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)
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