Biomedical Engineering Reference
In-Depth Information
The evolution equation determines the distribution of
v
, then the extracellular
potential distribution
u
e
is derived by solving the elliptic boundary value problem.
It is well known that if the two media have the same anisotropy ratio
σ
(
x
,
t
)
e
l
t
l
=
σ
=
t
σ
σ
n
σ
, then the Bidomain system reduces
to the Monodomain model (see [19, 36, 57]) with conductivity tensor and applied
current given by
=
λ
,i.e.
D
e
(
x
)=
λ
D
i
(
x
)
with constant
λ
n
σ
I
app
=
I
i
app
−
λ
I
app
/
(
(
)=
λ
(
)
/
(
+
λ
)
,
+
λ
)
.
D
m
x
D
i
x
1
1
(5.22)
We remark that this is not a physiological case, as clearly follows from well es-
tablished cardiac experimental evidence. Derivations of a reduced Bidomain model
which does not make such an assumption of
a priori equal anisotropy
assumption
have been developed in [19, 36, 65, 102], disregarding some source terms, i.e. the
projections of the total current flowing in the two media
J
tot
=
j
i
+
j
e
along and
across fibre directions
a
t
(
. In these derivations, the Monodomain model is
characterized by the following anisotropic conductivity tensor and applied current:
x
)
,
a
n
(
x
)
I
app
=
I
i
app
σ
l
/
(
σ
D
e
D
−
1
D
i
,
e
l
I
app
σ
i
e
l
i
l
D
m
=
−
+
σ
)
,
(5.23)
where
D
D
e
. We remark that it is possible to rescale the reaction-diffusion
equation (5.21) of the Monodomain model, obtaining a singular perturbation prob-
lem as in the Bidomain model (5.17). Performing formal asymptotic expansions (see
[8], Appendix A), it is possible to derive the anisotropic evolution law of the wave-
front. Up to terms of order
O
=
D
i
+
2
m
(
ε
)
, the normal velocity
θ
(
ν
)
is given by
m
m
m
ξ
(
θ
(
ν
)=
Φ
(
s
,
ν
)(
c
−
ε
div
Φ
s
,
ν
))
,
(5.24)
presenting the same structure of (5.20), but for the Monodomain model the
indicatrix
is
q
m
(
m
. This
indicatrix
does not coincide with the one derived
from the Bidomain model (5.19), since it is now defined in terms of the harmonic
mean of the conductivity tensors, i.e.
Φ
(
x
,
ξ
)=
x
,
ξ
)
T
D
m
(
D
e
D
−
1
D
i
.
q
m
(
x
,
ξ
)=
ξ
x
,
ξ
)
ξ
where
D
m
(
x
)=
(5.25)
Therefore the wavefront motion of the Monodomain model characterized by (5.25)
is different at the zero order from that of the Bidomain excitation wavefront, compare
(5.19) with (5.25).
In order to obtain an anisotropic Monodomain model, having the same anisotropic
law of motion of the excitation wavefront of the Bidomain model, the following
nonlinear diffusion term, i.e. a nonlinear conductivity tensor, was introduced in [8]
in the parabolic equation:
2
ξ
2
T
D
i
ξ
T
D
e
ξ
ξ
D
m
(
x
,
ξ
)
:
=
D
e
(
x
)+
D
i
(
x
)
.
(5.26)
T
D
T
D
ξ
ξ
ξ
ξ
