Biomedical Engineering Reference
In-Depth Information
n
T
u
i
∂Ω
i
\
Γ
m
and
n
T
u
e
=
∂Ω
e
\
Γ
m
,
∇
=
0on
∇
0on
(5.12)
v
ε
(
v
0
(
w
ε
(
w
0
(
c
ε
(
c
0
(
Γ
m
,
x
,
0
)=
x
)
,
x
,
0
)=
x
)
,
x
,
0
)=
x
)
su
(5.13)
Γ
m
pointing
where
R
=
τ
m
R
,
S
=
τ
m
S
and
n
=
ν
i
=
−
ν
e
denotes the normal to
Ω
e
. We note that the variables
v
ε
,
w
ε
,
c
ε
and
I
m
are defined only on the
towards
Γ
m
. Problem
P
ε
is not a standard parabolic homogenization prob-
lem and its main difficulties are associated with the fact that the evolution term de-
pends explicitly on
membrane surface
Ω
i
,
e
could be quite
ε
,itis
degenerate
and that the boundaries of
irregular.
Formal two-scale homogenization.
The two-scale method of homogenization (see
[12, 82, 118]) can be applied to the previous current conservation equations (5.9),
(5.10), where we assume that the cells are distributed according to an ideal peri-
odic organization similar to a regular lattice of interconnected cylinders. We de-
note by
E
i
,
3
E
e
:
= R
\
E
i
, two open, connected and periodic reference subsets of
3
R
by
Y
the elementary pe-
riodicity region, composed of the intra- and extracellular volumes
Y
i
,
e
=
with common Lipschitz boundary
Γ
m
:
=
∂
E
i
∩
∂
E
e
,
E
i
,
e
representing a reference volume box containing a cellular configuration
Y
i
with cell
membrane surface
S
m
=
Γ
m
∩
Y
∩
occupied by the cardiac tis-
sue is decomposed into the intra- and extracellular domains
Y
i
. The physical region
Ω
Ω
i
,
e
, obtained as
Ω
i
=
Γ
m
=
∂Ω
i
∩
∂Ω
e
=
Ω
∩
εΓ
m
models the cellular membrane. Since the cardiac tissue exhibits a number of sig-
nificant inhomogeneities, such as those related to cell-to-cell communications, the
conductivity tensors are considered dependent on both slow and fast variables, i.e.
σ
i
,
e
(
E
i
,
Ω
e
Ω
∩
ε
=
Ω
∩
ε
E
e
. The common boundary
x
ε
)
x
ε
models the inclusion of gap-junction effects.
We then define the rescaled symmetric conductivity matrices
x
,
. The dependence of
σ
i
on
x
ε
)
,
σ
i
,
e
(
x
)=
σ
i
,
e
(
x
,
3
are continuous functions satisfying the usual
uniform ellipticity and periodicity conditions.
The following macroscopic bidomain model can be formally derived by taking
the average of a cellular model in the periodic case, see for details [27, 86].
3
×
where
σ
i
,
e
(
x
,
ξ
)
:
Ω
×
E
i
,
e
→
M
Dimensionless averaged model P.
For a periodic network of interconnected cells,
let
be the ratio between the surface membrane and the volume of the
reference cell and let
β
=
|
S
m
|/|
Y
|
. Then the governing dimensionless equations
of the macroscopic intra and extracellular potentials of zero order in
β
i
,
e
=
|
Y
i
,
e
|/|
Y
|
ε
are given by
⎧
⎨
∂
v
div
D
i
(
x
)
∇
x
u
i
=
β
t
+
I
ion
(
v
,
w
,
c
)
∂
∂
v
(5.14)
div
D
e
(
x
)
∇
x
u
e
=
−
β
t
+
I
ion
(
v
,
w
,
c
)
⎩
∂
∂
w
∂
c
v
=
u
i
−
u
e
,
t
−R
(
v
,
w
)=
0
,
t
−S
(
v
,
w
,
c
)=
0
,
∂
∂
