Nonlinear dynamics of chemotherapeutic resistance

Abstract

We use a three-component replicator dynamical system with healthy cells, sensitive cells, and resistant cells, with a prisoner's dilemma payoff matrix from evolutionary game theory to understand the phenomenon of competitive release, which is the main mechanism by which tumors develop chemotherapeutic resistance. By comparing the phase portraits of the system without therapy compared to continuous therapy above a certain threshold, we show that chemotherapeutic resistance develops if there are pre-exisiting resistance cells in the population. We examine the basin boundaries of attraction associated with the chemo-sensitive population and the chemo-resistant population for increasing values of chemo-concentrations and show their spiral intertwined structure. We also examine the fitness landscapes both with and without continuous therapy and show that with therapy, the average fitness as well as the fitness functions of each of the subpopulations initially increases, but eventually decreases monotonically as the resistant subpopulation saturates the tumor.

On the design of treatment schedules that avoid chemotherapeutic resistance

“…Matrix entries are consistent with those from [32]. As discussed in [32], the prisoner's dilemma payoff matrix ensures: (i) Gompertzian growth of the cancer cells; (ii) a reduction in overall fitness of the population as the tumor grows; and (iii) a fitness cost associated with resistance.…”

“…As discussed in [32], the prisoner's dilemma payoff matrix ensures: (i) Gompertzian growth of the cancer cells; (ii) a reduction in overall fitness of the population as the tumor grows; and (iii) a fitness cost associated with resistance. In our previous paper [32] we studied the nonlinear dynamics associated with eqns (1)- (3) for constant values of the chemotherapy parameter 0 ≤ C ≤ 1 to demonstrate the mechanism of competitive release when C ≥ 1/3. In this paper, we investigate piecewise constant time-dependent functions C(t) to show how to avoid the evolution of resistance of the tumor.…”