Biomedical Engineering Reference
In-Depth Information
material objective measure of the fluid velocity with respect to the solid velocity, we
introduce the seepage velocity
w
FS
, which describes the difference in velocity be-
tween the fluid phase
x
F
x
S
. In connection
with the fluid volume fraction this leads to the definition of the filtration veloc-
ity n
F
w
FS
. An extended explanation of the kinematics of porous media is given in
de Boer (
2000
) or Ehlers (
2002
).
and the solid phase
x
S
x
F
−
Sas
w
FS
=
20.2.1.2 Assumptions
In general, we distinguish between four different types of biphasic materials: com-
pressible (both components are compressible), hybrid type I (compressible solid and
incompressible fluid), hybrid type II (incompressible solid and compressible fluid),
and incompressible type (both components are incompressible). Since in most bi-
ological tissues the compressibility of the overall solid porous material is much
higher in comparison with the material compressibility of the tissue material itself,
see, e.g., Humphrey (
2002
), we assume a material incompressible solid tissue ma-
trix (
(ρ
SR
)
S
=
0) saturated by an incompressible pore fluid (
(ρ
FR
)
F
=
0), i.e., the
incompressible type. The volumetric deformation of the mixture body results from
the change of pore space, namely a change of the volume fraction n
α
, which leads
to a macroscopic volumetric deformation. In liver tissue a volume deformation due
to a physiologically hydrostatic pressure can also be observed.
With the assumption of material incompressibility on the solid skeleton, a com-
paction point must be introduced defining the state where all fluid is pressed out and
all pores are closed so that no further compression occurs. The compaction point
cannot be achieved physiologically since in biological tissues, neither the intracel-
lular nor the interstitial fluid can be pressed out completely. Therefore, we used
the Helmholtz free energy function formulated in Bluhm (
2002
), where an energy
function based on Simo and Pister (
1984
) is extended to describe the compaction
effect.
Since no mass or volumetric changes occurred during the remodeling of dilated
sinusoids, we assume that mass exchanges between the solid and the fluid con-
stituent is negligible. In general, remodeling requires cell proliferation as well as cell
death. In the event of homeostatis, the balance of addition and depletion is equalized
and no residual mass or volume change occurs over time. Thus, it is convenient to
separate the phenomena of growth and remodeling for the modeling approach, see,
e.g., Garikipati et al. (
2006
) or Kuhl and Holzapfel (
2007
).
Due to constant thermal conditions during the remodeling process, we further
assume that interchanges of temperature and energy can be neglected among the
constituents. Hence, energy supplies (
e
α
) between the phases were not taken into
account and we assumed an isothermal process with an equal temperature
θ
for
solid (
θ
S
) and fluid phase (
θ
F
). Lastly, accelerations for all phases (
x
α
=
ˆ
o
)were
excluded.
