Biomedical Engineering Reference
In-Depth Information
and
ε
0
ε
1
ε
0
ε
1
ε
0
ε
1
−
5
−
1
AAAAAA
10
======
sec,
rr
rr
zz
zz
rz
rz
E
0
E
1
−
15
−
1
−
15
−
1
AA
10
== =
Vm/sec
10
NC/sec
r
r
The initial inner and outer radii are assumed to be
a
0
= 25 mm,
b
0
= 35 mm
and
e
0
= 0 is assumed. In the calculation,
u
r
(
t
) <<
a
0
is assumed for the sake of
simplicity; that is,
a
(
t
) and
b
(
t
) may be approximated by
a
0
and
b
0
.
3.6.1 A Hollow, Homogeneous Circular Cylindrical
Bone Subjected to Various External Loads
To analyze remodeling behavior affected by various loading cases we distin-
guish the following five loading cases:
• Case 1:
p
(
t
) =
n
× 2 MPa (
n
= 1, 2, 3, and 4),
P
(
t
) = 1500 N, with no other
type of load applied
• Table 3.1 lists the results at some typical instances of time,
obtained by both the analytical and the semianalytical solutions.
The semianalytical solution is obtained by dividing the bone
into
N
(= 10, 20, 40) sublayers. It is evident from the table, and
also from other extensive comparisons that are not shown here,
that the solutions have excellent agreement on the change rate of
porosity
e.
Hence, for the numerical results presented later, no
references are given regarding which method is used to obtain
the solution, unless otherwise stated. It is also evident from the
table that the numerical results will gradually converge to the
exact value as the layer number
N
increases.
• The extended results for this loading case are shown in Figure 3.3,
demonstrating the effect of external pressure on the bone remod-
eling process. It is evident that there is a critical value,
p
r
0
, above
TABLE 3.1
Comparison of Porosity
e
Obtained by Analytical and Semianalytical
Solutions (
P
=
1500 N,
p
=
2 Mpa)
Time (sec)
500,000
1,000,000
1,500,000
2,000,000
Semianalytical
N = 10
7.283 × 10
−5
1.533 × 10
−4
2.423 × 10
−4
3.406 × 10
−4
N = 20
7.294 × 10
−5
1.536 × 10
−4
2.427 × 10
−4
3.412 × 10
−4
N = 40
7.297 × 10
−5
1.536 × 10
−4
2.428 × 10
−4
3.413 × 10
−4
Analytical
7.298 × 10
−5
1.536 × 10
−4
2.428 × 10
−4
3.414 × 10
−4
