Biomedical Engineering Reference
In-Depth Information
5.4.2 Numerical Examples
In the following analysis, Qu et al. [2] considered a cubic bone section sub-
jected to uniaxial compressive pressure
P
and pulsed electromagnetic loads.
The side length
a
of the cubic section was assumed to be 1 cm. They also
assumed that the strain ε
ij
,
the electric field
E
i
,
and the magnetic field
H
i
all return to zero at the end of each load cycle, so their ranges and peaks
are the same. The model is given an initial porosity of 4.43% because this
allows equilibrium between the Haversian canals removed and added by
new BMUs [38,63]. This porosity produces an initial modulus of 17.8 GPa as
determined by Equation (5.1). A time increment of 0.5 day is examined to
integrate Equation (5.2). The state variables and constants used in Qu et al.
are shown in Table 5.1.
Bone resorption and formation can reach a proper equilibrium during the
remodeling process, which keeps the bone mass unchanged. It should be
mentioned that environmental factors affect the bone remodeling process.
TABLE 5.1
Model Stat
e Variables and Constants
State Variables
E
Elastic modular (MPa)
p
Porosity
N
R
Number of resorbing BMUs (BMUs/mm
3
)
N
F
Number of refilling BMUs (BMUs/mm
3
)
f
a
BMU activation frequency (BMUs/mm
3
/day)
ε
ij
Strain (με)
Φ
Environmental stimulus (cpd)
k
F
Correlation coefficient of the refilling and resorbing process
Constants
Description
Values used in Qu et al.
V
Volume of bone tissues (mm
3
)
1,000
Q
R
Mean resorbing rate for each BMU (mm
3
/day)
2.0 × 10
−4
Q
F
Mean refilling rate for each BMU (mm
3
/day)
1.0 × 10
−6
f
a
(max)
Maximum BMU activation frequency
(BMUs/mm
3
/day)
0.8
k
r
Activation frequency dose-response coefficient
−1.6
R
L
Mechanical loading rate (cpd)
3,000
C
ij
, C
i
, G
i
Damage rate coefficients
0.04, 0.08, 0.04
f
e
Frequency of electromagnetic field (Hz)
100
N
R
0
Number of naturally timeworn osteocytes
(BMUs/mm
3
)
0.4
c
0
Value of
k
F
during remodeling process
1.0
c
1
Value of
k
F
during modeling process
1.2
c
2
Value of
k
F
in disuse-mode remodeling
0.1
p
0
Porosity of unresorbable bone tissues
0.5
