Nonlinear adaptive control of competitive release and chemotherapeutic resistance

We use a three-component replicator system with healthy cells, sensitive cells, and resistant cells, with a prisoner's dilemma payoff matrix from evolutionary game theory, to model and control the nonlinear dynamical system governing the ecological mechanism of competitive release by which tumors develop chemotherapeutic resistance. The control method we describe is based on nonlinear trajectory design and energy transfer methods first introduced in the orbital mechanics literature for Hamiltonian systems. For continuous therapy, the basin boundaries of attraction associated with the chemo-sensitive population and the chemo-resistant population for increasing values of chemo-concentrations have an intertwined spiral structure with extreme sensitivity to changes in chemo-concentration level as well as sensitivity with respect to resistant mutations. For time-dependent therapies, we introduce an orbit transfer method to construct continuous families of periodic (closed) orbits by switching the chemo-dose at carefully chosen times and appropriate levels to design schedules that are superior to both maximum tolerated dose (MTD) schedules and low-dose metronomic (LDM) schedules, both of which ultimately lead to fixation of sensitive cells or resistant cells. By keeping the three subpopulations of cells in competition with each other indefinitely, we avoid fixation of the cancer cell population and regrowth of a resistant tumor. The method can be viewed as a way to dynamically shape the average population fitness landscape of a tumor to steer the chemotherapeutic response curve. We show that the method is remarkably insensitive to initial conditions and small changes in chemo-dosages, an important criterion for turning the method into an actionable strategy.