Environmental Engineering Reference
In-Depth Information
link both problems together. In the next sections, we explain each model and how
they are coupled.
2.1.1 Electrophysiological Models
The activation potential propagation is modelled using an excitable media model,
D
ij
∂ˆ
ʱ
∂
∂ˆ
ʱ
∂
∂
∂
=
+
L
(ˆ
ʱ
).
(1)
t
x
i
x
j
which consists of a diffusion equation with anisotropic diffusion tensor
D
ij
(Rubart
and Zipes
2001
; Coghlan et al.
2006
) and local non-linear terms. The anisotropy
comes from the fact that the cardiac tissue is made of muscular fibers with different
diffusion along or transversal to the fibers. The non-linear term
L
represents
the ionic current
I
ion
cell model, ranging from simple phenomenological schemes
(FitzHugh
1961
) up to more complex and physiologically meaningful models (Tuss-
cher et al.
2004
). Then, the electrophysiology modelling equations are labelled with
Greek subindices for the activation potentials involved. We focus here in the so-called
monodomain models, where intra- and extra-cellular potentials are modelled with
one single equation, so
(ˆ
ʱ
)
1. The electromechanical impulse, that is to say the
electrical activation which drives muscular contraction starts at the junctions of the
Purkinje network. In our model, synthetic networks are generated for each geometry
(Sebastian et al.
2012
).
ʱ
=
2.1.2 Mechanical Models
The myocardium is modelled as a compressible solid, with three-dimensional ele-
ments. The material model is hyper-elastic, with anisotropic behaviour ruled again
by the fiber structure. In this work, we use a transversally isotropic version of
a Holzapfel-Ogden material (Holzapfel and Ogden
2009
), already presented by
Lafortune et al. (
2012
). We briefly describe it in this section.
In a total-Lagrangian formulation, the governing equations are written as:
2
u
i
∂
ˁ
o
∂
∂
P
iJ
=
X
J
+
ˁ
o
B
i
.
(2)
t
2
∂
J
−
1
PF
T
, is defined through the Piola-Kirchoff
P
iJ
,the
The Cauchy stress
σ
=
∂
x
i
deformation gradient
F
iJ
=
and its Jacobian
J
. Stress is a combination of
∂
X
J
active and passive parts:
Ca
2
+
]
)
σ
=
σ
pas
+
˃
act
(ʻ,
[
f
⊗
f
(3)
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