Biomedical Engineering Reference
In-Depth Information
(of the order of 0.1 mm and 0.01 msec). This fact constrains 3-D simulations to
limited blocks with dimensions of a few centimetres and with small duration of the
evolution, see e.g. [23, 29, 58, 148] and the recent survey [149]. For large scale sim-
ulations involving the whole ventricles, computer memory and time requirements
become excessive, and less demanding approximations have been developed, such
as Eikonal and Monodomain models.
Eikonal models
We note that during the excitation phase of the heart beat the main feature, at a
macroscopic level, is the excitation wavefront configuration and its motion. In order
to describe the excitation sequence and avoid the high computational costs of the full
Bidomain model, in [8, 64, 66, 67] the
eikonal
curvature models
were developed.
With these models, the simulation of the activation sequence in large volumes of
cardiac tissue has become computationally practical, since they do not require a fine
spatial and temporal resolution.
During the excitation phase, the Bidomain model in dimensionless form (see [22]
for details on the suitable scaling) can be written as a singularly perturbed R-D sys-
tem
−
v
ε
∂
v
ε
∂
∂
1
ε
, −
∂
1
ε
v
ε
)
−
ε
u
i
)=
v
ε
)
−
ε
u
e
)=
t
+
g
(
div
(
D
i
∇
0
t
−
g
(
div
(
D
e
∇
0
,
(5.17)
u
e
and
g
is a scaled cubic-like form of an instantaneous ionic current-voltage relationship
i
ion
(
is of the order 10
−
3
10
−
2
,
v
ε
=
u
i
−
where the dimensionless parameter
ε
−
v
p
the three zeroes of
g
representing the resting, threshold and excited transmembrane values respectively,
and we assume that
v
p
v
r
v
)
related to the excitation phase. We denote by
v
r
<
v
th
<
g
(
v
)
dv
<
0. Given the previous singular perturbation struc-
ture,
u
i
,
u
e
diffuse quite slowly, while the reaction takes place much faster; hence,
the development of a moving layer associated with a traveling wavefront solution is
to be expected. Assuming that the excitation propagates in fully recovered tissue, a
monotonic temporal behavior of
v
is expected, thus we define the
activation time
as
the time instant
t
ε
ε
=
ψ
(
x
)
at which
v
(
x
,
t
)=(
v
r
+
v
p
)
/
2. Then the excitation wave-
front
S
ε
(
t
)
is represented by the level surface of the activation time at the time instant
t
, i.e.:
S
ε
(
t
)=
{
x
∈
Ω
H
,
ψ
(
x
)=
t
}.
(5.18)
,
ξ
)=
q
We then introduce the indicatrix function
Φ
(
x
(
x
,
ξ
)
,
where
T
D
i
(
T
D
e
(
ξ
x
)
ξ
ξ
x
)
ξ
1
q
(
x
,
ξ
)=
=
(5.19)
T
,
ξ
)
−
1
,
ξ
)
−
1
q
i
(
x
+
q
i
(
x
ξ
(
D
i
(
x
)+
D
e
(
x
))
ξ
is the conductivity of the bulk medium at a point
x
along the unit vector
ξ
, expressed
T
D
i
,
e
(
as the harmonic mean of the quadratic forms
q
i
,
e
(
associated with
the conductivity tensors
D
i
,
e
. The bulk medium is composed by coupling in series
the media (i) and (e).
x
,
ξ
)=
ξ
x
)
ξ