Adjoint method for a tumor invasion PDE-constrained optimization problem in 2D using adaptive finite element method

Abstract

In this paper we present a method for estimating an unknown parameter that appears in a two dimensional non-linear reaction-diffusion model of cancer invasion. This model considers that tumor-induced alteration of micro-environmental pH provides a mechanism for cancer invasion. A coupled system reaction-diffusion describing this model is given by three partial differential equations for the 2D non-dimensional spatial distribution and temporal evolution of the density of normal tissue, the neoplastic tissue growth and the excess H+ ion concentration. Each of the model parameters has a corresponding biological interpretation, for instance, the growth rate of neoplastic tissue, the diffusion coefficient, the re-absorption rate and the destructive influence of H+ ions in the healthy tissue. The parameter is estimated by solving a minimization problem, in which the objective function is defined in order to compare both the real data and the numerical solution of the cancer invasion model. The real data can be obtained by, for example, fluorescence ratio imaging microscopy. We apply a splitting strategy joint with the adaptive finite element method to numerically solve the model. The minimization problem (the inverse problem) is solved by using a gradient-based optimization method, in which the functional derivative is provided through an adjoint approach.