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.
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