A parallel solver for anisotropic cardiac models

Abstract

IntroductionKnowledge of the rules that govern the full cardiac cycle is a prerequisite for understanding and interpreting abnormal sequences that occur in conduction disturbances. The excitation and recovery phases are influenced by the direction of the myocardial fibers and by the anisotropic conductivity of the intra and extracellular media. While the former phase has been examined in considerable detail both experimentally and numerically (see [1,2,3]), much less is known concerning the latter (see [4,5,6,7]). During a normal heartbeat, the time course of the ventricular transmembrane potential displays mainly three phases having different time and space scales. At first in the excitation phase, a moving layer associated with the upstroke of the action potential sweeps the entire cardiac domain. Subsequently, small spatial and temporal potential variations are observed in the long plateau phase and finally smooth changes, in space and time, are associated with the repolarization phase. To tackle the high computational costs involved in large scale simulations of a full cardiac cycle in a three dimensional domain, adaptive and parallel tools are required. 2.Mathematical modelsThe Bidomain model. In the Bidomain approach, the anisotropy of the two averaged continuous media, the intra and the extracellular medium, are characterized by the conductivity tensors D i (x) and D e (x) related to the arrangement of the cardiac fibers which rotate counterclockwise from the epicardium to the endocardium, (see [8]). Moreover, from [9], the cardiac tissue has a laminar organization and may be conceived of as a set of muscle sheets running radially from epi to endocardium. Therefore, at any point x, it is possible to identify a triplet of orthonormal principal axes a l (x), a t (x), a n (x), with a l (x) parallel to the local fiber direction, a t (x) and a n (x) tangent and orthogonal to the radial laminae respectively and both being transversal to the fiber axis. Denoting by σi,e l , σi,e t , σi,e n the conductivity coefficients measured along the corresponding directions, then the conductivity tensors D i (x) and D e (x) related to orthotropic anisotropy of the media are given by:n a n a T n while for axially isotropic media, i.e. σi,e n = σ i,e t , we haveThe intra and extracellular electric potentials u i , u e in the Bidomain model are described by a reaction-diffusion system coupled with a system of ODEs for ionic gating variables w. Given the applied currents per unit volume I i,e app , satisfying the compatibility condition H I i app dx = H I e app dx, the initial conditions v 0 , w 0 , then, for an insulated cardiac domain H, u i , u e , w satisfy the system:where ∂ t = ∂ /∂t, c m = χ * C m , I ion = χ * i ion , with χ the ratio of membrane area per tissue volume, C m the surface capacitance and i ion the ionic current of the membrane per unit area. The system uniquely determines v, while the potentials u i and u e are defined only up to a same additive time-dependent constant related to the reference pot...

Mathematical Cardiac Electrophysiology

“…A limited number of simulations studies used the full Bidomain Model in three dimensions. Some histological findings (see LeGrice [292,293]) have supported the idea that the cardiac tissue anisotropy could be orthotropic, but most simulations have been performed for axisymmetric anisotropy, with the exception of Colli et al [121], Hooks et al [234] for the excitation phase and Colli Franzone et al [120] for both the excitation and recovery phases.…”