Effect of mobile phase amine additives on enantioselectivity for phenylalanine analogs

Curve fitting seems to be one of the best methods for the evaluation of chromatographic signals. As it is known, in this case mathematical function is fitted to digitized measured points. The most important task is to find the best mathematical function, which corresponds perfectly to the peak shape, and then to determine the parameters of the equation using a computerized least-squares method of approximation. In this work, a new mathematical function was sought for with the purpose of describing different chromatographic signals and it was fitted to the digitized measured points. The fitted curve is suitable for a quick evaluation of chromatographic information, noise filtering and correction of baseline drift. The fitting of gas chromatographic and high-performance liquid chromatographic signals were completed. The mathematical function, the generated chromatographic curves, the application of the function for describing real signals and the fitting process will be demonstrated in this study.     

Resolution of Overlapping Chromatographic Peaks Using a Genetic Algorithm

ABSTRACT

A genetic algorithm for resolution of overlapping chromatographic peaks (GAROCP) using real-number coding, non-uniform mutation and arithmetical crossover methods is described in this paper. It was applied to resolution of highly overlapped multicomponent high-performance liquid chromatographic peaks by fitting experimental chromatogram to the exponentially modified Gaussian (EMG) model. The genetic algorithm was used to find the minimum of fitting error to optimize the parameters in the EMG functions which determine the shape and area of each peak. The applicability of the method was investigated with both simulated signals calculated by EMG functions and experimental multicomponent overlapping chromatograms.     

Analysis of peak asymmetry in chromatography

The knowledge of the symmetry of chromatographic peaks is extremely important regarding the digital signal processing. The significant deviation of the peak shape from the symmetrical peak makes hardly possible the acquisition of chromatographic signal information, such as the retention time, the peak area, the peak width at half peak height, the peak overlapping, etc. In the literature one can find many methods for the determination of the asymmetry factor. For example it is suitable to calculate the skewness from the third central moment. However in case of noisy baseline the value of the skewness oscillates highly depending on the number of points used for the mathematical calculation. In this work a new method is presented for the determination peak shape asymmetry. We order mathematical function to the chromatographic peaks by fitting, and then symmetrical curve is generated with the same peak maximum position and height, the peak width is fitted. The difference of the two functions is constituted and areas of the data differences are calculated, which are really characteristics of the peak asymmetry. Correlation between the area of the difference signal and the asymmetry factor is established. The method was applied for different types of chromatographic peak shapes and the results were interpreted.     

Study of elution behaviour with gradient voltage in CEC using methacrylate monolithic columns

A theoretical study on the retention behaviour and chromatographic performance of neutral solutes using a lauryl methacrylate‐based monolithic column under voltage gradient mode in CEC was carried out. Through a flexible mathematical function based on a modified Gaussian model, the peak shape of compounds was firstly fitted under constant and gradient voltage. Using the peak shape parameters and retention time, the estimation of global chromatographic performance, efficiency and peak capacity under several voltage conditions was performed. The influence of voltage gradient on the separation efficiency is discussed and simple equations are presented to calculate retention and peak widths under voltage gradient conditions. A comparison in terms of chromatographic performance of a test mixture of neutral solutes under constant and gradient voltage modes was also carried out. The experiments carried out under gradient voltage showed better efficiencies (172 000 plates/m) and lower peak widths than those obtained under constant voltage (52 000 plates/m).     

The role of the theoretical plate concept and gamma density functions in studies on mass transport and mass loadability of chromatographic columns

Summary The validity of the use of the plate theory in transport processes with non-Gaussian peak shapes is discussed. It is shown that the application of the plate theory implies the assumption of a peak with a gamma density shape which, however, converges rapidly to symmetrical Gaussian shape for large plate numbers. A proposal for an extension of the plate theory, based on a gamma density function is given. This approach results in an extra parameter which characterizes the asymmetry of a peak. Use of the suggested fitting function permits an estimate of the moments of the peak, even without the use of a computer. The proposed model can be used to determine a transfer function for the chromatographic system, allowing a system theoretical approach to determine the influence of, for example, amplifiers, filters and detectors on the peak shape. The exponentially modified Gauss (EMG) is one of the possible peak shapes included in the model.     

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

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

Simulation of Asymmetric Peak-Shaped Analytical Signals by the Frame Representation of Their Shape Using Stripping Voltammetry as an Example

To simulate peak-shaped analytical signals, the shape characteristics and geometrical properties of four asymmetric basic functions (the derivative of the logistic function, the Gaussian function, the Cauchy function modified by Fraser and Suzuki, and the Poisson function with the Stromberg correction) are systematically studied. The frame representation, which was proposed earlier by the two authors of this paper, is used to characterize the peak shape. Some regularities are found in peak properties expressed in terms of the frame parameters and considered as functions of the coefficient of skewness b. Peak shapes are compared using two approaches to obtaining dimensionless frame parameters: one using relative parameters and another using parameters of the unit frame. It is shown that the frame representation of the peak shape can be used in the phenomenological simulation of metal peaks in stripping voltammetry. The practical use of information on the shape of the analytical signal in the solution of various problems of analytical chemistry is discussed.     

Estimation of chromatographic peak shape parameters in Fourier domain

A number of models in chromatography have analytical solutions in the Laplace or Fourier domain. Often, the moments of the Laplace domain solutions are calculated to characterize the peak shape. Nonlinear fitting in the Fourier domain can be performed to exploit the entire peak shape rather than the moments only. Curve fitting in the Fourier domain offers an attractive alternative for parameter estimation. In this study we will show - with some simple applications - the possibilities of estimation of chromatographic peak shape parameters in Fourier domain. Various models are fitted to different transient signals.     

Prediction of peak shape in hydro-organic and micellar-organic liquid chromatography as a function of mobile phase composition

A simple model is proposed that relates the parameters describing the peak width with the retention time, which can be easily predicted as a function of mobile phase composition. This allows the further prediction of peak shape with global errors below 5%, using a modified Gaussian model with a parabolic variance. The model is useful in the optimisation of chromatographic resolution to assess an eventual overlapping of close peaks. The dependence of peak shape with mobile phase composition was studied for mobile phases containing acetonitrile in the presence and absence of micellised surfactant (micellar-organic and hydro-organic reversed-phase liquid chromatography, RPLC). In micellar RPLC, both modifiers (surfactant and acetonitrile) were observed to decrease or improve the efficiencies in the same percentage, at least in the studied concentration ranges. The study also revealed that the problem of achieving smaller efficiencies in this chromatographic mode, compared to hydro-organic RPLC, is not only related to the presence of surfactant covering the stationary phase, but also to the smaller concentration of organic solvent in the mobile phase.     

Semi-differential analysis of irreversible voltammetric peaks

Voltammetric peaks obtained by simulation of electrochemical reactions under conditions of linear semi-infinite diffusion with an irreversible electron transfer process are analysed using a semi-differentiation procedure. Obtained semi-derivative peaks, separated or overlapped, are fitted with appropriate mathematical functions. The functions used for data fitting include a function describing symmetrical peaks, proposed by several authors for fitting irreversible semi-derivative peaks, and two alternative functions that express asymmetric shape of the irreversible semi-derivative signals. When applied to the overlapped irreversible semi-derivative peaks, the latter two functions allow calculating certain electrochemical parameters with a better accuracy as compared with the function derived for the symmetrical peaks.     

On the equations describing chromatographic peaks and the problem of the deconvolution of overlapped peaks

The problem of the appropriate choice of the function that describes a chromatographic peak is examined in combination with the deconvolution of overlapped peaks by means of the non-linear least-squares method. It is shown that the majority of the functions proposed in the literature to describe chromatographic peaks are not suitable for this purpose. Only the polynomial modified Gaussian function can describe almost every peak but it is mathematically incorrect unless it is redefined properly. Two new functions are proposed and discussed. It is also shown that the deconvolution of an overlapping peak can be done with high accuracy using a non-linear least-squares procedure, like Microsoft Solver, but this target is attained only if we use as fitted parameters the position of the peak maximum and the peak area (or height) of every component in the unresolved chromatographic peak. In case we use as fitted parameters all the parameters that describe each single peak enclosed in the multi-component peak, then Solver leads to better fits, which though do not correspond to the best deconvolution of the peak. Finally, it is found that Solver gives much better results than those of modern methods, like the immune and genetic algorithms.     

分辨伏安分析重叠峰的研究

研究了一种处理伏安(极谱)重叠峰的数学模型。将几类具峰状的极谱电流公式归纳成一般的关系式,提出了一个通用的拟合函数,经非线性最小二乘法处理,可得到重叠组份的蜂高、峰电位和半峰宽等参数。本法适用于示差脉冲极谱、交流极谱、方波极谱、一阶导数卷积伏安法及其反向溶出伏安重叠峰的分离。已用于示差脉冲极谱和交流极谱重叠峰的分辨,得到满意结果。     

Improved computer algorithm for characterizing skewed chromatographic band broadening : II. Results and comparisons

A newly developed method using an exponentially modified Gaussian peak shape model produces results that are more precise and less subject to baseline noise than previous methods for characterizing chromatographic band broadening. The method requires only precisely measurable experimental peak parameters: peak retention time, peak height, peak area, and peak centroid (first moment). Accuracy and precision of the new method were compared with other digital approaches by using computer-synthesized peaks and experimental chromatographic data from many HPLC columns. The proposed method offers a reasonable compromise between accuracy, precision, and convenience. A rapid visual estimate of peak skew can be made by inspecting peak shape and referring to a calibration plot involving peak parameters. Peak variance and skew data from this method are also useful for finding column dispersion corrections in size-exclusion chromatography calibrations.     

Parametric studies of matched filters to enhance the signal-to-noise ratios of LC-MS-MS peaks

Chromatographic parameters of reference signals employed in matched filter methods have been studied using numerical experiments to improve the signal-to-noise (S/N) ratios of small liquid chromatography (LC) peaks obtained with electrospray tandem mass spectrometers (MS-MS). These parameters include the width, shape, and S/N ratios of chromatographic peaks used as the reference signal profiles. Our results show the effect of reference peak widths on improving the S/N ratio of chromatographic peaks; the influence of reference peak shapes is negligible. To verify simulation results, various reference signals, including analyte peaks of high concentration standards, internal standard peaks, and artificial Gaussian peaks of different widths, have been employed to enhance signal peaks on real liquid chromatography-tandem mass spectrometry (LC-MS-MS) chromatograms via matched filter methods. Our experimental results demonstrate that the S/N ratio enhancement of chromatographic peaks agree with the simulation predictions. These findings, therefore, suggest that regardless of peak shape, a well-smooth peak with a width close to that of the analyte peak is an adequate reference signal, when matched filter methods are used to improve LC-MS-MS chromatograms. Nevertheless, all methods processed LC-MS-MS peaks in this study do not achieve the ideal improvement ratios estimated with simulation results. We attribute this deficiency to spike-like noise, which have considerable low frequency components riding on LC-MS-MS chromatograms. Matched filtering, which works as a low-pass filter in the frequency domain, cannot effectively eliminate low frequency flicker noise contributed by these spikes. In addition, simple median filtering does not provide adequate improvement despite being able to smooth out most spikes in the chromatograms.     

Automated alignment of one-dimensional chromatographic fingerprints

A general framework for the automatic alignment of one-dimensional chromatographic signals is presented in this article. The alignment of signals was achieved by explicitly modeling the warping function. Its shape was estimated using a linear combination of several B-spline functions. The coefficients of the spline functions were found in the course of an optimization procedure to maximize the Pearson's correlation coefficient between a target chromatogram and aligned chromatogram(s). The computational requirements of the method are discussed with respect to the correlation optimized warping method, frequently used for the alignment of chromatographic signals. As illustrated with two sets of one-dimensional chromatographic fingerprints, the automatic alignment approach performs well even when non-linear peak shifts need to be corrected. It can be applied in an on-the-fly manner since the alignment of signals is rapid.     

Rapid characterization of anthocyanins in red raspberry fruit by high-performance liquid chromatography coupled to single quadrupole mass spectrometry

This paper describes the results of a comparison of four peak functions in describing real chromatographic peaks. They are the empirically transformed Gaussian, polynomial modified Gaussian, generalized exponentially modified Gaussian and hybrid function of Gaussian and truncated exponential functions. Real chromatographic peaks of different shapes (fronting. symmetric, and tailing) are obtained by various separation conditions of reversed-phase liquid chromatography. They are then fitted to the peak functions via the Marquardt-Levenberg algorithm, a nonlinear least-squares curve-fitting procedure, by Microsoft Solver. The qualities of the fits are evaluated by the sum of the squares of the residuals. It is concluded in the study that the empirically transformed Gaussian function offers the highest flexibility (best fits) to all shapes of chromatographic peaks, including extremely asymmetric tailing peaks with a peak asymmetry of up to 8. The flexibility of this function should improve our ability to process chromatographic peaks such as deconvolution of overlapped peaks and smoothing noisy peaks for the determination of statistical moments.     

Monolithic silica columns for high-efficiency chromatographic separations

A deconvolution methodology for overlapped chromatographic signals is proposed. Several single-wavelength chromatograms of binary mixtures, obtained in different runs at diverse concentration ratios of the individual components, were simultaneously processed (multi-batch approach), after being arranged as two-way data. The chromatograms were modelled as linear combinations of forced peak profiles according to a polynomially modified Gaussian equation. The fitting was performed with a previously reported hybrid genetic algorithm with local search, leaving all model parameters free. The approach yielded more accurate solutions than those found when each experimental chromatogram was fitted independently to the peak model (single-batch approach). The improvement was especially significant for those chromatograms where the peaks were severely affected by the tails of the preceding compounds. Peak shifts among chromatograms, which are a usual source of non-bilinearity, were modelled in a continuous domain instead of in a discrete way, which avoided some drawbacks associated with latent variable methods. An experimental design involving simulated chromatograms was applied to check the method performance. Five main factors affecting the deconvolution were examined: concentration pattern, chromatographic resolution, number of batches and replicates, and noise level, which were evaluated using first- and second-order figures of merit. The method was also tested on three real samples containing compounds showing different overlap. Four multi-batch deconvolution methods were considered differing in the nature of the processed information and kind of peak matching among chromatograms. In all cases, the multi-batch deconvolution yielded better performance than the single-batch approach.     

Recursive Wavelet Peak Detection of Analytical Signals

A novel algorithm, entitled recursive wavelet peak detection (RWPD), is proposed to detect both normal and overlapped peaks in analytical signals. Recursive peak detection is based on continuous wavelet transforms (CWTs), which can be used to obtain initial peak positions even for overlapped peaks. Genetic algorithm (GA) and Gaussian fitting are used to refine peak parameters (peak positions, widths, and heights). Finally, area of peaks can be calculated by numeric integration. Simulated and ultrahigh performance liquid chromatographic ion trap time-of-flight mass spectrometry (UPLC-IT-TOF-MS) data sets have been analyzed by RWPD, MassSpecWavelet, and peakfit package by Tom O’Haver. Results show that RWPD can obtain more accurate positions and smaller relative fitting errors than MassSpecWavelet and peakfit, especially in overlapped peaks. RWPD is a convenient tool for peak detection and deconvolution of overlapped peaks, and it has been developed in R programming language and is available at https://github.com/zmzhang/RWPD.