**Topics of interest**

- Realization of a hierarchy of electro-physiological models

- Characterization of the electro-mechanical interaction

- Benchmarking and model adaptation

**Project description**

Depolarization waves in the heart are an important electrophysiological phenomenon described by the basic model for the cardiac reaction-diffusion system: The bidomain equations for electrical potentials in and between cells and the spread of the depolarization waves. The cardiac electrophysiological system triggers contractions of the heart muscle, which is modeled as a time-varying elastic body. The electrical cellular transmembrane voltages are given as solutions to nonlinear reaction-diffusion equations on a manifold. Depending on the accuracy of the model, this is coupled with up to 50 ODEs to determine the transmembrane currents.

There are different types of depolarization waves: A single wavefront (e.g., created in the sinus node) which propagates across the heart and then vanishes naturally; a periodic stable wave pattern existing for some period of time (e.g., atrial flutter); small regular patterns that live for a few seconds and then disappear; or chaotic dynamics not building stable patterns (e.g., atrial or ventricular fibrillation). Our goal is to develop, to analyze, and to numerically realize a fully coupled model for the cardiac electromechanical system, to describe the dynamics of depolarization waves in this system, and to develop and evaluate a hierarchy of reduced models which allows a qualitatively correct prediction of the main types of depolarization waves.

Beginning with an investigation of depolarization waves on a fixed heart geometry with a simplified model for the transmembrane currents, we want to numerically find criteria to characterize the dynamical patterns. This will then be combined with the finite deformation of the cardiac tissue. Step by step extensions to more realistic numerical models will be included. A future perspective is to analyze the wellposedness of the corresponding dynamical system on the time-dependent manifold of the deformed heart geometry. The interplay of numerical simulations and qualitative analysis will lead to a better understanding of heart models and finally support new therapeutic approaches.