Biomedical Engineering Reference
In-Depth Information
sG
sG
sG
i
e
b
T
+
T
=
T
on
E
H
,
[20]
i
e
b
s
n
s
n
s
n
where the T's are the conductivities normal to the interface, and the 0/0
n
is the
normal derivative operator. In order for the problem to be well posed, a third
boundary condition on E
H
is required. While the first two follow necessarily for
all electrical phenomena, there are a number of ways to formulate the third
boundary condition. Typically, we specify:
sG
i
T
=
0 on
E
H
,
[21]
i
s
n
which has the physical interpretation that at the heart/body interface all intracel-
lular current must flow first through the extracellular space before it flows into
the surrounding tissue. A boundary condition at E
B
for the Laplace equation in
G
b
is also required. Given that air is a poor conductor, this is simply
sG
T
b
=
0 on
E
B
.
[22]
b
s
n
Finally, an initial condition on the transmembrane voltage must be speci-
fied,
v
(
x
,
t
=0) =
V
(
x
). Then from this, initial conditions on G
e
(
x
,
t
=0) and G
b
(
x
,
t
=0)
can be found by solving the appropriate elliptic equation. Equations [16]-[22]
specify the bidomain problem.
Under some restrictive assumptions, the bidomain equations can be simpli-
fied dramatically. If the surrounding tissue is taken to be a good insulator, then
T
b
= 0 in
B
. Then we have
G
e
= 0 on E
H
,
[23]
sG
e
sI
=
0 on
E
H
,
[24]
sG
i
s
=
0 on
E
H
,
[25]
n
and the Laplace equation for G
b
need not be considered. Additionally, under the
assumption of equal anisotropy, namely, that
