Biomedical Engineering Reference
In-Depth Information
The last constitutive relation describing the electrical source term of the
Fitzhugh-Nagumo-type excitation Eq. (
13.6
) is additively decomposed into the
excitation-induced purely electrical part
I
e
(Φ,r)
and the stretch-induced mechano-
I
m
(
g
electrical part
;
F
,Φ)
,i.e.
I
φ
=
I
e
(Φ,r)
+
I
m
(
g
;
F
,Φ).
(13.12)
The former describes the effective current generation due to the inward and outward
flow of ions across the cell membrane. This ionic flow is triggered by a perturbation
of the resting potential of an excitable cardiac muscle cell beyond some physical
threshold upon the arrival of the depolarization front. The latter, on the other hand,
incorporates the opening of ion channels under the action of deformation, see Kohl
et al. (
1999
).
Apart from the primary field variables, the recovery variable
r
, which describes
the repolarization response of the action potential, appears among the arguments of
I
e
in Eq. (
13.12
). Evolution of the recovery variable
r
chiefly determines the shape
and duration of the action potential locally inherent to each cardiac cell and may
change throughout the heart. For this reason, evolution of the recovery variable
r
is commonly modeled by a local ordinary differential equation
=
f
r
(Φ,r)
.From
an algorithmic point of view, the local nature of this evolution equation allows us
to treat the recovery variable as an internal variable. This is one of the key features
of the proposed formulation that preserves the modular global structure of the field
equations as set out in our recent works (Göktepe and Kuhl,
2009
,
2010
; Göktepe et
al.,
2010
; Wong et al.,
2011
).
˙
r
13.3 Model Problem of Cardiac Electromechanics
In this section, we present the specific constitutive equations that are utilized in
the analysis of the representative numerical example in Sect.
13.4
. In particular, we
identify the specific expressions for the Kirchhoff stress
τ
, the potential flux
ˆ
q
, and
ˆ
I
φ
.
the current source
13.3.1 Active and Passive Stress Response
The ventricular myocardium can be conceived as a continuum with a hierarchi-
cal architecture where uni-directionally aligned muscle fibers are interconnected in
the form of sheets. Loosely connected by perimysial collagen, these approximately
four-cell-thick sheets can easily slide along each while being stiffest in the direction
of the large coiled perimysial fibers aligned with the long axes of the cardiomy-
ocytes, as depicted in Fig.
13.3
. To model the passive response of myocardium,
we employ the orthotropic model of hyperelasticity recently proposed by Holzapfel
