Biomedical Engineering Reference
In-Depth Information
5.3.1 Relationship between Elastic Modulus and Bone Porosity
This model was developed based on the relationship between Young's
modulus
E
and porosity
p
obtained by fitting a polynomial to the results
of Curry [48] and Rho, Ashman, and Turner [49]. By assuming a linear rela-
tionship between apparent density (ρ) and porosity (
p
), Hazelwood et al. [1]
obtained a relationship between modulus and bone porosity as
56
65
64
E
=
(8.83
×
10 )
p
−
(2.99
×
10 )
p
+
(3.99
×
10 )
p
4
(5.1)
63
52
5
−
(2.64
×
10 )
p
+
(9.08
×
10 )
p
−
(1.68
× +×
10 )
p
2.37
10
in which
E
is in the unit of megapascals.
5.3.2 Porosity Changes
Based on the work of Martin [50], Hazelwood et al. [1] assumed that the rate
of change of porosity (
p
) is a function of the mean bone resorbing (
Q
R
) and
refilling (
Q
F
) rates for each BMU, and the density of resorbing (
N
R
) and refill-
ing (
N
F
) (the unit is BMUs/area)
=
pQNQN
RR
−
FF
(5.2)
where the resorption (
Q
R
=
A
/
T
R
) and refilling (
Q
F
=
A
/
T
F
) rates are assumed
to be linear in time, with
A
representing the area of bone resorbed by each
BMU.
T
R
and
T
F
are the resorption and refilling periods, respectively.
In a cortical bone, the BMU forms a cylindrical canal about 2000 μm long
and 150-200 μm wide. It gradually burrows through the bone at a speed of
20-40 μm/day. At the tip, on the order of 10 osteoclasts dig a circular tun-
nel (cutting cone) in the dominant loading direction. An activated osteoclast
is able to resorb 200,000 μm
3
/day [50]. Then, several thousand osteoblasts
will fill the tunnel (closing cone) to produce a (secondary) osteon of renewed
bone. In this way, between 2% and 5% of cortical bone is remodeled each
year [51]. In a trabecular bone, BMUs resorb and refill trenches rather than
tunnels, but the process is similar and
A
is assumed to be the same for both
cortical and trabecular bones.
The populations NR and NF were found by integrating over the appropri-
ate resorption (
T
R
), reversal (
T
I
), and refilling (
T
F
) time intervals of the BMU
activation frequency (
f
a
) history. For a histologic section,
T
R
is defined as the
period from the moment the osteoclasts of a resorbing BMU enter the section
to the time that resorption by these cells ceases in the section. The reversal
or inactive period,
T
I
,
follows and is the transition from osteoclastic to osteo-
blastic activity. Osteoblasts of the BMU then form new bone in the section
during the refilling period,
T
F
.
N
R
is found by integrating
f
a
from time
t
-
T
R
to
t,
where
t
is the present time
