Numerical solution of metastatic tumor growth models with treatment

In this paper we introduce an efficient numerical method in order to solve Volterra integral equations (VIE) of the second type. We are motivated by the fact that the coupled PDE-ODE model, used to describe the metastatic tumor growth, can be reformulated in terms of VIE, whose unknowns are biological observables, such as the cumulative number of metastases and the total metastatic mass. Here in particular we focused our attention on the 2D non autonomous case, where also the treatment is considered. After reformulating the model as a VIE and introducing and studying the numerical method, we first compare it with a method previously introduced by the authors for the 1D case, and extended to the 2D case only for the sake of comparison, in term of efficiency in the run time execution. Secondly, we present numerical results on the effectiveness of different treatment protocols on the total cumulative number of metastases and the total metastatic mass.