Dynamics of Chemotherapy Models with Variable Infusion and Time Delays

Abstract

Chemotherapy is a fundamental and commonly used form of cancer treatment, usually done with the application of a chemotherapy agent to the infected individual. The chemotherapy agent targets fast-growing cells including cancer cells as well as other fast-growing normal cells such as those of the skin, hair and bone marrow, and hence may cause severe side effects to the body of the patient. To better understand the trade-offs between reducing cancer cells and impacting normal cells, mathematical models have been used extensively to study the effectiveness of chemotherapy treatments. In particular, Pinho et al. proposed an autonomous dynamical system with time delays which modeled the interaction between the normal cells and cancer cells with metastasis and used to study the effect of the metastasis. Based on the idea of Pinho’s work, a nonautonomous dynamical system that models the interactions among cancer cells, normal cells and the chemotherapy agent under time varying environmental conditions was developed and studied by Xiaoying Han in 2017. It is well justified in the existing literature that time delays often exist in chemotherapy treatments, yet the effect of delays is not fully understood. For example, Pinho et al. conjectured that time delays are critical for the global stability of the tumor-free equilibrium but did not provide further evidence. To the best of our knowledge there are no solid results elaborat- ing how time delays affect dynamics of chemotherapy models. The goal of this dissertation is to investigate both analytically and numerically the effects of time delays and time-varying environmental conditions on the stability of steady states of chemotherapy models. To this end two mathematical models of chemotherapy cancer treatment are studied and compared, one modeling the chemotherapy agent as the predator and the other modeling the chemotherapy agent as the prey. In both models constant delay parameters are introduced to incorporate the time lapsed from the instant the chemotherapy agent is injected to the moment it starts to be effective. For each model, the existence and uniqueness of non-negative bounded solutions are first established. Then both local and Lyapunov stability for all steady states are investigated. In particular, sufficient conditions dependent on the delay parameters under which each steady state is asymptotically stable are constructed. Numerical simulations are presented to illustrate the theoretical results. Furthermore, another non-autonomous mathematical model of chemotherapy cancer treatment with time-dependent infusion concentration of the chemotherapy agent is developed and studied. In particular, a mutual inhibition type model is adopted to describe the interactions between the chemotherapy agent and cells, in which the chemotherapy agent is modeled as the prey being consumed by both cancer and normal cells, thereby reducing the population of both. Properties of solutions and detailed dynamics of the nonautonomous system are investigated, and conditions under which the treatment is successful or unsuccessful are established. It can be shown both theoretically and numerically that with the same amount of chemotherapy agent infused during the same period of time, a treatment with variable infusion may over perform a treatment with constant infusion.