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.     

Evaluation of variation matrix arrays by parallel factor analysis

PARAFAC is one of the most widely used algorithms for trilinear decomposition. The uniqueness properties of the PARAFAC model are very attractive regardless of whether one is interested curve resolution or not. The fact that PARAFAC provides one unique solution simplifies interpretation of the model. But in three‐way data arrays the uniqueness condition can only be expected when k_A + k_B + k_C ≥ 2F + 2, where F is the number of components and k's are the Kruskal ranks of loadings A to C. As much as second order instruments produce data of varying complexity depending upon the nature of the analytical techniques being combined, with some three‐way data it is possible for patterns generated by the underlying sources of variation to have sufficient independent effects in two modes, yet nonetheless be proportional in a third mode. For example, in three‐way data for spectrophotometric titrations of weak acids or bases (pH‐wavelength‐sample), a rank deficiency may occur in two modes, that is closure rank deficiency in the pH mode and proportionality rank deficiency in the sample direction because each analyte will have acidic and basic forms that are linear combinations in the sample mode. The goal of the present paper is to overcome the non‐uniqueness problem in the second order calibration of monoprotic acids mixtures. The solution contains two steps: first each pH‐absorbance matrix is pretreated by subtraction of the first spectrum from each spectrum in the data matrix. This pretreated data matrix is called the variation matrix. Second, by stacking the variation matrices, a three‐way trilinear variation data array will be obtained without the proportional linear dependency problem that can be resolved uniquely by PARAFAC. It is shown, although unique results are not guaranteed by the Kruscal's condition for the original three‐way data, this condition is fulfilled for pretreated three‐way data. Hence, the variation array may be uniquely decomposed by the PARAFAC algorithm. Studies on simulated as well as real data array reveal the applicability of the proposed method to this kind of problem in the second order calibration of monoprotic acids. Copyright © 2008 John Wiley & Sons, Ltd.     

Fast quantitative analysis of four tyrosine kinase inhibitors in different human plasma samples using three‐way calibration‐ assisted liquid chromatography with diode array detection

A simple method has been developed by combining high‐performance liquid chromatography with diode array detection with the alternating trilinear decomposition method for simultaneous determination of four tyrosine kinase inhibitors in different human plasma samples. Chromatographic separation of the analytes was performed on a reversed‐phase column with methanol (65%, v/v, A) and 0.1% aqueous solution of formic acid (35%, v/v, B). Analysis time was 5.0 min per run and analytes could be completely eluted within 2.8??3.8 min. The calibration concentration ranges of vandetanib, pazopanib, afatinib and dasatinib were designed as 0.50–6.10, 0.50–6.10, 0.70–7.00 and 0.70–7.00 μg·mL^?1, respectively. The intra‐ and inter‐day RSDs ranged between 0.1 and 8.9%. Quantitative information could be extracted from the unsegregated interferences of different human plasma samples with the aid of the “second‐order advantage” of three‐way (second‐order) calibration methods. All results demonstrated that the proposed method for direct quantitative analysis of four tyrosine kinase inhibitors in different complex systems possessed good characteristics of rapidity, sensitivity and efficiency, and it is expected to be an attractive choice in the fast analysis of clinical samples.     

Improvement of residual bilinearization by particle swarm optimization for achieving the second‐order advantage with unfolded partial least‐squares

The combination of unfolded partial least‐squares (U‐PLS) with residual bilinearization (RBL) provides a second‐order multivariate calibration method capable of achieving the second‐order advantage. RBL is performed by varying the test sample scores in order to minimize the residues of a combined U‐PLS model for the calibrated components and a principal component model for the potential interferents. The sample scores are then employed to predict the analyte concentration, with regression coefficients taken from the calibration step. When the contribution of multiple potential interferents is severe, particle swarm optimization (PSO) helps in preventing RBL to be trapped by false minima, restoring its predictive ability and making it comparable to the standard parallel factor (PARAFAC) analysis. Both simulated and experimental systems are analyzed in order to show the potentiality of the new technique. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Excitation-emission-kinetic fluorescence coupled with third-order calibration for quantifying carbaryl and investigating the hydrolysis in effluent water

A novel method for determination of carbaryl in effluent was proposed in this study. The kinetic evolution of excitation-emission matrix fluorescence (EEM) for the pesticide were recorded and come into being a four-way data array. The four-way fluorescence data were analyzed using the parallel factor analysis (PARAFAC). The methodology exploits the second-order advantage of three-order calibration based on quadrilinear parallel factor analysis, allowing analyte concentrations to be estimated even in the presence of an uncalibrated fluorescent background. It gave the satisfactory results for determination of the carbaryl in effluent samples. In addition, the kinetic study of degradation of carbaryl was performed according to the kinetic profile provided by the calibration.     

Fast and simultaneous determination of 12 polyphenols in apple peel and pulp by using chemometrics‐assisted high‐performance liquid chromatography with diode array detection

In this work, a smart chemometrics‐enhanced strategy, high‐performance liquid chromatography, and diode array detection coupled with second‐order calibration method based on alternating trilinear decomposition algorithm was proposed to simultaneously quantify 12 polyphenols in different kinds of apple peel and pulp samples. The proposed strategy proved to be a powerful tool to solve the problems of coelution, unknown interferences, and chromatographic shifts in the process of high‐performance liquid chromatography analysis, making it possible for the determination of 12 polyphenols in complex apple matrices within 10 min under simple conditions of elution. The average recoveries with standard deviations, and figures of merit including sensitivity, selectivity, limit of detection, and limit of quantitation were calculated to validate the accuracy of the proposed method. Compared to the quantitative analysis results from the classic high‐performance liquid chromatography method, the statistical and graphical analysis showed that our proposed strategy obtained more reliable results. All results indicated that our proposed method used in the quantitative analysis of apple polyphenols was an accurate, fast, universal, simple, and green one, and it was expected to be developed as an attractive alternative method for simultaneous determination of multitargeted analytes in complex matrices.     

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.     

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.     

Chemometrics‐assisted liquid chromatography with full scan mass spectrometry for the interference‐free determination of glucocorticoids illegally added to face masks

A smart chemometrics‐assisted strategy that combines the full scan mode of liquid chromatography with mass spectrometry with second‐order calibration method based on alternating trilinear decomposition algorithm was developed for the rapid determination of 15 glucocorticoids including the epimers betamethasone and dexamethasone illegally added into face masks. Fifteen glucocorticoids were rapidly eluted (11 min) under a simple elution program. By means of the second‐order calibration method, 15 target analytes were successfully quantified in the presence of peak overlaps, unknown interferences and baseline drifts. Notably, the epimers, namely, betamethasone and dexamethasone, were simultaneously quantified by the proposed method under a simple elution program. The average spiked recoveries for all target analytes ranged from 87.3 ± 2.2 to 119.4 ± 5.8%. The validation parameters including sensitivity, selectivity, limit of detection, limit of quantitation, and precision were calculated to validate the accuracy of the proposed method, and the quantitative analysis results were further confirmed by liquid chromatography with tandem mass spectrometry. All results proved that the proposed chemometrics‐assisted liquid chromatography with mass spectrometry strategy was an accurate and fast method to determine epimers and multiple glucocorticoids in complex face mask samples.     

Second‐Order Data Obtained from Differential Pulse Voltammetry: Determination of Lead in River Water Using Multivariate Curve Resolution‐Alternating Least‐Squares (MCR‐ALS)

Multivariate curve resolution by alternating least squares (MCR‐ALS) with the aim of achieving the electrochemical second order advantage has been applied to potential‐time second‐order data. In this work a simple way is reported as a first approach towards generation of the instrumental electrochemical second‐order data by differential pulse voltammetry (DPV). A linear dependency exists in the pulse duration profiles of the electroactive species in the mixture samples. Rank deficiency of the mixture data matrix is broken by matrix augmentation. Due to existence of potential shift in the obtained data, MCR‐ALS could not be achieved the convergence on the augmented data. So this shift was corrected with potential shift correction algorithm. Results of MCR‐ALS after shift correction show that the proposed method could be efficiently used for determination of Pb²⁺ in the presence of unexpected interferents in the river water sample.     

Determination of Riboflavin in Human Plasma by Excitation‐Emission Matrix Fluorescence and Multi‐Way Analysis

Direct determination of riboflavin (Fig. 1), a vitamin, in human plasma was accomplished based on excitation‐emission matrix (EEM) fluorescence measurements and multi‐way chemometrics method based on parallel factor analysis (PARAFAC). The PARAFAC trilinear model, without restrictions and using one factor was used in the data analysis. The excitation wavelength range was from 380 to 460 nm and the emission was recorded from 480 to 600 nm. The calibration set was constructed with sixteen standard solutions in a concentration range of 0.02–0.38 μg mL^?1 for riboflavin. The capabilities of the method for the analysis were evaluated by determination of riboflavin in synthetic and real samples with satisfactory results. The accuracy of the methods, evaluated through the root mean square error of prediction (RMSEP), was 0.0059 for riboflavin by the PARAFAC model. Also, partial least squares (PLS) model was built at one excitation wavelength and used to determine a set of synthetic and real samples. The best model was obtained with PARAFAC. This result shows that molecular fluorescence spectroscopy can be used for the development of robust analytical methods for the direct determination of riboflavin in complex backgrounds such as human plasma.     

Hard–soft modeling parallel factor analysis to solve equilibrium processes

PARAFAC model is the most famous model for analyzing three‐way data. However, this method does not converge to chemically meaningful solutions when applied to three‐way problems involving rank overlap profiles at least in one mode. Rank overlap can be simply found where components have similar spectral profiles or analytes appearing in identical proportions throughout an experiment. However, an appropriate selection of the initial parameters and constraints such as non‐negativity and unimodality can still make PARAFAC model useful in this regard. Although such constraints reduce rotational freedom in PARAFAC solution, they are generally insufficient to wholly eliminate the rotational problem. The goal of the present paper is to incorporate hard modeling constraint in the soft‐modeled PARAFAC algorithm to overcome non‐uniqueness problem in the equilibrium processes involving linearly dependent factors at least in one mode. The hard constraint is introduced to force some or all of the concentration profiles to fulfill an equilibrium model that is refined at each iteration cycle of the optimization process of PARAFAC. The proposed approach is called hard–soft PARAFAC (HSPARAFAC). When the rank overlap species obeys equilibrium model in HSPARAFAC, the unique results are obtained even in the presence of non‐modeled interferences. The new modification in the treatment of equilibrium data sets yields more satisfactory results than the exclusive PARAFAC algorithm. Simulated and real examples with rank overlap problem are used to confirm this statement. Copyright © 2011 John Wiley & Sons, Ltd.     

Second‐order calibration serves as a remedial measure for the simultaneous determination of andrographolide and dehydroandrographolide in Andrographis paniculata and its preparations by HPLC without complete baseline separation

In this work, high‐performance liquid chromatography with diode array detection was applied for the simultaneous determination of andrographolide and dehydroandrographolide in Andrographis paniculata and its preparations. As a result of the incomplete baseline separation caused by complex backgrounds, the classical univariate calibration method failed to determine accurate contents of the analytes. On this occasion, chemometric second‐order calibration based on the well‐known alternating trilinear decomposition algorithm was then explored to serve as a post‐experimental remedial tool to solve this problem. By using the intelligent “mathematical separation” of alternating trilinear decomposition, the peak areas of the analytes do not need to be directly measured and the predictive results become accurate. The contents of andrographolide and dehydroandrographolide were determined to be (7.95 ± 0.15) and (1.85 ± 0.02) μg/mL for Andrographis paniculata, (1.34 ± 0.01) and (5.53 ± 0.04) μg/mL for its preparations, which was in agreement with those obtained by a reference liquid chromatography with mass spectrometry method. This study showed the superiority of second‐order calibration method over classical univariate calibration method for simultaneous determination of multi‐analytes in complex samples. It also proved that second‐order calibration may be a good choice for remedying incomplete baseline separation problem, with the accompanied reduction of experimental burden and toxic organic solvents as well as analysis time and cost.     

Comparison of PARAFAC and PARALIND in modeling three‐way fluorescence data array with special linear dependences in three modes: a case study in 2‐naphthol

Owing to excited‐state proton transfer in 2‐naphthol solutions, the fluorescence excitation–emission matrices (EEMs) have factors that are highly dependent in three modes. For the first time such EEMs are used to compare the capability of PARALIND (PARAFAC with linear dependence) and conventional PARAFAC in modeling three‐way EEMs having linearly dependent factors in three modes. Two primary conclusions have been drawn. First, the results indicate that a 3‐factor PARAFAC model fit the data better than two PARALIND models (type 1 and 2) in this case while equally well with a specially PARALIND model (type 3); second, a negative core consistency (CC) in the 3‐factor PARAFAC model is reported but the type 3 PARALIND model reports a nearly 100 CC. This work has demonstrated that a properly constrained PARALIND can fit the very special EEMs of 2‐naphthol. The presence of negative CC associated with a perfect PARAFAC model would imply the presence of very special linear dependences in EEMs, which would be used as an “alarm” for the investigators to interpret the data more carefully when dealing with complicated environmental EEMs in the absence of a priori knowledge. Copyright © 2010 John Wiley & Sons, Ltd.     

Three‐way analysis of a designed compost experiment using near‐infrared spectroscopy and laboratory measurements

Composts are complex organic systems that undergo batch fermentation processes. Traditional monitoring of such processes is usually based on measuring important chemical (physical) laboratory parameters but the common trend includes using more rapid and non‐destructive methods like near‐infrared (NIR) spectroscopy. A lab‐scale designed (simplex mixture) experiment with nine compost batches, including three repeated centre point batches, was monitored over 5 weeks by NIR spectroscopy (900–1700 nm) and by wet chemical and physical measurements: pH, energy content, moisture content, NH₃/NH and temperature. The data were organized in three‐way data arrays and different three‐way methods were used for analysis: (1) PARAFAC, (2) Tucker3 and (3) PARAFAC2. The present paper stresses the advantages and the possibilities of three‐way methods compared to traditional two‐way analysis methods such as principal component analysis (PCA). Two‐way methods have a tendency to mix variables and produce, from a parsimony point of view, more complex models which are hard to interpret. The results from the three‐way methods reproduced the mixture triangle, gave common time profiles (PARAFAC and Tucker3) for all compost batches and rate constants (half‐lives) could be calculated: 6.9 days for the PARAFAC loadings from the chemical/physical parameters and between 6 and 10 days for the PARAFAC loadings from the NIR data. PARAFAC2 includes the possibility of getting individual time profiles for each compost batch. The results show that chemical/physical data and the NIR data give similar interpretations. The conclusion is that three‐way methods can be used to monitor composts batches over time. Copyright © 2011 John Wiley & Sons, Ltd.     

Determination of Sudan I in chilli powder from solvent components gradual change-visible spectra data using second order calibration algorithms

It was discovered that a second order spectra data matrix of Sudan I produced from the solvent components gradual change-visible absorption spectra can be expressed as the combination of two bilinear data matrices. Based on this discovery, a new method for the determination of Sudan I in gray systems using second order calibration algorithms has been developed. The second order calibration algorithms were based on the popular parallel factor analysis (PARAFAC) and rank annihilation factor analysis (RAFA), respectively. In the method described here, the components of the solvent were changed gradually by adding ethanol into cyclohexane, the absorption spectra of Sudan I and chilli samples in a series of cyclohexane-ethanol mixed solvents with various ethanol volume fractions were recorded, and then the second order data were obtained from the solvent components gradual change-visible absorption spectra. Thus, the concentration of Sudan I in a gray system could be determined from the spectra matrices using second order calibration algorithms. This method is simple, convenient and dependable. The method has been used to determine Sudan I in chilli powder with satisfactory results.     

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.     

The Successive Projections Algorithm for interval selection in trilinear partial least-squares with residual bilinearization

In this work the Successive Projection Algorithm is presented for intervals selection in N-PLS for three-way data modeling. The proposed algorithm combines noise-reduction properties of PLS with the possibility of discarding uninformative variables in SPA. In addition, second-order advantage can be achieved by the residual bilinearization (RBL) procedure when an unexpected constituent is present in a test sample. For this purpose, SPA was modified in order to select intervals for use in trilinear PLS. The ability of the proposed algorithm, namely iSPA-N-PLS, was evaluated on one simulated and two experimental data sets, comparing the results to those obtained by N-PLS. In the simulated system, two analytes were quantitated in two test sets, with and without unexpected constituent. In the first experimental system, the determination of the four fluorophores (l-phenylalanine; l-3,4-dihydroxyphenylalanine; 1,4-dihydroxybenzene and l-tryptophan) was conducted with excitation-emission data matrices. In the second experimental system, quantitation of ofloxacin was performed in water samples containing two other uncalibrated quinolones (ciprofloxacin and danofloxacin) by high performance liquid chromatography with UV–vis diode array detector. For comparison purpose, a GA algorithm coupled with N-PLS/RBL was also used in this work. In most of the studied cases iSPA-N-PLS proved to be a promising tool for selection of variables in second-order calibration, generating models with smaller RMSEP, when compared to both the global model using all of the sensors in two dimensions and GA-NPLS/RBL.     

Tucker1 model algorithms for fast solutions to large PARAFAC problems

We describe a method of performing trilinear analysis on large data sets using a modification of the PARAFAC‐ALS algorithm. Our method iteratively decomposes the data matrix into a core matrix and three loading matrices based on the Tucker1 model. The algorithm is particularly useful for data sets that are too large to upload into a computer's main memory. While the performance advantage in utilizing our algorithm is dependent on the number of data elements and dimensions of the data array, we have seen a significant performance improvement over operating PARAFAC‐ALS on the full data set. In one case of data comprising hyperspectral images from a confocal microscope, our method of analysis was approximately 60 times faster than operating on the full data set, while obtaining essentially equivalent results. Copyright © 2008 by John Wiley & Sons, Ltd.