Analysis of Impulsive Differential Equation Models of Cell Populations Undergoing Radiation Therapy

Radiation therapy is a mode of treatment which is implemented for approximately 50% of cancer patients. Treatment needs to be able to kill cancer cells, but also do minimal damage to surrounding healthy tissue. We propose two main impulsive differential equation models of radiation therapy to capture the periodic nature of the treatment. These models build off of previous studies using clinical data to ensure biological relevance. The first model incorporates only cancer cell populations, and we provide parameter relationships which theoretically ensure treatment outcomes of cancer eradication, cancer approaching a carrying capacity, and cancer approaching a periodic solution. We then extend this model to include the dynamics of healthy cells through a Lotka-Volterra type competition system. Theoretically, we show conditions under which cancer is eradicated, cancer “wins” (healthy cells go extinct), and the cell populations coexist. Only certain portions of parameter space are able to be analyzed theoretically for this model, so we also perform numerical bifurcation analysis for varying doses and time between treatment fractions. Through our theoretical and numerical work, we suggest that the assumption that cancer will win in the absence of treatment may not always be reasonable, and that coexistence solutions in the absence of treatment should also be considered in future studies. We also investigate dynamics of FLASH Radiotherapy through bifurcation analysis, which serves as a starting point for population dynamics for this cutting-edge treatment. Advisors: Huijing Du and Bo Deng