A Mathematical Model of Cancer Treatment by Radiotherapy

Abstract

A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

On a multiobjective optimal control of a tumor growth model with immune response and drug therapies

“…Most of the models consider the two main population dynamics, effector cells and tumor cells. In general, they analyze the effect of immunity and some specific therapies, namely chemotherapy, immunotherapy, or radiotherapy (De Pillis et al., ; Isaeva and Osipov, ; Liu and Yang, ; Mamat et al., ). Another simple model of antitumor immune response considers only three populations, the tumor cells (represented by at time t ), cytotoxic effector cells (represented by ), and cytokine IL‐2 (represented by ) (Isaeva and Osipov, ).…”

“…Radiotherapy is a treatment procedure that uses radiation to kill rapidly reproducing cells such as those in malignant cell. Four different control mechanisms of radiation may be delivered: (a) continuous constant radiation, (b) continuous radiation that is proportional to the instantaneous tumor concentration, (c) continuous radiation that is proportional to the ratio of tumor to healthy cell concentration, and (d) periodic administration of radiation (Belostotski and Freedman, ; Liu and Yang, ).…”

“…Our model includes radiation activated at periodic times and only a periodic step function is analyzed. For example, in Belostotski and Freedman () and Liu and Yang (), we assume that R is the strategy of radiotherapy, a periodic step function with period w . Hence, when (treatment stage) and when (no treatment stage) for all , where w is the periodicity of treatment, D is the treatment duration, and is the rest period between radiation treatments.…”

“…To analyze the effect of radiotherapy, two different periodic radiation treatments with units day −1 are tested (this is a dose level inferior to those tested in Liu and Yang, ). We also use unit −1 as proposed in Liu and Yang ().…”

“…(), a system of five ordinary differential equations to simulate the interactions between normal, cancer, and endothelial cells, chemotherapy agent and antiangiogenic agent in tumor growth is developed. In Liu and Yang (), the radiotherapy is analyzed as the primary cancer strategy. The interactions between healthy and cancer cells are viewed as competitions for nutrients, space, blood supply, and the model includes a periodic administration of radiation under conditions that both tumor and healthy cells are affected by radiation.…”