Biomedical Engineering Reference
In-Depth Information
table 25.2
Constants in the Computational Simulation of trabecular bone remodeling
Constant
nominal values
T
r
T
i
T
f
f
a
Φ
0
f
a
(max)
kbob
kcob
kbfa
kcfa
Qrmax
Qfmax
Resorption period (days)
Reversal period (days)
Refilling period (days)
Initial BMU activation frequency (BMUs/mm
2
/day)
Initial mechanical stimulus (Pa)
Maximum BMU activation frequency (BMUs/mm
2
/day)
Dose-response coefficient (Pa
−1
) in refilling rate function
Dose-response coefficient (Pa) in refilling rate function
Dose-response coefficient (Pa
−1
) in activation frequency function
Dose-response coefficient (Pa) in activation frequency function
Maximum resorption rate (mm
2
/day)
Maximum formation rate (mm
2
/day)
1
a
4
a
33
a
0.00640
b
28.12
0.5
b
0.31
c
14.06
c
0.31
c
14.06
c
0.031
d
9.341×10-4
e
Note:
The nominal values are for the example in Section 25.2.2.1 about bone loss in the trabecular bone of the rat
femur.
a
Based on Baron et al.,
Anatomical Record
, 208, 237-45, 1984.
b
Based on Hazelwood et al.,
Journal of Biomechanics
, 34, 299-308, 2001.
c
Parametrical sensitivity analyses done for the coefficients to fit the curves within known experimental data ranges.
(From Lecoq et al.,
Joint Bone Spine
, 73, 189-95, 2006.)
d
Calculated local resorption rate in Hernandez et al.,
Bone
, 32, 357-63, 2003.
e
Calculated local formation rate in Hernandez et al.,
Bone
, 32, 357-63, 2003.
dP t t
was assumed to be a function of the bone resorption rate
Q
r
(
t
) and the bone refilling rate
Q
f
(
t
) for each BMU and the density of resorbing
N
R
(
t
) and refilling
N
F
(
t
) of the BMUs/area, as proposed by Hazelwood et al. (2001):
The rate of change of porosity ()/
dP t
dt
()
=
QtNt QtNt
r
() ()
−
() ()
(25.1)
R
f
F
where the resorption rate
Q
r
(
t
) was assumed to be constant, that is,
Q
r
(
t
)=
Q
r max
. The refilling rate
Q
f
(
t
) was determined by
Qt Q
()
=
/(1
+
e
kbob kcobt
(
−ϕ
( ))
))
for ()
ϕ<ϕ
(25.2)
f
f
max
0
to account for the reduced refilling on bone surfaces during disuse (Frost, 1998).
kbob
is defined as
the slope and
kcob
is the inflection point of the curve.
f
(
t
) is the mechanical stimulus described by
strain energy density (Mullender and Huiskes, 1995):
1
2
ϕ=
()
t Et
() ()
ε
2
t
(25.3)
where
E
(
t
) is elastic modulus,
e
(
t
) is mechanical strain, and
f
0
is the mechanical stimulus at
equilibrium:
Qt Q
f
()
=
for ()
ϕ≥ϕ
(25.4)
f
max
0
N
R
(
t
) and
N
F
(
t
) are the populations of resorbing BMUs and refilling BMUs, which are calculated as
t
∫
(25.5)
Nt
()
=
ftdt
a
(')'
R
tT
−
r
Search WWH ::
Custom Search