An alternative quadrilinear decomposition algorithm for four-way calibration with application to analysis of four-way fluorescence excitation–emission–pH data array

A novel quadrilinear decomposition algorithm for four-way calibration (third-order tensor calibration), which was called as regularized self-weighted alternating quadrilinear decomposition (RSWAQLD), has been developed in this work. It originates from the alternating trilinear decomposition (ATLD) algorithm, inherits the philosophy behind self-weighting operation from the self-weighted alternating trilinear decomposition (SWATLD) algorithm. The RSWAQLD algorithm is based on a nearby least-squares scheme, in which two extra terms are added to each loss function, making it more stable and flexible. Experiment shows that RSWAQLD has the features of fast convergence and being insensitive to the excess estimated factors in the model. Owing to its unique optimizing approach, RSWAQLD is much more efficient than four-way PARAFAC. Moreover, the performance of RSWAQLD is quit stable as the number of factors used in calculation varies (as long as it is no less than the true number of factors). Such a feature will simplify the analysis of four-way data arrays, since it is unnecessary to spend a lot of time and effort on accurately determining the appropriate number of factors in the matrix. In addition, the result of four-way fluorescence excitation–emission–pH data, as well as that of simulated data, illustrated that RSWAQLD can not only remain the “higher-order advantage” but also provide a satisfying result even in high collinear systems.     

A novel fourth-order calibration method based on alternating quinquelinear decomposition algorithm for processing high performance liquid chromatography–diode array detection– kinetic-pH data of naptalam hydrolysis

Five-way high performance liquid chromatography–diode array detection (HPLC–DAD)–kinetic-pH data were obtained by recording the kinetic evolution of HPLC–DAD signals of samples at different pH values and a new fourth-order calibration method, alternating quinquelinear decomposition (AQQLD) based on pseudo-fully stretched matrix forms of the quinquelinear model, was developed. Simulated data were analyzed to investigate the performance of AQQLD in comparison with five-way parallel factor analysis (PARAFAC). The tested results demonstrated that AQQLD has the advantage of faster convergence rate and being insensitive to the excess component number adopted in the model. Then, they have been successfully applied to investigate quantitatively the kinetics of naptalam (NAP) hydrolysis in two practical systems. Additionally, the serious chromatographic peak shifts were accurately corrected by means of chromatographic peak alignment method based on abstract subspace difference. The good recoveries of NAP were obtained in these samples by selecting the time region of chromatogram. The elution time, spectral, kinetic time and pH profiles resolved by the chemometric techniques were in good agreement with experimental observations. It demonstrates the potential for the utilization of fourth-order data for some complex systems, opening up a new approach to fourth-order data generation and subsequent fourth-order calibration.     

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.     

三线性分解算法对液相色谱-质谱联用仪多样本测定数据分辨适用性的比较

从三线性分解算法对液相色谱-质谱联用仪(LC-MS)多样本测定数据分辨的适用性角度入手,探讨了双线性和三线性分解算法的实际应用效果、存在的问题及其解决方案。本文选择含有低丰度肽段和高干扰背景信号的代表性测定数据进行测试。结果表明,双线性方法不具有分辨唯一性,不能分离LC-MS多样品测定数据存在的背景干扰,从而不适用于低丰度肽段问题的分析。常规的三线性分解算法难以满足质谱信号具有的数学特性--稀疏性,分辨结果并不完全可靠。本文提出了具有非负约束的交替三线性分解(non-negative alternating trilinear decomposition, NNATLD)新算法用于LC-MS多样本测定数据分辨及数学分离,能够很好地适应质谱的数学特性,且具有计算资源节约和收敛速度快等特点。     

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.     

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.     

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.     

Quantitative analysis of levodopa, carbidopa and methyldopa in human plasma samples using HPLC-DAD combined with second-order calibration based on alternating trilinear decomposition algorithm

An HPLC method combined with second-order calibration based on alternating trilinear decomposition (ATLD) algorithm has been developed for the quantitative analysis of levodopa (LVD), carbidopa (CBD) and methyldopa (MTD) in human plasma samples. Prior to the analysis of the analytes by ATLD algorithm, three time regions of chromatograms were selected purposely for each analyte to avoid serious collinearity. Although the spectra of these analytes were similar and interferents coeluted with the analytes studied in biological samples, good recoveries of the analytes could be obtained with HPLC-DAD coupled with second-order calibration based on ATLD algorithm, additional benefits are decreasing times of analysis and less solvent consumption. The average recoveries achieved from ATLD with the factor number of 3 (N = 3) were 100.1 ± 2.1, 96.8 ± 1.7 and 104.2 ± 2.6% for LVD, CBD and MTD, respectively. In addition, elliptical joint confidence region (EJCR) tests as well as figures of merit (FOM) were employed to evaluate the accuracy of the method.     

三维荧光光谱结合交替三线性分解二阶校正法同时测定血浆样中三种抗癌药物含量

姜黄素、阿霉素和白藜芦醇对多种恶性肿瘤的治愈皆有一定的疗效,且其联合用药对癌细胞的抑制效果更加显著．本文采用三维激发发射荧光光谱结合基于交替三线性分解（Alternating Trilinear Decomposition,ATLD）算法的化学计量学二阶校正法,同时对血浆样中姜黄素、盐酸阿霉素和白藜芦醇的含量进行了定量分析研究．当量测体系的组分数预估计取4时,ATLD方法解析得到的血浆样中姜黄素、盐酸阿霉素和白藜芦醇的平均回收率分别为（91．4±1．1）％、（104．5±2．8）％和（103．4±4．7）％．实验结果表明,此法能够解决干扰共存和荧光光谱重叠下血浆样中姜黄素、盐酸阿霉素和白藜芦醇的直接、快速、同时定量分析的难题．     

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.     

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.      

Direct determination of reserpine in urine using excitation-emission fluorescence combined with three-way chemometric calibration methodologies

The concentration of reserpine in urine was directly and quantitatively measured by using the excitation-emission fluorescence (EEM) combined with three-way calibration methodologies. Two calibration methods are based on the alternating trilinear decomposition (ATLD) and the self-weighted alternating trilinear decomposition (SWATLD) algorithms, respectively. These chemometric methodologies have the second-order advantage, which is the ability to get accurate concentration estimates of interested analyte(s) even in the presence of uncalibrated interferences. The satisfactory results on spiked urine samples are obtained, when the component number was chosen to 3 (N = 3) for both the methods. This experiment is easily carried out without time-consuming and complicated pretreatment. It has proved that the three-way calibration methodologies based on ATLD and SWATLD can be feasible to directly quantify the medical content of reserpine in urine. __________ Translated from Chemical Journal of Chinese Universities, 2007, 28 (5): 827–830 [译自: 高等学校化学学报]     

Trilinear decomposition method applied to removal of three-dimensional background drift in comprehensive two-dimensional separation data

A novel technique for removal of three-dimensional background drift in comprehensive two-dimensional (2D) liquid chromatography coupled with diode array detection (LCxLC-DAD) data is proposed. The basic idea is to perform trilinear decomposition on the instrumental response data, which is based on the alternating trilinear decomposition (ATLD) algorithm. In model construction, the background drift is modeled as one component or factor as well as the analytes of interest, hence, the drift is explicitly included into the calibration. The method involves performing trilinear decomposition on the raw data, then extracting the background component and subtracting this background data from the raw data, leaving the analytes' signal on a flat baseline. Simultaneous evaluation of three-dimensional background drift and true signals may improve the quality of the data. This method is applied to the determination and removal of three-dimensional background drifts in simulated multidimensional data as well as experimental comprehensive two-dimensional liquid chromatographic data. It is shown that this technique yield a good removal of background drift, without the need to perform a blank chromatographic run, and required no prior knowledge about the sample composition.     

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.     

Simultaneous determination of naphazoline and pyridoxine in eye drops using excitation–emission matrix fluorescence coupled with second-order calibration method based on alternating trilinear decomposition algorithm

A novel method is developed for the direct determination of naphazoline hydrochloride(NAP) and pyridoxine hydrochloride(VB6) in commercial eye drops. By using excitation–emission matrix(EEM)fluorescence coupled with second-order calibration method based on the alternating trilinear decomposition(ATLD) algorithm, the proposed approach can achieve quantitative analysis successfully even in the presence of unknown and uncalibrated interferences. The method shows good linearity for NAP and VB6 with correlation coefficients greater than 0.99. The results were in good agreement with the labeled contents. To further confirm the feasibility and reliability of the proposed method, the same batch samples were analyzed by multiple reaction monitoring(MRM) based on LC–MS/MS method.T-test demonstrated that there are no significant differences between the prediction results of the two methods. The satisfactory results obtained in this work indicate that the use of the second-order calibration method coupled with the EEM is a promising tool for industrial quality control and pharmaceutical analysis due to its advantages of high sensitivity, low-cost and simple implementation.     

三维荧光二阶校正法用于未知混合物中苯酚含量的测定

将三维荧光光谱技术与秩消失因子分析、广义秩消失因子分析和交替三线性分解3种二阶校正方法相结合,建立了测定未知混合物中苯酚含量的三维荧光二阶校正新方法。设定在激发波长240~280 nm和发射波长280~360 nm范围内测定未知混合物中苯酚的三维荧光光谱,构建三维响应数据阵,运用基于三线性分解的二阶校正算法进行解析。结果表明,当模拟样品的组分数为2时,秩消失因子分析、广义秩消失因子分析和交替三线性分解3种方法测定苯酚的预测均方根误差分别为0.33,1.18和0.15,平均回收率分别为101.6%,115.6%和101.9%;当组分数为3时,3种方法的预测均方根误差则分别为1.61,1.80和0.51,平均回收率分别为134.2%,133.9%和107.1%;将其分别应用于实际样品中苯酚的测定,结果满意,且交替三线性分解法的测定结果优于秩消失因子分析法和广义秩消失因子分析法。     

Interference-free determination of fluoroquinolone antibiotics in plasma by using excitation-emission matrix fluorescence coupled with second-order calibration algorithms

Fluoroquinolones or so-called second-generation quinolones, in particular, ofloxacin (OFL), norfloxacin (NOR), and enoxacin (ENO), with therapeutic advantages possess strongly overlapped fluorescence spectra. In this paper, two strategies were proposed for simultaneous direct determination of OFL, NOR and ENO in plasma by combining fluorescence excitation-emission matrix (EEM) with second-order calibration based on the alternating trilinear decomposition algorithm (ATLD) and parallel factor analysis (PARAFAC). The results showed that both algorithms could solve the problem of serious fluorescence spectral overlapping of the sought-for analytes even in the presence of uncalibrated interferents. However, ATLD has advantages of being insensitive to overestimated component number and fast convergence. The results by using ATLD with an estimated component number of five were reasonably acceptable for clinical analysis. The average recoveries of OFL, NOR and ENO in synthetic samples were 99.7 ± 2.4, 101.5 ± 2.4 and 97.3 ± 3.8%, respectively; the average recoveries of OFL, NOR and ENO in complex plasma were 94.3 ± 2.6, 85.6 ± 3.3 and 103.3 ± 3.0%, respectively.     

Standardization of near infrared spectra measured on multi-instrument

Calibration model transfer is essential for practical applications of near infrared (NIR) spectroscopy because the measurements of the spectra may be performed on different instruments and the difference between the instruments must be corrected. An approach for calibration transfer based on alternating trilinear decomposition (ATLD) algorithm is proposed in this work. From the three-way spectral matrix measured on different instruments, the relative intensity of concentration, spectrum and instrument is obtained using trilinear decomposition. Because the relative intensity of instrument is a reflection of the spectral difference between instruments, the spectra measured on different instruments can be standardized by a correction of the coefficients in the relative intensity. Two NIR datasets of corn and tobacco leaf samples measured with three instruments are used to test the performance of the method. The results show that, for both the datasets, the spectra measured on one instrument can be correctly predicted using the partial least squares (PLS) models built with the spectra measured on the other instruments.     

三维荧光二阶校正法快速测定块茎类蔬菜、土壤和污水中克百威残留量

克百威是一种高效内吸广谱氨基甲酸酯类杀虫剂.本文充分利用荧光光谱仪操作简单、灵敏度高,化学计量学二阶校正算法具有的"二阶优势",将三维荧光(EEM)与化学计量学交替三线性分解(ATLD)算法相结合,实现了红薯、土豆、红萝卜、土壤和污水5种实际复杂体系中克百威残留量的直接快速定量测定.当选取组分数为2时,用ATLD获得的平均回收率分别为(99.0±5.3)%、(97.2±4.2)%、(102.7±5.9)%、(101.1±3.8)%和(91.3±1.9)%.另外,还用椭圆置信区间(EJCR)测试和品质因子,如灵敏度(SEN)、选择性(SEL)、检测下限(LOD)和预测均方根误差(RSMEP)评估了该种算法的准确性.实验结果表明,该方法能以"数学分离"代替繁琐的"化学分离",成功地解决实际复杂体系中内源干扰物质与分析物光谱重叠所引起的难分辨的问题,可用于未知干扰共存下克百威含量的直接快速定量测定.     

Analytical solution and meaning of feasible regions in two-component three-way arrays

Although many efforts have been directed to the development of approximation methods for determining the extent of feasible regions in two- and three-way data sets; analytical determination (i.e. using only finite-step direct calculation(s) instead of the less exact numerical ones) of feasible regions in three-way arrays has remained unexplored. In this contribution, an analytical solution of trilinear decomposition is introduced which can be considered as a new direct method for the resolution of three-way two-component systems. The proposed analytical calculation method is applied to the full rank three-way data array and arrays with rank overlap (a type of rank deficiency) loadings in a mode. Close inspections of the analytically calculated feasible regions of rank deficient cases help us to make clearer the information gathered from multi-way problems frequently emerged in physics, chemistry, biology, agricultural, environmental and clinical sciences, etc. These examinations can also help to answer, e.g., the following practical question: “Is two-component three-way data with proportional loading in a mode actually a three-way data array?” By the aid of the additional information resulted from the investigated feasible regions of two-component three-way data arrays with proportional profile in a mode, reasons for the inadequacy of the seemingly trilinear data treatment methods published in the literature (e.g., U-PLS/RBL-LD that was used for extraction of quantitative and qualitative information reported by Olivieri et al. (Anal. Chem. 82 (2010) 4510–4519)) could be completely understood.