Biomedical Engineering Reference
In-Depth Information
Fig. 20.7
Developing
different angel
θ
ap
in the
a
F0
×
p
0
plain spanned by the
pressure gradient
p
0
and
preferred flow direction
a
F0
or
a
F0
see Fig.
20.7
. Thus, we postulate that the preferred flow direction
a
F
with
1
develops in the direction of the pressure gradient
p
0
by
a
F0
. According to Himpel
(
2008
) the merging process of both vectors is expressed by the relation
|
a
F
|=
a
F0
=
a
F0
+
a
F0
,
a
F0
·
a
F0
=
0
,
(20.9)
where
a
F0
denotes the incremental update of the preferred flow direction
a
F0
.The
incremental update can be expressed by a rigid body rotation of
a
F0
by
a
F0
=
ω
×
a
F0
,
(20.10)
where
ω
denotes the angular velocity of reorientation. Since
ω
is perpendicular to
the plane spanned by
a
F0
and
p
0
the condition
ω
=
δ
t
(
a
F0
×
p
0
)
(20.11)
must hold where
δ
t
denotes a virtual damping coefficient with respect to the time
dependent model. Inserting (
20.11
)in(
20.10
) yields the direct evaluation of the
incremental update with
a
F0
=
δ
t
[
a
F0
.
a
F0
×
p
0
]×
(20.12)
Thus, the new updated flow direction
a
F0
can be expressed directly by (
20.9
)
1
.
The
governing
weak
formulations
and
their
implementation
into
the
FE-
calculation program FEAP by Taylor (
2012
)aregiveninRickenetal.(
2010
).
20.3 Numerical Example: Recovery of Liver Perfusion After
Focal Outflow Obstruction
In the numerical example we consider the situation in a liver lobule (see Fig.
20.5
)
with the material parameters given in Table
20.1
. The blood supply of the lobule is
ensured via a small branch of the portal vein and the hepatic artery. The hepatic arte-
rial and the portal venous blood come together in the sinusoids that supply the liver
