Mathematical Modeling of Tumor Growth for Free-boundary Problem by Enhanced Finite Volume Method
Modeling tumor growth due to infiltration of immune cells presents several challenges in numerical computations. First, it involves multiple cell species whose total number should be a constant, due to the incompressibility assumption; second, by mapping the Eulerian coordinate of the free-boundary problem onto a fixed logical domain, geometric source terms appear and they need to be addressed properly in numerical methods. In this work, we use a simplified model that contains two species and prescribed infiltration velocity and to show that the conventional finite volume methods fail to preserve the trivial (constant) solutions. To this end, we introduce the totality conservation law (TCL) and the geometric law (GCL) as the two criterions to address the incompressibility and property of preserving constant solutions on changing Eulerian domains, respectively. The classical Godunov-type finite volume methods are enhanced to satisfy these conditions, and performance improvements are verified by numerical tests with arbitrary infiltration velocities.
Saleh, Mashriq Ahmed, "Mathematical Modeling of Tumor Growth for Free-boundary Problem by Enhanced Finite Volume Method" (2018). ETD Collection for University of Texas, El Paso. AAI10929664.