Biomedical Engineering Reference
In-Depth Information
The formulae for the variation of the radii with time
t
can be obtained by
substituting Equation (4.53) into Equation (4.17). Thus,
()
(
)
Bt
at
=+ε− η+ε
aa
aB t
e
1
0
0
∞
02
∞
∞
(
)
()
Bt
bt
=+η−
bb
1
e
(4.54)
1
0
0
∞
The final radii of the cylinder are then
()
()
a
=
lim
a t
=
a
(1),
+ε
b
=
lim
b t
=
b
(1
+η
)
(4.55)
∞
0
∞
∞
0
∞
t
→∞
t
→∞
All of these solutions are theoretically valid. However, the first is the most
likely solution to the problem, as it is physically possible when
t
→ ∞ [1].
Therefore, it can be used to calculate the bone surface remodeling.
4.3 Application of Semianalytical Solution to Surface
Remodeling of Inhomogeneous Bone
The semianalytical solution presented in Section 3.4 can be used to calculate
strains and stresses at any point on the bone surface. These results form the
basis for surface bone remodeling analysis. This section presents applica-
tions of solution (3.58) to the analysis of surface remodeling behavior in
inhomogeneous bone.
It is noted that surface bone remodeling is a time-dependent process.
The change in the radii (ε or η) can therefore be calculated by using
the rectangular algorithm of integral (see Figure 4.1). The procedure is
described here.
First, let
T
0
be the starting time and
T
be the length of time to be consid-
ered and divide the time domain
T
into
m
equal intervals Δ
T
=
T
/
m.
At the
time
t,
calculate the strain and electric field using Equations (3.32)-(3.36).
The results are then substituted into Equation (4.1) to determine the normal
rate of the surface bone remodeling. Assu
m
ing that Δ
T
is sufficiently small,
we can replace
U
with its mean value
U
at each time interval [
t,
t
+ Δ
T
].
The change in the radii (ε or η) at time
t
can thus be determined using
the results of surface velocity. Accordingly, the strain and electric field are
updated by considering the change in the radii. The updated strain and
electric field are in turn used to calculate the normal surface velocity at
the next time interval. This process is repeated up to the last time interval
[
T
0
+ (
m
− 1)Δ
T
,
T
0
+
T
]. Figure 4.1 shows the rectangular algorithm of inte
g
ral
when we replace
U
with its initial value
U
t
(rather than its mean value
U
) at
the time interval [
t,
t
+ Δ
T
].
