Analysis of a mathematical model for low-grade gliomas under chemotherapy as a dynamical system

Abstract

We analyze dynamical system properties of a 3-compartment mathematical model for the cell-cycle under chemotherapy with a phase non-specific drug. It is assumed that the drug damages both proliferating and quiescent cells, but possibly at different rates. While damage to proliferating cells is lethal, damaged quiescent cells may be repaired and can reenter the cell cycle. We prove that there exists a unique dosage $\tilde{u}$ (depending only on the parameters of the dynamics of the system) such that the tumor can be eradicated for $u\ge \tilde{u}$ as all trajectories converge to the tumor-free equilibrium point (global stability). For lower doses, $0\le u<\tilde{u}$, however, there exists a locally asymptotically stable equilibrium point with positive values and such doses are not sufficient to eradicate the tumor. Mathematically, for $u=\tilde{u}$, the positive and tumor-free equilibrium points are equal and a transcritical or exchange of stability bifurcation occurs. Our theoretical analysis is independent of specific values of the parameters. For the numerical illustration of the results we use clinically validated parameter values for low-grade glioma from Ribba et al., (2012).