Controllability in a class of cancer therapy models with co-evolving resistance

Adaptive therapy is a recent paradigm in cancer treatment aiming at indefinite, safe containment of the disease when cure is judged unattainable. In modeling this approach, inherent limitations arise due to the structure of the vector fields and the bounds imposed by toxic side-effects of the drug. In this work we analyze these limitations in a minimal class of models describing a cancer population with a slowly co-evolving drug resistance trait. Chemotherapeutic treatment is introduced as any bounded time-varying input, forcing the cells to adapt to a changing environment. We leverage the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria. We show that these controllable sets entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.