Biomedical Engineering Reference
In-Depth Information
Let
u
i
,
u
e
be the intra- and extracellular potentials and
J
i
.
e
=
−
Σ
i
,
e
∇
u
i
,
e
their cur-
rent densities. Let
ν
i
,
ν
e
denote the unit exterior normals to the boundary of
Ω
i
and
Ω
e
respectively, satisfying
ν
i
=
−
ν
e
on
Γ
m
.Let
v
:
=
u
i
|
Γ
m
−
u
e
|
Γ
m
denote the trans-
membrane potential, evidencing that
Γ
m
is a discontinuity surface for the potential
field. Under quasi-stationary conditions (see [93]), due to the current conservation
law, the normal current flux through the membrane is continuous, i.e.
J
i
·
ν
i
=
J
e
·
ν
i
,
and this common flux equals the total transmembrane current
I
m
per unit area (5.1).
If no currents are applied to the intra- and extracellular spaces, since the only active
source elements lie on the membrane
Γ
m
, in terms of potentials we have:
−
div
(
Σ
i
∇
u
i
)=
0
,
in
Ω
i
−
div
(
Σ
e
∇
u
e
)=
0 n
Ω
e
(5.7)
C
m
∂
v
T
i
T
e
I
m
=
−
ν
Σ
i
∇
u
i
=
ν
Σ
e
∇
u
e
=
t
+
I
ion
on
Γ
m
,
(5.8)
∂
where we supplement the equations in (5.7) with homogeneous Neumann boundary
conditions for
u
i
,
u
e
, assuming that the cellular aggregate is embedded in an insulated
medium, and we assign for (5.8) degenerate initial conditions.
We consider two characteristic length scales for the electric potentials
u
i
,
u
e
:a
micro
scale related to the typical cell dimensions
{
d
c
,
l
c
}
and a
macro
scale defined
below. Define ¯
)
over the cells' arrangement. Let
R
m
be an estimate of the passive membrane re-
sistance near the equilibrium point
v
r
, i. e. the resting transmembrane potential.
Then, we introduce the membrane time constant
μ
=
μ
i
+
¯
μ
e
, with ¯
¯
μ
i
, ¯
μ
e
being the average eigenvalues of
Σ
i
(
x
)
,
Σ
e
(
x
τ
m
=
R
m
C
m
, the length scale unit
Λ
=
l
c
¯
, i. e. the ratio between the
micro
and the
macro
length constants. We consider the following scalings of the
space and time variables
μ
R
m
and the dimensionless parameter
ε
=
l
c
/
Λ
/
Λ
,
t
/
τ
m
and define
u
i
(
,
t
u
i
(
Λ
,
τ
m
t
and
analogously
u
e
,
w
ε
,
c
ε
. Rescaling the Eqs. (5.7) and (5.8) in the intra- and extracel-
lular media and omitting the superscripts
x
=
x
=
t
x
)=
x
)
of the dimensionless variables, we obtain
the following model:
Dimensionless cellular model P
ε
.
Let
=
Ω
i
∪
Ω
e
∪
Γ
m
be the dimensionless
Ω
:
Γ
m
the surface cellular membrane,
¯
cardiac tissue volume,
I
ion
=
I
ion
R
m
the dimensionless conductivity matrices and ionic membrane current, respec-
tively. Then, the full Reaction-Diffusion system associated with the cellular model
in dimensionless form can be formulated as follows: the vector
σ
i
,
e
=
Σ
i
,
e
/
μ
and
u
i
,
u
e
,
w
ε
,
c
ε
)
(
, with
v
ε
=
u
i
−
u
e
, satisfies the problem
σ
i
(
u
i
Ω
i
, −
σ
e
(
u
e
=
Ω
e
−
)
∇
=
)
∇
div
x
0 n
div
x
0 n
(5.9)
I
m
=
−
σ
i
(
n
T
u
i
=
−
σ
e
(
n
T
u
e
x
)
∇
x
)
∇
v
ε
∂
=
ε
[
∂
v
ε
,
w
ε
,
c
ε
)]
,
Γ
m
+
I
ion
(
on
(5.10)
t
w
ε
∂
c
ε
∂
∂
∂
v
ε
,
w
ε
)=
v
ε
,
w
ε
,
c
ε
)=
Γ
m
t
−R
(
0
,
t
−S
(
0
,
on
(5.11)
