Page 109 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 79

2.4.1 3D hemodynamics using the lattice Boltzmann
method
CFD modeling of intra-cardiac flow uses either unstructured,
body-conforming grids (e.g. finite element methods [229–235]
or finite volume methods [236–239]), or static, non-conforming
grid (e.g. finite difference methods [131], the immersed bound-
ary method [240–246], and recently also the Lattice Boltzmann
method [247]). Given the significant domain distortion specific to
cardiac simulation (wall and valve motion and deformation), any
body-conformal grid method has to employ automatic remesh-
ing strategies, which can significantly increase the complexity and
cost of the simulation. In static grid methods the volume grid for
the flow simulation does not need to be deformed or remeshed to
conform to the deforming cardiac geometry, which makes them
appropriate for the fully automated simulation of cardiac flow.
The Immersed Boundary Method [248,249] is the oldest method
that was used successfully to compute full 3D cardiac flow within
complex moving geometries [250]. In contrast with finite volume
methods, which impose the boundary conditions directly on the
grid, the IBM introduces local body forces to achieve the same ef-
fect. While early on the method has been criticized for its inability
to preserve mass accurately, later studies starting with [251]have
shown that careful implementation can alleviate that. Further-
more, the Lattice Boltzmann method, whose efficiency in comput-
ing vessel blood flow has been established [247], is an interesting
new option to explore, and it is considered here in more detail.

The Lattice Boltzmann Method
The Lattice Boltzmann Method (LBM) describes physics of
fluid flow at a mesoscopic scale by taking into account molecular
interactions between flow particles. The LBM provides ultimately
the same solution as the Navier–Stokes based solvers [214], but it
is “naturally” highly parallelizable, which can enable an efficient
computation of two-way FSI. In the following, we provide a short
description of the LBM theory, together with implementation de-
tails.
LBM models the interaction of fluid particles using a mathe-
matical model based on the Boltzmann equation:
∂f
+ u ·∇f = K(f ). (2.31)
∂t
Here f = f(u,x,t) is a probability density function and it gives the
probability of a fluid particle to have the velocity u and to be at
position x at time t.The righthandsideofEq.(2.18)isknown