Balancing conduction velocity error in cardiac electrophysiology using a modified quadrature approach

Abstract

Conduction velocity error is often the main culprit behind the need for very fine spatial discretizations and high computational effort in cardiac electrophysiology problems. In light of this, a novel approach for simulating an accurate conduction velocity in coarse meshes with linear elements is suggested based on a modified quadrature approach. In this approach, the quadrature points are placed at arbitrary offsets of the isoparametric coordinates. A numerical study illustrates the dependence of the conduction velocity on the spatial discretization and the conductivity when using different quadrature rules and calculation approaches. Additionally, examples using the modified quadrature in coarse meshes for wave propagation demonstrate the improved accuracy of the conduction velocity with this method. This novel approach possesses great potential in reducing the computational effort required but remains limited to specific linear elements and experiences a reduction in accuracy for irregular meshes and heterogeneous conductivities. Further research can focus on developing an adaptive quadrature and extending the approach to other element formulations in order to make the approach more generally applicable.

Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology

“…We extend these results to more complex membrane models and geometrical domains. The dependency of the conduction velocity on the spatial resolution was reported in several studies 7,8 . We study the effect of both space and time discretization on this phenomenon in an unstructured anatomically motivated mesh.…”

“…Combining the exact solution (8) with the explicit Euler approximation (9) plus an explicit Euler approximation of the nonlinear part of ( 10) is called Rush-Larsen method; compare, references 5,20,21.…”

“…Here, are disjoint sets, are amplitude functions, is the starting time of the ‐th stimulus and is its duration (). In many applications the amplitude functions of the external stimulus are chosen to be constant functions, that is, in (3); compare 8,10,14. In this case, however, the function is discontinuous both in time and space, and the missing regularity has the effect that convergence of the numerical methods can only be observed for very fine discretizations.…”

“…The dependency of the conduction velocity on the spatial resolution was reported in several studies. 7,8 We study the effect of both space and time discretization on this phenomenon in an unstructured anatomically motivated mesh. More recently, the convergence and accuracy of time integrators were studied on 1d and 2d regular rectangular domains with simplified external stimuli, for example by Roy et al 9 and Woodworth et al 10 However, as Krishnamoorthi et al 11 already observed, the irregularity of the mesh has an immense impact on physiological properties and will thus be considered in the present study.…”

Fast and reliable reduced‐order models for cardiac electrophysiology

“…The problem of cardiac electrophysiology often requires a fine spatial and temporal discretization and, therefore, one might need to deal with a large number of equations whose solution could be time consuming and exhaustive. Therefore, developing efficient solution techniques is of great importance and is one of the essential goals in the virtual modeling of the heart electrical activity [15,62,63].…”