Page 81 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 51

the heterogeneous and anisotropic nature of the myocardial tis-
sue. The following sections describe in more details some possible
modeling choices, discussing their advantages and potential im-
plementations.

2.2.1 LBM-EP: efficient solver for the monodomain
problem

Numerically, cardiac electrophysiology (EP) models are usually
solved using finite difference methods (FDM) or finite element
methods (FEM) [209]. In FDM, the computational domain is spa-
tially discretized into a Cartesian grid, and the spatial derivatives
occurring in the differential equation are approximated using dif-
ferences over neighboring nodes [209]. These methods tend to be
easier to implement, and have the advantage of speed thanks to
efficient implementations based on the structured connectivity
of the grid nodes. However, the boundary conditions can be dif-
ficult to impose over complex geometries. Furthermore, Clayton
et al. [209] showed that the implementation of anisotropic con-
ductivity can be difficult, with the propagation speed depending
significantly on the fiber orientation with respect to the Cartesian
grid axes.
Finite element methods [210,211] on the other hand solve
a weak formulation of the governing equation derived by the
Galerkin approach. The computational domain is divided into a
set of simple elements, and the weak formulation is solved us-
ing (typically) low order polynomial interpolation inside the ele-
ments. This method has the advantage of being able to adapt to
complex geometries easily. The implicit and semi-implicit formu-
lations typically used for time discretization require the solution
of a linear system of equations at every timestep, and can be com-
putationally slow. For this reason, explicit time stepping schemes
using lumped mass matrices have become popular, enabling fast
computations [212]. However, the solution accuracy is sensitive to
the way in which mass-lumping has been performed [213].
Recently, the Lattice-Boltzmann Method (LBM) has gained
interest for solving complex reaction–diffusion–advection equa-
tions, in particular the Navier–Stokes equation in Computational
Fluid Dynamics. This model has had great success in the fluid dy-
namics community over the last couple of decades (see [214,215]
for comprehensive reviews), and has also been applied to other
problems with a reaction–diffusion nature [216]. Complex geome-
tries can be handled easily using LBM by level-set-based extrapo-
lation techniques, enabling second-order accurate boundary con-
ditions. Furthermore, LBM depends on highly localized computa-