High-order spectral/hp element discretisation for reaction–diffusion problems on surfaces: Application to cardiac electrophysiology

Abstract

We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium.

Nektar++: An open-source spectral/hp element framework

“…Additionally, σ captures the potentially heterogeneous and anisotropic nature of the tissue which governs the speed of electrical propagation. While full 3D simulations of myocardium are traditionally performed, the left atrium is sufficiently thin that it can be reasonably represented as a two-dimensional manifold embedded in three dimensions and solved at significantly reduced computational cost [38]. An example of conduction propagation over the left atrium using the monodomain equations is illustrated in Fig.…”

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

“…In full-heart human electrophysiology simulations such "overkill" meshes lead to systems of hundreds of millions of degrees of freedom. Approaches to improving the front approximation without excessive global refinement that have been suggested in the literature include modifying the quadrature rule for the ionic currents [43], applying mesh adaptivity near the front [9,18,61], and more recently using high-order Spectral Element discretizations [12]. In this work, we investigate an approach similar to the latter, except that we replace orthogonal polynomials with B-splines or Non-Uniform Rational B-splines (NURBS) [52] in the context of Isogeometric Analysis (IGA) in the framework of the Galerkin method [23,38].…”

Influence of left atrial geometry on rotor core trajectories in a model of atrial fibrillation

“…In addition, human atria exhibit fibrosis and scar outside the pulmonary veins (PV) to a considerable degree, which may contribute to the initiation mechanisms for AF in many patients [8]. A variety of computational models propose that fibre orientation, fibrosis and the geometry of LA together can play an important role for the generation of anisotropic conductivity [9,10].…”

“…In addition, human atria exhibit fibrosis and scar outside the pulmonary veins (PV) to a considerable degree, which may contribute to the initiation mechanisms for AF in many patients [8]. A variety of computational models propose that fibre orientation, fibrosis and the geometry of LA together can play an important role for the generation of anisotropic conductivity [9,10].Clinical studies show that LA size and diameter and the PV structural characteristics in isolation [11][12] are related with the outcome of catheter ablation and are useful as independent predictors of AF recurrence. In this paper, we investigate, using a computational model, and anatomically accurate LA geometries derived from persistent AF patients, the effect of geometry in isolation on the maintenance of localised fibrillatory propagation, the spatiotemporal evolution of rotors over time and their association with anatomical characteristics.…”

“…The mitral valve and the PV sleeves were opened in order to produce a topologically accurate finite element mesh of the atrial chamber and PV sleeves. Gmsh [14], a high-order three-dimensional finite element mesh generator, was used to remesh the surface using a coarser uniform triangulation, which consisted of elements of approximately the same characteristic length of a size appropriate for use with a high-order spectral element discretisation [10]. A second mesh, which was used for generating a two-dimensional unwrapping of the surface, was created by adding cuts on either side of the atrium from the mitral valve up to the ends of the veins.…”