Bifurcation analysis of the tangential switching mode in a non-monotonic tumor growth model

Abstract

The study of tumor growth is a classic topic in mathematical biology. However, the analysis and modeling of non-monotonic growth is a relatively less studied subtopic of mathematical oncology. Nevertheless, an emerging “implicit approach” in this field for modeling cancer therapies is to consider the underlying dynamics of untreated tumors described by ordinary differential equations whose change in parameters describes the application of one or more oncological therapies. Accordingly, here we use piecewise smooth vector fields to address the non-monotonic tumor growth revealed in experiments with untreated mice, and how such in vivo growth is affected by hypothetical treatments whose application is controlled by a critical threshold value of tumor cell number. As a result, we obtain bifurcation scenarios involving the so-called tangential sliding vector field in a model presenting 3D folds and cusps in the switching manifold. Based on our mathematical analysis we find conditions for attractive scenarios where, for example, the tumor volume is kept stabilized. Our findings offer a theoretical framework for modeling isolated and combined therapies in mathematical oncology, including intermittent strategies that emulate adaptive therapy. Moreover, an extensive list of bifurcations is presented. These bifurcations arise from variations in the parameters that control angiogenesis and resource availability.