In this paper we present a method for estimating unknown parameter thatappear in a two dimensional nonlinear reaction-diffusion model of cancerinvasion. This model considers that tumor-induced alteration ofmicroenvironmental pH provides a mechanism for cancer invasion. A coupledsystem reaction-diffusion describing this model is given by three partialdifferential equations for the 2D non-dimensional spatial distribution andtemporal evolution of the density of normal tissue, the neoplastic tissuegrowth and the excess concentration of H+ ions. Each of the model parametershas a corresponding biological interpretation, for instance, the growth rate ofneoplastic tissue, the diffusion coefficient, the re-absorption rate and thedestructive influence of H+ ions in the healthy tissue. After solving thedirect problem, we propose a model for the estimation of parameters by fittingthe numerical solution with real data, obtained via in vitro experiments andfluorescence ratio imaging microscopy. We define an appropriate functional tocompare both the real data and the numerical solution using the adjoint methodfor the minimization of this functional. We apply a splitting strategy jointwith Adaptive Finite Element Method (AFEM) to solve the direct problem and theadjoint problem. The minimization problem (the inverse problem) is solved byusing a trust-region-reflective method including the computation of thederivative of the functional.
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