Chemotherapeutic resistance via the mechanism of competitive release of resistant tumor cell subpopulations is a major problem associated with cancer treatments and one of the main causes of tumor recurrence. Often, chemoresistance is mitigated by using multidrug schedules (two or more combination therapies) that can act synergistically, additively, or antagonistically on the heterogeneous population of cells as they evolve. In this paper, we develop a three-component evolutionary game theory model to design two-drug adaptive schedules (timing and dose levels associated with C1(t) and C2(t)) that mitigate chemoresistance and delay tumor recurrence in an evolving collection of tumor cells with two resistant subpopulations: R1 (sensitive to drug 1, resistant to drug 2), and R2 (sensitive to drug 2, resistant to drug 1). A key parameter, e, takes us from synergistic (e > 0), to additive (e = 0), to antagonistic (e < 0) drug interactions. In addition to the two resistant populations, the model includes a population of chemosensitive cells, S that have higher baseline fitness but are not resistant to either drug. Using the nonlinear replicator dynamical system with a payoff matrix of Prisoner’s Dilemma (PD) type (enforcing a cost to resistance), we investigate the nonlinear dynamics of the three-component system (S, R1, R2), along with an additional tumor growth model whose growth rate is a function of the fitness landscape of the tumor cell populations. We show that antagonistic drug interactions generally result in slower rates of adaptation of the resistant cells than synergistic ones, making them more effective in combating the evolution of resistance. We then design closed loops in the three-component phase space by shaping the fitness landscape of the cell populations (i.e. altering the evolutionary stable states of the game) using appropriately designed time-dependent schedules (adaptive therapy), altering the dosages and timing of the two drugs using information gleaned from constant dosing schedules. We show that the bifurcations associated with the evolutionary stable states are transcritical, and we detail a typical antagonistic bifurcation that takes place between the sensitive cell population S and the R1 population, and a synergistic bifurcation that takes place between the sensitive cell population S and the R2 population for fixed values of C1 and C2. These bifurcations help us further understand why antagonistic interactions are more effective at controlling competitive release of the resistant population than synergistic interactions in the context of an evolving tumor.
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