Biomedical Engineering Reference
In-Depth Information
Fig. 20.6
Remodeling of a sinusoid to a vessel (Dirsch et al.,
2008
)
The base permeability can either be described by using the initial Darcy permeabil-
ity k
0S
[
and the specific weight
γ
FR
N
/
m
3
m
/
s
]
[
]
, or by use of the intrinsic solid
permeability k
0S
[
m
2
and the effective shear viscosity
μ
FR
Ns
/
m
2
]
[
]
, where
n
F
1
m
n
F
1
m
(
n
F
)
2
α
F0
=
k
0S
k
0S
γ
FR
=
μ
FR
,
(20.8)
n
0S
n
0S
−
−
where m denotes a dimensionless material parameter; see also, e.g., Eipper (
1998
).
In (
20.8
), the term related to the volume fraction n
F
is related to the change of
permeability caused by the change of pore space, where n
0S
denotes the reference
solid volume fraction. Hence, an increasing fluid volume fraction is connected to a
decrease of the permeability and vice versa.
20.2.2.2 Remodeling of Preferred Flow Direction
AsshowninDirschetal.(
2008
), liver tissue undergoes a vascular remodeling pro-
cess in response to focal outflow obstruction. After blocking the outflow, the de-
pendent liver territory develops congestion with a clear demarcation line between
properly drained areas and outflow obstructed areas (see Fig.
20.1
a and Fig.
20.6
a).
Starting from universal sinusoidal dilatation in the border zone (Fig.
20.6
b), single
dilated sinusoids are transformed into vascularized sinusoidal canals which drain
the congested area and therefore the obstructed zone recovers (Fig.
20.6
c).
For the continuum mechanical description we consider a local pressure-modu-
lated modeling approach. Since the time scale of the remodeling of the sinusoids is
significantly higher than for the fluid flow, an artificial modeling time step is intro-
duced in the definition of a remolding evolution relation for the preferred flow di-
rection
a
F
. Similar modeling approaches can be found, e.g., in Hariton et al. (
2007
)
where a collagen fiber stress state depending on a reorientation approach in arterial
walls is discussed, or in Himpel (
2008
) where a general kinematics-based reorien-
tation approach can be found.
Here, we assume that the preferred flow direction develops in the liver lobule
in order to minimize the flow dissipation, i.e., in order to find the optimal state
with least resistance to blood flow. Therefore, the sinusoid tends to be oriented in
the direction of the (normalized) pressure gradient
p
0
=
Grad
λ
with
|
p
0
|=
1;
