Generalized Minimal Principle for Rotor Filaments

Abstract

To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G ¼ detðDÞD −1 . The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes.

Dynamics of cardiac re-entry in micro-CT and serial histological sections based models of mammalian hearts

“…Theory (Wellner et al, 2002;Dierckx et al, 2015) predicts strong effects of anisotropy on dynamics of singular filaments, which must act in addition to purely geometric forces. An iteresting confirmation of the theoretical prediction is the case of isotropic vs anisotropic conductance in the sheep atria, which resulted in the very different lengths and orientation of the final pinned filaments shown in Figure 6(b) and in Figure 6(c).…”

“…with the kinetics parameters α = 0.3, β = 0.71, γ = 0.5, which in perfectly homogeneous infinite medium correspond to a rigidly rotating vortex with positive filament tension . The specific forces generated by anisotropy (Wellner et al, 2002;Dierckx et al, 2015;Biktasheva et al, 2018) and by small fuctuations of thickness in the layer (Biktasheva et al, 2015;Kharche et al, 2015a) are only two types out of a variety of specific forces acting on cardiac re-entry in the heart (Biktashev et al, 2011). For this study, we have intentionally chosen the simplified FHN model with the parameters corresponding to the simplest possible rigidly rotating positive filament tension re-entry, in order to eliminate effects of e.g.…”