Biomedical Engineering Reference
In-Depth Information
tTT
−+
(
)
∫
r
i
Nt
()
=
ftdt
a
(')'
(25.6)
F
tTTT
−++
(
)
r
i
f
where
f
a
(
t
) is BMU activation frequency.
A BMU activation frequency threshold
akfa
(
t
) was introduced by Gong et al. (2006) to obtain a
mechanical-biological factor coupled BMU activation frequency function:
f
a
kbfa
(max)
(
ft
()
=
S
A
(25.7)
a
t
akfa t
ϕ
()
()
−
kcfa
)
1
+
e
where
f
(
t
) is the mechanical stimulus,
f
a
(Fyhrie and Carter, 1986;
Hazelwood et al., 2001), and
kbfa
and
kcfa
are coefficients defining the slope and inflection point
of the curve. Sensitivity analyses were done for the coefficients in these functions to fit the curves
within known experimental data ranges (Lecoq et al., 2006).
akfa
(
t
) is the BMU activation threshold
and
S
A
is the specific surface area, as mentioned by Hazelwood et al. (2001).
In this chapter, bone loss in trabecular bone of a rat femur due to mechanical disuse and/or estro-
gen deficiency was taken as an example to illustrate the implementation of this method. The elastic
modulus was found to be 7 GPa for cortical bone in the rat femur and 0.9 GPa for cancellous bone
(Westerlind et al., 1997; Ferretti et al., 1993). Assuming a typical bone volume fraction of 0.28 for
cancellous bone and 1.0 for cortical bone (Westerlind et al., 1997), and a linear relationship between
porosity
P
(
t
) and elastic modulus
E
(
t
) (Keaveny et al., 2001), the elastic modulus can be expressed as
= 0.5 BMUs/mm day
2
(max)
Et
()
=
(8472
×
(1
−
Pt
())
−
1472)MPa
(25.8)
In this analysis, the equilibrium strain level was set to 250 με for cancellous bone. The system was
initiated without any active BMUs in an equilibrium state at
t
= 0 before beginning the experiment.
The model was given an initial porosity
P
(0) = 72% (Westerlind et al., 1997). The cross-sectional
area for the load was 6 mm
2
. Hence, a compressive force of
F
(0)
=σ⋅
A
=⋅ε⋅
EA
=
1.35
N
can
provide the mechanical stimulus for the bone resorption and formation to be in equilibrium.
25.2.1.2 Computational Simulation of Cortical bone remodeling
A computational simulation of cortical endosteal surface remodeling was developed at the BMU
level (Gong and Zhang, 2010). Six state variables and nine constants included in the model are
listed in Tables 25.3 and 25.4, respectively. The remodeling analysis was performed on a represen-
tative rectangular slice of the cross-section of the cortical bone volume, as shown schematically in
Figure 25.1.
An imbalance between bone resorption and refilling leads to change in cortical volume. The
rate of change in cortical volume was assumed to be a function of the bone resorption rate (
Q
r
(
t
))
and bone refilling rate (
Q
f
(
t
)) for each BMU, and the density of resorbing and refilling BMUs/area
(
N
R
(
x
,
t
) and
N
F
(
x
,
t
), respectively):
d
ht l
hl
dt
()
⋅
⋅
0
=
QtNt QtNt
r
() ()
−
() ()
(25.9)
R
f
F
where the resorption rate
Q
r
(
t
) and the refilling rate
Q
f
(
t
) were assumed to be linear in time:
=
A
T
A
T
Qt
()
Qt
()
=
and
with
A
representing the area of bone resorbed by each BMU.
h
(
t
) was
r
f
r
f
the cortical thickness at time
t
,
l
was the length of the representative rectangular slice, and
h
0
was
the initial cortical thickness, as shown in Figure 25.1.
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