Mathematical models for tumor growth can aid researchers in studying the evolution of tumors within the body and the effects of various drug treatments. Such models incorporate a variety of factors, including different cell populations, the presence of a drug and/or nutrient, and advection due to flow within the tumor.

We consider a chemotaxis fluid model for multiphase tumor growth. Our model assumes that the tumorous cells undergo chemotaxis in response to the presence of a nutrient, a consideration which is neglected in many other models. Additionally, we assume that the flow of cells through the extracellular matrix is modeled as flow through a porous medium, using either Darcy's Law or Brinkman's equation. Furthermore, we consider the model on a moving (time-dependent) domain in order to allow the shape of the tumor to evolve in time. These assumptions present several challenges for the analysis of the model.

We prove that there exist weak solutions to this model. The proof of existence relies on constructing an approximating system by means of time-discretization and an Arbitrary Lagrangian Eulerian (ALE) mapping. We then prove that the solutions to these approximating schemes converge to a solution to the original problem.

We also construct a convergent finite element scheme for this model. In the case of Darcy's Law, such a scheme can be constructed on either a fixed domain or a moving, polygonal domain, while for Brinkman's equation we focus only on the case of a fixed, polygonal domain. This numerical scheme can be used to simulate the evolution of a tumor under various assumptions on parameters, and the results of various numerical experiments are included here. These results illustrate the impact of the chemotactic effects, the moving domain, and different parameter choices.