Biomedical Engineering Reference
In-Depth Information
elliptic problem
div
D
0
(
n
T
D
0
∇
x
)
∇
u
0
(
x
,
t
)=
0in
Ω
0
,
u
0
(
x
,
t
)=
0on
Γ
0
n
T
D
0
(
n
T
D
e
(
u
0
(
x
,
t
)=
u
e
(
x
,
t
)
on
Γ
H
,
x
)
∇
u
0
=
x
)
∇
u
e
on
Γ
H
.
Defining
D
i
(
u
e
(
x
)+
D
e
(
x
)
,
x
∈
Ω
H
x
,
t
)
,
x
∈
Ω
H
D
=
D
(
x
)=
u
(
x
,
t
)=
D
0
(
x
)
,
x
∈
Ω
0
u
0
(
x
,
t
)
,
x
∈
Ω
0
,
then the extracellular and extracardiac potential field
u
satisfies the following elliptic
problem
⎧
⎨
div
J
v
(
x
,
t
)
in
Ω
H
div
D
∇
u
(
x
,
t
)=
0
in
Ω
0
(5.6)
D
⎩
n
T
n
T
J
v
[[
(
,
) ]
Γ
H
=
,
[[
(
,
) ]
Γ
H
=
(
,
)
u
x
t
0
∇
u
x
t
x
t
n
T
D
0
(
,
)=
,
∇
u
x
t
0
on
Γ
0
where
J
v
(
x
,
t
)=
−
D
i
∇
v
(
x
,
t
)
and
[[
Φ
]]
S
=
Φ
S
+
−
Φ
S
−
denotes the jump through
and
n
T
J
v
(
a surface
S
,
)
act as
impressed
current and
current source density
. Thus, if we assume known the
transmembrane potential distribution
v
Φ
|
S
±
. We remark that the right-hand sides div
J
v
(
x
,
t
)
x
,
t
is known, the above elliptic problem
fully characterizes the extracellular and extracardiac field
u
(
x
,
t
)
(
x
,
t
)
up to an additive
constant.
5.2.3 Modelling cardiac cells arrangements
The Bidomain model introduced in the previous section will now be derived from a
family of cellular problems by using homogenization techniques, which will solidly
justify its macroscopic meaning. The cardiac tissue is composed of a collection of
elongated cardiac cells having roughly a cylindrical form with diameter
d
c
≈
10
μ
m
and length
l
c
≈
m. The cells are coupled together in end-to-end, mainly, and
also in side-to-side apposition by gap junctions [62, 114]. The end-to-end contacts
produce the fibres structure of the cardiac muscle, whereas the presence of lateral
junctions establishes a connection between the elongated fibres. The cellular struc-
ture of the cardiac tissue can be roughly viewed as composed by two ohmic con-
ducting media
100
μ
Ω
i
(intracellular space) and
Ω
e
(extracellular space) separated by the
active membrane
=
∂Ω
i
∩
∂Ω
e
. The effects of the microstructure on the current
flow are described by the conductivity tensors
Γ
m
:
(
)
,
Σ
(
)
reflecting the local vari-
ations of conductances because of the presence of structural intra- and extracellular
inhomogeneities of resistance associated with e. g. gap junctions, connective tissue,
collagen, blood vessel. Due to the presence of gap junctions connecting the cardiac
cells end-to-end and side-to-side,
Σ
x
x
i
e
Ω
i
and
Ω
e
are regarded as two simply-connected
3
.
open sets of
R
