Biomedical Engineering Reference
In-Depth Information
tem of non-linear equations at each time step (see, e.g., [10, 29, 44]). Attempts to
reduce this computational complexity (without compromising numerical stability
too much) consist in introducing some sort of explicit treatment within the time-
marching procedure. For instance, by considering semi-implicit (see, e.g., Sect. 4.4
and [3, 15, 23, 39, 63]) or operator-splitting (see, e.g., [31, 64, 71]) schemes. All
these approaches uncouple the ODE system (ionic state and non-linear reaction
terms) from the electro-diffusive components (transmembrane and extracellular po-
tentials). A few articles [3, 39, 63, 71] considered, without analysis, a decoupled
(
Gauss-Seidel
-like) time-marching of the three fields.
It can be shown that the Gauss-Seidel and the Jacobi electro-diffusive splittings
do not compromise the stability of the resulting scheme. They simply alter the en-
ergy norm, and the time-step restrictions are uniquely dictated by the semi-implicit
treatment of the ODE system and the non-linear reaction terms. Let us consider
the semi-discretization in time of (4.18) obtained by combining a first-order semi-
implicit treatment of the ionic current with an explicit (Gauss-Seidel or Jacobi-like)
treatment of the electro-diffusive coupling, as detailed in Algorithm 1.
Algorithm 1
Decoupled time-marching for the bidomain equation.
1. Ionic state: find
w
n
+
1
∈
L
∞
(
Ω
H
)
such that
∂
τ
w
n
+
1
)
ξ
=
0
+
g
(
V
m
,
w
n
+
1
Ω
H
L
2
;
2. Transmembrane potential: find
V
n
+
1
m
for all
ξ
∈
(
Ω
H
)
H
1
∈
(
Ω
H
)
such that
χ
m
Ω
H
∂
τ
V
n
+
1
V
n
+
1
m
u
e
·
∇φ
φ
+
Ω
H
σ
i
∇
·
∇φ
+
Ω
H
σ
i
∇
m
I
n
+
1
app
)
φ
V
m
,
w
n
+
1
=
−
I
ion
(
Ω
H
φ
∈
H
1
(
Ω
H
)
;
3. Extracellular potential: find
u
n
+
1
e
for all
∈
H
1
(
Ω
H
)
∩
L
0
(
Ω
H
)
,
u
n
+
1
e
V
m
·
∇ψ
=
Ω
H
(
σ
+
σ
)
∇
·
∇ψ
+
Ω
H
σ
∇
0
i
e
i
(
Ω
H
)
∩
L
0
(
Ω
H
)
;
4. Go to next time-step.
for all ψ
∈
H
1
m
and
u
n
+
e
are implic-
itly coupled and, therefore, steps 2 and 3 of Algorithm 1 have to be performed
simultaneously (as in Sect. 4.4). On the other hand, for
u
e
,
V
m
)=(
u
n
+
1
V
n
+
1
, the unknown potentials
V
n
+
1
(
,
)
For
e
m
u
e
,
V
m
)=(
u
e
,
V
n
+
1
m
(
)
or
u
e
,
V
m
)=(
u
e
,
V
m
)
(
, the electro-diffusive coupling becomes explicit and therefore
these steps can be performed separately: either sequentially (Gauss-Seidel) or in
parallel (Jacobi).
