Mathematical modeling and analysis of therapies for metastatic cancers

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We introduce a mathematical model for the evolution of a cancer disease at the organism scale, taking into account for the metastases and their sizes as well as action of several therapies such as primary tumor surgery, chemotherapy and anti-angiogenic therapy. The mathematical problem is a renewal equation with bi-dimensional structuring variable. Mathematical analysis and functional analysis of an underlying Sobolev space are performed. Existence, uniqueness, regularity and asymptotic behavior of the solutions are proven in the autonomous case. A lagrangian numerical scheme is introduced and analyzed. Convergence of this scheme proves existence in the non-autonomous case. The effect of concentration of the boundary data into a Dirac mass is also investigated. Possible applications of the model are numerically illustrated for clinical issues such as the failure of anti-angiogenic monotherapies, scheduling of combined cytotoxic and anti-angiogenic therapies and metronomic chemotherapies. In order to give mathematical answers to these clinical problems an optimal control problem is formulated, analyzed and simulated.