Biomedical Engineering Reference
In-Depth Information
moduli, which can be obtained through the Doyle-Ericksen formula, the reader is
referred to Göktepe et al. (
2011
).
For the active part of the free-energy function (
13.9
), we assume the following
transversely isotropic function
Ψ
a
g
;
F
e
,
M
=
2
η
I
4m
−
1
2
,
1
(13.18)
in terms of the material parameter
η
and the invariant
I
4m
:=
g
:
m
e
with
m
e
:=
F
e
MF
e
T
. This leads to us to the active part of the Kirchhoff stress tensor (
13.10
)
τ
a
g
F
e
,
M
=
2
η
I
4m
−
1
m
e
.
;
(13.19)
Calculation of the active stress tensor necessitates the knowledge of the elastic
part of the deformation gradient, which depends on the active part of the deformation
gradient. As introduced in Eq. (
13.2
),thelatterisassumedtobefunctionofthe
transmembrane potential
Φ
through the following ansatz
+
λ
a
1
M
.
F
a
=
1
−
(13.20)
The active fiber stretch
λ
a
is considered to be function of the normalized intracellular
calcium concentration
c
through the following relationship
ξ
λ
a
1
)
λ
a
max
,
=
(13.21)
1
+
f(
c)(ξ
¯
−
where the functions
f
and
ξ
of the normalized calcium concentration
c
¯
:=
c/c
R
are
defined as
¯
−
1
2
+
1
π
arctan
(β
ln
f(
c
0
)
1
¯
:=
¯
:=
f(
c)
c)
and
ξ
,
(13.22)
λ
a
max
f(
c
0
)
¯
−
respectively. The evolution of the normalized calcium concentration
c
is modeled
¯
by the following ordinary differential equation, Pelce et al. (
1995
),
q
Φ
Φ
−
¯
c
=
+
k
c
with
¯
c(t
0
)
¯
=¯
c
0
.
(13.23)
In the algorithmic setting, this evolution equation is integrated locally by using the
implicit Euler scheme. Having the active part of the deformation gradient at hand,
the elastic part of the deformation gradient can be obtained as
F
e
FF
a
−
1
yielding
=
the following closed-form expression
−
1
λ
a
−
1
Ff
0
⊗
F
e
=
F
−
f
0
.
(13.24)
13.3.2 Spatial Potential Flux
We have already introduced the spatial potential flux
q
in Eq. (
13.11
) as a function
ˆ
of the conduction tensor
D
and the potential gradient
∇
x
Φ
. For the model problem,
