Biomedical Engineering Reference
In-Depth Information
where Φ is defined as the mechanical stimulus and
k
D
is a damage rate coef-
ficient. For the sake of simplicity, Hazelwood et al. assumed that the strain,
ε, was the principal compressive strain and that it returns to zero at the end
of each load cycle, so that the strain range and peak strain are synonymous.
The value for the exponent
q
was set at a nominal value of four based on the
results of Whalen and Carter [54].
Then, Hazelwood et al. assumed that when BMUs and damage are ran-
domly distributed in the bone, the damage removal rate is
Df
a
A.
However, it
was assumed that damage initiates BMU activation [55,56], so the efficiency
of damage removal is greater than that of random remodeling. To allow for
this, a damage removal specificity factor,
F
s
,
is included in the equation for
the damage removal rate
DDfAF
R
s
(5.8)
a
where
F
s
was set to five based on the frequency with which microcracks were
associated with new resorption cavities in the experiments of Mori and Burr
[56]. In a state of equilibrium,
This provides an estimate of the dam-
DD
.
F
R
age rate coefficient
k
D
as
k
D
=
D
0
f
a
0
AF
s
/Φ
0
(5.9)
where the subscript 0 indicates the initial equilibrium values assigned at the
start of the simulation.
Using an average crack length of 0.088 mm [57] and the average crack den-
sity for a 40-year-old person [58], Hazelwood et al. obtained
D
0
as 0.0366 mm/
mm
2
. They also obtained an initial activation frequency,
f
a
0
= 0.00670 BMU/
mm
2
/day, from averaging several studies of cortical bone [52,53,59]. The ini-
tial mechanical stimulus (Φ
0
) was estimated from cyclic strain levels nec-
essary to “maintain” cortical bone mass in equilibrium [60,61]. A person
walking 4.5 km/day with a modest 1.5 m stride (i.e., two steps) experiences
about 3,000 cycles per day (cpd) (or roughly 1 million cycles per year) of
lower extremity loading. Assuming
R
L
= 3,000 cpd to be typical, the results
presented by Beaupre, Orr, and Carter [10] indicate that an equivalent cyclic
strain of approximately 500 με would constitute an equilibrium condition for
cortical bones, with an initial mechanical stimulus of Φ
0
= 1.875 × 10
-10
cpd.
Substituting these values into Equation (5.9) yielded
k
D
= 1.85 × 10
5
mm/mm
2
.
5.3.5 Disuse
To formulate the BMU response to disuse, Hazelwood et al. [1] adopted the
“daily stress stimulus” approach of Carter, Fyhrie, and Whalen [62]. In their
work, disuse was defined as stimulus values Φ below the equilibrium stimulus
Φ
0
. Thus, they used Φ both to calculate the damage formation rate (
=Φ
D
F D
)
and to quantify disuse as Φ
0
- Φ. Also, in disuse (Φ
0
> Φ), the refilling rate
