Invariant analysis of the two-cell tumor growth model in the brain

Abstract

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.