Page 111 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 2 Implementation of a patient-specific cardiac model 81

eq
The collision term K i,j (f i (x,t) − f (x,t)) is formulated as a relax-
i
ation towards a thermodynamic equilibrium, where K is the colli-
sion matrix (or collision operator), containing the relaxation fac-
tors. A popular formulation is the multiple-relaxation-time (MRT)
collision operator [252].

Turbulence modeling
Cardiac flow may enter physical regimes characterized by high
Reynolds numbers, which require turbulence modeling to be
added to the regular LBM description. The Smagorinsky model,
for example, is a well established approach for simulating tur-
bulent flows. It consists of modeling sub-grid scale effects as an
additional viscosity ν t , called turbulent viscosity. In our experience
the Smagorinsky turbulence model, while simple to implement,
was not stable enough for the high Reynolds numbers that can
occur during cardiac flow physics computation. It should be men-
tioned that one of the most important limitations of the LBM is
numerical instability that appears when viscosity is too low (or,
equivalently, Reynolds number is too large).
An approach to improve the stability is the entropic Lattice-
Boltzmann method (ELBM) [253], which is based on the Boltz-
mann’s H theorem, and which consists of changing the collision
model so that it satisfies the second law of thermodynamics. The
equilibrium distribution function is changed as follows:

√
D % & '% 2u d + ' e id
$
f eq (x,t) = w i ρ 2 − 1 + 3u 2 1 + 3u d .
d
1 − u d
d=1
Accordingly, the collision operation becomes (with D the space di-
mension, 3 in our case):
eq
f i (x + e i δt,t + δt) = f i (x,t) + αβ(f (x,t) − f i (x,t)),
i
where β = 2/τ is the equivalent of the LBGK relaxation time [252]
and is related to the fluid viscosity. The additional factor α is spe-
cific to the ELBM model and it is computed for each lattice node
at each iteration by solving a non-linear equation:

H(f ) = H(f + α(f eq − f )), (2.35)
N % '
f i
#
H(f ) = f i ln . (2.36)
w i
i=0
Eq. (2.35) is usually solved iteratively using a Newton method.
Note that if α = 2 then the LBGK collision is recovered. In the