Biomedical Engineering Reference
In-Depth Information
the second-order conduction tensor is split into the isotropic and anisotropic parts
D
=
d
iso
g
−
1
+
d
ani
m
,
(13.25)
in terms of the scalar conduction coefficients
d
iso
and
d
ani
, where the latter accounts
for the faster conduction along the myofiber directions.
13.3.3 Current Source
In order to complete the description of the model problem, we need to specify the
constitutive equations for the electrical source term
I
φ
. In phenomenological elec-
trophysiology, it is common practice to set up the model equations and parameters
in the non-dimensional space. For this purpose, we introduce the non-dimensional
transmembrane potential
φ
and the non-dimensional time
τ
through the following
conversion formulas
Φ
=
β
φ
φ
−
δ
φ
and
t
=
β
t
τ.
(13.26)
The non-dimensional potential
φ
is related to the physical transmembrane potential
Φ
through the factor
β
φ
and the potential difference
δ
φ
, which are both in milli-
volt (mV). Likewise, the dimensionless time
τ
is converted to the physical time
t
by multiplying it with the factor
β
t
in millisecond (ms). Thus, the conversion for-
mulae in Eq. (
13.26
) imply the equality
=
(β
φ
/β
t
)
ˆ
i
φ
and the additive split of
I
φ
, introduced in Eq. (
13.12
) Sect.
13.2.3
, becomes
ˆ
i
φ
I
φ
=
ˆ
i
e
+
ˆ
i
m
, which denote
the purely electrical current source
ˆ
i
e
and the stretch-induced mechano-electrical
current source
ˆ
i
m
in the non-dimensional setting.
In this model problem, we use the celebrated two-parameter model of Aliev and
Panfilov (
1996
), which favorably captures the characteristic shape of the action po-
tential in excitable ventricular cells,
ˆ
i
e
=
cφ(φ
−
α)(
1
−
φ)
−
rφ,
(13.27)
where
c,α
are material parameters. The evolution of the recovery variable
r
is
driven by the specific source term
γ
+
−
r
−
cφ(φ
−
b
−
1
)
.
μ
1
r
μ
2
+
φ
ˆ
i
r
=
(13.28)
Analogous to the algorithmic update of
c
, we use the backward Euler integration to
calculate the current value of
r
. For the stretch-induced current generation
f
m
,we
adopt the formula proposed by Panfilov et al. (
2005
) and Keldermann et al. (
2007
)
ˆ
i
m
=
ϑG
s
(
¯
λ
−
1
)(φ
s
−
φ),
(13.29)
