Physiologic Parameter Estimation Using Inverse Problems

Abstract

The estimation of in vivo physiologic parameters is an important, but difficult, issue in bio-medicine. Therefore the development of mathematical techniques predicting these parameter values is very relevant. In a previous work we have proposed a convection-diffusion-shape model, which correlates colonic crypt patterns with the cellular kinetics occurring inside the crypts (this correlation is significant in the context of colorectal cancer). This model involves several physiologic parameters, for which only qualitative information is available in the literature, such as the birth rate of proliferative cells. In this paper we present a framework for estimating this birth rate parameter, in a colonic crypt, assuming that the distribution of proliferative cells is known in that crypt. More precisely, we resolve an inverse problem, where the unknown coefficient field, the birth rate, is connected to the observed measurements, the proliferative cell density, through a partial differential equation. This inverse problem is a PDE-constrained optimization problem, highly nonlinear and time-dependent, which is solved by an inexact Newton method. Some test simulations illustrate the efficacy of the proposed parameter inversion and forecast its application with real patient data.