Biomedical Engineering Reference
In-Depth Information
of tissue as a function of oxygen transport to
and from the cells within that volume, the
boundary condition, i.e., the fl u x of ox ygen into
the tissue from surrounding tissues, needs to be
specifi ed. If, on the other hand, the presence of
injury or disease causes the control volume to
become disconnected functionally from its sur-
rounding tissue, then a sealed boundary con-
dition with a constant zero or low fl ux may be
more appropriate. The mass balance of oxygen
within the model will be signifi cantly affected
by which boundary condition is chosen. In the
studies at hand, we defi ned a stiffness and
permeability of the material at all edges (bound-
aries). To model a long bone (Fig
Interestingly, it was shown that variation in
periosteal permeability exerts the greatest
infl uence on pore pressure distribution, which
drives fl ow within the cortex (Fig.
). This
was particularly surprising, given that the
periosteum is often assumed to be a “sealed
surface” in modeling bone as a poroelastic
material. [
10
.
7
].
Four-point bending loads were applied to the
model (Fig.
12
,
26
D), with the loading conditions
the same as those applied in vivo. Using the
equations of poroelasticity [
10
.
6
] embedded in the
fi nite element program, we calculated pressure
gradients that are shown in Fig.
4
10
.
6
E. Each
), we defi ned
the endosteum, i.e., the boundary between the
cortex and the medullary cavity, as material
that exhibits low stiffness but high permeabil-
ity. This defi nition refl ects the high degree
of vascularization and permeability of the
endosteum, but does not signifi cantly refl ect
the structural strength of bone. In contrast, the
bone cortex contributes signifi cantly to the
structural strength and stiffness of bone, but is
less permeable than the soft endosteal tissue,
because it is made up of relatively impermeable
mineralized matrix. Finally, the periosteum
or outer surface of the bone was assumed to
exhibit both low stiffness and permeability. A
parametric study was carried out to determine
the degree of infl uence that each of these vari-
ables has on the fl ow fi eld within the bone cortex.
10
.
7
cross section of the tibia (in Fig.
E) was then
depicted with one aspect under compression
and one under tension, with the neutral axis in
between (Fig.
10
.
6
F).
We then calculated mass transport with the
aid of the heat transfer package of the fi nite
element software. Mass and heat transport are
governed by the same equations, provided
inertial terms can be neglected, as here. This
calculation led to the magnitudes (Fig.
10
.
6
A)
and directions for every velocity vector at every
element in the model. We had expected that
fl uid would be squeezed out of segments under
compression and taken up by segments under
tension. This was not the case in our original
model (Fig.
10
.
9
B) and led us to examine criti-
cally the defi nition of the material parameters
in the new model (see below). To calculate mass
10
.
9
60
1E-14
50
1E-13
40
1E-12
30
1E-11
20
1E-10
10
1E-9
0
1E-8
-10
Endosteum
Open
0
10
20
30
40
periosteum
endosteum
Figure 10.7.
Effect of boundary conditions on the development of pore pressure in the cortex. The finite element mesh is divided
into three concentric sections to define independently material properties of the endosteum (yellow, sheath closest to the medul-
lary cavity, low stiffness and high permeability), the cortex (orange, high stiffness and medium permeability), and the periosteum
(red, outer sheath, low stiffness, permeability varied). The pore pressure (kPa, y-axis) in the cortex between the surface closest to
the endosteum and periosteum, respectively, is plotted as a function of periosteal sheath permeability (colored lines represent
permeability, as defined in the sidebar). Reprinted from Journal of Theoretical Biology, Volume 220, R. Steck, P. Niederer, and
M.L. Knothe Tate, “A finite element analysis for the prediction of load-induced fluid flow and mechanochemical transduction in
bone,” p. 252, with permission from Elsevier.
