Biomedical Engineering Reference
In-Depth Information
Table 20.1
Parameters of
Parameter
Value
Unit
Parameter
Value
Unit
liver lobule
μ
S
10
6
k
0S
10
7
1
·
Pa
1
·
m
/
s
λ
S
10
4
γ
FR
10
4
N
/
m
3
1
·
Pa
1
·
J
cp
0
.
0
-
α
1
0
.
0
Pa
ρ
SR
0S
10
3
kg
/
m
3
1
·
α
2
0
.
0
-
ρ
FR
0S
10
3
kg
/
m
3
n
0S
1
·
0
.
5
-
n
0S
α
F
0
.
90
-
0
.
5
-
δ
t
0
.
25
-
cell plates. As mentioned in Sect.
20.2.2.2
, sinusoids are oriented in the direction of
the pressure gradient between portal tract and central vein.
In order to capture this physiological situation in the numerical model, we started
with the unphysiological assumption that the direction of all sinusoids is horizontal.
In Fig.
20.8
a the different angle between pressure gradient and the direction of the
sinusoids is plotted in the left column, the pressure is in the middle and on the right-
hand side, the development of the norm of velocity is observed, each at three time
steps: time
=
1
,
5 and 50, where in this approach the time has a virtual scale. At
t
1, the large difference angle causes a high resistance against the blood flow
which causes a high pressure amplitude as well as local zones with high velocities.
As time goes on, the sinusoids orient in the direction of the pressure gradient and
the different angles, pressure, and the maximum of velocity all become smaller.
Finally, the sinusoids are oriented parallel to the pressure gradient and no further
reorientation occurs. This state can be defined as the optimal state and a reasonable
physiological starting configuration.
In the next step, an outflow obstruction of the left lower vein is considered, see
the black mark in Fig.
20.8
b. Now the drainage of the liver lobule will only be
realized with the remaining outflows. The obstruction first causes an increase of
the pressure magnitude and then a new distribution of the pressure gradient. Again,
the sinusoids follow the direction of the pressure gradient which can be seen at
time t
=
100, the sinusoids are oriented to the remaining outflow
gates, so that a new optimized drainage has been established where the pressure and
velocity magnitude are as low as possible.
We repeat this procedure by closing two more veins as given in Fig.
20.8
c. Again,
the same reorientation mechanism can be observed.
=
55. Finally, at t
=
20.4 Discussion
Proceeding from the theory of porous media, the model equations for a fluid-
saturated liver have been derived. In order to model the deformation of the solid, the
diffusion flow of the fluid and the internal change of sinusoidal distribution com-
prise a two phase solid-fluid model that includes an evolutional transverse isotropic
