Self-modeling Curve Resolution Method Research Articles

A simple self modelling curve resolution (SMCR) method for two-component systems

Multivariate self-modeling curve resolution (SMCR) methods are the best choice for analyzing chemical data when there is not any prior knowledge about the chemical or physical model of the process under investigation [[1Q3: The reference ‘1’ is only cited in the abstract and not in the text. Please introduce a citation in the text.]]. However, the rotational ambiguity is the main problem of SMCR methods, yielding a range of feasible solutions. It is, therefore, important to determine the range of all feasible solutions of SMCR methods. Different methods have been presented in the literature to find feasible solutions of two, three, and four component systems. Here, a novel simple SMCR method is presented for calculating the boundaries of feasible solutions of two-component systems.At first, the simple strategy is presented for calculating the feasible solutions of two-component systems. Next, four different experimental two-component systems are analyzed in detail for calculating the boundaries of feasible solutions in both spaces, including complex formation equilibrium, keto-enol tautomerization kinetic, lipidomics data, and a case for quantification of an analyte in gray systems. In all cases, the boundaries of range of feasible solutions are properly determined by the proposed simple strategy.

A systematic study on the effects of multi-set data analysis on the range of feasible solutions

The objective of self modeling curve resolution (SMCR) methods is to decompose a second-order bilinear data matrix into a range of chemically meaningful matrices without any knowledge about the chemical or physical model describing the considered system. In addition, SMCR methods are efficient approaches to deeply investigate data structures by finding not only one of the solutions but all possible ones.Multi-set data analysis can be a powerful tool to decrease the range of feasible solutions in the absence of appropriate conditions for unique resolution. Using SMCR methods, we have investigated the impact of multi-set data analysis on the accuracy of soft modeling results. Interestingly, the feasible regions of individual and simultaneous analysis are compared in a common abstract space. It is demonstrated how such global analysis can result in the reduction of rotational ambiguity in soft modeling analysis. Moreover, as a systematic study, different factors are considered in order to discover the advantages and limitations of multi-set data analysis and lead to a proper design for more accurate results.

Data-driven signal-resolving approaches of infrared spectra to explore the macroscopic and microscopic spatial distribution of organic and inorganic compounds in plant.

The nondestructive and label-free infrared (IR) spectroscopy is a direct tool to characterize the spatial distribution of organic and inorganic compounds in plant. Since plant samples are usually complex mixtures, signal-resolving methods are necessary to find the spectral features of compounds of interest in the signal-overlapped IR spectra. In this research, two approaches using existing data-driven signal-resolving methods are proposed to interpret the IR spectra of plant samples. If the number of spectra is small, "tri-step identification" can enhance the spectral resolution to separate and identify the overlapped bands. First, the envelope bands of the original spectrum are interpreted according to the spectra-structure correlations. Then the spectrum is differentiated to resolve the underlying peaks in each envelope band. Finally, two-dimensional correlation spectroscopy is used to enhance the spectral resolution further. For a large number of spectra, "tri-step decomposition" can resolve the spectra by multivariate methods to obtain the structural and semi-quantitative information about the chemical components. Principal component analysis is used first to explore the existing signal types without any prior knowledge. Then the spectra are decomposed by self-modeling curve resolution methods to estimate the spectra and contents of significant chemical components. At last, targeted methods such as partial least squares target can explore the content profiles of specific components sensitively. As an example, the macroscopic and microscopic distribution of eugenol and calcium oxalate in the bud of clove is studied.

Performance assessment of chemometric resolution methods utilized for extraction of pure components from overlapped signals in gas chromatography–mass spectrometry

Multivariate resolution technique is a set of mathematical tools that uncovers the underlying profiles from a set of measurements of time evolving chemical systems. This technique was proposed for resolving the overlapping GC–MS peaks into pure chromatogram and mass spectra. In this paper, several common resolution chemometric techniques in GC–MS resolution such as mean field-independent component analysis (MF-ICA), multivariate curve resolution-alternating least squares (MCR-ALS), and multivariate curve resolution-objective function minimization (MCR-FMIN) were investigated. The obtained solutions using chemometric methods are assessed by lack of fit (LOF) and R2. Results show that all solutions by fulfilling the same constraints have same performance in resolving high overlapping peaks. Also, the differences obtained in each case should be related to the unresolved rotational ambiguity.Among the different ambiguities such as intensity, permutation and rotation in resolution methods, rotational ambiguity is the most difficult and critical one. Because of rotational ambiguity, there is a set of feasible MCR solutions, which explain equally well the observed experimental data, and fulfill sufficiently the imposed constraints of the system. So in these methods, a range of feasible solutions exist. The rotational ambiguities of the profiles are a challenging fact which complicates the development of stable and universal self-modeling curve resolution (SMCR) algorithms. The relative component contribution (RCC) function values for the component profiles obtained by the different methods are calculated by MCR-BANDS. The values of RCC for these three methods are equivalent. Rotational ambiguities of the solutions of SMCR methods can be reduced by applying suitable constraints. The obtained results show, using data sets, which are arranged in a single augmented data matrix could be the best solution for reducing or removing of rotational ambiguity.

Newer developments on self-modeling curve resolution implementing equality and unimodality constraints

Analytical self-modeling curve resolution (SMCR) methods resolve data sets to a range of feasible solutions using only non-negative constraints. The Lawton–Sylvestre method was the first direct method to analyze a two-component system. It was generalized as a Borgen plot for determining the feasible regions in three-component systems. It seems that a geometrical view is required for considering curve resolution methods, because the complicated (only algebraic) conceptions caused a stop in the general study of Borgen’s work for 20 years. Rajkó and István revised and elucidated the principles of existing theory in SMCR methods and subsequently introduced computational geometry tools for developing an algorithm to draw Borgen plots in three-component systems. These developments are theoretical inventions and the formulations are not always able to be given in close form or regularized formalism, especially for geometric descriptions, that is why several algorithms should have been developed and provided for even the theoretical deductions and determinations. In this study, analytical SMCR methods are revised and described using simple concepts. The details of a drawing algorithm for a developmental type of Borgen plot are given. Additionally, for the first time in the literature, equality and unimodality constraints are successfully implemented in the Lawton–Sylvestre method. To this end, a new state-of-the-art procedure is proposed to impose equality constraint in Borgen plots. Two- and three-component HPLC-DAD data set were simulated and analyzed by the new analytical curve resolution methods with and without additional constraints. Detailed descriptions and explanations are given based on the obtained abstract spaces.

Assessment of the co-elution problem in gas chromatography-mass spectrometry using non-linear optimization techniques

Multivariate curve resolution based on the minimization of an objective function (MCR-FMIN) defined directly from the non-fulfillment of constraints was applied for the first time as a deconvolution method to separate co-eluted gas chromatographic-mass spectrometric (GC-MS) signals. Simulated and real (standard real mixture and limon oil) GC-MS data were used to evaluate the feasibility of this method. The MCR-FMIN solutions have been obtained based on the rotation of principal component analysis (PCA) solutions using the non-linear optimization algorithms. Calculation of the initial values of R rotation matrix using model free analysis methods such as fixed-size moving window-evolving factor analysis (FSMW-EFA), evolving latent projective graphs (ELPGs), and heuristic evolving latent projection (HELP) was proposed for faster convergence and avoiding to be stuck in local minima in MCR-FMIN algorithm. The band boundaries of feasible solutions (MCR-BANDS) obtained using MCR-FMIN were calculated for simulated data to assess the reliability of the method. In addition, the results of this method were compared with those of two most common self-modeling curve resolution (SMCR) methods of multivariate curve resolution-alternating least square (MCR-ALS) and HELP. A reasonable result can be obtained by selecting proper constraints, such as non-negativity, unimodality, normalization, and selectivity. However, when the number of components or the level of noise in each peak cluster increase, the convergence of algorithm becomes difficult and the results are not reliable. For quick and accurate analysis of co-eluted multi-component problematic GC-MS data MCR-FMIN can be considered as an alternative method to the MCR-ALS and HELP methods.

Self-modeling curve resolution method applied for the evaluation of dissolution testing data: A case study of meloxicam–mannitol binary systems

This paper introduces some chemometric methods, i.e., self-modeling curve resolution (SMCR), multivariate curve resolution-alternating least squares (MCR-ALS) and parallel factor analysis (PARAFAC and PARAFAC2), which are used to evaluate in vitro dissolution testing data detected by a UV–vis spectrophotometer on meloxicam–mannitol binary systems. These systems were chosen because of their relative simplicity to apply as part of the validation process illustrating the effectiveness of the developed and applied chemometric method. The paper illustrates the failure of PARAFAC methods used before for pharmaceutical data evaluations as well, and we suggest application of the feasible band form given by SMCR as a more general procedure. Steps to improve the dissolution behavior of drugs have become among the most interesting aspects of pharmaceutical technology, and our results show that a larger particle size of meloxicam is advantageous for dissolution. Instead of the use of only one characteristic wavelength, appropriate chemometric methods can furnish more information from dissolution testing data, i.e., the individual dissolution rate profiles and the individual spectra for all the components can be obtained without resorting to any separation techniques such as HPLC.

Studies on the adaptability of different Borgen norms applied in self‐modeling curve resolution (SMCR) method

AbstractLawton and Sylvestre, and later Borgen et al. provided first the analytical solution for determining feasible regions of self‐modeling curve resolution (SMCR) method for two‐ and three‐component systems, respectively. After 20 years, Rajkó and István recently revitalized Borgen's method given a clear interpretation and algorithm how to draw Borgen plots using computer geometry tools; later Rajkó proved the existence of the natural duality in minimal constrained SMCR. In both latter cases, 1‐norm was used to normalize raw data; however Borgen et al. introduced a more general class of normalization. In this paper, the definition and detailed descriptions of Borgen norms are given firstly appearing in the chemical literature. Some theoretical and practical studies on the adaptability of some Borgen norms used for SMCR method are also provided. Copyright © 2009 John Wiley & Sons, Ltd.

Characterization of Cu 2+-binding modes in the prion protein by visible circular dichroism and multivariate curve resolution

Visible circular dichroism (CD) spectra from the copper(II) titration of the metal-binding region of the prion protein, residues 57–98, were analyzed using the self-modeling curve resolution method multivariate curve resolution–alternating least squares (MCR-ALS). MCR-ALS is a set of mathematical tools for estimating pure component spectra and composition profiles from mixture spectra. Model-free solutions (e.g., soft models) are produced under the assumption that pure component profiles should be nonnegative and unimodal. Optionally, equality constraints can be used when the concentration or spectrum of one or more species is known. MCR-ALS is well suited to complex biochemical systems such as the prion protein which binds multiple copper ions and thus gives rise to titration data consisting of several pure component spectra with overlapped or superimposed absorption bands. Our study reveals the number of binding modes used in the uptake of Cu 2+ by the full metal-binding region of the prion protein and their relative concentration profiles throughout the titration. The presence of a non-CD active binding mode can also be inferred. We show that MCR-ALS analysis can be initialized using empirically generated or mathematically generated pure component spectra. The use of small model peptides allows us to correlate specific Cu 2+-binding structures to the pure component spectra.

Application of a self-modeling curve resolution method for studying the photodegradation kinetics of nitrendipine and felodipine

Dihydropyridine (DHP) derivatives, as calcium channel blockers with cardiovascular activity, are highly photosensitive and converted in the presence of light to compounds that are inactive. In this work, a self-modeling curve resolution method was applied to study the photodegradation kinetics of nitrendipine and felodipine by spectrophotometric method. The methanolic solutions of drugs were separately exposed to UV and daylight, respectively. A fully soft-modeling multivariate curve resolution method based on the combination of iterative target transformation and Kubista methods were used to analyze the recorded absorbance data, extracting the concentration profiles and pure spectra of the drugs and their photodegradation products. By fitting the concentration profiles of the studied DHP drugs to different kinetic equations, it was found that at the beginning of lighting, the reaction is zero-order and in the case of nitrendipine it changes to a first-order kinetic when the concentration of products exceeded than that of the initial compounds.

Combining Self-Modeling Curve Resolution Methods and Partial Least Squares to Develop a Quantitative Reaction Monitoring Method with Minimal Reference Data

An example of combining self-modeling curve resolution (SMCR) methods and partial least squares (PLS) to construct a quantitative model using minimal reference data is presented. The objective was to construct a quantitative calibration model to allow real-time in situ ultraviolet-attenuated total reflection (UV/ATR) measurements to determine the end-point during a chlorination reaction. Time restrictions for development combined with difficult reaction sampling conditions required the method to be developed using only a few key reference measurements. Utilizing evolving factor analysis (EFA) and the orthogonal projection approach (OPA), initial estimates of the concentration and spectral profiles for the intermediate and product were obtained. Further optimization by multivariate curve resolution-alternating least squares (MCR-ALS) led to refined estimates of the concentration profiles. A PLS2 model was then constructed using the calculated concentration profiles and the preprocessed UV spectra. Using a standard PLS model compatible with the spectrometer's standard process software facilitated real-time predictions for new batches. This method was applied to five 45 liter batches in a large-scale laboratory facility. The method successfully predicted the product concentration of batch 1 but exhibited larger prediction error for subsequent batches. The largest prediction error was attained during batch 3, for which a final concentration of 0.22 mole L(-1) was predicted, while the true measured value was 0.271 mole L(-1) (an error of 18.8%). However, the qualitative real-time profiles proved to be extremely useful as they allowed the end-point to be determined without sampling or performing off-line analysis. Furthermore, the concentration profile of the intermediate species, which could not be observed by the offline method, could also be observed in real-time and gave further confidence that the process was approaching the end-point. Another benefit of real-time reaction profiles was encountered during the manufacture when the formation of product in batch 3 appeared to be progressing slower than was observed in previous batches. This prompted a check of the batch temperature and it was found to be 10 degrees C lower than the required set-point. The temperature was corrected and the batch successfully reached completion in the expected time.

Application of regularized alternating least squares and independent component analysis to HPLC-DAD data of Haematococcus pluvialis metabolites

The analysis of data from analytical equipment will be an important factor in the execution of metabolomics. Self-modeling curve resolution (SMCR) is one of the theoretical techniques of chemometrics and has recently been applied to the data of hyphenated chromatography techniques. Alternating least squares (ALS) is a classical SMCR method. In ALS, however, different solutions are produced depending on randomly chosen initial values. Simulation in the present study showed that the use of a normalized constraint in calculating ALS was effective in avoiding this problem. We also improved the ALS algorithm by adding a regularized term (regularized ALS: RALS). Independent component analysis (ICA) is a comparatively new method and has been discussed very actively by information science researchers, but has still been applied only in very few cases to curve resolution problems in chemometrics studies. We applied RALS with a normalized constraint and ICA to the HPLC-DAD data of Haematococcus pluvialis metabolites and obtained a high accuracy of peak detection, suggesting that these curve resolution methods are useful for identification of metabolites in metabolomics.

Local resolution of two-way data from multicomponent equilibria

A self-modeling curve resolution (SMCR) method is proposed to calculate concentration and spectral profiles for the two-way spectral data from an equilibrium containing several chemical components. The proposed method has three main distinctive steps: (i) fixed size moving window evolving factor analysis (FSMWEFA) is used to identify the selective and zero concentration regions for a desired component, (ii) orthogonal projection resolution (OPR) is used to calculate its concentration profile and (iii) the component striping is done directly to resolve other components. The results of simulated and real polyprotic acid dissociation equilibria showed that the proposed combined method performs well even in situation when the successive stepwise equilibrium constants are close to each other. The applicability of method for resolving the triprotic acid system with rank deficiency due full spectral overlapping of two involved chemical species also is shown.

Analytical solution for determining feasible regions of self‐modeling curve resolution (SMCR) method based on computational geometry

AbstractDuring 20 years the analytical solution for determining feasible regions of self‐modeling curve resolution (SMCR) method was only slightly researched. After publishing of the famous papers of Borgen et al. in 1985 and 1986, pure generalizations have not been given and even investigated. The reason might have been that the published algorithms and descriptions have been hard to interpret (to code) and to understand respectively. In this paper, several theoretical reasoning is revised giving clearer and more general proofs for principles of SMCR, that is the normalized responses are embraced in the (N−1)‐dimensional simplex with the vertices being the N‐normalized pure profiles. For the first time in the chemometric literature, computational geometry tools are introduced instead of, for example, linear programming tools, for developing our algorithm to draw Borgen plots of any three‐component systems. Numbers of illustrations are given for the sake of clarity. As it has turned out, the highly cited and used data matrix published by Lawton and Sylwestre has not two but more components as it is presented in detail. For a simulated three‐component system, Borgen plots are drawn and some explanations are given. Copyright © 2006 John Wiley & Sons, Ltd.

Enhancement of Infrared Spectral Images for Maximizing Chemical Information by Minimizing Baseline Interferences

The popularity of spectral images in many areas of analysis has greatly increased during the last decade due to the development of charge-coupled device (CCD) and infrared sensitive cameras. Large amounts of spatial information can be obtained in short periods of time. The general goal in analytical chemistry is to convert spectral images into chemical images, which show the spatial locations of various chemical components. Self-modeling multivariate curve resolution methods can be used to extract pure component spectra from the mixture spectra in images and produce chemical images. However, there is a difficulty in processing infrared spectral images due to large pixel-to-pixel baseline variations. Herein, a method for minimizing baseline interferences using fast Fourier transform (FFT) filtering in both the spectral and spatial domains is discussed. The methodology is demonstrated on a microscopic sample of butter contaminated with non-pathogenic E. coli and on a cross-sectional sample of rabbit aorta containing plaque. The processing to reduce baseline effects improved the spatial resolution without compromising the spectral resolution.

Electrochemistry of Conductive Polymers. 34. Two-Dimensional Correlation Analysis of Real-Time Spectroelectrochemical Data for Aniline Polymerization

Electrochemical polymerization of aniline has been studied using real-time spectroelectrochemical experiments conducted concurrently with potentiodynamic scans in a nitric acid electrolyte with its pH adjusted to 0.50. Two-dimensional correlation spectral analysis and subsequent extraction of pure component spectra as well as their relative concentration profiles from complex spectroelectrochemical data employing the alternating least-squares regression iteration-based self-modeling curve resolution method led to positive identification of intermediate species. A conclusion was reached on the polymerization reaction mechanism based on these intermediate species and how their concentrations increased or decreased during the potential scans. The mechanism thus concluded represents a summary of a current understanding of the aniline polymerization reaction.

Resolution of two-way data from on-line Fourier-transform Raman spectroscopic monitoring of the anionic dispersion polymerization of styrene and 1,3-butadiene by parallel vector analysis (PVA) and window factor analysis (WFA)

On-line Fourier-transform Raman spectroscopic monitoring of the anionic dispersion polymerization of styrene and 1,3-butadiene has been implemented with a noninvasive optic fiber Raman probe. After suitable pretreatment of the Raman spectra obtained, the resulting two-way data are analyzed using two self-modeling curve resolution (SMCR) techniques, parallel vector analysis (PVA) proposed in the preceding paper, as well as window factor analysis (WFA), coupled with a slightly modified principal component analysis (PCA) procedure. The idea of PVA is to construct a set of subspaces comprising only one common (spectral) component, and then find a vector that is in parallel with a series of vectors coming from different subspaces. This procedure offers a versatile avenue to approach the unique resolution of spectral profiles. The modified PCA procedure is a useful approach to eliminate the interference from nonreacting species and extract the spectral information concerning only the active reactions. The results reveal that there are three Raman spectrally active species in the system, i.e., styrene, 1,3-butadiene and poly(butadiene), and the product, poly(styrene), turns out to have no Raman signals in the investigated spectral region. The spectral and the concentration profiles of the three species are resolved uniquely using the SMCR methods. The resolved spectral profiles exhibit only small discrepancies compared to the spectra measured for pure species, and the estimated concentration profiles coincide with the results predicted by previous copolymerization theory. These results demonstrate that the proposed PVA method and the modified PCA procedure are competitive approaches for the resolution of two-way data from multivariate spectroscopic monitoring of reactions.

Band-target entropy minimization. A robust algorithm for pure component spectral recovery. Application to complex randomized mixtures of six components.

A newly developed self-modeling curve resolution method, band-target entropy minimization (BTEM), is described. This method starts with the data decomposition of a set of spectroscopic mixture data using singular value decomposition. It is followed by the transformation of the orthonormal basis vectors/loading vectors into individual pure component spectra one at a time. The transformation is based in part on some seminal ideas borrowed from information entropy theory with the desire to maximize the simplicity of the recovered pure component spectrum. Thus, the proper estimate is obtained via minimization of the proposed information entropy function or via minimization of derivative and area of the spectral estimate. Nonnegativity constraints are also imposed on the recovered pure component spectral estimate and its corresponding concentrations. As its name suggests, in this method, one targets a spectral feature readily observed in loading vectors to retain, and then combinations of the loading vectors are searched to achieve the global minimum value of an appropriate objective function. The major advantage of this method is its one spectrum at a time approach and its capability of recovering minor components having low spectroscopic signals. To illustrate the application of BTEM, spectral resolution was performed on FT-IR measurements of very highly overlapping mixture spectra containing six organic species with a two-component background interference (air). BTEM estimates were also compared with the estimates obtained using other self-modeling curve resolution techniques, i.e., SIMPLISMA, IPCA, OPA-ALS, and SIMPLISMA-ALS.

Chemometric resolution of ATR-IR spectra data for polycondensation reaction of bis(hydroxyethylterephthalate) with a combination of self-modeling curve resolution (SMCR) and local rank analysis.

Self-modeling curve resolution (SMCR) methods, simple-to-use interactive self-modeling mixture analysis (SIMPLISMA) and alternating least squares (ALS) were used to calculate pure concentration profiles and pure spectra for the two-way spectral data collected during the on-line polycondensation reaction of bis(hydroxyethylterephthalate) with an ATR-FT-IR spectrometer. In order to improve the resolution results, SIMPLISMA was combined with local rank analysis method, fixed size moving window evolving factor analysis (FSMWEFA) to search for selective regions of various components and then look for the purest wavenumber variables in the selective regions. Such combination allows more accurate determination of the number of chemical components in the reaction system and the calculations of more accurate concentration profiles and spectra.

Dynamic Monte Carlo self-modeling curve resolution method for multicomponent mixtures

A new algorithm for performing self-modeling curve resolution (SMCR) on second-order bilinear data sets is described. The new method, called Dynamic Monte Carlo SMCR (DMC-SMCR), seeks to define boundaries of allowable pure component profiles (spectra, concentrations, etc.) in mixture analysis. The algorithm employs a directed Monte Carlo approach to search for valid solutions with high efficiency. The parameters for the search (direction, step size) are set through bootstrap estimates of the geometry of the solution space. Step sizes are also continuously adjusted to provide a success rate of 50%, ensuring that boundary regions are fully explored. The algorithm employs the usual non-negativity assumptions, but adapts to problems arising from measurement noise by accepting as valid some solutions which may exist slightly outside the principal components subspace. The DMC-SMCR algorithm was successfully applied to four different data sets: (1) UV–VIS spectra from the oxidation of oxalic acid by permanganate (three components), (2) infrared spectra from nitric acid aerosols (three components), (3) fluorescence spectra from a mixture of four polycyclic aromatic hydrocarbons (PAHs), and (4) a simulated six-component mixture. In all cases, the algorithm produced boundaries in good agreement with the known or estimated pure component profiles and calculation times were typically under 5 min on a standard laboratory computer.