Page 84 - Artificial Intelligence for Computational Modeling of the Heart

54 Chapter 2 Implementation of a patient-specific cardiac model

where c = δx/δt is the speed of the particles in lattice units. This
process is usually represented as two distinct steps: collision and
streaming. The collision steps updates the distribution function at
each location to mimic the scattering of particles due to molecu-
lar collisions and due to applied external sources (currents). The
post-collision distribution function is given as:

f (x,t) = f(x,t) − M −1 SM(f − f (0) ) + δts(x,t). (2.4)
∗
The streaming step merely propagates each component of the dis-
tribution function towards the nearest node along its lattice direc-
tion

∗
f i (x + ce i δt,t + δt) = f (x,t). (2.5)
i
This process is depicted in the schematic in Fig. 2.12.Itcan be
shown that with the proper definition of S the dynamics of the
distribution functions tends asymptotically to the solution of the
partial differential equations of the monodomain model.

Figure 2.12. Description of the LBM-EP algorithm on a 2-D slice. The first image
shows the pre-collision distribution in a node at the start of the step. The collision
step redistributes the distribution function values (middle figure) and finally the
post-collision values stream to the corresponding neighbors.

LBM-EP evaluation
Computational models of electrophysiology are extremely hard
to validate, owing to the lack of analytical solutions. To this end,
recently a benchmark study was conducted [221]tocompare the
predictions of eleven existing computational solvers. To eliminate
geometric complexity, the problem domain was chosen to be a
cuboidal piece of tissue of size 20 × 7 × 3 mm. All the solvers
were required to use the epicardial variant of the ten Tusscher
and Panfilov cell model with prescribed model parameters. The
computations were required to be done at prescribed spatial res-
olutions of 0.5 mm, 0.2 mm and 0.1 mm and the computations
at each spatial resolution were run at three different time-steps of