On the design of treatment schedules that avoid chemotherapeutic resistance

We introduce a method of designing treatment schedules for a model three-component replicator dynamical system that avoids chemotherapeutic resistance by controlling and managing the competitive release of resistant cells in the tumor. We use an evolutionary game theory model with prisoner’s dilemma payoff matrix that governs the competition among healthy cells, chemo-sensitive cells, and chemo-resistant cells and the goal is to control the evolution of chemo-resistance via the competitive release mechanism. The method is based on nonlinear trajectory design and energy transfer methods first introduced in the orbital mechanics literature for Hamiltonian systems. By using the structure of the trajectories defined by solutions of the replicator system for different constant chemotherapeutic concentrations (which produces a curvilinear coordinate system spanning the full region), we construct 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 either sensitive cells or resistant cells. By keeping the three sub-populations of cells in competition with each other, neither the sensitive cell population nor the resitant cell population are able to dominate as we balance the populations indefinitely (closed periodic orbits), thereby avoiding fixation of the cancer cell population and re-growth of a resistant tumor. The schedules we design have the feature that they maintain a higher average population fitness than either the MTD or the LDM schedules. PACS numbers: 87.23.Kg; 87.55.de; 87.19.Xj; 87.19.lr