Eikonal Formulation of the Minimal Principle for Scroll Wave Filaments

Abstract

Recently, Wellner et al. [Proc. Natl. Acad. Sci. U.S.A. 99, 8015 (2002)] proposed a principle for predicting a stable scroll wave filament shape as a geodesic in a 3D space with a metric determined by the inverse diffusivity tensor of the medium. Using the Hamilton-Jacobi theory we show that this geodesic is the shortest path for a wave propagating through the medium. This allows the use of shortest path algorithms to predict filament shapes, which we confirm numerically for a medium with orthotropic anisotropy. Our method can be used in cardiac tissue experiments since it does not require knowledge of the tissue anisotropy.

A geometric theory for scroll wave filaments in anisotropic excitable media

“…A suitable expansion parameter in the plane transverse to the filament is κ = d 2 R 1212 . Our ansatz thus reads u(ρ A , σ , τ ) = u 0 (ρ A ) + λũ(ρ A , σ , τ ) + O(λ 2 , κ) (33) with ∂ σũ /∂ ρũ = O(λ).…”

“…(1). Ten Tusscher and Panfilov later reformulated this principle using the Hamilton-Jacobi theory and showed that the geodesic is equivalent to the shortest path for wave propagation through the medium [33]. The minimal principle was verified for specific examples of anisotropy in cardiac tissue [32,33].…”

Effective dynamics of twisted and curved scroll waves using virtual filaments

“…Particular successes were achieved with this method to describe filament drift in excitable media with anisotropic diffusion, which has an important application in cardiac tissue modeling [40][41][42][43][44]. After Wellner's discovery that such anisotropic systems can be efficiently treated by a curved-space formalism [22,43,[45][46][47], more results on wave dynamics under generic local anisotropy followed [22,23,25,34,35]. For simplicity, we choose not to consider filament dynamics in anisotropic media here; such generalization can be found in Ref.…”

Dynamics of Filaments of Scroll Waves

“…direction of the maximal diffusivity of the transmembrane potential. As ten Tusscher and Panfilov [2004] noted, the metric defining the steady-state geometries of filaments, can be conveniently formulated in terms of excitation wave propagation time: given its end points, the filament will follow the quickest path connecting the end points. This hypothesis was confirmed by Verschelde et al [2007] who generalized the equation (10) for the anisotropic media, under the assumption of det(D ij ) = const.…”