Biomedical Engineering Reference
In-Depth Information
namics and a time-independent term
I
0
(
v
,
c
)
. The gating variables
w
:
=(
w
1
,...,
w
M
)
regulate the conductances of the various ionic fluxes and
c
:
are vari-
ables regulating the intracellular concentrations of the various ions. The dynamics
of the gating variables
w
is given by a first-order kinetic model, while the ionic con-
centrations
c
satisfy differential equations associated to ion channels, pumps and
exchanger currents that are carrying the same ionic species.
Simplified models of lower complexity, with only 1 or 2 gating variables and no
ionic concentrations, have been proposed and employed for analytical and numerical
studies. The simplest model with only one gating variable
w
is the FitzHugh-Nagumo
(FHN) model [50] and its variants [1, 101], the latter yielding better approximation of
a typical cardiac action potential. A simplified ionic model with two gating variables
(M = 2) was proposed in [48, 49], and for human ventricular cell see [137].
=(
c
1
,...,
c
Q
)
5.2.2 The anisotropic macroscopic Bidomain model
The cardiac ventricular tissue is conceived at a macroscopic level as an arrange-
ment of cardiac fibres organized in toroidal layers nested within the ventricular wall
and rotating counterclockwise from epi- to endocardium (see e.g. [130]). More re-
cently, [71, 72] have shown that this fibre structure has an additional laminar or-
ganization modelled as a set of muscle sheets running radially from epi- to endo-
cardium. Therefore, at any point
x
, it is possible to identify a triplet of orthonormal
principal axes
a
l
(
x
)
,
a
t
(
x
)
,
a
n
(
x
)
, with
a
l
(
x
)
parallel to the local fibre direction,
a
t
(
tangent and orthogonal to the radial laminae, respectively, and both
transversal to the fibre axis [38, 72]. Denoting by
x
)
and
a
n
(
x
)
e
n
the conductivity
coefficients in the intra and extracellular media measured along the corresponding
directions
a
l
,
i
,
e
i
,
e
i
,
σ
,
σ
,
σ
t
l
a
t
,
a
n
, the anisotropic conductivity tensors
D
i
(
x
)
and
D
e
(
x
)
related to
the
orthotropic anisotropy
of the media are given by
e
l
a
l
(
i
,
e
t
a
t
(
i
,
a
l
(
a
t
(
e
n
a
n
(
i
,
a
n
(
D
i
,
e
(
x
)=
σ
x
)
x
)+
σ
x
)
x
)+
σ
x
)
x
)
.
(5.3)
From the macroscopic point of view the cardiac tissue is represented as a
bidomain
(see e.g. [22, 53, 57, 69, 76, 92, 103, 145]), i. e. the superposition of two anisotropic
continuous media, the intra- and extracellular media (i) and (e), coexisting at ev-
ery point of the tissue and separated by a distributed continuous cellular membrane.
Let
Γ
H
=
∂Ω
H
denote the volume and the heart surface, respectively. The
intra and extracellular electric potentials
u
i
,
Ω
H
and
u
e
in the anisotropic Bidomain model
are described by a reaction-diffusion system coupled with a system of ODEs for
ionic gating variables
w
and ion concentrations
c
.Let
I
e
,
i
app
be an applied extracellu-
lar and intracellular current per unit volume, satisfying the compatibility condition
Ω
H
(
I
app
+
I
i
app
)
u
i
,
e
the intra- and extracellular
current density. Due to the current conservation law, it follows
dx
=
0, and denote by
j
i
,
e
=
−
D
i
,
e
∇
c
m
∂
v
I
i
app
,
I
app
,
div
j
i
=
−
J
m
+
div
j
e
=
J
m
+
J
m
=
χ
I
m
=
t
+
i
ion
(
v
,
w
,
c
)
,
∂
where
χ
is the ratio of membrane area per tissue volume and
J
m
,
c
m
=
χ
C
m
and
i
ion
=
χ
I
ion
are the transmembrane current, the membrane capacitance and the ionic
