Biology Reference
In-Depth Information
where the s
's
are the conductivities normal to the interface, and the
∂
n
is the normal derivative operator.
In order for the problem to be well posed, a third boundary condition
on d
H
is required. While the first two follow necessarily for all electri-
cal phenomena, there are a number of ways to formulate the third
boundary condition. Typically, we specify:
/
∂
∂
∂
f
i
n
(14)
s
=
0 on
H
δ
i
which has the physical interpretation that at the heart/body interface,
all intracellular current must flow first through the extracellular space
before it flows into the surrounding tissue. A boundary condition at d
B
for the Laplace equation in f
b
is also required. Given that air is a poor
conductor, this is simply
∂
∂
f
(15)
b
n
s
=
0on
δ
B
b
Finally, an initial condition on the transmembrane voltage must be
specified,
v
(
x
,t
= 0) =
V
(
x
). Then from this, initial conditions on f
e
(
x
,t
= 0)
and f
b
(
x
,t
=0) can be found by solving the appropriate elliptic equation.
Equations (9)-(15) specify the bidomain problem.
Under some restrictive assumptions, the bidomain equations can be
simplified dramatically. If the surrounding tissue is taken to be a good
insulator, then s
b
=
0 in
B
. Then we have
f
=
0
e
∂
∂
f
h
e
=
0
(16)
∂
∂
f
i
n
=
0
on
H
δ
and the Laplace equation for f
b
need not be considered. Additionally,
under the assumption of equal anisotropy, namely, that
1
(17)
M
()
x
=
M
()
x
i
e
k
where
k
is called the anisotropy ratio, eqs. (9) and (10) uncouple, requiring
then only solution of the parabolic equation,
∂
∂
v
t
1
1
k
7
9
:
?
(18)
(,)
x
t
=
−
I
(,)
x
t
−
I
(,)
x
t
+
(
)
∇
(()(, )
Mv
xx
∇
t
ion
app
i
C
b
k
+
1
m
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