Biomedical Engineering Reference
In-Depth Information
Exploiting the singular perturbation approach, we can derive formally anisotropic
geometric evolution laws capturing the asymptotic behavior of traveling wavefront
solutions of the R-D system (5.17) (see [8, 9, 64, 66, 67]). The propagation of the
wavefront
S
ε
(
2
t
)
behaves, up to
O
(
ε
)
terms, as a hypersurface
S
(
t
)
, propagating with
the anisotropic geometric law with velocity
θ
(
ν
)
, along the Euclidean unit vector
ν
normal to the wavefront and oriented toward the resting tissue, (see e.g. [8], Ap-
pendix B), given by:
θ
(
ν
)=
Φ
(
,
ν
)(
−
ε
Φ
ξ
(
,
ν
))
.
s
c
div
s
(5.20)
Equations of this type are also called
eikonal-curvature models
since
K
Φ
=
div
n
Φ
=
div
is the
anisotropic
mean curvature with respect to a suitable Finsler
metric; see [8, 9] for definitions and details. Moreover,
c
is the velocity of a 1-D
traveling wave solution , i.e. (c,
a
) is the unique bounded solution of the eigenvalue
problem:
a
+
c
a
−
g
(
a
)=
0
a
(
−
∞
)=
v
p
,
Φ
ξ
(
s
,
ν
))
a
(
∞
)=
v
r
,
a
(
0
)=(
v
p
+
v
r
)
/
2
.
An eikonal model, equivalent to (5.20) up to second-order terms of
and called
eikonal-diffusion
equation, is also formally derived in [21, 22] and used in studies of
the anisotropic 3-D propagation of the excitation wavefronts, see e.g. [23, 28, 139].
The rigorous justification of the connection between the evolution of a suitable
level set of
v
and the surface flowing under a geometric evolution law remains to our
knowledge an open problem. A partial rigorous characterization of the anisotropic
curvature term was obtained for the stationary Bidomain model in [2].
ε
Relaxed linear and nonlinear anisotropic Monodomain model
Many large scale simulations have been performed using the so-called Monodomain
model, in order to avoid the high computational costs of the full Bidomain model.
This model is a
relaxed Bidomain model
described by a system of a parabolic equa-
tion coupled with an elliptic equation, but, unlike the Bidomain model, the former
evolution equation is fully uncoupled to the elliptic one. An anisotropic Mono-
domain model consists in a single parabolic reaction-diffusion equation for
v
, with
conductivity tensor
D
m
and applied current
I
app
, coupled with gating and ionic
concentration systems and with the elliptic problem:
⎧
⎨
(
)
x
c
m
∂
v
∂
t
−
I
app
div
(
D
m
(
x
)
∇
v
)+
i
ion
(
v
,
w
,
c
)=
in
Ω
H
×
(
0
,
T
)
∂
w
∂
c
t
−
R
(
v
,
w
)=
0
,
t
−
S
(
v
,
w
,
c
)=
0
in
Ω
H
×
(
0
,
T
)
∂
∂
n
T
(
D
m
(
x
)
∇
v
=
0
in
Γ
H
×
(
0
,
T
)
⎩
v
(
x
,
0
)=
v
0
(
x
)
,
w
(
x
,
0
)=
w
0
(
x
)
,
c
(
x
,
0
)=
c
0
(
x
)
in
Ω
H
.
I
i
app
+
I
app
−
div
((
D
i
+
D
e
)
∇
u
e
)=
div
(
D
i
∇
v
)+
in
Ω
H
,
n
T
n
T
D
i
∇
−
(
D
i
+
D
e
)
∇
u
e
=
v
on
Γ
H
.
(5.21)
