir model had an optimum fit for a porosity that corresponds well with the value found experimentally. The adsorption energy distribution (AED) calculations and error analysis confirmed a bimodal energy distribution. It was also found that band profiles can be accurately predicted with a quite arbitrary chosen porosity, under prerequisite that a wrong but flexible adsorption model is chosen instead of the correct one. The latter result is very useful for quick optimizations of preparative separations where the exact value of the column porosity is not available.     

Systematic errors in the measurement of adsorption isotherms by frontal analysis Impact of the choice of column hold-up volume, range and density of the data points

Besides the accuracy and the precision of the measurements of the data points, several important parameters affect the accuracy of the adsorption isotherms that are derived from the data acquired by frontal analysis (FA). The influence of these parameters is discussed. First, the effects of the width of the concentration range within which the adsorption data are measured and of the distribution of the data points in this range are investigated. Systematic elimination of parts of the data points before the calculation of the nonlinear regression of the data to the model illustrates the importance of the numbers of data points (1) within the linear range and (2) at high concentrations. The influence of the inaccuracy of the estimate of the column hold-up volume on each adsorption data point, on the selection of the isotherm model, and on the best estimates of the adsorption isotherm parameters is also stressed. Depending on the method used to measure it, the hold-up time can vary by more than 10%. The high concentration part of the adsorption isotherm is particularly sensitive to errors made on t(0,exp) and as a result, when the isotherm follows bi-Langmuir isotherm behavior, the equilibrium constant of the low-energy sites may change by a factor 2. This study shows that the agreement between calculated and experimental overloaded band profiles is a necessary condition to validate the choice of an adsorption model and the calculation of its numerical parameters but that this condition is not sufficient.     

Investigation of the tetramer-dimer equilibrium in haemoglobin solutions by high-performance size-exclusion chromatography on a diol column

High-performance size-exclusion chromatography on a diol grafted column was applied to study the tetramer-dimer equilibrium in haemoglobin solution. Human haemoglobin A, isolated alpha A and beta A subunits and haemoglobin variants with structural modifications located at the interface between subunits were used as models. The elution volume of the subunits was found to deviate strongly from that expected from only a gel permeation mechanism and therefore ionic interactions are likely to participate in the protein retention. Experimental results and computer simulation indicate that the individual elution bands corresponding to the discrete components (tetramer, dimer, monomer) can be resolved under certain conditions. In general both the equilibrium and kinetic interconversion parameters must be considered. Single tetramer elution bands were observed from haemoglobin in the concentration range measurable, although the influence of dimers could be seen in the shape and shift of the profile. beta Chains showed resolved peaks for the tetramer-dimer forms.     

Impact of an error in the column hold-up time for correct adsorption isotherm determination in chromatography I. Even a small error can lead to a misunderstanding of the retention mechanism

The impact of a realistic error in the column hold-up time on the determination of the adsorption isotherm model was systematically investigated. Frontal analysis and the inverse method were used for the accurate determination of the adsorption isotherm. The true retention times of the breakthrough curves were used with a known hold-up time as reference. The adsorption isotherms were calculated using the same procedure that is used for real experimental adsorption isotherms, where the true hold-up time is unknown. The raw data were analyzed with calculations of adsorption energy distributions (AEDs), Scatchard plots, fitting to different rival adsorption models and finally their ability to predict true profiles. The results show that for a true Langmuir and bi-Langmuir model with an underestimated hold-up time the error may lead to a more heterogeneous model and for overestimated cases false adsorption processes like multi-layer adsorption or solute-solute interaction are assumed. The Scatchard plots for data obtained using a Langmuir adsorption isotherm are nonlinear and the AEDs show clear deviations from Langmuir behavior already at small deviations from the true hold-up time at a moderate surface coverage. The inverse method confirms the result that was obtained from the frontal analysis procedure.     

Determination of competitive adsorption isotherm parameters of pindolol enantiomers on alpha1-acid glycoprotein chiral stationary phase

In this paper, inverse method (IM) was used to determine the binary competitive adsorption isotherm of pindolol enantiomers by a least-square fitting of the proposed model to the experimentally measured elution curves of racemic pindolol. The isotherm parameters were determined by minimizing the least-square error using an adaptation of genetic algorithm, non-dominated sorting genetic algorithm with jumping genes (NSGA-II-JG). An equilibrium dispersive (ED) model combined with bi-Langmuir isotherm was used in predicting the elution profiles. The determined parameters show good agreement with the experimental profiles at various experimental conditions such as sample volume, concentration and flow rates of the racemic mixture. Robustness and validity of the isotherm parameters were also verified by frontal analyses at various step inputs. Results from both the pulse tests and the frontal analysis indicate that adsorption isotherm derived from the inverse method is quite reliable. This method requires relatively less number of experiments to be performed and therefore, lower experimental costs confirming that inverse method is an attractive alternative approach of experimental technique in determining the competitive adsorption isotherm for binary systems.     

Martin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatography

An alternative method, called the Martin-Synge algorithm, is introduced to calculate numerical solutions of the equilibrium-dispersive (ED) model. The developed algorithm is based on the earlier work of Friday and Levan and on the continuous plate model of Martin and Synge. The column is divided evenly into a series of virtual vessels in which a simplified mass balance equation is solved accurately by the Runge-Kutta-Fehlberg method and the elution profile is given by the numerical solution for the last vessel. The dispersion of the compound during the elution process is controlled by adjusting the number of virtual vessels into which the column is divided. Solving the ED model under linear conditions with this method gives exactly the same profile as the analytical solution of the Martin-Synge plate model. The Martin-Synge method gives better results than the Rouchon method (1) when the isotherms involved are sigmoidal or anti-Langmuir; and, more importantly, (2) in the case of multi-component problems. Finally, the Martin-Synge method proves to be more robust and faster than the OCFE method that, until now, was considered to be one of the most robust and accurate algorithms. The developed algorithm was used for the calculation of the coefficients of the isotherm of butyl benzoate by the inverse method, using a simplex optimization algorithm.     

Influence of pressure on the chromatographic behavior of insulin variants under nonlinear conditions

The effect of pressure on the chromatographic behavior of two insulin variants in RPLC was investigated on a YMC-ODS C18 column, under nonlinear conditions. The adsorption isotherm data of porcine insulin and Lispro were measured at average column pressures ranging from 52 to 242 bar. These data fit well to the Toth and the bi-Langmuir isotherm models. The saturation capacity increases rapidly with increasing pressure while the affinity (or equilibrium) constant and the parameter characterizing the surface heterogeneity decrease. It is noteworthy that the distribution coefficient of the insulin variants increases with increasing pressure whereas their equilibrium constant b decreases for porcine insulin and increases for Lispro. The association constant b(ds), which characterizes the adsorption and desorption equilibrium of insulin in the system, increases with increasing pressure. The excellent agreement between the experimental overloaded profiles recorded under different pressures and those calculated using the POR model suggests that the chromatographic behavior of insulin is controlled more by equilibrium thermodynamics than by the mass transfer kinetics. The latter seems to be nearly independent of the average column pressure. Thus, increasing the average column pressure is an efficient, albeit costly, way to increase the loading capacity of the column, hence the production rate in preparative chromatography.