Biomedical Engineering Reference
In-Depth Information
∇
x
c
via the deformation dependent,
anisotropic diffusion tensor
d
(
ϕ
)
. Based on the microstructure of SM tissue the
diffusion tensor
relates to the calcium concentration gradient
n
d
aniso
n
Z
i
d
(
ϕ
)
=
d
iso
I
+
(5.10)
i
=
1
is additively decomposed into an isotropic (related to the elastin and matrix mate-
rial) and an anisotropic part (related to the collagen fibers in SMCs) including the
appropriate diffusion coefficients
d
iso
and
d
aniso
, respectively. The number of con-
sidered directions inside the SM tissue is controlled by
n
and
I
denotes the identity
tensor. Further,
FM
i
FM
i
Z
i
=
RZ
i
R
T
=
FM
i
|
⊗
(5.11)
|
|
FM
i
|
are the rotated structural tensors without the stretch component
U
of the deformation
gradient
F
RU
, whereby
R
is the rotation tensor.
Analogously to the momentum balance, the calcium field equation also uses cor-
responding essential and natural boundary conditions
=
c
=
c
on
∂
B
c
and
q
=
q
on
∂
B
q
.
(5.12)
The diffusion surface flux term
q
is related to the spatial flux vector through the
¯
Cauchy-type formula
q
¯
:=
q
·
n
.
5.2.3 An Active Artery Model
In this section we give a short review over the governing constitutive equations for
the active artery model. Thus, the used strain-energy function for the media layer
Ψ(
ϕ
)
=
Ψ
e
+
Ψ
c
+
Ψ
s
(5.13)
is additively decomposed in the three components: the load-bearing proteins elastin
(
Ψ
e
) and collagen (
Ψ
c
) and the active, contractile SMCs (
Ψ
s
).
5.2.3.1 Elastin
The first component of the strain-energy function
Ψ
e
stays for elastin, a protein used
to build up load-bearing structures in creature tissue. As flexible elastin molecules
are randomly arranged in a three-dimensional network, the isotropic neo-Hookean
material model
μ
e
2
(I
1
−
Ψ
e
=
3
)
(5.14)