Phase-field models for tumour growth with viscoelasticity, relaxation and growth
In the last years, modelling tumour growth has become a popular research topic. However, it is very challenging to find mathematical models that account for a realistic description of the evolution of tumours, as many biological processes are very complex or still not fully understood. In this work, we present a phase-field approach for tumour growth with a diffuse interface separating a tumour from the surrounding host tissue. In our model, we include biological effects like chemotaxis or active nutrient uptake, such as transport processes by an internal velocity field. We include viscoelastic effects with a general Oldroyd-B type description with stress relaxation and stress generation by growth. For specific variants of the model, we establish existence of weak solutions with the help of stable, converging, fully-discrete finite element schemes. Finally, we illustrate properties of solutions with the help of numerical simulations.