Modern therapies deal with increasingly complex technical equipment and require a deep understanding of biophysical processes in order to design effective treatments. A detailed and precise prediction of the effects of changing therapy parameters is often far beyond the skills even of experienced physicians. Mathematical models help to understand, predict, and design such treatments.

• modeling to translate medical problems into the language of mathematics
• simulation to gain insight into processes and to predict effects of therapies
• identification to obtain model parameters from noisy measurements
• optimization for individual treatment planning

On the mathematical side this involves e.g. adaptive finite element methods for solving various partial differential equations, inexact Newton methods for tackling nonlinear problems, and interior point methods for solving optimal control problems.