Biomedical Engineering Reference
In-Depth Information
Fig. 20.5
Permeability in dependence on vessel and sinusoid distribution
change of the shape of the sinusoids. Although this connection is not straight linear,
we mark an average major direction out, neglecting the waviness of the sinusoids.
This major direction is neither fixed nor unique but underlies a varying dispersion as
showninFig.
20.5
. The major direction, dispersion, and diameter of the sinusoids
are variable in time and space. Depending on boundary conditions, the liver tissue is
able to change all three properties. In order to give a continuum representation of the
sinusoidal orientation, we introduce a generalized structure tensor
M
F
=
a
F
⊗
a
F
,
where the arbitrary vector
a
F
=
F
S
a
F0
, related to the arbitrary reference unit vector
with
1, represents the major distribution direction of the sinusoids. Due to
the fact that the motions of both solid and fluid are connected by the interaction
forces
|
a
F0
|=
p
F
p
S
ˆ
=−ˆ
and considering the thermodynamic restriction
p
F
λ
grad n
F
ˆ
=
−
S
F
w
FS
(20.4)
(see (
20.3
) and Ricken and Bluhm
2009
,
2010
), we propose the representation of
the anisotropic intrinsic permeability of the liver tissue as
α
F0
(
1
α
F
M
F
−
1
.
S
F
=
−
α
F
)
I
+
(20.5)
The parameter
α
F
∈[
0
,
1
]
defines the range between the fully isotropic state
(
α
F
=
1). This parameter is
suitable to adjust the direction distribution of sinusoids. In the case of a parallel dis-
tribution of all sinusoids,
α
F
becomes one, whereas for a random distribution, the
condition
α
F
=
0) and the complete transverse isotropic state (
α
F
=
0 is assumed.
We derive the balance equation of momentum for the fluid phase with
div
−
n
F
λ
I
+
ρ
F
b
−
α
F0
(
1
+
α
F
M
F
−
1
w
FS
=
+
λ
grad n
F
−
α
F
)
I
0
,
(20.6)
see Ricken et al. (
2010
). In order to obtain a determination of the filter velocity
n
F
w
FS
, we rearrange (
20.6
)into
(
n
F
)
2
α
F0
(
1
α
F
M
F
−
ρ
FR
b
.
n
F
w
FS
=
−
α
F
)
I
+
grad
λ
+
(20.7)
