Modeling of tumor radiotherapy with damage and repair processes
School of Mathematical Sciences, Heilongjiang University, 150080, Harbin, Heilongjiang, People’s Republic of China
2 Department of Mathematical Sciences, New Mexico State University, 88001, Las Cruces, NM, USA
Accepted: 2 March 2022
Published online: 13 May 2022
Under irradiation, some cells are damaged permanently and die while some damaged cells can be self-repaired and become normal cells. The same situation happens in tumor radiotherapy. There are several models to calculate the probability of cell survival after irradiation, and several mathematical models for tumor radiotherapy which incorporate cell survival probability. However, there is no detailed studies about how both radiation damage process and cell repair process impact outcomes of tumor radiotherapy. This study focuses on impacts of these two processes in tumor radiotherapy. The study employs mathematical modeling including mathematical analysis and numerical simulations. Based on established mathematical models for tumor growth and for irradiation, a functional reaction diffusion system for tumor radiotherapy is proposed. The model has the tumor cell population and damaged tumor cell population and tracks their movements in the tumor site. The model considers the repair time of damaged tumor cells as a delay parameter. It is the first mathematical model to incorporate cell repair process. Detailed analysis is conducted while numerical simulations are performed with brain tumor glioma data. We obtain the functional radiation threshold which combines the tumor growth rate, the damaged cell death rate, and the damaged cell repair rate. The functional radiation threshold is a increasing function of the tumor growth rate and the damaged cell repair rate which is a decreasing function of radiation dose while the radiation damage rate is a increasing function of radiation dose. The radiation damage rate, the functional radiation threshold, and repair time roughly determine the outcomes of radiotherapy. Given radiation dose, when the radiation damage rate is greater than the functional radiation threshold, radiotherapy may destroy the tumor, or two tumor cell populations oscillate at low levels if the damaged cell repair rate is greater than the damaged tumor cell death rate and the damaged cell repair time is long enough, or Turing instability occurs if diffusion coefficients of two tumor cells are bounded each other. When the radiation damage rate is less than the functional radiation threshold, radiotherapy may control tumor growth and the tumor load decreases as the radiation dose increases if the damaged cell repair time of damaged tumor cells is less than a critical time, or two tumor cell populations oscillate at high levels if the repair time of the damaged tumor cells is beyond its critical time. The damaged tumor cell repair process increases the functional radiation threshold and complicates outcomes of radiotherapy. Our results have some medical implications or applications in precise radiotherapy. The functional radiation threshold can be computed according to particular tumor growth rate and average life time of damaged tumor cells. Based on the functional radiation threshold, appropriate radiation doses can be found under which the tumor can be destroyed or controlled. Those results may help to designed precise radiation procedures for different types of tumors in different patients.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022