A new mathematical function for describing electrophoretic peaks

A new model is proposed for characterizing skewed electrophoretic peaks, which is a combination of leading and trailing edge functions, empirically modified to get a rapid recovery of the baseline. The peak model is a sum of square roots and is called thereby "combined square roots (CSR) model". The flexibility of the model was checked on theoretical and experimental peaks with asymmetries in the range of 0-10 (expressed as the ratio of the distance between the center and the trailing edge, and the center and the leading edge of the chromatographic peak, measured at 10% of peak height). Excellent fits were found in all cases. The new model was compared with other three models that have shown good performance in modelling chromatographic peaks: the empirically transformed Gaussian, the parabolic Lorentzian-modified Gaussian, and the Haarhoff-van der Linde function. The latter model was proposed recently to describe electrophoretic peaks. The CSR model offered the highest flexibility to describe electrophoretic peak profiles, even those extremely asymmetrical with long tails. The new function has the advantage of using measurable parameters that allow the direct estimation of peak areas, which is useful for quantitative purposes.     

Approaches to characterise chromatographic column performance based on global parameters accounting for peak broadening and skewness

Peak broadening and skewness are fundamental parameters in chromatography, since they affect the resolution capability of a chromatographic column. A common practice to characterise chromatographic columns is to estimate the efficiency and asymmetry factor for the peaks of one or more solutes eluted at selected experimental conditions. This has the drawback that the extra-column contributions to the peak variance and skewness make the peak shape parameters depend on the retention time. We propose and discuss here the use of several approaches that allow the estimation of global parameters (non-dependent on the retention time) to describe the column performance. The global parameters arise from different linear relationships that can be established between the peak variance, standard deviation, or half-widths with the retention time. Some of them describe exclusively the column contribution to the peak broadening, whereas others consider the extra-column effects also. The estimation of peak skewness was also possible for the approaches based on the half-widths. The proposed approaches were applied to the characterisation of different columns (Spherisorb, Zorbax SB, Zorbax Eclipse, Kromasil, Chromolith, X-Terra and Inertsil), using the chromatographic data obtained for several diuretics and basic drugs (β-blockers).     

Exponentially Modified Gaussian functions—A good model for chromatographic peaks in isocratic HPLC?

Summary The Exponentially Modified Gaussian (EMG) function is fitted to a variety of typical peaks from isocratic HPLC, to assess its validity for peak characterization. This model has previously been applied, but with simplifying assumptions, allowing estimation of peak parameters from simple front and rear half-width measurements. The full EMG function is employed in this work. The results of the simplified approach have been compared to the EMG fitted results. Comparison is also made to fitted Gaussian functions. Results show that the EMG function is a good model for typical LC peaks, under a range of conditions, and that the simplified approaches give close correlation for a range of peak assymetries.     

三元重叠色谱峰面积的定量方法

本从色谱峰的ＥＭＧ模型出发，通过对重叠色谱峰的模拟和回归分析，提出了一种三元重叠色谱峰的面积的定量方法，三元重叠色谱峰的每一个峰面积可以由峰面积比和总面积求得，此法所需的数据都由实验色谱图上测得，峰面积计算结果的相对误差小于±５％，适用于相对峰谷为５０％－９５％的三元重叠色谱峰面积的定量。     

The peak width of nearly Gaussian peaks

Summary In chromatography the peak shape is often described by the method of statistical moments and therefore the second central moment is considered as the correct measure of peak width. Using the exponentially modified Gaussian peak model as an example and a criterion more related to chromatography, the extent of separation, it is shown that for nearly Gaussian peaks the width measured at 1/8 of the peak height is a more meaningful width parameter to evaluate the efficiency of chromatography and the resolution. The second central moment gives too much weight to the remote parts of the tail of the usual, somewhat skewed, peak.     

Fitting Lorentzian peaks with evolutionary genetic algorithm based on stochastic search procedure

A global search technique for curve fitting based on evolutionary random search was modified and applied for quantifying a combination of Gaussian and Lorentzian peaks. This stochastic search procedures based on randomized operators is a modified Monte Carlo method. The proposed method tested on self obtained several overlapped Lorentzian peaks with random noise, Lennard particles in three dimensions and discrete mathematical functions previously used for optimization in literature. It was found to be the proposed method is suitable for complex and large scale optimization. The results of the new method have been compared with those obtained by two peak fitting programs. Developed method was found to be very fast and thus it is time saving.     

Temporally convoluted Gaussian equations for chromatographic peaks

Both spatial and temporal peaks that are produced by the discrete parcel model can be mathematically approximated by Gaussian functions, but the transformation from a spatial pattern to a temporal image requires a convolution treatment. A first-order convolution is given for temporal peaks under a linear isotherm, whereas a second-order convolution is proposed for those under non-linear isotherms. Numerical tests show that the peak shapes generated by the proposed temporally convoluted Gaussian equations (TCG) match perfectly with those obtained by the discrete parcel model. Although the full TCG equation may be quite complicated, it can be made easier by a recursion calculation technique, and a group of peak curves can be plotted simultaneously on computer worksheet. The results also suggest that the temporal distortion effect should be predominately considered, in addition to those known-to-exist spatial effects, for explaining the peak asymmetry.     

A study of the precision and accuracy of peak quantification in comprehensive two-dimensional liquid chromatography in time

Simulated chromatographic data were used to determine the precision and accuracy in the estimation of peak volumes (i.e., peak sizes) in comprehensive two-dimensional liquid chromatography in time (LC × LC). Peak volumes were determined both by summing the areas in the second dimension chromatograms and by fitting the second dimension areas to a Gaussian peak. The Gaussian method is better at predicting the peak volume than the moments method provided there are at least three second dimension injections above the limit of detection (LOD). However, when only two of the second dimension signals are substantially above baseline, the accuracy and precision of the Gaussian fit method become quite poor because the results from the fitting algorithm become indeterminate. Based on simulations in which the modulation ratio (M_R = 4¹σ/t_s) and sampling phase (?) were varied, we conclude for well-resolved peaks that the optimum precision in peak volumes in 2D separations will be obtained when the M_R is between two and five, such that there are typically four to ten second dimension peaks recorded over the eight σ width of the first dimension peak. This sampling rate is similar to that suggested by the Murphy–Schure–Foley criterion. This provides an RSD of approximately 2% for the signal-to-noise ratio used in the present simulations. The precision of the peak volume of experimental data was also assessed, and RSD values were in the range of 4–5%. We conclude that the poorer precision found in the LC × LC experimental data as compared to LC may be due to experimental imprecision in sampling the effluent from the first dimension column.     

基于非接触式电导信号的土壤速效钾含量检测方法

该文针对土壤中速效钾含量采用传统测定法操作复杂、检测时效滞后的问题,建立了基于非接触式电导检测信号快速检测土壤速效钾含量的分析方法。采用高效毛细管电泳/非接触式电导检测仪获取河南潮土的非接触式电导检测信号,并使用导数法与高斯曲线拟合法相结合进行初步峰谱识别,按条件进行峰值过滤后,引入基于Levenberg-Marquardt(L-M)的高斯分峰拟合算法,实现了单峰和重叠峰的拟合计算,得到高斯峰和相应的特征参数,包括峰位、峰高、半峰宽和峰面积;最后将拟合得到的高斯峰及相应的特征参数表征原始非接触式电导检测信号离子峰谱信息结合偏最小二乘法(PLS),确定特征参数与土壤速效钾含量的关系,建立模型,实现了对土壤中速效钾含量的预测。结果表明,将基于L-M的高斯分峰拟合算法结合偏最小二乘法应用于非接触式电导检测信号测定土壤速效钾含量时具有较高精度,回归模型决定系数(R~2)为0.856 4,相对分析误差(RPD)为2.639,适用于土壤速效钾的快速检测分析。     

Evaluation of peak overlap in migration-time distributions determined by organelle capillary electrophoresis: Type-II error analogy based on statistical-overlap theory

Organelles commonly are separated by capillary electrophoresis (CE) with laser-induced-fluorescence detection. Usually, it is assumed that peaks observed in the CE originate from single organelles, with negligible occurrence of peak overlap. Under this assumption, migration-time and mobility distributions are obtained by partitioning the CE into different regions and counting the number of observed peaks in each region. In this paper, criteria based on statistical-overlap theory (SOT) are developed to test the assumption of negligible peak overlap and to predict conditions for its validity. For regions of the CE having constant peak density, the numbers of peaks (i.e., intensity profiles of single organelles) and observed peaks (i.e., maxima) are modeled by probability distributions. For minor peak overlap, the distributions partially merge, and their mergence is described by an analogy to the Type-II error of hypothesis testing. Criteria are developed for the amount of peak overlap, at which the number of observed peaks has an 85% or 90% probability of lying within the 95% confidence interval of the number of peaks of single organelles. For this or smaller amounts of peak overlap, the number of observed peaks is a good approximation to the number of peaks. A simple procedure is developed for evaluating peak overlap, requiring determination of only the peak standard deviation, the duration of the region occupied by peaks, and the number of observed peaks in the region. The procedure can be applied independently to each region of the partitioned CE. The procedure is applied to a mitochondrial CE.     

A simple method to extract spectral parameters using fractional derivative spectrometry

The nonlinear fitting method, based on the ordinary least squares approach, is one of several methods that have been applied to fit experimental data into well-known profiles and to estimate their spectral parameters. Besides linearization measurement errors, the main drawback of this approach is the high variance of the spectral parameters to be estimated. This is due to the overlapping of individual components, which leads to ambiguous fitting. In this paper, we propose a simple mathematical tool in terms of a fractional derivative (FD) to determine the overlapping band spectral parameters. This is possible because of several positive effects of FD connected with the behavior of its zero-crossing and maximal amplitude. For acquiring a stable and unbiased FD estimate, we utilize the statistical regularization method and the regularized iterative algorithm when a priori constraints on a sought derivative are available. Along with the well-known distributions such as Lorentzian, Gaussian and their linear combinations, the Tsallis distribution is used as a model to correctly assign overlapping bands. To demonstrate the power of the method, we estimate unresolved band spectral parameters of synthetic and experimental infra-red spectra.     

Correction method for quantitative area determination of overlapping chromatographic peaks based on the exponentially modified Gaussian (EMG) model

A simple method is presented for peak area correction of overlapping peaks. This correction is necessary for the normal approach of dealing with overlapping peaks by a vertical line at the valley point. The relative area errors caused by this vertical line are calculated as the correction factors in three dimensions of peak separation, peak ratio, and peak tailing skew. The calculation is based on the exponentially modified Gaussian asymmetric peak model.     

Simultaneous deconvolution and re-construction of primary and secondary overlapping peak clusters in comprehensive two-dimensional gas chromatography

In this study, simultaneous deconvolution and reconstruction of peak profiles in the first ((1)D) and second dimension ((2)D) of comprehensive two-dimensional (2D) gas chromatography (GC×GC) is achieved on the basis of the property of this new type of instrumental data. First, selective information, where only one component contributes to the peak elution window of a given modulation event, is employed for stepwise stripping of each (2)D peak with the help of pure components corresponding to that compound from the neighbouring modulations. Simulation based on an exponentially modified Gaussian (EMG) model aids this process, where the EMG represents the envelope of all (2)D peaks for that compound. The peak parameters can be restricted by knowledge of the pure modulated (2)D GC peaks derived from the same primary compound, since it is modulated into several fractions during the trapping and re-focusing process of the cryogenic modulation system according to the modulation period. Next, relative areas of all pure (2)D components of that compound are considered for reconstruction of the primary peak. This strategy of exploitation of the additional information provided by the second dimension of separation allows effective deconvolution of GC×GC datasets. Non-linear least squares curve fitting (NLLSCF) allows the resolved 2D chromatograms to be recovered. Accurate acquisition of the pure profiles in both (1)D and (2)D aids quantification of compositions and prediction of 2D retention parameters, which are of interest for qualitative and quantitative analysis. The ratio between the sum of squares of deconvolution residual and original peak response (R(rr)) is employed as an effective index to evaluate the resolution results. In this work, simulated and experimental examples are used to develop and test the proposed approach. Satisfactory performance for these studies is validated by minimum and maximum R(rr) values of 1.34e-7% and 1.09e-2%; and 1.0e-3% and 3.0e-1% for deconvolution of (1)D and (2)D peaks, respectively. Results suggest that the present technique is suitable for GC×GC data processing.     

粉末衍射峰形拟合程序包CPOWDER

开发了一个Ｘ射线和中子粉末衍射峰形拟合程序包ＣＰＯＷＤＥＲ,它是由多个最小二乘峰形拟合程序与蒙特卡罗分峰程序组成的。最小二乘峰形拟合程序可用于约束条件下的峰形拟合分析,蒙特卡罗分峰程序可用于寻找全局最优解,为最小二乘拟合程序提供优质的峰形参数初始值,两者结合使用特别适合于多个严重重叠峰的分离。     

Achieving high peak capacity production for gas chromatography and comprehensive two-dimensional gas chromatography by minimizing off-column peak broadening

By taking into consideration band broadening theory and using those results to select experimental conditions, and also by reducing the injection pulse width, peak capacity production (i.e., peak capacity per separation time) is substantially improved for one dimensional (1D-GC) and comprehensive two dimensional (GC×GC) gas chromatography. A theoretical framework for determining the optimal linear gas velocity (the linear gas velocity producing the minimum H), from experimental parameters provides an in-depth understanding of the potential for GC separations in the absence of extra-column band broadening. The extra-column band broadening is referred to herein as off-column band broadening since it is additional band broadening not due to the on-column separation processes. The theory provides the basis to experimentally evaluate and improve temperature programmed 1D-GC separations, but in order to do so with a commercial 1D-GC instrument platform, off-column band broadening from injection and detection needed to be significantly reduced. Specifically for injection, a resistively heated transfer line is coupled to a high-speed diaphragm valve to provide a suitable injection pulse width (referred to herein as modified injection). Additionally, flame ionization detection (FID) was modified to provide a data collection rate of 5kHz. The use of long, relatively narrow open tubular capillary columns and a 40°C/min programming rate were explored for 1D-GC, specifically a 40m, 180μm i.d. capillary column operated at or above the optimal average linear gas velocity. Injection using standard auto-injection with a 1:400 split resulted in an average peak width of ～1.5s, hence a peak capacity production of 40peaks/min. In contrast, use of modified injection produced ～500ms peak widths for 1D-GC, i.e., a peak capacity production of 120peaks/min (a 3-fold improvement over standard auto-injection). Implementation of modified injection resulted in retention time, peak width, peak height, and peak area average RSD%'s of 0.006, 0.8, 3.4, and 4.0%, respectively. Modified injection onto the first column of a GC×GC coupled with another high-speed valve injection onto the second column produced an instrument with high peak capacity production (500-800peaks/min), ～5-fold to 8-fold higher than typically reported for GC×GC.     

The divergence from Gaussian of peak shapes in gamma-ray spectra observed with germanium detectors

A formula for the fitting of gamma-ray peaks produced from solid state detectors is suggested, which takes account of possible distortion from a pure Gaussian shape. Comparative analyses of peaks using the formula, and an assumed Gaussian shape are presented showing the improvement in the fitting. When accurate peak energies are required in order to identify unknown nuclides, it is recommended that the expression provided is used, since the calculated peak positions are influenced by the extent of the distortion present.     

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.     

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.