Biomedical Engineering Reference
In-Depth Information
the voltage field is reproduced, is that ionic species are not well resolved. Neverthe-
less, these types of electrical models are able to provide some specific information
of interest, such as contractility.
Systems of ordinary differential equations (ODEs) of the form
∂
t
v
−
I
ion
(v,
w
)
=
0
,
∂
t
w
−
m
(v,
w
)
=
0
,
(14.1)
are employed to describe these cellular models, without any spatial detail. Here
v
denotes the transmembrane potential field,
w
contains all gating variables and
concentration of ionic species, and
I
ion
and
m
drive the kinetics of the system, its
specific form depending on the chosen cellular model.
Models at the cell level can be incorporated into macroscopic descriptions for
the propagation of electrical excitation throughout the cardiac muscle in the sim-
plest way assuming homogeneous diffusion of ionic species on the microstructure
of the substrate. The texture of the cardiac tissue can be incorporated observing that
cellular and extracellular components are characterized by different diffusivities.
A homogenization process yields the so-called bidomain equations (Tung,
1978
):
I
app
,
I
app
,
(14.2)
∂
t
v
−∇·
(
D
e
∇
u
e
)
+
I
ion
=
∂
t
v
+∇·
(
D
i
∇
u
i
)
+
I
ion
=
where
u
i
and
u
e
are the intra- and extracellular electric potentials (both defined in
every point of the domain), and
(I
app
,I
app
)
are possible externally applied stimuli.
The cardiomyocytes are organized in fibers that originate the anisotropic conduc-
tivity in the electrophysiology of the heart. The myofiber angle varies continuously
from about
60° (inverse circumferential axis) at the epicardium, to about 70° at the
endocardium. From the apical region, the myofibers that conform the tissue follow a
right helical orientation towards the subendocardium and a left helical path parallel
to the wall on the subepicardium. On the mid-wall region, cardiac fibers exhibit a
circumferential orientation, and on the basal site fibers cross from subendocardial
to the subepicardial region. Myocardial propagation velocities in the parallel and
perpendicular myofiber directions can differ up to an order of magnitude. These
geometrical features are encoded in the anisotropic conductivity tensors
D
i
and
D
e
(Colli Franzone and Pavarino,
2004
).
−
14.2.2 Mechanical Response of the Myocardium
The characterization of the material properties of the cardiac tissue requires precise
experimental settings that should reproduce physiological conditions as close as
possible. Usual tests include uniaxial and biaxial tension experiments, as well as
shear tests, from which it is possible to recover stress-strain relations on the different
directions of the anisotropic medium (fiber, sheets, and sheet-normal axes).
The usual kinematics descriptors of a continuum medium placed in
Ω
o
⊂ R
3
in its reference configuration are the deformation gradient of its motion
F
and the
