Biexponential fitting for noisy data with error propagation

Biexponential time-series models commonly find use in biophysics, biochemistry and pharmacokinetics. Indeed, reactions that are described by biexponential functions are typical for many biological processes. The kinetics of these reactions are modelled by transcendental irrational equations interconnecting the reagent concentrations, time and rate constants. The biexponential is apparently a case of nonlinear regression, and as such, very often the estimate of its parameters is obtained with methods and software tools performing nonlinear fits. The first problem that the user encounters when using these techniques consists in having to provide the software with a not too inaccurate estimate of the intervals within which the parameters can vary. Providing arbitrary initial guesses on the parameter ranges to the nonlinear fit procedure causes its nonconvergence. The second problem is the need to obtain an estimate of the parameters with an error interval due to the propagation of the experimental error that affects the measurements of the dependent variable. In this study, we propose an extension of a well-established mathematical method based on linearization techniques of integral equations for the efficient and unsupervised estimation of the parameters of a biexponential function. Our extension consists in the integration of a model of error propagation from the measurements of the dependent variable to the parameter estimates. There are three main innovative contributions of this work: (1) having made the unsupervised regression method available in experimental practical applications; (2) having provided methods for error propagation in complex operations, such as integration, matrix inversion and multiplication; (3) the calibration of the biexponential dynamics of (i) water desorption of a small ligand from a surface where two types of binding sites are present and (ii) of the decrease of a determinant of viability of organs sustaining ischaemic injury before transplantation.