Biology Reference
In-Depth Information
Equation (18) is referred to as the monodomain equation.
Conductivity tensors at each point within the heart are specified by
fiber orientation and by specific conductivities in each of the local coor-
dinate directions. The conductivity tensor in the local coordinate
system,
G
i
(
x
)
,
is defined as
7
s
:
1
,
i
9
9
9
?
?
?
G
i
()
x
=
(19)
s
2
,
i
s
8
;
3
,
i
where s
1
,i
is the longitudinal and s
2
,i
and s
3
,i
are the transverse intra-
cellular conductivities, respectively. This local tensor may be expressed
in global coordinates to give the conductivity tensor of eq. (18) using
the transformation
M
i
(
x
) =
P
(
x
)
G
(
x
)
P
T
(
x
)
(20)
where
P
(
x
) is the coordinate transformation matrix from local to global
coordinates.
P
(
x
) is in turn determined by the underlying fiber organi-
zation of the heart, and is obtained using DTMRI as described
previously.
Figure 9.8 shows the results of applying these methods to the
analysis of conduction in a normal canine heart. Figure 9.8a shows
activation time (color bar, in ms) measured experimentally in response
to a stimulus pulse applied at the epicardial locations marked by the
silver balls positioned on the left aspect of the heart. Epicardial conduc-
tion was measured using electrode arrays consisting of a nylon mesh
with 256 electrodes and electrode spacing of ~5 mm sewn around its
surface. Bipolar epicardial twisted-pair pacing electrodes were sewn
onto the right atrium (RA) and the right ventricular (RV) free wall.
Four to ten glass beads filled with gadolinium-DTPA (~5 mM) were
attached to the sock as localization markers, and responses to different
pacing protocols were recorded. After all electrical recordings were
obtained, the animal was euthanized with a bolus of potassium chlo-
ride, and the heart was scanned with high-resolution T1-weighted
imaging in order to locate the gadolinium-DTPA-filled beads in scan-
ner coordinates. Following electrical mapping, this heart was excised,
imaged using DTMRI, and an FEM was then fitted to the resulting
geometry and fiber orientation data sets. Figure 9.8a shows experimen-
tally measured activation times displayed on this FEM. The stimulus
wave front can be seen to follow the orientation of the epicardial fibers,
which is indicated by the dark line segments in figure 9.8a. Figure 9.8b
shows results of simulating conduction using a computational model
of the very same heart that was mapped electrically in figure 9.8a.
Results can be seen to agree qualitatively; however, model conduction
is more rapid in the region where the right and left ventricles join.
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