Computation of the range of feasible solutions in self-modeling curve resolution algorithms

Self-modeling curve resolution (SMCR) describes a set of mathematical tools for estimating pure-component spectra and composition profiles from mixture spectra. The source of mixture spectra may be overlapped chromatography peaks, composition profiles from equilibrium studies, kinetic profiles from chemical reactions and batch industrial processes, depth profiles of treated surfaces, and many other types of materials and processes. Mathematical solutions are produced under the assumption that pure-component profiles and spectra should be nonnegative and composition profiles should be unimodal. In many cases, SMCR may be the only method available for resolving the composition profiles and pure-component spectra from these measurements. Under ideal circumstances, the SMCR results are accurate quantitative estimates of the true underlying profiles. Although SMCR tools are finding wider use, it is not widely known or appreciated that, in most circumstances, SMCR techniques produce a family of solutions that obey nonnegativity constraints. In this paper, we present a new method for computation of the range of feasible solutions and use it to study the effect of chromatographic resolution, peak height, spectral dissimilarity, and signal-to-noise ratios on the magnitude of feasible solutions. An illustration of its use in resolving composition profiles from a batch reaction is also given.