Soft Trilinearity Constraints in Second Order Calibration

Multivariate Curve Resolution (MCR) methods work by determining the pure component response profiles and their estimated concentrations when no prior information is available about the composition of the mixture. Experimental measurements generated by analytical instruments are so called second-order data when there is one matrix per sample and one measurement mode modulates the other, for example as in HPLC-DAD. If the second-order data follows a trilinear mathematical model this information can be applied as a constraint in MCR. Second-order data are obtained when measurements in a third-mode modulate the response matrices for each sample. In practice, data analysis of trilinear data may be complicated by the fact that there are shifts in retention times of specific analytes from sample to sample. In such cases, a trilinear model is not valid and every analyte may not have the same elution profile in every sample, a strict requirement for trilinear behavior. Moreover, the sensitivity of the spectral response can change in the presence of different chemical matrix effects from sample to sample. In such cases the unique profiles normally obtained by strictly enforced “hard” trilinearity constraints may not necessarily represent the true profiles because the data set as a whole does not follow trilinear behavior. “Soft” constraints allow small deviations from constrained values and were first introduced by Gemperline [1]. He introduced a new algorithm using least squares penalty functions to implement constraints during alternating least squares steps. Later, Rahimdoust et.al visualized the effect of “soft” constraints on the accuracy of self-modeling curve resolution methods [2]. In this study, the use of soft trilinearity constraints rather than hard trilinearty constraints is investigated. For example, in the analysis of HPLC-DAD data, soft trilinearity constraints allow small deviations in HPLC peak shape or retention time. Simulated chromatography data sets are presented to evaluate the performance of the new algorithm, and a real case of an HPLC-DAD chromatogram of a three-compound system with two identified pesticides (azinphos-ethyl and fenitrothion) and one unknown interferent is analyzed. The results show that by imposing soft trilinearity constraints, a range of possible solutions is calculated that follow the trilinearity criteria, whereas by imposing hard trilinearity constraints, the unique solution that is calculated is not a correct solution and the result suffers from the presences of active constraints.