Biomedical Engineering Reference
In-Depth Information
1
M
()
x
=
Mx
()
,
[26]
i
e
L
where
k
is called the anisotropy ratio, Eqs. [16] and [17] uncouple, then requir-
ing only solution of the parabolic equation
1
¯
sO
1
L
.
[27]
(,)
xt
=
¡
I
(,)
xt
I
(,)
xt
+
(
)
¸
(
M x
()
O
(,)) on
xt
°
H
¢
±
ion
app
i
s
t
C
C L
+
1
m
Equation [27] is referred to as the monodomain equation.
The conductivity tensors at each point within the heart are specified by fiber
orientation and by specific conductivities in each of the local coordinate direc-
tions. The conductivity tensor in the local coordinate system,
G
i
(
x
), is defined as
¯
T
¡
°
1,
i
¡
°
()
x
G
i
=
,
[28]
¡
°
T
¡
2,
i
°
¡
°
T
¡
°
¢
±
3,
i
where T
1,
i
is the longitudinal and T
2,
i
and T
3,
i
are the transverse intracellular con-
ductivities, respectively. This local tensor may be expressed in global coordi-
nates to give the conductivity tensor of Eq. [27] using the transformation
M
i
(
x
) =
P
(
x
)
G
(
x
)
P
T
(
x
),
[29]
where
P
(
x
) is the coordinate transformation matrix from local to global coordi-
nates.
P
(
x
) is in turn determined by the underlying fiber organization of the
heart, and is obtained using DTMRI as described in §3.2. If only fiber direction
information is available, then it is appropriate that conductivities transverse to
the fiber long-axis be assumed equal (T
2,
i
= T
3,
I
).
Figure 5B shows the results of applying these methods to the analysis of
conduction in a normal canine heart. As described previously, Figure 5A shows
activation time (color bar, in msec) measured experimentally in response to an
RV stimulus pulse applied at the epicardial locations marked in red. Following
electrical mapping, this heart was excised, imaged using DTMRI, and an FEM
was then fit to the resulting geometry and fiber orientation data sets. Figure 5A
shows activation time displayed on this FEM. The stimulus wavefront can be
seen to follow the orientation of the epicardial fibers, which is indicated by the
dark line segments in Figure 5A. Figure 5B shows results of simulating conduc-
tion using a computational model of the very same heart that was mapped elec-
trically in Figure 5A. Results can be seen to agree qualitatively; however, model
conduction is more rapid in the region where the RV and LV join. Nonetheless,
