Biomedical Engineering Reference
In-Depth Information
Remark 4.
An alternative can be to neglect the boundary coupling in (4.17)
2
while
keeping
u
e
and
u
T
fully coupled (see [52]). In a pure propagation framework (i.e.,
without extracellular pacing) numerical experiments suggest that this approach can
provide accurate ECG signals.
Conclusion
According to the above discussion, we can conclude that cell heterogeneity has to be
taken into account to get realistic ECGs. If we are only interested in a qualitatively
reasonable ECG, heart-torso uncoupling can be considered, but for quantitatively
precise results, it seems necessary to solve the coupled problem. It seems that re-
placing the bidomain equations by its monodomain approximation does not affect
significantly the ECGs. Numerical simulations show that the geometries of the heart
and the torso, the anisotropy in the heart and the heterogeneity of the conductivity
in the torso also have an important impact on ECGs. We refer to [7, 72] for more
details and further discussions about these effects and a sensitivity analysis.
4.5 Fully decoupled time-marching schemes
In this section, we introduce and analyze some time-marching schemes for the nu-
merical approximation of the isolated bidomain model (Sect. 4.5.1) and the heart-
torso system (Sect. 4.5.2). The particularity of these schemes is that they all allow
for an uncoupled computation of the fields involved (ionic state, transmembrane po-
tential, extracellular and torso potentials). The original ideas presented here were
originally proposed in [25].
In what follows, the quantity
∂
τ
x
n
denotes the first-order backward difference
x
n
x
n
−
1
(
−
)
/
τ
.
4.5.1 Isolated bidomain model
The isolated bidomain system (4.4)-(4.5) can be cast into weak form as follows: for
t
L
∞
(
Ω
H
)
H
1
H
1
L
0
(
Ω
H
)
>
0, find
w
(
·,
t
)
∈
,
V
m
(
·,
t
)
∈
(
Ω
H
)
and
u
e
(
·,
t
)
∈
(
Ω
H
)
∩
,
such that
∂
t
w
)
ξ
=
+
g
(
V
m
,
w
0
,
Ω
H
χ
m
∂
t
V
m
+
)
φ
+
I
ion
(
V
m
,
w
Ω
H
σ
i
∇
(
V
m
+
u
e
)
·
∇φ
=
I
app
φ
,
(4.18)
Ω
H
Ω
H
Ω
H
(
σ
i
+
σ
e
)
∇
u
e
·
∇ψ
+
Ω
H
σ
i
∇
V
m
·
∇ψ
=
0
)
.
The rapid dynamics of the ODE system (4.18)
1
, acting on the reaction terms
(4.18)
2
, detects a sharp propagating wavefront (see Fig. 4.3 (right)), which often
requires fine resolutions in space and in time. Fully implicit time-marching is, there-
fore, extremely difficult to perform since it involves the resolution of a large sys-
)
×
H
1
L
2
H
1
L
0
(
Ω
(
ξ
,
φ
,
ψ
)
∈
(
Ω
)
×
(
Ω
(
Ω
)
∩
for all
H
H
H
H