Parallel factor analysis of HPLC-DAD data for binary mixtures of lidocaine and prilocaine with different levels of chromatographic separation

A set of 17 samples containing a constant amount of lidocaine (667 μM) and a decreasing amount of prilocaine (667-0.3 μM) was analysed by LC-DAD at three different levels of separation, followed by parallel factor analysis (PARAFAC) of the data obtained. In Case 1 no column was connected, the chromatographic resolution (R_s) therefore being zero, while Cases 2 and 3 had partly separated peaks (R_s=0.7 and 1.0). The results showed that in Case 1, analysed without any separation, the PARAFAC decomposition with a model consisting of two components gave a good estimate of the spectral and concentration profiles of the two compounds. In Cases 2 and 3, the use of PARAFAC models with two components resolved the underlying chromatographic, spectral and concentration profiles. The loadings related to the concentration profile of prilocaine were used for regression and prediction of the prilocaine content. The results showed that prediction of prilocaine content was possible with satisfactory prediction (RMSEP<0.01). This study shows that PARAFAC is a powerful technique for resolving partly separated peaks into their pure chromatographic, spectral and concentration profiles, even with completely overlapping spectra and the absence or very low levels of separation.     

Classification of weathered petroleum oils by multi-way analysis of gas chromatography-mass spectrometry data using PARAFAC2 parallel factor analysis

The application of multi-way parallel factor analysis (PARAFAC2) is described for the classification of different kinds of petroleum oils using GC-MS. Oils were subjected to controlled weathering for 2, 7 and 15 days and PARAFAC2 was applied to the three-way GC-MS data set (MSxGCxsample). The classification patterns visualized in scores plots and it was shown that fitting multi-way PARAFAC2 model to the natural three-way structure of GC-MS data can lead to the successful classification of weathered oils. The shift of chromatographic peaks was tackled using the specific structure of the PARAFAC2 model. A new preprocessing of spectra followed by a novel use of analysis of variance (ANOVA)-least significant difference (LSD) variable selection method were proposed as a supervised pattern recognition tool to improve classification among the highly similar diesel oils. This lead to the identification of diagnostic compounds in the studied diesel oil samples.     

On uniqueness and selectivity in three-component parallel factor analysis

Unambiguous recovery of profiles is a distinguishable advantage of Parallel Factor Analysis (PARAFAC) as a trilinear model and has made it a promising exploratory tool for data analysis. Linear dependency in profiles destroys trilinearity and will increase ambiguity in the curve resolution of three-way data sets. PARAFAC uniqueness deteriorates totally or partially in data sets with linearly dependent loadings. Exploiting a reliable method for determination and direct visualization of feasible bands in the PARAFAC model can be helpful not only in full characterization of uniqueness conditions but also in the investigation of the effects of constraints on the PARAFAC feasible solutions. The purpose of this paper is twofold. First, the calculation of rotational ambiguity in the PARAFAC model extends to three components system. The principle behind the algorithm is described in detail and tested for simulated and real data sets. Completely general and thoroughly investigated results are presented for the three component cases. Secondly, the effects of selective regions in the profiles on the resolution of systems that suffered from the rank deficiency problem, due to rank overlap, are emphasized. In the case of two-way data sets the effect of selectivity constraint on the unique recovery of profiles was investigated and applied. However, to our knowledge, in this report, for the first time, the effect of the presence of selective windows in the profiles, on the unique resolution of three-way data sets has been systematically investigated.     

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.     

Three-way Resolution of the Overlapping Ultrahigh-performance Liquid Spectrochromatograms for the Analysis of a Quaternary Mixture Using Parallel Factor Analysis

This paper presents a new application of three-way parallel factor analysis (3W-PARAFAC) model to the coeluting spectrochromatograms for the quantitative resolution of a quaternary mixture system consisting of paracetamol, propyphenazone, and caffeine with aspirin as an internal standard. Spectrochromatograms of calibration standards, validation sets, and unknown samples were recorded as a function of retention time and wavelength in the range of 0.0–2.5?min and 200–400?nm, respectively, using ultra-performance liquid chromatography with photodiode array detection (UPLC-PDA). Three-way UPLC-PDA data array X (retention time?×?wavelength?×?sample) was obtained from the data matrices of the spectrochromatograms. 3W-PARAFAC decomposition of three-way UPLC-PDA data array provided three loading matrices corresponding to chromatographic mode, spectral mode, and relative concentration mode. Quantitative estimation of paracetamol, propyphenazone, and caffeine in analyzed samples was accomplished using the relative concentration mode obtained by the deconvolution of the UPLC-PDA data set. The validity and ability of 3W-PARAFAC model were checked by analyzing independent test samples. It was observed from analyses that 3W-PARAFAC method has potential to uniquely resolve strongly overlapping peaks of analyzed compounds in a spectrochromatogram, which was obtained under experimental conditions consisting of the lower flow rate, short run time, and simple mobile phase composition. The proposed three-way chemometric approach was successfully applied to the simultaneous quantification of paracetamol, propyphenazone, and caffeine in tablets. Experiments showed that the determination results were in good agreement with label amount in commercial pharmaceutical preparation.     

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.     

Comprehensive analysis of chromatographic data by using PARAFAC2 and principal components analysis

The most straightforward method to analyze an obtained GC–MS dataset is to integrate those peaks that can be identified by their MS profile and to perform a Principal Component Analysis (PCA). This procedure has some important drawbacks, like baseline drifts being scarcely considered or the fact that integration boundaries are not always well defined (long tails, co-eluted peaks, etc.). To improve the methodology, and therefore, the chromatographic data analysis, this work proposes the modeling of the raw dataset by using PARAFAC2 algorithm in selected areas of the GC profile and using the obtained well-resolved chromatographic profiles to develop a further PCA model. With this working method, not only the problems arising from instrumental artifacts are overcome, but also the detection of new analytes is achieved as well as better understanding of the studied dataset is obtained. As a positive consequence of using the proposed working method human time and work are saved. To exemplify this methodology the aroma profile of 36 apples being ripened were studied. The benefits of the proposed methodology (PARAFAC2 + PCA) are shown in a practitioner perspective, being able to extrapolate the conclusions obtained here to other hyphenated chromatographic datasets.     

Second-order data by flow injection analysis with spectrophotometric diode-array detection and incorporated gel-filtration chromatographic column

A flow injection analysis (FIA) system furnished with a gel-filtration chromatographic column and with photodiode-array detection was used for the generation of second-order data. The system presented is a model system in which the analytes are blue dextran, potassium hexacyanoferrate(III) and heparin. It is shown that the rank of the involved sample data matrices corresponds to the number of chemical components present in the sample. The PARAFAC (parallel factor analysis) algorithm combined with multiple linear regression and tri-PLS (tri-linear partial least-squares regression), which allows unknown substances to be present in the sample, are implemented for FIA systems and it is illustrated how these three-way algorithms can handle spectral interferents. The prediction ability of the two methods for pure two-component samples and also the predictions ability in the presence of unknown interferents are satisfactory. However, the predictions obtained by tri-PLS are slightly better than those obtained using PARAFAC regression algorithm.     

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。     

Chemometric processing of second-order liquid chromatographic data with UV–vis and fluorescence detection. A comparison of multivariate curve resolution and parallel factor analysis 2

Second-order liquid chromatographic data with multivariate spectral (UV–vis or fluorescence) detection usually show changes in elution time profiles from sample to sample, causing a loss of trilinearity in the data. In order to analyze them with an appropriate model, the latter should permit a given component to have different time profiles in different samples. Two popular models in this regard are multivariate curve resolution-alternating least-squares (MCR-ALS) and parallel factor analysis 2 (PARAFAC2). The conditions to be fulfilled for successful application of the latter model are discussed on the basis of simple chromatographic concepts. An exhaustive analysis of the multivariate calibration models is carried out, employing both simulated and experimental chromatographic data sets. The latter involve the quantitation of benzimidazolic and carbamate pesticides in fruit and juice samples using liquid chromatography with diode array detection, and of polycyclic aromatic hydrocarbons in water samples, in both cases in the presence of potential interferents using liquid chromatography with fluorescence spectral detection, thereby achieving the second-order advantage. The overall results seem to favor MCR-ALS over PARAFAC2, especially in the presence of potential interferents.     

Application of parallel factor analysis to total synchronous fluorescence spectrum of dilute multifluorophoric solutions: Addressing the issue of lack of trilinearity in total synchronous fluorescence data set

In recent years, total synchronous fluorescence (TSF) spectroscopy has become popular for the analysis of multifluorophoric systems. Application of PARAFAC, a popular deconvolution tool, requires trilinear structure in the three-way data array. The present work shows that TSF based three-way array data set of dimension sample × wavelength × Δλ does not have trilinear structure and hence it should not be subjected to PARAFAC analysis. This work also proposes that a TSF data set can be converted to an excitation–emission matrix fluorescence (EEMF) like data set which has trilinear structure, so that PARAFAC analysis can be performed on it. This also enables the retrieval of PARAFAC-separated component TSF spectra.     

Vertex vector sequential projection for the resolution of three-way data

Usually, the PARAFAC2 method is utilized for handling retention time shifts in resolving chromatographic three-way data. It requires all profiles shift the same amount, which, unfortunately, seems unlikely the case in the practice of chromatographic analysis of multi-component samples. The present authors deal with the problem by unfolding the three-way data array along a certain direction into one matrix and setting up a multi-bilinear model. Then, a new method called vertex vector sequential projection (VVSP) is proposed to select pure variables and then the alternating least squares (ALS) procedure is used to iteratively improve the fit of the data to the multi-bilinear model. With a good estimate that is as close as possible to the pure variables, a fast convergence can be expected. Moreover, no prerequisite on the shifting is required and the multi-bilinear model provides a plausible manner to make use of the multi-sample information. An additional advantage is that the present fitting procedure is easier to adjust when constraints such as non-negativity, unimodality, etc., are to be imposed on the loading matrix. The proposed method is evaluated with simulated and real chemical data sets. Satisfactory resolution results are obtained, which demonstrates the performance of the proposed method.     

Determination of phenolic compounds and authentication of PDO Lambrusco wines by HPLC-DAD and chemometric techniques

This work proposes a fast and simple method for detection and quantification of phenolic compounds in PDO Lambrusco wines using HPLC-DAD and chemometric techniques. Samples belonging to three different varieties of Lambrusco (Grasparossa, Salamino and Sorbara) were analyzed to provide a methodology appropriate for routine analysis. Given the high complexity of the sample and the coelution among chromatographic peaks, the use of chemometric techniques to extract the information of the individual eluting compounds was needed. Multivariate curve resolution-alternating least squares (MCR-ALS) allowed the resolution of the chromatographic peaks obtained and the use of this information for the quantification of the phenolic analytes in the presence of interferences. Use of multiset analysis and local rank/selectivity information was proven to be crucial for the correct resolution and quantification of compounds. The quantitative data provided by MCR-ALS about the phenolic targets and additional compounds present in the samples analyzed provided wine composition profiles, which were afterwards used to distinguish among wine varieties. Principal component analysis applied to the wine profiles allowed characterizing the wines according to their varieties.     

Multivariate curve resolution of pH gradient flow injection mixture analysis with correction of the Schlieren effect

Multivariate curve resolution using alternating least squares (MCR-ALS) was used to quantify ascorbic (AA) and acetylsalicylic (ASA) acids in four pharmaceutical samples using a flow injection analysis (FIA) system with pH gradient and a diode array (DAD) spectrometer as a detector. Four different pharmaceutical drugs were analyzed, giving a data array of dimensions 51 x 291 x 61, corresponding respectively to number of samples, FIA times and spectral wavelengths. MCR-ALS was applied to these large data sets using different constraints to have optimal resolution and optimal quantitative estimations of the two analytes (AA and ASA). Since both analytes give an acid-basic pair of species contributing to the UV recorded signal, at least four components sholuld be proposed to model AA and ASA in synthetic mixture samples. Moreover, one additional component was needed to resolve accurately the Schlieren effect and another additional component was also needed to model the presence of possible interferences (like caffeine) in the commercial drugs tablets, giving therefore a total number of 6 independent components needed. The best quantification relative errors were around 2% compared to the reference values obtained by HPLC and by the oxidation-reduction titrimetric method, for ASA and AA respectively. In this work, the application of MCR-ALS allowed for the first time the full resolution of the FIA diffusion profile due to the Schlieren effect as an independent signal contribution, suggesting that the proposed MCR-ALS method allows for its accurate correction in FIA-DAD systems.     

Quantification from highly drifted and overlapped chromatographic peaks using second-order calibration methods

For determining low levels of pesticides and phenolic compounds in river and wastewater samples by high performance liquid chromatography (HPLC), solid phase extraction (SPE) is commonly used before the chromatographic separation. This preconcentration step is not necessarily selective for the analytes of interest and it may retain other compounds of similar characteristics as well. In this case, we present, humic and fulvic acids caused a large baseline drift and overlapped the analytes to be quantified. The inaccurate determinations of the area of the peaks of these analytes made it difficult to quantify them with univariate calibration. Here we compare three second-order calibration algorithms (generalized rank annihilation method (GRAM), parallel factor analysis (PARAFAC) and multivariate curve resolution-alternating least squares (MCR-ALS)) which efficiently solve this problem. These methods use second-order data, i.e., a matrix of responses for each peak, which is easily obtained with a high performance liquid chromatography-diode array detector (HPLC-DAD). With these methods, the area does not need to be directly measured and predictions are more accurate. They also save time and resources because they can quantify analytes even if the peaks are not resolved. GRAM and PARAFAC require trilinear data. Biased and imprecise concentrations (relative standard deviation, %R.S.D. = 34) were obtained without correcting the time-shift. Hence, a time-shift correction algorithm to align the peaks was needed to obtain accurate predictions. MCR-ALS was the most robust to the time-shift. All three algorithms provided similar mean predictions, which were comparable to those obtained when sulfite was added to the samples. However, the predictions for the different replicates were more similar for the second-order algorithms (%R.S.D. = 3) than the ones obtained by univariate calibration after the sulfite addition (%R.S.D. = 13).     

Implementation and efficiency of an automatic peak-purity-control procedure in HPLC-UV-VIS-coupling based on principal component analysis

Summary Based on a thorough knowledge of the actual system precision significance testing of the primary eigen values, resulting from principal component analysis of the two-dimensional data array of HPLC with photodiode-array detection, is a powerful means to uncover unresolved chromatographic peaks. The implementation of this chemometric technique for assuring peak homogeneity and results showing the efficiency for two-component peaks in regard to spectral characteristics, chromatographic resolution and absorbance ratio of the investigated compounds are presented.     

The resolution of overlapping chromatographic peaks by evolving factor analysis

A completely model-free method for the resolution of overlapping chromatographic peaks is presented. Evolving factor analysis enhances the power of classical factor analysis by exploiting the additional information contained in the response data through the intrinsic order of the elution time. The results are the elution profiles and the normalized spectra of the components.     

Handling within run retention time shifts in two-dimensional chromatography data using shift correction and modeling

The use of PARAFAC for modeling GC × GC-TOFMS peaks is well documented. This success is due to the trilinear structure of these data under ideal, or sufficiently close to ideal, chromatographic conditions. However, using temperature programming to cope with the general elution problem, deviations from trilinearity within a run are more likely to be seen for the following three cases: (1) compounds (i.e., analytes) severely broadened on the first column hence defined by many modulation periods, (2) analytes with a very high retention factor on the second column and likely wrapped around in that dimension, or (3) with fast temperature program rates. This deviation from trilinearity is seen as retention time-shifted peak profiles in subsequent modulation periods (first column fractions). In this report, a relaxed yet powerful version of PARAFAC, known as PARAFAC2 has been applied to handle this shift within the model step by allowing generation of individual peak profiles in subsequent first column fractions. An alternative approach was also studied, utilizing a standard retention time shift correction to restore the data trilinearity structure followed by PARAFAC. These two approaches are compared when identifying and quantifying a known analyte over a large concentration series where a certain shift is simulated in the successive first column fractions. Finally, the methods are applied to real chromatographic data showing severely shifted peak profiles. The pros and cons of the presented approaches are discussed in relation to the model parameters, the signal-to-noise ratio and the degree of shift.     

Data handling in HPLC-diode array UV-spectrometry

Summary The mathematical resolution of overlapped chromatographic peaks obtained in HPLC with a diode array UV detector has received considerable attention recently. The purpose of the proposed methods is the quantitation and identification of unknown solutes in complex mixtures by an efficient use of all available data. Basically two approaches have been proposed: the first one resolves the concentration profiles and calculates the pure spectra by applying a minimal number of assumptions, which is denoted self-modeling-curve resolution. The second one is based on a match of pure spectra available in a spectral library with the measured mixture spectra. In this paper both approaches are evaluated with respect to their performance to provide the pure spectra and an accurate quantitation of the concentrations.