Biomedical Engineering Reference
In-Depth Information
where the effective conductivity tensors are given by
,
ξ
)
I
T
d
1
w
i
,
e
D
i
,
e
(
x
)=
Y
i
,
e
σ
i
,
e
(
x
−
∇
ξ
(
)
ξ
(5.15)
|
Y
|
w
i
,
e
1
w
i
,
e
2
w
i
,
e
3
and
w
i
,
e
T
are solutions of the following cellular problems
=(
,
,
)
⎧
⎨
,
ξ
)
∇
ξ
w
k
=
(
,
div
ξ
σ
x
0
in
Y
i
,
e
i
,
e
(5.16)
⎩
,
ξ
)
∇
ξ
w
i
,
e
n
T
ξ
σ
i
,
e
(
x
=
n
ξ
k
,
on
S
,
k
=
1
,
2
,
3
.
k
The tensors
D
i
,
e
are symmetric and positive definite matrices.
The abstract variational framework of the cellular and averaged models share the
same structural properties, see [27]. It was shown in [27] that both the cellular and the
averaged models are well posed, assuming the FitzHugh-Nagumo membrane model
[50], where the ionic current is a cubic-like function in
v
, linear in the recovery
variable
w
,i.e.
I
ion
(
v
,
w
)=
g
(
v
)+
α
w
,
and
R
(
v
,
w
)=
η
v
−
γ
,
α
,
η
,
γ
>
0
.
Theorem 1 (model P
ε
well-posedness).
Assume
Γ
ε
regular, suppose that the ini-
v
0
,
w
0
)
∈
L
2
(
Γ
m
)
×
L
2
(
Γ
m
)
V
ε
=
tial data satisfy
(
and define the quotient space
H
1
(
Ω
e
)
/
{
(
γ
,
γ
)
(
Ω
i
)
×
H
1
L
2
(
Γ
m
)
}
×
:
γ
∈
R
. Then there exists a unique solu-
tion U
ε
=(
u
i
,
u
e
,
w
ε
)
∈
C
0
V
ε
)
of the variational formulation of Problem
P
ε
(]
0
,
T
]
;
v
ε
/
∂
w
ε
/
∂
L
2
T
;
L
2
(
Γ
m
))
,
and u
¯
ε
:
u
i
,
u
e
)
with
∂
t
,
∂
t
∈
(
0
,
=(
solution of the differ-
ential problem
P
ε
in the standard distributional sense.
L
2
L
2
Theorem 2 (model P well-posedness).
Let
(
v
0
,
w
0
)
∈
(
Ω
)
×
(
Ω
)
and define
=
H
1
(
Ω
)
/{
(
γ
,
γ
)
H
1
L
2
the quotient space
V
:
(
Ω
)
×
:
γ
∈
R
}×
(
Ω
)
. Then there
C
0
exists a unique solutionU
=(
u
i
,
u
e
,
w
)
∈
(]
0
,
T
]
;
V)
of the variational formulation
L
2
T
;
L
2
of the averaged Problem
P
with
∂
v
/
∂
t
,
∂
w
/
∂
t
∈
(
0
,
(
Ω
))
and u
¯
:
=(
u
i
,
u
e
)
solution of the differential problem
P
in the standard distributional sense.
Existence results for global solutions of the averaged Bidomain model with other
simplified membrane models have been obtained using different techniques in [10,
14, 15]. Extension of the well-posedness results for the cellular and the averaged
models with more complex ionic current membrane dynamics have been obtained
recently in [146, 147], both for the classical Hodgkin-Huxley model of the nerve
action potential [59] and for the Luo-Rudy Phase I model [74].
The derivation of the Bidomain model based on the two-scale method is only
formal. The following convergence result for the Bidomain model with FHN gating
has been developed in [86] using homogenization techniques and
Γ
−
convergence
theory.