Biomedical Engineering Reference
In-Depth Information
We recall that the macroscopic electrical properties of the cardiac muscle are
markedly anisotropic and recent studies have evidenced a laminar organization of
the fibres (see e.g. [71]), yielding two preferred transverse fibre directions, one tan-
gent and the other orthogonal to the laminae, respectively. This geometrical fibre
architecture yields orthotropic properties of the effective macroscopic conductivity
tensors, see e.g. [61, 144].
We also recall that the heart can be stimulated by a unipolar extracellular elec-
trode (i.e.
I
i
app
=
0and
I
app
=
0 in (5.4)) by applying a cathodal (
I
app
<
0) or anodal
(
I
app
>
0) pulse in four distinct ways: by turning a negative current on (cathode
make) or off (cathode break), or by turning a positive current on (anode make) or
off (anode break). In [104, 105] the bidomain model with unequal anisotropic ratio
was first proposed and used to establish that the stimulation by a unipolar extracel-
lular cathodal or anodal electrode produces a characteristic transmembrane pattern
called
virtual electrodes response
. Only bidomain models with unequal anisotropy
ratios of the intra- and extracellular media are able to generate regions known as
virtual electrodes
, see e.g. [124, 153]. The effects and features of make and break
excitation mechanism, have been explained in terms of the underlying
virtual elec-
trodes polarization
, and have been subsequently investigated by simulation studies
in [45, 91, 100, 106, 107, 108, 109, 110, 116, 117, 128] and by experimental studies
in [43, 44, 125, 154]; see also the recent surveys [111, 155].
Bidomain simulations of anode make mechanism.
Here we focus on the 3D inves-
tigation of the excitation mechanisms associated with an anodal stimulation, which
can be generated only in tissues with unequal anisotropy ratio.
The cardiac domain
H
considered is a Cartesian slab of dimensions 0
.
96
×
0
.
96
×
32 cm
3
, modelling a portion of the left ventricular wall and we use as cardiac mem-
brane model the Luo-Rudy LRd model, see [47], augmented with the funny and elec-
troporation currents; see [33] for more details. The conductivity coefficients used are
given in Table 5.2 and yield local conductivity velocities of 0
0
.
015 cm/ms
along the fibres, and tangent and orthogonal to the laminae, respectively. These val-
ues are in partial agreement with the conductivity coefficient estimates of [61] de-
rived from experimental measurements. The same table reports also the associated
unequal anisotropy ratios
.
065, 0
.
03, 0
.
e
tn
. In order to compare the predicted response in
unequal and equal anisotropy tissues we also considered an equal anisotropy tissue
with conductivity coefficients given in the same Table 5.2 that yield the same con-
servative values of conductivity velocities, i.e. 0
i
,
e
i
,
ρ
,
ρ
lt
015 cm/ms. The car-
diac fibres are parallel on each intramural plane parallel to the epicardium and rotate
linearly with depth of 120
◦
starting from
.
065, 0
.
03, 0
.
45
◦
on the epicardium. In order to inves-
tigate the 3D shape of the virtual electrode response, we apply an anodal stimulus to
a slab initially at rest. Tha anodal stimulus is applied with a duration of 10
ms
and
an amplitude of 0
−
03 cm
3
.
0648
mA
in a small region of dimensions 0
.
06
×
0
.
06
×
0
.
(i.e.
I
i
app
=
600 mA/cm
3
) at the center of the epicardial face.
Figs. 5.2 and 5.3 show the epicardial distribution of the intracellular potential
(top), extracellular potential (middle) and transmembrane potentials (bottom) 2 ms
after the anodal stimulus elicited in a slab without and with transmural fibre rota-
tion, respectively. In each figure, the panels on the right column refer to a tissue
0 mA/cm
3
,
I
app
=
