Eikonal formulation of the minimal principle for scroll wave filaments

Area of Science:

• Computational physics
• Wave propagation dynamics
• Biomedical modeling

Background:

• Scroll waves are complex dynamic structures crucial in various physical and biological systems, including cardiac tissue.
• Predicting the stable filament shape of scroll waves has been a significant challenge in understanding their behavior.
• Previous work proposed a principle linking filament shape to geodesics defined by the medium's inverse diffusivity tensor.

Purpose of the Study:

• To mathematically validate that the geodesic path predicted by Wellner et al. represents the shortest path for wave propagation.
• To establish a theoretical foundation for using shortest path algorithms in predicting scroll wave filament shapes.
• To demonstrate the practical applicability of this method in anisotropic media, relevant to biological tissues.

Main Methods:

• Application of Hamilton-Jacobi theory to analyze the wave propagation dynamics.
• Derivation of the relationship between the geodesic and the shortest path.
• Numerical simulations using shortest path algorithms on a medium with orthotropic anisotropy.

Main Results:

• The study confirms that the geodesic path determined by the inverse diffusivity tensor is indeed the shortest path for wave propagation.
• Numerical simulations successfully predicted scroll wave filament shapes in an orthotropic anisotropic medium.
• The developed method does not require prior knowledge of the specific tissue anisotropy for accurate predictions.

Conclusions:

• The geodesic principle provides a robust theoretical framework for predicting stable scroll wave filament shapes.
• Shortest path algorithms offer an efficient computational tool for determining these filament shapes.
• This approach has significant implications for experimental studies in cardiac electrophysiology, simplifying the analysis of wave dynamics.