Biomedical Engineering Reference
In-Depth Information
where
G
s
and
φ
s
denote the maximum conductance and the resting potential of the
stretch-activated channels, separately. This contribution to the current source term
is due to the opening of ion channels, and, therefore, exists only when myofibers are
under tension.
13.4 Numerical Example: Excitation-Contraction of the Heart
This section is devoted to the coupled electromechanical analysis of a biventricu-
lar generic heart model that favorably illustrates the main physiological features of
the overall response of the heart. For the material parameters that govern the pas-
sive stress response, we used the values given in Table 1 of Göktepe et al. (
2011
).
The parameters governing the potential flux and the current source, outlined in
Sects.
13.3.2
and
13.3.3
, have been adopted from Table 4 of Göktepe and Kuhl
(
2010
). The material parameters governing the active part in Eqs. (
13.18
)-(
13.23
)
are taken as
λ
a
max
=
3
.
5s
−
1
.
The solid model of a biventricular generic heart is constructed by means of two
truncated ellipsoids. The generic heart model whose dimensions and spatial dis-
cretization are depicted in Fig. 5 of Göktepe and Kuhl (
2010
) is meshed with 13 348
four-node coupled tetrahedral elements connected at 3 059 nodes. The unevenly dis-
tributed average orientation of contractile myocytes
f
0
is depicted with yellow lines
in Fig.
13.4
. This fiber organization is consistent with the myofiber orientation in
the human heart where the fiber angle ranges from approximately
2
(
mVs
)
−
1
0
.
70,
η
=
0
.
05 MPa,
β
=
6,
q
=
and
k
=
70
◦
−
in the epi-
70
◦
in the endocardium with respect to the longitudinal plane. Dis-
placement degrees of freedom on the top base surface are restrained and the whole
surface of the heart is assumed to be flux-free.
To initiate the excitation, the elevated initial value
Φ
0
=−
cardium to
+
10 mV of the trans-
membrane potential is assigned to the nodes located at the upper part of the septum
as indicated by the partially depolarized region in the first panel in Fig.
13.4
.The
initial transmembrane potential at the remaining nodes is set to its resting value
Φ
0
=−
80 mV. The excitation at the top of the septum generates the depolarization
front traveling from the location of stimulation throughout the entire heart, thereby
resulting in the contraction of the myocytes, see the snapshots in Fig.
13.4
corre-
sponding to systole. At first glance, we observe that the contraction of myocytes
gives rise to the upward motion of the apex. More importantly, we also note that
the upward motion of the apex is accompanied by the physiologically observed wall
thickening and the overall twisting of the heart. To appreciate these phenomena bet-
ter, the two slices are presented in the complementary images shown in the lower
rows of Figs.
13.4
and
13.5
. Undoubtedly, it is the heterogeneous distribution of
myocyte orientation that yields this physiological response through the non-uniform
contraction of myofibers. The panels in Fig.
13.5
illustrate the relaxation of the heart
during the course of repolarization.
