Multivariate curve resolution algorithms

In the ATR FTIR study of the synthesis of cyclopentyl silsesquioxane 7F3, in situ ATR FTIR spectra of the reaction mixture were collected every 2 min during the reaction. The spectra obtained were plotted as a function of reaction time (Fig. 9.11). Pure component spectra and relative concentration profiles were subsequently recovered using a multivariate curve resolution (MCR) [59] technique based on a modified target factor analysis algorithm [60]. 

Multivariate curve resolution is the main topic of Malinowski s book [23]. The author is a physical chemist and so the book is oriented towards that particular audience, and especially relates to the spectroscopy of mixtures. It is well known because the first edition (in 1980) was one of the first major texts in chemometrics to contain formal descriptions of many common algorithms such as principal components analysis. 

Multivariate curve resolution-alternating least squares (MCR-ALS) is an algorithm that fits the requirements for image resolution [71, 73-75]. MCR-ALS is an iterative method that performs the decomposition into the bilinear model D = CS by means of an alternating least squares optimization of the matrices C and according to the following steps ... 

Ramos et al. [13] first separated the individual spectra of benzo[b]fluoranthene and benzo[k]fluoranthene or chrysene and benz[a]anthracene from each other in a purposely created co-eluting peak of mixtures of each of the two pairs of PAHs. They were then able to deconvolute the individual spectra from a mixture of the isomers benzo[e]pyrene, benzo[b]fluoranthene, and benzo[k]fluoranthene in a similar fashion. In all cases they purposely generated peaks with severe overlap (greater than 90 % of each peak co-eluting with the other components) to show the power of deconvolution. Tauler et al. described another algorithm to deconvolute the individual spectra in co-eluting peaks and reviewed similar efforts by others. Multivariate curve resolution has been used as an alternative approach for peak deconvolution. It can identify minor impurity peaks and yield the true retention times [14, 15]. 

Apart from discrete modelling of relaxation processes, ID and 2D Inverse Laplace Transformation (ILT) is gaining more and more interest. Moreover, soft and hard modelling data processing tools like PLS or multivariate curve resolution (MCR) are applied to low field NMR data. Special algorithms were developed for the needs in relaxation modelling, for example DOUBLESLICING l... 

Multivariate Curve Resolution used in the quantitative mode was compared to PLS predichon results [25]. The MCR algorithm used a correlation constraint that related the values from the calculated concentrahon profiles to the known concentrahon of the active and other excipients [24]. 

Llamas et al. developed other spectrophotometric methods for the determination of Amaranth, Sunset Yellow, and Tartrazine in beverages [32]. The spectra of the samples (simply filtered) were recorded between 359 and 600 nm, and mixtures of pure dyes, in concentrations between 0.01 and 1.8 mg/L for Amaranth, 0.08 and 4.4 mg/L for Sunset Yellow, and 0.04 and 1.8 mg/L for Tartrazine, were disposed in a column-wise augmented data matrix. This kind of data structure, analyzed by multivariate curve resolution-alternating least squares (MCR-ALS), makes it possible to exploit the so-called second-order advantage. The MCR-ALS algorithm was applied to the experimental data under the nonnegativity and equality constraints. As a result, the concentration of each dye in the sample and their corresponding pure spectra were obtained. 

Multivariate curve resolution (MCR) [14] is the latest of the methods that seek to constrain the rotation of the matrices by forcing the C and S matrices to obey certain restrictions. One such restriction is that all absorbances in the spectra must be greater than or equal to zero. This restriction was also imposed in SAO-ITTFA [21]. In MCR it can be imposed that none of the concentrations will be less than zero. Other restrictions can result from chemical mass balance equations to determine composition in a dynamic system, or knowledge of some of the pure component spectra can be fed into the method algorithm to resolve other pure component spectra. One major advantage of MCR (and SAO-ITTFA) is to determine concentration profiles from kinetic systems as well as to determine the spectra of transient species. 

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