A review of recent methods for the determination of ranges of feasible solutions resulting from soft modelling analyses of multivariate data

Quantification of Uranium in Complex Acid Media: Understanding Speciation and Mitigating for Band Shifts

In situ and real-time analysis of chemical systems, or online monitoring, has numerous benefits in all fields of chemistry. A common challenge can be found in matrix effects, where the addition of a new chemical species causes chemical interactions and changes the fingerprints of other chemical species in the system. This is demonstrated here by looking at the Raman and visible spectra of the uranyl ion within combined nitric acid and hydrofluoric acid media. This system is not only highly important to nuclear energy, a green and reliable option for energy portfolios, but also provides a clear chemistry example that can be applied to other chemical systems. The application of optical spectroscopy is discussed, along with the application and comparison of both multivariate curve resolution and HypSpec to deconvolute and understand speciation. Finally, the use of chemical data science in the form of chemometric modeling is used to demonstrate robust quantification of uranium within a complex chemical system where potential matrix effects are not known a priori.

A new constraint to model background signals when processing chromatographic-spectral second-order data with multivariate curve resolution

BACKGROUND

the chemometric processing of second-order chromatographic-spectral data is usually carried out with the aid of multivariate curve resolution-alternating least-squares (MCR-ALS). When baseline contributions occur in the data, the background profile retrieved with MCR-ALS may show abnormal lumps or negative dips at the position of the remaining component peaks.

RESULTS

The phenomenon is shown to be due to remaining rotational ambiguity in the obtained profiles, as confirmed by the estimation of the boundaries of the range of feasible bilinear profiles. To avoid the abnormal features in the retrieved profile, a new background interpolation constraint is proposed and described in detail. Both simulated and experimental data are employed to support the need of the new MCR-ALS constraint. In the latter case, the estimated analyte concentrations agreed with those previously reported.

SIGNIFICANCE

The developed procedure helps to reduce the extent of rotational ambiguity in the solution and to better interpret the results on physicochemical grounds.

Estimation of confidence intervals for quantitation of coeluted peaks in liquid chromatography-Photodiode array detection through a combination of multivariate curve resolution-alternating least-square and Bayesian inference techniques

There is a dramatic increase in drug candidates that exhibit complex structures and do not comply with Lipinski's rule of five. One of the most critical and complex technical challenges in the quality control of such drug candidates is the control of analogous substances contained in active pharmaceutical ingredients and related formulations. Although the development of ultrahigh-performance liquid chromatography and high-performance columns has improved efficiency per unit time, the difficulty of peak separation to quantify impurities with similar structures and physicochemical properties continues to rise, and so does the probability of failure to achieve the necessary separation. Coeluting peaks observed in the case of high-performance liquid chromatography (HPLC) with photodiode array detection can be separated using the multivariate curve resolution-alternating least-square (MCR-ALS) method exploiting differences in analyte UV spectra. However, relatively large quantitation errors have been observed for coeluting analogous substances, and the reliability of the corresponding quantitative data requires improvement. Herein, Bayesian inference is applied to separation by the MCR-ALS method to develop an algorithm assigning a confidence interval to the quantitative data of each analogous substance. The usefulness and limitations of this approach are tested using two analogs of telmisartan as models. For this test, a simulated two-component HPLC-UV dataset with an intensity ratio (relative to the main peak) of 0.1-1.0 and a resolution of 0.5-1.0 is used. The developed algorithm allows the prediction confidence interval, including the true value, to be assigned to the peak area in almost all cases, even when the intensity ratio, resolution, and signal-to-noise ratio are changed. Finally, the developed algorithm is also evaluated on a real HPLC-UV dataset to confirm that reasonable prediction confidence intervals including true values are assigned to peak areas. In addition to allowing the separation and quantitation of substances such as impurities challenging to separate by HPLC in a scientifically valid manner, which is impossible for conventional HPLC-UV detection, our method can assign confidence intervals to quantitative data. Therefore, the adopted approach is expected to resolve the issues associated with assessing impurities in the quality control of pharmaceuticals.

A new ingenious combination of rank annihilation factor analysis (RAFA) and self-modeling to enhance the unambiguous resolution of multicomponent spectra

In the article, a novel chemometric method for resolution of three-component spectral mixtures is presented and thoroughly scrutinized. Its main core is based on the concept of 'soft' data self-modeling using the SVD factorization (SVD-SM). Each spectrum is then represented as an element of the geometric object known as a simplex which, in this case, takes a form of a triangle. Since its vertices are defined by the spectra of the mixture's pure components, the carried out analysis comes down to a proper determination of the vertices locations. As opposed to the conception of doing so by estimating the areas of feasible solutions (AFS), the idea of unambiguous identification of the simplex's sides is introduced. This may be achieved through a neat application of the rank annihilation factor analysis (RAFA), which allows for generation of two-component difference spectra, whose point representations mark the edges of the searched triangle. Consequently, the obtained final results remain highly unique. In the paper, basics and details of the outlined hybrid RAFA-SVD-SM method are critically discussed from both conceptual and practical points of view. Finally, effectiveness of the proposed algorithm (written in MATLAB) is (successfully) validated on a few series of experimental quenched fluorescence spectra. The high enough recovery rate of the individual component spectra qualifies the proposed approach to be a justifiable alternative for the several well-established chemometric methods tested in the study, including GRAM and PARAFAC.

Leveraging visible and near-infrared spectroelectrochemistry to calibrate a robust model for Vanadium(IV/V) in varying nitric acid and temperature levels

Spectroelectrochemistry and optimal design of experiments can be used to rapidly build accurate models for species quantification and enable a greater level of process awareness. Optical spectroscopy can provide vital elemental and molecular information, but several hurdles must be overcome before it can become a widely adopted analytical method for remote analysis in the nuclear field. Analytes with varying oxidation state, acid concentration, and fluctuating temperature must be efficiently accounted for to minimize time and resources in restrictive hot cell environments. The classic one-factor-at-a-time approach is not suitable for frequent calibration/maintenance operations in this setting. Therefore, a novel alternative was developed to characterize a system containing vanadium(IV/V) (0.01-0.1 M), nitric acid (0.1-4 M), and varying temperatures (20-45 °C). Spectroelectrochemistry methods were used to acquire a sample set selected by optimal design of experiments. This new approach allows for the accurate analysis of vanadium and HNO₃ concentration by leveraging UV-Vis-NIR absorption spectroscopy with robust and accurate chemometric models. The top model's root mean squared error of prediction percent values were 3.47%, 4.06%, 3.40%, and 10.9% for V(IV), V(V), HNO₃, and temperature, respectively. These models, efficiently developed using the designed approach, exhibited strong predictive accuracy for vanadium and acid with varying oxidation states and temperature using only spectrophotometry, which advances current technology for real-world hot cell applications. Additionally, Nernstian analysis of the V(IV/V) standard potential was performed using traditional absorbance methods and multivariate curve resolution (MCR). The successful tests demonstrated that MCR Nernst tests may be valuable in highly convoluted spectral systems to better understand the redox processes' behavior.

A multi-method chemometric analysis in spectroelectrochemistry: Case study on molybdenum mono-dithiolene complexes

Spectroelectrochemical (SEC) analyses combine spectroscopic measurements with electrochemical techniques and can provide deep insight into complex multi-component chemical reaction systems. SEC experiments typically produce large amounts of spectroscopic data. Chemometric techniques are required for the data analysis and aim at extracting the underlying pure component information. Here we analyze spectroelectrochemically gained UV-vis data from five molybdenum mono-dithiolene complexes with changing redox states. SEC enables an electrochemical control of the mixture composition which supports the application of chemometric curve resolution techniques. The factor ambiguity problem is addressed by a multi-method approach combining chemometric tools from the evolving factor analysis (EFA) and from the area of feasible solutions (AFS) methodology in combination with factor duality arguments. EFA enables a subsystem analysis. Two subsystems with three species each are identified, which belong to a reductive and to an oxidative region. A joint species is contained in both regions. A complete pure component decomposition becomes possible in a final step.

Chromatographic Applications in the Multi-Way Calibration Field

In this review, recent advances and applications using multi-way calibration protocols based on the processing of multi-dimensional chromatographic data are discussed. We first describe the various modes in which multi-way chromatographic data sets can be generated, including some important characteristics that should be taken into account for the selection of an adequate data processing model. We then discuss the different manners in which the collected instrumental data can be arranged, and the most usually applied models and algorithms for the decomposition of the data arrays. The latter activity leads to the estimation of surrogate variables (scores), useful for analyte quantitation in the presence of uncalibrated interferences, achieving the second-order advantage. Recent experimental reports based on multi-way liquid and gas chromatographic data are then reviewed. Finally, analytical figures of merit that should always accompany quantitative calibration reports are described.

Achieving the analytical second-order advantage with non-bilinear second-order data

Multi-way calibration based on second-order data constitutes a revolutionary milestone for analytical applications. However, most classical chemometric models assume that these data fulfil the property of low rank bilinearity, which cannot be accomplished by all instrumental methods. Indeed, various techniques are able to generate non-bilinear data, which are all potentially useful for the development of novel second-order calibration methodologies. However, the achievement of the second-order advantage in these cases may be severely limited, since methods for comprehensive modelling of non-bilinear second-order data remain only partially explored. In this research, the analytical performance of three well-known second-order models, namely non-bilinear rank annihilation (NBRA), unfolded partial least-squares with residual bilinearization (U-PLS-RBL) and multivariate curve resolution - alternating least-squares (MCR-ALS) is systematically assessed through sets of simulated and experimental non-bilinear second-order data, involving one analyte and one interferent. Although it is not possible to establish a single strategy to model any type of non-bilinear second-order data with the studied methods, each approach may lead to successful predictions under certain circumstances. It is shown that the prediction capacity is severely affected by data properties such as the level of instrumental noise, the rank of the response matrices and the signal selectivity pattern of the analyte.

Workflow for the Supervised Learning of Chemical Data: Efficient Data Reduction-Multivariate Curve Resolution (EDR-MCR)

A new method termed efficient data reduction-multivariate curve resolution (EDR-MCR) has been devised for classification of high-dimensional data. The method introduces the coupling of EDR and MCR as a new strategy for data splitting, variable selection, and supervised classification of high dimensionality data. The method reduces data dimensionality and selects the training set using principal component analysis (PCA) and convex geometry prior to data classification. Then, the reduced data are categorized using an MCR model, in which numerical constraints are imposed to resolve the data into classes and readily interpretable pure component signal weights. The performance of the EDR and supervised MCR methods were tested for their ability to enable discrimination between the constituents of two benchmark and two high-dimensional data sets. The results were compared with the output of the application of different data splitting methods including iterative random selection (IRS), Kennard-Stone (KS), and discrimination methods including partial least-squares-discriminant analysis (PLS-DA) and the ensemble-learning frameworks of linear discriminant analysis (LDA), k-nearest neighbors (KNN), classification and regression trees (CART), and support vector machine (SVM). Overall, EDR resulted in comparable results with other data splitting methods despite the small size of the training set samples that it created. The proposed MCR approach, in comparison with other commonly used supervised techniques, has the advantages of speed in implementation, tuning of fewer parameters, flexibility in the analysis of data characterized by low sample numbers and class imbalances, improved accuracy from the inclusion of additional system information in the form of numerical constraints, and the ability to resolve pure components signal weights.

A down-to-earth analyst view of rotational ambiguity in second-order calibration with multivariate curve resolution - a tutorial

Rotational ambiguity is a phenomenon with the potential of generating an uncertainty in the estimation of analyte concentrations in protocols based on matrix instrumental data processed by multivariate curve resolution - alternating least-squares (MCR-ALS). This is particularly relevant when the second-order advantage is to be achieved, i.e., when selected analytes are determined in unknown samples having unexpected constituents, not considered in the calibration set of samples. It is therefore imperative that analytical chemists developing second-order multivariate calibration methods using MCR-ALS acknowledge the relevance of this issue, and more importantly, have access to the required tools to size the relative impact of this potential source of uncertainty on the estimated analyte concentrations. The purpose of this tutorial is to provide a down-to-earth view of rotational ambiguity, by studying in detail a synthetic example mimicking a typical chromatographic-spectral experiment, where a set of calibration samples is joined with an unknown sample having an uncalibrated interference. After explaining the background information needed to understand the origin of the phenomenon, the available tools for the estimation of the feasible MCR-ALS solutions and the derived uncertainty on analyte predictions will be discussed. A multi-component experimental system will also be discussed, stressing the fact that rotational ambiguity uncertainties, however small, should always be estimated and reported.

Exploring the potential of combining chemometric approaches to model non-linear multi-way data with quantitative purposes - A case study

Second-order based calibration methods have been widely investigated capitalizing on the inherent benefits of the data structure and the decomposition models, demonstrating that second-order advantage is a property that conspires to a high likelihood success in the resolution of systems of varying complexity. This work aims to demonstrate the applicability of a combined chemometric strategy to solve non-linear multivariate calibration systems in the presence of non-multilinear multi-way data. The determination of histamine by differential pulse voltammetry at different pH is presented as case study. The experimental system has the outstanding difficulty arisen from the large displacement along the potential axis by the pH, which was successfully overcome by implementation of the presented combined strategy. For data modeling, MCR-ALS, U-PLS/RBL and U-PCA/RBL-RBF were used. MCR-ALS allowed unraveling the non-linear behavior between the signal and the concentration, and extracting the underlying profiles of the constituent. Quantitative analysis was performed through the three models, and a comparative evaluation of the predictive performance was done. The best results were achieved with U-PCA/RBL-RBF (mean recovery = 101%) whereas, MCR-ALS yield the lowest mean recovery for all samples (70%).

Chemometric Strategies for Spectroscopy-Based Food Authentication

In the last decades, spectroscopic techniques have played an increasingly crucial role in analytical chemistry, due to the numerous advantages they offer. Several of these techniques (e.g., Near-InfraRed—NIR—or Fourier Transform InfraRed—FT-IR—spectroscopy) are considered particularly valuable because, by means of suitable equipment, they enable a fast and non-destructive sample characterization. This aspect, together with the possibility of easily developing devices for on- and in-line applications, has recently favored the diffusion of such approaches especially in the context of foodstuff quality control. Nevertheless, the complex nature of the signal yielded by spectroscopy instrumentation (regardless of the spectral range investigated) inevitably calls for the use of multivariate chemometric strategies for its accurate assessment and interpretation. This review aims at providing a comprehensive overview of some of the chemometric tools most commonly exploited for spectroscopy-based foodstuff analysis and authentication. More in detail, three different scenarios will be surveyed here: data exploration, calibration and classification. The main methodologies suited to addressing each one of these different tasks will be outlined and examples illustrating their use will be provided alongside their description.

On the analysis of chromatographic biopharmaceutical data by curve resolution techniques in the framework of the area of feasible solutions

Monitoring preparative protein chromatographic steps by in-line spectroscopic tools or fraction analytics results in medium or large sized data matrices. Multivariate Curve Resolution (MCR) serve to compute or to estimate the concentration values of the pure components only from these data matrices. However, MCR methods often suffer from an inherent solution ambiguity which underlies the factorization problem. The typical unimodality of the chromatographic profiles of pure components can support the chemometric analysis. Here we present the pure components estimation process within the framework of the area of feasible solutions, which is a systematic approach to represent the range of all possible solutions. The unimodality constraint in combination with Pareto optimization is shown to be an effective method for the pure component calculation. Applications are presented for chromatograms on a model protein mixture containing ribonuclease A, cytochrome c and lysozyme and on a two-dimensional chromatographic separation of a monoclonal antibody from its aggregate species. The root mean squared errors of the first case study are 0.0373, 0.0529 and 0.0380 g/L compared to traditional off-line analytics. The second case study illustrates the potential of recovering hidden components with MCR from off-line reference analytics.

Multiblock modelling on the study of the kinetic degradation of rosuvastatin calcium in the presence of retention time shifts and rank deficiency

In pharmaceutical development, forced degradation studies are mandatory before the commercialization of any drug product. They aim at identifying the possible degradation routes and the potential products that may be formed during drug product shelf life. The most widely used techniques for monitoring this in the pharmaceutical industry are hyphenated techniques such as Liquid Chromatography coupled to ultraviolet diode array detector (LC-DAD). There are however some drawbacks, such as long analysis times required for the elution of all compounds and coelution, which is not easily detected since degradation products usually have spectra very similar to that of the drug. Chemometrics methods applied to LC-DAD data are capable of solving this issue, but the approaches described in the literature first require peak alignment to solve the rank deficiency problem, which is a delicate preprocessing method for high order data. The present work describes another approach where extra information - the kinetic degradation profiles - is included for the modelling, generating a third-order data set for each sample, resulting in a four-way array (sample x retention times x spectra x degradation profile). This approach has the advantage of using the information in the third mode to solve the peak co-elution problem without the need for peak alignment among samples. With the proposed approach, it was possible to study the degradation of calcium rosuvastatin, a modern cholesterol lowering drug, using a 2 min-run, despite all the challenges in the modelling of this data. The proposed strategy was compared to an approach based on augmenting the matrix in the spectral/kinetic modes (second order modelling strategy).

Spectral Decomposition of X-ray Absorption Spectroscopy Datasets: Methods and Applications

X-ray absorption spectroscopy (XAS) today represents a widespread and powerful technique, able to monitor complex systems under in situ and operando conditions, while external variables, such us sampling time, sample temperature or even beam position over the analysed sample, are varied. X-ray absorption spectroscopy is an element-selective but bulk-averaging technique. Each measured XAS spectrum can be seen as an average signal arising from all the absorber-containing species/configurations present in the sample under study. The acquired XAS data are thus represented by a spectroscopic mixture composed of superimposed spectral profiles associated to well-defined components, characterised by concentration values evolving in the course of the experiment. The decomposition of an experimental XAS dataset in a set of pure spectral and concentration values is a typical example of an inverse problem and it goes, usually, under the name of multivariate curve resolution (MCR). In the present work, we present an overview on the major techniques developed to realize the MCR decomposition together with a selection of related results, with an emphasis on applications in catalysis. Therein, we will highlight the great potential of these methods which are imposing as an essential tool for quantitative analysis of large XAS datasets as well as the directions for further development in synergy with the continuous instrumental progresses at synchrotron sources.

Unraveling VEALYL Amyloid Formation Using Advanced Vibrational Spectroscopy and Microscopy

Intermediate species are hypothesized to play an important role in the toxicity of amyloid formation, a process associated with many diseases. This process can be monitored with conventional and two-dimensional infrared spectroscopy, vibrational circular dichroism, and optical and electron microscopy. Here, we present how combining these techniques provides insight into the aggregation of the hexapeptide VEALYL (Val-Glu-Ala-Leu-Tyr-Leu), the B-chain residue 12-17 segment of insulin that forms amyloid fibrils (intermolecularly hydrogen-bonded β-sheets) when the pH is lowered below 4. Under such circumstances, the aggregation commences after approximately an hour and continues to develop over a period of weeks. Singular value decompositions of one-dimensional and two-dimensional infrared spectroscopy spectra indicate that intermediate species are formed during the aggregation process. Multivariate curve resolution analyses of the one and two-dimensional infrared spectroscopy data show that the intermediates are more fibrillar and deprotonated than the monomers, whereas they are less ordered than the final fibrillar structure that is slowly formed from the intermediates. A comparison between the vibrational circular dichroism spectra and the scanning transmission electron microscopy and optical microscope images shows that the formation of mature fibrils of VEALYL correlates with the appearance of spherulites that are on the order of several micrometers, which give rise to a "giant" vibrational circular dichroism effect.

Soft-trilinear constraints for improved quantitation in multivariate curve resolution

Soft trilinearity constraints give a range of feasible solutions (grey) that envelop the true solution (blue). PARAFAC2 (green) and MCR-ALS results (black) are shown for comparison.

Contribution to second-order calibration based on multivariate curve resolution with and without previous chromatographic synchronization

Rotational ambiguity in the bilinear solutions provided by multivariate curve resolution - alternating least-squares (MCR-ALS) leads to an additional source of uncertainty in the estimation of analyte concentrations by second-order multivariate calibration. The phenomenon is particularly important when measuring matrix instrumental data derived from liquid chromatography with spectral detection, where elution time profiles usually vary from sample to sample both in position and shape. This makes the data non-trilinear, precluding the use of unique trilinear decomposition models. The present report compares some analytical results achieved by: (1) the usual MCR-ALS analysis of augmented matrices built from raw matrix data and (2) a previously reported procedure based on synchronizing the MCR-ALS elution time profiles using correlation optimized warping (COW), reconstructing the augmented matrix with the spectra and the aligned chromatograms, and then applying MCR-ALS again with the trilinearity constraint, leading to unique solutions, which is possible because the warping process restores the trilinearity of the data. We show that alternative (2) does not solve the rotational ambiguity issues and artificially modifies the original data, without significant improvements in analytical performance. In the simulated systems, the best average errors for alternative (1) were about 2%, whereas for alternative (2) they were in the range 4-11%. For the experimental system, the corresponding errors were 2-3% and 3-4% respectively, i.e. with no significant improvement in going to alternative (2). All efforts should be directed to reduce the degree of rotational ambiguity by applying a full battery of chemically reasonable constraints.

Fast Decomposition of Three-Component Spectra of Fluorescence Quenching by White and Grey Methods of Data Modeling

‘White’ and ‘grey’ methods of data modeling have been employed to resolve the heterogeneous fluorescence from a fluorophore mixture of 9-cyanoanthracene (CNA), 10-chloro-9-cyanoanthracene (ClCNA) and 9,10-dicyanoanthracene (DCNA) into component individual fluorescence spectra. The three-component spectra of fluorescence quenching in methanol were recorded for increasing amounts of lithium bromide used as a quencher. The associated intensity decay profiles of differentially quenched fluorescence of single components were modeled on the basis of a linear Stern-Volmer plot. These profiles are necessary to initiate the fitting procedure in both ‘white’ and ‘grey’ modeling of the original data matrices. ‘White’ methods of data modeling, called also ‘hard’ methods, are based on chemical/physical laws expressed in terms of some well-known or generally accepted mathematical equations. The parameters of these models are not known and they are estimated by least squares curve fitting. ‘Grey’ approaches to data modeling, also known as hard-soft modeling techniques, make use of both hard-model and soft-model parts. In practice, the difference between ‘white’ and ‘grey’ methods lies in the way in which the ‘crude’ fluorescence intensity decays of the mixture components are estimated. In the former case they are given in a functional form while in the latter as digitized curves which, in general, can only be obtained by using dedicated techniques of factor analysis. In the paper, the initial values of the Stern-Volmer constants of pure components were evaluated by both ‘point-by-point’ and ‘matrix’ versions of the method making use of the concept of wavelength dependent intensity fractions as well as by the rank annihilation factor analysis applied to the data matrices of the difference fluorescence spectra constructed in two ways: from the spectra recorded for a few excitation lines at the same concentration of a fluorescence quencher or classically from a series of the spectra measured for one selected excitation line but for increasing concentration of the quencher. The results of multiple curve resolution obtained by all types of the applied methods have been scrutinized and compared. In addition, the effect of inadequacy of sample preparation and increasing instrumental noise on the shape of the resolved spectral profiles has been studied on several datasets mimicking the measured data matrices.

Graphical Abstract