Comparing partial differential equations and agent-based simulations in spatio-temporal modeling of cancer growth and shape

Spatio-temporal partial differential equations (PDEs) models of cancer make several ad-hoc assumptions about the dynamics on the unknown boundary of the tumor in order to solve the PDE system of equations and, at the same time, determine also the boundary of the tumor. In this paper, we developed a different computation approach using the agent-based simulation (ABS) modeling method. We consider a simple cancer model, and use, in the ABS model, the same parameters that express the rates of interactions among cells and proteins, as in the PDE model, but we make no ad-hoc assumptions on the dynamics of the unknown tumor boundary. However, ABS is a stochastic process, hence, in order to get a good approximation of the tumor boundary, we need to repeat the simulations many times, and then take the average. We show, in the spherical case, that the tumor volume computed by the ABS is in increasingly good agreement with the PDE volume as the number of ABS repetitions is increased. We next use the ABS model to compute several non-symmetric shapes that are commonly seen in non-invasive and invasive cancers.