Maximum Bound Principle and Non-Negativity Preserving ETD Schemes for a Phase Field Model of Prostate Cancer Growth with Treatment

Prostate cancer (PCa) is a significant global health concern that affects the male population. In this study, we present a numerical approach to simulate the growth of PCa tumors and their response to drug therapy. The approach is based on a previously developed model, which consists of a coupled system comprising one phase field equation and two reaction–diffusion equations. To solve this system, we employ the fast second-order exponential time differencing Runge–Kutta (ETDRK2) method with stabilizing terms. This method is a decoupled linear numerical algorithm that preserves three crucial physical properties of the model: a maximum bound principle (MBP) on the order parameter and non-negativity of the two concentration variables. Our simulations allow us to predict tumor growth patterns and outcomes of drug therapy over extended periods, offering valuable insights for both basic research and clinical treatments.