Design of two-stage multi-drug chemotherapy schedules using replicator game dynamics
We use a replicator evolutionary game in conjunction with control theory to design a two-stage multidrug chemotherapy schedule where each stage has a specific design objective. In the first stage, we use optimal control theory that minimizes a cost function to design a transfer orbit which takes any initial tumor-cell frequency composition and steers it to a state-space region of three competing clonal subpopulations in which the three populations co-exist with a relatively equal abundance (high-entropy co-existence region). In the second stage, we use adaptive control with continuous monitoring of the subpopulation balance to design a maintenance orbit which keeps the subpopulations trapped in the favorable co-existence region to suppress the competitive release of a resistant cell population in order to avoid the onset of chemoresistance. Our controlled replicator dynamics model consists of a chemo-sensitive cell phenotype S, which is sensitive to both drugs, and two resistant cell phenotypes, R1 and R2, which are sensitive to drugs 1 and 2 respectively, but resistant to drug 2 and 1. The 3 x 3 payoff matrix used to define the fitness function associated with the interactions of the competing populations is a prisoners dilemma matrix which ensures that in the absence of chemotherapy, the S population (defectors) has higher fitness (reproductive prowess) than the two resistant cell populations, reflecting an inherent cost of resistance which our chemotherapy design methodology seeks to exploit. In our model, the two drugs C1 and C2 can act synergistically, additively, or antagonistically on the populations of cells as they compete and evolve under natural and artifical selection dynamics. Our model brings to light the inherent trade-offs between navigating to the maintenance orbit in minimal time vs. arriving there using the least total drug dose and also that the optimal balance of synergystic or antagonistic drug combinations depends the frequency balance of the populations of cells.