Exact solutions through symmetry reductions for a high-grade brain tumor model with response to hypoxia

Mathematical biology models can simulate cell behavior scenarios, specifically for tumor proliferation. In this paper, we study a continuous model describing the evolution of high-grade gliomas from the point of view of the theory of symmetry reductions in partial differential equations (PDEs). Malignant gliomas are the most frequent and deadly type of brain tumor. Over the last few years, complex mathematical models of cancerous growths have been developed increasingly, especially on solid tumors, in which growth primarily comes from abnormal cellular proliferation. The presented PDE system includes two different cellular phenotypes, depending on their oxygenation level. Furthermore, this mathematical model assumes that both phenotypes differ in migration and proliferation rates. Specifically, it includes the possibility of hypoxic cells diffusing into well-oxygenated areas of a tumor. Our main findings are obtained through the classical symmetries admitted by the proposed system, and transformation groups are used to reduce the PDE system to ordinary differential equations. By these means, we provide not only exact solutions but also capture a 3-dimensional representation of the biological phenomenon. The simulations provided show the relationship between normoxic and hypoxic phenotypes in high-grade gliomas.