Reactive-Advective-Diffusive Models for the Growth of Gliomas Treated with Radiotherapy

Abstract

Gliomas are malignant brain tumors responsible for 50% of primary human brain cancer cases. They have a combination of rapid growth and invasiveness, and high fatality rates with a median survival time of one year. Mathematical models that describe its growth have helped to improve treatment. In this paper, a combined model formed by terms of two other models known in the literature is analyzed. The combined model is a Reactive-Advective-Diffusive partial differential equation, which is solved by combining the finite difference method, the Crank-Nicolson method and the upwind method. Logistic growth is used for cell proliferation ensuring a saturation threshold for glioma growth, which is crucial to properly estimate patient survival time. The well-known linear-quadratic radiobiological model is used to describe cell death due to radiotherapy treatment. Two initial conditions are compared in the simulations, indicating the need for further studies to have a model as close as possible to reality. Simulation results are shown for four scenarios: no radiotherapy, application of a single dose, and two dose fractionation schemes.

Keywords

• tumor growth
• finite differences
• glioblastoma
• radiotherapy
• mathematical models