A new algorithm is proposed for simultaneously finding multiple solutions of an underdetermined inverse problem. The algorithm was developed for an ODE parameter identification problem in pharmacokinetics for which multiple solutions are of interest. The algorithm proceeds by computing a cluster of solutions simultaneously, and is more efficient than algorithms that compute multiple solutions one-by-one because it fits the Jacobian in a collective way using a least squares approach. It is demonstrated numerically that the algorithm finds accurate solutions that are suitably distributed, guided by a priori information on which part of the solution set is of interest, and that it does so much more efficiently than a baseline Levenberg-Marquardt method that computes solutions one-by-one. It is also demonstrated that the algorithm benefits from improved robustness due to an inherent smoothing provided by the least-squares fitting.
Introduction.Since the information we can obtain clinically from a live patient going through treatment is often much less extensive than the complexity of the internal activity in a patient's body, underdetermined inverse problems naturally appear in the field of mathematical medicine. Our interest in underdetermined inverse problems of this kind was initiated by the parameter identification problem of a pharmacokinetics model for the anticancer drug CPT-11 (also known as Irinotecan) [2]. This pharmacokinetics model is an ODE-based mathematical model for the transportation, metabolization, and excretion of the drug in a human body. In this problem, a large set of parameters needs to be estimated from a very small number of measurements that correspond to integrated (accumulated) quantities (area-underthe-curve) at a single final time Konagaya has proposed a framework called "virtual patient population convergence" [12] (see [13] for the English translation), whose essential idea is to estimate the parameters of a whole body pharmacokinetics model from the clinically observed patient data. The essential difference of this framework with other approaches is that instead of finding a single set of parameters that is suitable for the pharmacokinetics model to reproduce the clinically observed data, its aim is to find multiple sets of such parameters, because the ranges and extremal values of