Biomedical Engineering Reference
In-Depth Information
bone at three organizational levels: tissue, cell,
and molecule. Stochastic models lend them-
selves to the study of effects of structural and
compositional changes on the fl ow of intersti-
tial fl uid through the pericellular network. We
applied this approach, which is used exten-
sively in chemical engineering [
where
d
is the bond diameter,
l
is the distance
between the two nodes, and
m
is the fl uid vis-
cosity [
]. The pressure at each node is calcu-
lated by solving a system of linear equations for
the fl ow balance at each node. When the pres-
sure at each node is known, the fl ow through
the entire network can be calculated,
15
], to develop
a stochastic network model to simulate fl ow
through the pericellular network and through
the matrix microporosity, and to determine the
infl uence of decreasing osteocyte density on
cortical bone permeability [
15
, and, by
using Darcy's law, the permeability
κ
of the
network can be determined:
Q
tot
∆
In order to demonstrate the effect of osteocyte
density on tissue permeability, we utilized data
that quantify the change in osteocyte density
in trabecular bone of patients
κ =
p
].
Network modeling involves two steps. First,
the random network of nodes and connecting
bonds is constructed for optimal representa-
tion of the structure to be simulated, in this
case the cellular network of bone (Fig.
24
30
to
60
years old
A
and B). Second, the fl ow through this network
is calculated. Both steps are repeated several
times until statistical signifi cance is achieved.
In the fi rst step, a three-dimensional, cubic-
lattice network model, with the dimensions L
10
.
10
[
]. The permeability is calculated as a
mean value from the outcome of
20
,
21
calcula-
tions of the model for every osteocyte density
(Fig.
20
D). Whereas the osteocyte density is
assumed to vary almost linearly [
10
.
10
], the
loss in permeability must be approximated
with a power law (R
2
=
0
20
,
21
×
L
), is developed according to methods
described by Meyers and Liapis [
×
L (L
=
15
).
These calculations illustrate the profound
effect of declining osteocyte density on tissue
permeability. The data predict that a
.
98
]; this simu-
lates the properties of the matrix microporos-
ity. Two different bond diameters, representing
the pores between the apatite crystals and the
pores between the collagen fi bers, respectively,
are distributed randomly with defi ned proba-
bilities across the network. This maintains the
overall porosity of the matrix. Next, osteocytes
are distributed randomly across the nodes of
the network. For every osteocyte, the distance
to the neighboring osteocytes is determined. If
the distance is smaller than a predefi ned
threshold value, the osteocytes are connected
by a canaliculus. Finally, since the network rep-
resents the tissue and is not an isolated entity,
periodic boundary conditions
are implemented
for the microporosity bonds and the canaliculi
(Fig.
15
%
decrease in osteocyte density between the ages
of
5
years will decrease bone permea-
bility by almost
30
and
40
%. Such a reduction is likely
to have a marked effect on transport to and
from bone cells.
On the basis of microscopic observations, a
logical next step in model development is to
determine the infl uence of osteocyte connec-
tivity on tissue permeability. Osteocytes in
close proximity to each other are typically con-
nected by canaliculi that decrease in number
with increasing distance from the blood supply;
they also decrease in the presence of bone
disease. Furthermore, by taking into account
the preferred spatial orientation of the lacuno-
canalicular network, it is possible to detect
anisotropic differences in the permeability of
bone tissue, which will be important for the
development of more accurate, continuum-
level fi nite element models. Finally, by exclud-
ing pores that are too small to allow the passage
of a given molecule, we have been able to simu-
late the molecular sieving properties of bone
tissue in preliminary studies. Our discrete
models were designed to bridge the level
between tissue and cell, but also to bridge to the
molecular level. This approach is therefore
useful to examine the transport of specifi c
50
B and C). In the second step, the
actual fl ow through the network is calculated.
The driving force for this fl ow is a pressure
gradient
p
10
.
10
p
out
between the upper and
the lower surfaces of the network. Therefore,
all nodes on these surfaces are assigned either
p
in
or
p
out
. The fl ow rate,
Q
ij
through the bond
between two nodes can be calculated as a func-
tion of the pressure gradient between the two
nodes:
=
p
in
−
(
)
3
ppd
l
d
−
i
j
Q
=
ij
128
⎛
⎜
⎛
⎜
⎞
⎟
⎞
⎟
+
24
µ
π
