Positivity-Preserving Segregate-Flux Method for Infiltration Dynamics in Tumor Growth Models
We study the positivity preserving property and an incompressibility condition in a recently proposed tumor growth model as well as its numerical simulations. In this model, the biological process is described by a free-boundary problem of hyperbolic equations that govern the in-tumor motion of cancer cells and the infiltration of immune cells. Particularly, due to an assumption that cells take constant volume (the incompressibility condition), the tumor growth/shrinkage is closely correlated to the magnitude of infiltration of immune cells into the tumor. Despite the fact that previous simulation results largely reproduced experimental data, there remain unanswered questions that are crucial for the justification of such models. In this thesis, we make a first step to address two such questions, namely preserving the positivity of variables that represent cell number densities and the incompressibility condition, from both a mathematical perspective and a numerical point of view. In particular, we first show that under certain assumptions, the analytic solutions to the mathematical model must preserve positive cell number densities as well as maintaining the cell incompressibility. Then we examine a recently proposed segregate-flux finite volume method, which is designed preserves numerical cell incompressibility, and show that it also positivity-preserving under minor modifications.
Acheampong, Gilbert Danso, "Positivity-Preserving Segregate-Flux Method for Infiltration Dynamics in Tumor Growth Models" (2020). ETD Collection for University of Texas, El Paso. AAI28089416.