Biomedical Engineering Reference
In-Depth Information
4.2.1 Isolated heart modelling
The bidomain equations, originally derived in [68], are the most widely accepted
mathematical model of the macroscopic electrical activity of the heart (see, e.g., the
monographs [53, 61]). This macroscopic model is based on the assumption that, at
the cell scale, the cardiac tissue can be viewed as partitioned into two ohmic con-
ducting media, separated by the cell membrane: the intracellular medium, made of
the cardiac cells, and the extracellular one which represents the space between them.
After a homogenization process (see [46, 49]), the intra- and extracellular domains
are supposed to occupy the whole heart volume
H
(this also applies to the cell
membrane). Hence, the averaged intra- and extracellular densities of current,
j
i
and
j
e
, the conductivity tensors,
Ω
σ
i
and
σ
e
, and the electric potentials,
u
i
and
u
e
,are
defined in the whole heart domain
Ω
H
. The electrical charge conservation becomes
div
(
j
i
+
j
e
)=
0
,
(4.1)
and the homogenized equation of the electrical activity of the cell membrane is given
by
A
m
C
m
∂
t
V
m
+
)
+
i
ion
(
V
m
,
w
div
(
j
i
)=
I
app
,
(4.2)
complemented with the Ohm's laws
j
i
=
−
σ
i
∇
u
i
,
j
e
=
−
σ
e
∇
u
e
.
def
=
Here
V
m
stands for the transmembrane potential, defined as
V
m
u
e
,
A
m
is a
constant representing the rate of membrane area per volume unit and
C
m
the mem-
brane capacitance per area unit. The reaction term
i
ion
(
u
i
−
represents the ionic
current across the membrane and
I
app
a given applied current stimulus. In general,
the ionic variable
w
(possibly vector-valued) satisfies a system of ODEs of the type:
V
m
,
w
)
∂
t
w
+
g
(
V
m
,
w
)=
0
.
(4.3)
The definition of the functions
g
and
I
ion
depends on the cell membrane ionic model
considered (see [53, 61] and the references therein). A very large number of ionic
models have been proposed in the literature with different degrees of complex-
ity and realism. The ionic models typically fall into one of the following cate-
gories (see [53, Chap. 3]): phenomenological (FitzHugh-Nagumo [26, 45], Aliev-
Panfilov [2], Roger-McCulloch [57], van Capelle-Durrer [69], Fenton-Karma [24],
Mitchell-Schaeffer [43]) or physiological (e.g., Beeler-Reuter [4], Luo-Rudy I[41],
Luo-Rudy II [40], Noble-Varghese-Kohl-Noble [48], Djabella-Sorine [19]).
To sum up, the system of equations modelling the electrical activity within the
heart (in terms of
V
m
and
u
e
) consists of a coupled system of ODEs, (4.3), a nonlinear
reaction-diffusion equation, (4.2), and an elliptic equation, (4.1):
⎧
⎨
⎩
∂
t
w
+
g
(
V
m
,
w
)=
0 nΩ
H
×
(
0
,
T
)
,
χ
m
∂
t
V
m
+
I
ion
(
V
m
,
w
)
−
div
(
σ
i
∇
V
m
)
−
div
(
σ
i
∇
u
e
)=
I
app
in
Ω
H
×
(
0
,
T
)
,
(4.4)
div
(
σ
i
+
σ
e
)
∇
u
e
−
−
div
(
σ
i
∇
V
m
)=
0 n
Ω
H
×
(
0
,
T
)
,
