Biomedical Engineering Reference
In-Depth Information
20.2.2 Constitutive Modeling
With respect to thermodynamically consistent material laws we consider the entropy
inequality, see Ricken et al. (
2010
). Thus, the thermodynamic restrictions
2
ρ
S
F
S
∂ψ
S
T
S
n
S
λ
I
∂
C
S
F
S
,
T
F
n
F
λ
I
=−
+
=−
(20.2)
for the solid (
T
S
) and fluid (
T
F
) stress, respectively, as well as for the fluid interac-
tion force
p
F
p
S
ˆ
=−ˆ
with
p
F
λ
grad n
F
=
−
S
F
w
FS
,
(20.3)
must hold. In (
20.2
) and (
20.3
),
S
F
denotes a positive definite material parameter
tensor for permeability and
λ
represents the pressure.
On the one hand, the enlarged entropy inequality (
20.2
) forms the basic struc-
ture for the thermodynamically consistent constitutive modeling and provides the
necessary restrictions for the material laws. On the other hand, biological tissues
show highly coupled material behavior including anisotropy deformation response,
viscoelasticity, anisotropic poroelasticity, osmosis, remodeling, and growth. For the
description of the liver we confined ourself to a model including the anisotropy de-
formation response, anisotropic poroelasticity, and modeling of the poroelasticity.
Although not all characteristics of the material are represented in the current model,
we were able to describe the vascular behavior of the hepatic tissue in the case of
outflow obstruction. The modeling of the anisotropic material behavior for stress
which includes the compaction point is given in Ricken et al. (
2010
). Here, we will
focus on the filter velocity, the transversely isotropic permeability, and the flow re-
orientation.
20.2.2.1 Filter Velocity and Transversely Isotropic Permeability
For the liver tissue we consider that the fluid flow is affected by two major mecha-
nisms. Firstly, on the macroscale, blood is transported inside the liver from the main
vessels (liver artery and portal vein; see Fig.
20.2
) via smaller vessels into the lobar
vessels. From here, the second mechanism starts: a micro-filter flow through the si-
nusoids; see Fig.
20.5
. This filter flow forms the connection between the branches
of the portal vein and liver artery with the branches of the liver vein. In the normal
liver evenly and clearly directed, mostly straight perfusion of the sinusoids is seen
by employing Orthogonal Polarization Spectroscopy (OPS) (Dirsch et al.,
2008
and
Dahmen et al.,
2007
). The 'sponge-like property' of the liver became more appar-
ent, when observing the redirected blood flow of the liver with outflow obstruction
using OPS (Dirsch et al.,
2008
). Redirection of flow leads not only to an actual
change of direction—aiming for liver territories in which outflow is preserved—but
also to a visible change in flow characteristics, like flow velocity, as well as a visible
