Harald Garcke (University of Regensburg)
Phase field approaches for tumour growth
Modelling of tumour growth is one of the challenging frontiers of applied mathematics. In the last years, phase field models for tumour growth have been studied intensively. Alike classical free boundary models they use a continuum approach to describe the growth of tumours. However, an advantage to free boundary models is that phase field models allow for topology changes like break up and coalescence. In addition, phase field methods can be used numerically without an explicit tracking of the interface which is necessary for free boundary models. In my talks I will introduce several macroscopic models for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. The resulting evolution equation is a Cahn-Hilliard equation taking source and sink terms into account. In addition, nutrient diffusion is incorporated by a coupling to a reaction-equation diffusion. I will show existence, uniqueness and regularity results. In addition, several continuous dependence results are shown. I will also discuss how to couple the system to an internal velocity field which either solves a Darcy-type or a (Navier-)Stokes system or a viscoelastic system. Using matched asymptotics, the phase-field systems are related to classical free boundary problems. Finally, using optimisation theory and reduced order modelling I will describe how parameter in the system can be estimated in a patient specific way. In all lectures properties of solutions will be illustrated with the help of numerical simulations.