Biomedical Engineering Reference
In-Depth Information
permeability could be much larger than this value. The PV permeabilities reported
in the past are thought to be significant underestimates because these previous
values represent PLC and PV lumped measurements rather than a PV measurement
alone. These lumped measurements also compromise the measurement of the
permeability of the PLC as pointed out [50] in the discussion of the reported PLC
permeabilities [51, 52]. Estimates for the ratio of the permeabilities of the PV to
those of the PLC are of the order of 10
10
. These ratio estimates and the approximate
ratio of the diameters as 167 are measures of the significant size difference, pore
pressure difference, and relaxation times in these two distinct pore size porosities.
9.7.2
The Lacunar-Canalicular Porosity (PLC)
The PLC consists of the fluid spaces surrounding the osteocytes and their pro-
cesses (Figure 9.7), less the volume of the soft tissue structures in these fluid
spaces. The pore size estimate of the effective radii is of the order 0.1-0
m.
There are four reported estimates of the permeability of the PLC. A theory-based
estimate of the intrinsic permeability [53] of the PLC is
k
.
2
µ
=
1
.
47
×
10
−
20
m
2
(this
m
4
N
−
1
=
/
=
.
×
10
−
17
s
−
1
).
corresponds to a hydraulic permeability of
k
k
m
1
47
10
−
22
m
2
An experiment-based estimate [54] is
k
=
2
.
2
×
(this corresponds to
10
−
19
m
4
N
−
1
s
−
1
). More recently a
nanoindentation technique was combined with a poroelastic analysis to provide
an estimate of
k
=
4
.
1
×
10
−
24
m
2
(this corresponds to a hydraulic permeability
of
k
=
k
/
m
=
4
.
1
×
10
−
21
m
4
N
−
1
s
−
1
) [55]. The assumption of an incompressible
fluid in the analysis of experimental data renders the estimate [55] smaller than it
should be. In the incompressible model, all of the hydrostatic stress is transferred
from the solid matrix material to the pore fluid pressure; in the compressible
model, only a fraction of the hydrostatic stress is transferred from the solid matrix
material to the pore fluid pressure. The reason for this is that the compressible
model applied to the situation in which the solid matrix material is much stiffer
than the bulk modulus of the fluid and the solid shields the fluid from a fraction
of its hydrostatic stress (see Table 9.1). A measurement of the PLC permeability
reported in [56] based on the analytical model of Gailani and Cowin [57] yielded
values on the order of 10
−
24
a hydraulic permeability of
k
=
k
/
m
=
2
.
2
×
−
10
−
25
m
2
. Each of the latter estimates above is 2
orders of magnitude smaller than the previous one for the permeability of the PLC.
9.7.3
The Collagen-Hydroxyapatite Porosity (PCA)
The collagen-hydroxyapatite porosity (PCA) has the smallest pore size (approxi-
mately 10 nm diameter) [7, 58]. The interstitial pore fluid in the collagen-apatite
porosity has been shown [59] to be bound to the solid structure, and it is not of
interest in the present considerations of interstitial flow for that reason. This por-
tion of the interstitial fluid is considered to be part of the collagen-hydroxyapatite
structure.
