Minimal principle for rotor filaments

Abstract

Three-dimensional rotors, or scroll waves, provide essential insight into the activity of excitable media. They also are a suspected cause in the formation and maintenance of ventricular fibrillation, whose lethality is well known. It is therefore of considerable interest to find out what configurations can be adopted by such pathologies. A scroll's behavior is embodied in its organizing center or filament, a largely quiescent tube about which the scroll rotates. Predicting filament shape has normally required computer-intensive simulations of the whole scroll in time. We have found a fast and robust principle that yields the prediction for stationary filaments on a purely geometrical basis, blind to the reaction parameters of the medium. The procedure is to calculate the filament shape as a minimal path. We work in singly diffusive media whose diffusivity tensor-and no other feature-varies spatially. Mathematical and numerical evidence is presented for the proposition that a stable filament is a geodesic in a three-dimensional space whose metric is given by the inverse diffusivity tensor of the medium. Away from the boundaries, a stable filament is unaffected by the reaction parameters. The algorithmic aspects of this work are subsidiary to our main purpose of drawing attention to the universal and unexpectedly exact fit of an elementary geodesic principle within reaction-diffusion theories.

Is Ablation to Termination the Best Strategy for Ablation of Persistent Atrial Fibrillation?

“…Therefore, endocardial breakthroughs may not be readily identified. 53 It is also possible that rotors may represent bystander activation in response to adjacent primary focal discharges.…”

Effective dynamics of twisted and curved scroll waves using virtual filaments

“…A second approach consists of mapping known spiral solutions to the local neighborhood of the filament curve to find the leading order dynamics. 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].…”

Eikonal Formulation of the Minimal Principle for Scroll Wave Filaments

“…Recently, Wellner et al proposed a ''minimal principle for rotor filaments'' [12]. They showed that the stable filament shape of a scroll wave in a 3D anisotropic medium is described by a geodesic in a 3D space with a metric given by the inverse diffusivity tensor of the medium, thus providing an elegant analytical formulation for predicting filament shapes.…”

“…Furthermore, to specify boundary conditions for the geodesic equation, a priori knowledge of the position of filament ends on boundaries is required. Therefore, Wellner et al suggest that future research in this direction should make use of variational principles in geometric contexts [12].…”