Analysis and computations of a stochastic Cahn–Hilliard model for tumor growth with chemotaxis and variable mobility

In this work, we present and analyze a system of PDEs, which models tumor growth by taking into account chemotaxis, active transport, and random effects. Tumor growth may undergo erratic behaviors such as metastases that cannot be predicted simply using deterministic models. Moreover, random perturbations are evident in models accounting for therapeutic treatment in terms of therapy uncertainty or parameter identification problems. The stochasticity of the system is modeled by Wiener noises that appear in the tumor and nutrient equations. The volume fraction of the tumor is governed by a stochastic phase-field equation of Cahn–Hilliard type, and the mass density of the nutrients is modeled by a stochastic reaction-diffusion equation. We allow a variable mobility function and nonincreasing growth functions, such as logistic and Gompertzian growth. Via approximation and stochastic compactness arguments, we prove the existence of a probabilistic weak solution and, in the case of constant mobilities, the well-posedness of the model in the strong probabilistic sense. Lastly, we propose a numerical approximation based on the Galerkin finite element method in space and the semi-implicit Euler–Maruyama scheme in time. We illustrate the effects of stochastic forcings in tumor growth in several numerical simulations.