Stochastic modelling of slow-progressing tumors: Analysis and applications to the cell interplay and control of low grade gliomas

Tumor-normal cell interplay defines the course of a neoplastic malignancy. The outcome of this dual relation is the ultimate prevailing of one of the cells and the death or retreat of the other. In this paper we study the mathematical principles that underlay one important scenario: that of slow-progressing cancers. For this, we develop, within a stochastic framework, a mathematical model to account for tumor-normal cell interaction in such a clinically relevant situation and derive a number of deterministic approximations from the stochastic model. We consider in detail the existence and uniqueness of the solutions of the deterministic model and study the stability analysis. We then focus our model to the specific case of low grade gliomas, where we introduce an optimal control problem for different objective functionals under the administration of chemotherapy. We derive the conditions for which singular and bang-bang control exist and calculate the optimal control and states.