A flexible trilinear decomposition algorithm for three-way calibration based on the trilinear component model and a theoretical extension of the algorithm to the multilinear component model

There is a great deal of interest in decompositions of multilinear component models in the field of multi-way calibration, especially the three-way case. A flexible novel trilinear decomposition algorithm of the trilinear component model as a modification of an alternating least squares algorithm for three-way calibration is proposed. The proposed algorithm (constrained alternating trilinear decomposition, CATLD) is based on an alternating approximate least-squares scheme, in which two extra terms are added to each loss function, making it more efficient and flexible. The analysis of simulated three-way data arrays shows that it converges fast, is insensitive to initialization, and is insensitive to the overestimated number of components used in the decomposition. The analysis of real excitation–emission matrix (EEM) fluorescence and real high performance liquid chromatography–photodiode array detection (HPLC–DAD) data arrays confirms the results of the simulation studies, and shows that the proposed algorithm is favorable not only for EEMs but also for HPLC–DAD data. The three-way calibration method based on the CATLD algorithm is very efficient and flexible for direct quantitative analysis of multiple analytes of interest in complex systems, even in the presence of uncalibrated interferents and varying background interferents. Additionally, a theoretical extension of the proposed algorithm to the multilinear component model (constrained alternating multilinear decomposition, CAMLD) is developed.     

Simultaneous determination of metronidazole and tinidazole in plasma by using HPLC–DAD coupled with second-order calibration

<正>A method using HPLC-DAD coupled with second-order calibration was developed to simultaneously determine metronidazole and tinidazole in plasma samples in this paper.The second-order calibration method based on APTLD(alternating penalty trilinear decomposition) algorithm was proposed to analyze the three-way HPLC-DAD data from both standard and prediction samples, which makes it possible that calibration can be performed even in the presence of unknown interferences with a simple and green chromatographic condition and short analysis time.The results showed that good recoveries were obtained although the chromatographic and spectral profiles of the analytes of interest as well as background were partially overlapped with each other in plasma samples.     

三维(二阶)算法在液相色谱分析中的应用

使用"垂线法"、"切线法"或"三角形法"等传统方法对液相色谱重叠峰分辨时经常会遇到误差过大的情况,而使用三维(二阶)算法对重叠和拖尾峰分辨可以最大限度地降低这种因几何分割而人为产生的误差。这样改进的色谱解析方法具有自动化程度高、抗干扰能力强、对重叠/拖尾峰定量准确等优点,甚至可以减少样品前处理和色谱条件优化。该方法的核心是基于化学计量学三维(二阶)算法抽取有效信息和建模的思想,三维色谱数据按照对三线性模型的符合程度有"三线性数据"和"非三线性数据"的区别,相应地将三维(二阶)算法分为"三线性算法"和"非三线性算法"。本文综述了近10年来国内外三维(二阶)算法在复杂体系液相色谱分析中的应用进展,侧重于样品前处理、辅助算法、校正算法间的联用和对比等问题。     

Enhancing the selectivity of liquid chromatography–mass spectrometry by using trilinear decomposition on LC‐MS data: An application to three‐way calibration of coeluting analytes in human plasma

The high selectivities of liquid chromatography and mass spectrometry make liquid chromatography–mass spectrometry one of the most popular tools for quantitative analysis in complex chemical, biological, and environmental systems, while the potential mathematical selectivity of liquid chromatography–mass spectrometry is rarely investigated. This work discussed the mathematical selectivity of liquid chromatography–mass spectrometry by three‐way calibration based on the trilinear model, with an application to quantitative analysis of coeluting aromatic amino acids in human plasma. By the trilinear decomposition of the constructed liquid chromatography–mass spectrometry‐sample trilinear model and individual regression of the decomposed relative intensity versus concentration, the proposed three‐way calibration method successfully achieved quantitative analysis of coeluting aromatic amino acids in human plasma, even in the presence of uncalibrated interferent(s) and a varying background. This analytical method can ease the requirements for sample preparation and complete chromatographic separation of components, reduce the use of organic solvents, decrease the time of chromatographic separation, and increase the peak capacity of liquid chromatography–mass spectrometry. As a “green analytical method”, the liquid chromatography–mass spectrometry three‐way calibration method can provide a promising tool for direct and fast quantitative analysis in complex systems containing uncalibrated spectral interferents, especially for the situation where the coelution problem is difficult to overcome.     

三线性分解方法用于激发-发射矩阵荧光数据中一阶瑞利散射的扣除

由于荧光分析具有检测灵敏度高、数据容易获得等优点,近年来二阶张量校正方法与激发-发射矩阵荧光光谱技术的联用正受到人们越来越多的关注.但是,在三维荧光分析中,经常出现的一阶瑞利散射干扰往往容易导致建立的三线性模型存在较大的偏离,进而直接影响复杂体系中感兴趣组分的定性、定量分析.针对该问题,我们提出了一种基于对组分数不敏感的三线性分解算法扣除一阶瑞利散射干扰的新思路.该方法的特点是根据一阶瑞利散射分别在水平切片矩阵和侧面切片矩阵所处位置相同,沿I-模和J-模同时构建含一阶瑞利散射的三维数据阵,利用三线性分解算法对此各自建模,将一阶瑞利散射当作一个响应组分或因子拟合后从三维数据阵中扣除掉.通过对模拟和实际三维激发发射矩阵荧光光谱实验数据进行讨论,结果表明该方法能有效地扣除体系中的一阶瑞利散射干扰.改进后的方法不仅操作简单,而且不受组分数选取不当的困扰.另外,由于同时从两个方向进行一阶瑞利散射扣除,因此不会出现因边缘瑞利散射峰形不完整而扣除不完全的情况.该方法为三维荧光光谱的无损分析提供了新思路,为进一步进行三维荧光光谱的定量分析奠定了良好的基础.     

Multivariate selectivity as a metric for evaluating comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry subjected to chemometric peak deconvolution

Two-dimensional gas chromatography (GC x GC) coupled to time-of-flight mass spectrometry (TOFMS) [GC x GC-TOFMS)] is a highly selective technique well suited to analyzing complex mixtures. The data generated is information-rich, making it applicable to multivariate quantitative analysis and pattern recognition. One separation on a GC x GC-TOFMS provides retention times on two chromatographic columns and a complete mass spectrum for each component within the mixture. In this report, we demonstrate how GC x GC-TOFMS combined with trilinear chemometric techniques, specifically parallel factor analysis (PARAFAC) initiated by trilinear decomposition (TLD), results in a powerful analytical methodology for multivariate deconvolution. Using PARAFAC, partially resolved components in complex mixtures can be deconvoluted and identified without requiring a standard data set, signal shape assumptions or any fully selective mass signals. A set of four isomers (iso-butyl, sec-butyl, tert-butyl, and n-butyl benzenes) is used to investigate the practical limitations of PARAFAC for the deconvolution of isomers at varying degrees of chromatographic resolution and mass spectral selectivity. In this report, multivariate selectivity was tested as a metric for evaluating GC x GC-TOFMS data that is subjected to PARAFAC peak deconvolution. It was found that deconvolution results were best with multivariate selectivities over 0.18. Furthermore, the application of GC x GC-TOFMS followed by TLD/PARAFAC is demonstrated for a plant metabolite sample. A region of GC x GC-TOFMS data from a complex natural sample of a derivatized metabolic plant extract from Huilmo (Sisyrinchium striatum) was analyzed using TLD/PARAFAC, demonstrating the utility of this analytical technique on a natural sample containing overlapped analytes without selective ions or peak shape assumptions.     

三维荧光校正法直接测定尿液中的利血平

利用化学计量学三维数据校正方法中的交替三线性分解算法(ATLD)和自加权交替三线性分解算法(SWATLD), 不经化学分离, 对采用激发-发射矩阵荧光法所得到的三维响应数据阵进行三线性成分分解, 再基于标样已知浓度, 利用简单回归法直接测定尿液中利血平(Reserpine)的含量. 结果表明, 当体系的主要组分数取3时, 两种方法均可迅速、 快捷地得到待测物的浓度, 有效地解决了荧光法定量测定时未知背景及干扰物光谱严重重叠而引起的问题.     

Comprehensive two-dimensional liquid chromatography of polymers

The need for and the emergence of comprehensive two-dimensional liquid chromatographic separations of synthetic polymers are reviewed in this paper. LC×SEC is shown to be a particularly valuable two-dimensional technique in this domain. An improved (symmetrical) configuration based on a single 10-way switching valve is described. The use of LC×SEC to understand and optimize one-dimensional separations is illustrated, as well as the potential of the technique for the separation and characterization of functional polymers and copolymers.     

Use of pseudo-sample extraction and the projection technique to estimate the chemical rank of three-way data arrays

Determining the rank of a trilinear data array is a first step in subsequent trilinear component decomposition. Different from estimating the rank of bilinear data, it is more difficult to decide the significant number of component to fit the trilinear decompositions exactly. General methods of rank estimation utilize the information contained in the singular values but ignore information from eigenvectors. In this paper, a rank estimating method specifically for trilinear data arrays is proposed. It uses the idea of direct trilinear decomposition (DTLD) to compress the cube matrix into two pseudo sample matrices which are then decomposed by singular value decomposition. Two eigenvectors combined with the projection technique are used to estimate the rank of trilinear data arrays. Simulated trilinear data arrays with homoscedastic and heteroscedastic noise, different noise levels, high collinearity, and real three-way data arrays have been used to illustrate the feasibility of the proposed method. Compared with other factor-determining methods, for example use of the factor indication function (IND), residual percentage variance (RPV), and the two-mode subspace comparison approach (TMSC), the results showed that the new method can give more reliable answers under the different conditions applied.      

三线性直接分解法分析高维灰色体系

对于由多个两维测量数据组成的三维阵，本文提出一种新三线性直接分解方法。采用高维ＰＣＡ分解，从三维阵中直接提取抽象光谱和抽象浓度，再结合ＱＺ算法，唯一地确定混合物中各组分光谱的浓度。该方法可以排除其它未知组分的干扰，适用于高维灰色体系定性定量分析和多点校准。用模拟数据讨论了光谱分离度对该方法的影响，应用于混合维生素Ｂ１、Ｂ２和Ｂ６的荧光分析，求得的光谱和浓度与实验值吻合很好。     

Image background removal in comprehensive two-dimensional gas chromatography

This paper describes a new technique for removing the background level from digital images produced in comprehensive two-dimensional gas chromatography (GCxGC). Background removal is an important first step in the larger problem of quantitative analysis. The approach estimates the background level across the chromatographic image based on structural and statistical properties of GCxGC data. Then, the background level is subtracted from the image, producing a chromatogram in which the peaks rise above a near-zero mean background. After the background level is removed, further analysis is required to determine the quantitative relationship between the peaks and chemicals in the sample. The algorithm is demonstrated experimentally to be effective at determining and removing the background level from GCxGC images. The algorithm has several parametric controls and is incorporated into an interactive program with graphical interface for rapid and accurate detection of GCxGC peaks.     

Two‐way and three‐way approaches to ultra high performance liquid chromatography–photodiode array dataset for the quantitative resolution of a two‐component mixture containing ciprofloxacin and ornidazole

Two‐way and three‐way calibration models were applied to ultra high performance liquid chromatography with photodiode array data with coeluted peaks in the same wavelength and time regions for the simultaneous quantitation of ciprofloxacin and ornidazole in tablets. The chromatographic data cube (tensor) was obtained by recording chromatographic spectra of the standard and sample solutions containing ciprofloxacin and ornidazole with sulfadiazine as an internal standard as a function of time and wavelength. Parallel factor analysis and trilinear partial least squares were used as three‐way calibrations for the decomposition of the tensor, whereas three‐way unfolded partial least squares was applied as a two‐way calibration to the unfolded dataset obtained from the data array of ultra high performance liquid chromatography with photodiode array detection. The validity and ability of two‐way and three‐way analysis methods were tested by analyzing validation samples: synthetic mixture, interday and intraday samples, and standard addition samples. Results obtained from two‐way and three‐way calibrations were compared to those provided by traditional ultra high performance liquid chromatography. The proposed methods, parallel factor analysis, trilinear partial least squares, unfolded partial least squares, and traditional ultra high performance liquid chromatography were successfully applied to the quantitative estimation of the solid dosage form containing ciprofloxacin and ornidazole.     

Multivariate curve resolution based chromatographic peak alignment combined with parallel factor analysis to exploit second-order advantage in complex chromatographic measurements

In the present contribution, a new combination of multivariate curve resolution-correlation optimized warping (MCR-COW) with trilinear parallel factor analysis (PARAFAC) is developed to exploit second-order advantage in complex chromatographic measurements. In MCR-COW, the complexity of the chromatographic data is reduced by arranging the data in a column-wise augmented matrix, analyzing using MCR bilinear model and aligning the resolved elution profiles using COW in a component-wise manner. The aligned chromatographic data is then decomposed using trilinear model of PARAFAC in order to exploit pure chromatographic and spectroscopic information. The performance of this strategy is evaluated using simulated and real high-performance liquid chromatography-diode array detection (HPLC-DAD) datasets. The obtained results showed that the MCR-COW can efficiently correct elution time shifts of target compounds that are completely overlapped by coeluted interferences in complex chromatographic data. In addition, the PARAFAC analysis of aligned chromatographic data has the advantage of unique decomposition of overlapped chromatographic peaks to identify and quantify the target compounds in the presence of interferences. Finally, to confirm the reliability of the proposed strategy, the performance of the MCR-COW-PARAFAC is compared with the frequently used methods of PARAFAC, COW-PARAFAC, multivariate curve resolution-alternating least squares (MCR-ALS), and MCR-COW-MCR. In general, in most of the cases the MCR-COW-PARAFAC showed an improvement in terms of lack of fit (LOF), relative error (RE) and spectral correlation coefficients in comparison to the PARAFAC, COW-PARAFAC, MCR-ALS and MCR-COW-MCR results.     

Rapid Determination of Costunolide and Dehydrocostuslactone in Human Plasma Sample and Chinese Patent Medicine Xiang Sha Yang Wei Capsule Using HPLC‐DAD Coupled with Second‐order Calibration

A novel methodology that combines high performance liquid chromatography with photodiode‐array detector (HPLC‐DAD) coupled with second‐order calibration method based on alternating trilinear decomposition (ATLD) algorithm was used in determination of the effective constituents such as costunolide and dehydrocostuslactone, in plasma sample and Chinese patent medicine Xiang Sha Yang Wei (XSYW) capsule. Complicated systems such as plasma and Chinese patent medicine which have intricate components are tedious to isolate and purify. The problem that chromatographic peaks are heavily overlapped among the analytes and interferents from the background matrices can be resolved, and the satisfactory quantification results have been gained with the help of the ATLD algorithm which utilized "mathematical separation" instead of partial "physical or chemical separation". Meanwhile, HPLC‐MS/MS method was used to validate the accuracy of the proposed determination method.     

Data analysis programs for comprehensive two-dimensional chromatography

User-friendly and easy-to-use laboratory-written programs for visualisation and interpretation of comprehensive two-dimensional chromatographic data were developed. The programs that are not tied to any particular commercial instrument, and data obtained either by comprehensive two-dimensional liquid (LC × LC) or gas (GC × GC) chromatography can be analysed. Operations of the programs allow visualisation of 2D and 3D plots, comparison of two 2D plots at a time, as well as determination of retention times and peak heights and volumes.     

Simultaneous determination of psoralen and isopsoralen in plasma and Chinese medicine Xian Ling Gu Bao capsule by using HPLC-DAD coupled with alternating trilinear decomposition algorithm

A method using high performance liquid chromatography with photodiode-array detection (HPLC-DAD) coupled with alternating trilinear decomposition (ATLD) algorithm was proposed for simultaneous determination of psoralen and isopsoralen in plasma and Chinese medicine “Xian Ling Gu Bao” capsule (XLGBC). In this paper, the application of ATLD algorithm into traditional chromatographic method can handle this problem that the chromatographic and spectral peaks are heavily overlapped among the analytes and even between the analytes and interferences from the background matrices. A simple improvement of chromatographic condition like mobile phase is not enough to realize effective separation for the two isomeric compounds, especially in the presence of interferences. However, the ATLD algorithm utilized “mathematical separation” instead of partial “physical or chemical separation” to directly determine the spectral profiles of the analytes of interests in complex system. The satisfactory quantification results have been gained with simple mobile phase. In the analysis of real Chinese medicine samples, the accuracy of the concentrations which were obtained by ATLD was also validated by HPLC-MS method.     

Algorithm combination strategy to obtain the second‐order advantage: simultaneous determination of target analytes in plasma using three‐dimensional fluorescence spectroscopy

Although a number of algorithms have established to obtain the well‐known second‐order advantage that quantifies analytes of interest in the presence of interferents, each has associated problems. In this work, for the first time, the optimization procedure of trilinear decomposition has been divided into three subparts, and a novel strategy is developed for assembling the advantages of the alternating trilinear decomposition (ATLD) algorithm, the self‐weighted alternating trilinear decomposition (SWATLD) algorithm, and the parallel factor analysis (PARAFAC) algorithm. The performance of the proposed strategy was evaluated using a simulated data set, a published fluorescence data set together with a new fluorescence data set that simultaneously quantifies procaine and tetracaine in plasma. Results show that the novel method can accurately and effectively estimate the qualitative and quantitative information of analytes of interest. Besides, the resolved profiles are very stable with respect to the number of components as long as the employed number is chosen to be equal or larger than the underlying one. Additionally, the study confirms that better prediction can be obtained by the new strategy when compared with ATLD, SWATLD, and PARAFAC as well as the strategy that employs direct trilinear decomposition method as initial values for PARAFAC. Moreover, the strategy can be directly extended to third‐order or higher‐order data analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimating the chemical rank of three-way data arrays by a simple linear transform incorporating Monte Carlo simulation

Estimating an appropriate chemical rank of a three-way data array is very important to second-order calibration. In this paper, a simple linear transform incorporating Monte Carlo simulation approach (LTMC) to estimate the chemical rank of a three-way data array was suggested. The new method determines the chemical rank through performing a simple linear transform procedure on the original cube matrix to produce two subspaces by singular value decomposition. One of two subspaces is derived from the original three-way data array itself and the other is derived from a new three-way data array produced by the linear transformation of the original one. Projection technique incorporating the Monte Carlo approach acts as distinguishing criterion to choose the appropriate component number of the system. Simulated three-way trilinear data arrays with different noise types (homoscedastic and heteroscedastic), various noise level as well as high collinearity are used to illustrate the feasibility of the new method. The results have shown that the new method could yield accurate results with different conditions appended. The feasibility of the new method is also confirmed by two real arrays, HPLC-DAD data and excitation-emission fluorescent data. All the results are compared with the other three factor-determining methods: factor indicator function (IND), core consistency diagnostic (CORCONDIA) and two-mode subspace comparison (TMSC) approach. It shows that the newly proposed algorithm can objectively and quickly determine the chemical rank to fit the trilinear model.     

Factors that affect quantification of diode array data in comprehensive two-dimensional liquid chromatography using chemometric data analysis

There has been a tremendous increase in research on comprehensive two dimensional LC (LC×LC); however, to date, the central analytical issue, quantification, has received only minimal attention. It is vital to the further development of LC×LC that a greater understanding of the specific factors affecting peak quantification in LC×LC be attained. This work focuses on the following factors: data complexity, retention time shifting, dynamic range issues, chromatographic and spectral peak overlap and difficulties related to background signal removal. The above mentioned factors that affect peak quantification are investigated using fourteen replicate analyses of a urine sample, representing the effects of such factors when analyzing samples in complex matrices. We demonstrate that quantification of LC×LC data is improved following implementation of chemometric techniques that minimized the deleterious effects on quantification due to chromatographically overlapped peaks, retention time shifting and background signal interference. The chemometrically resolved data shows a 2.5-fold increase in precision of quantification over the quantification of the raw data. It is also demonstrated that the method quantifies sixteen peaks that were not visually evident prior to chemometric analysis. The purpose of this paper is to determine the impact of these issues on the effectiveness of LC×LC as a technique for the quantitative analysis of complex samples.