Biomedical Engineering Reference
In-Depth Information
cylinder, the volume fraction of bone matrix materials varies from the inner
to the outer surface. To solve this problem, a semianalytical model was pre-
sented in Qin and Ye [19]. The following discussion is a brief summary of
this work.
Considering Equations (3.1)-(3.3) and assuming a constant longitudinal
strain, the following first-order differential equations can be obtained [19]:
c
cr
1
12
−
u
u
∂
∂
c
r
r
11
11
=
r
σ
ψ
(/
cc
r
−
1)
σ
r
12
11
r
2
r
(3.45)
λ
c
c
1
13
−
c
cc
r
11
11
+
ε+
T
z
(
)
c
(1
−
c
/)
c
/1
−λ
13
12
11
12
11
1
r
2
where
ψ= −
11
. In this equation, the effect of electrical potential is
absent because it is independent of
u
r
and σ
r
. The contribution of electrical
field can be calculated separately, as described in the previous section, and
then included in the remodeling rate equation.
Assuming that a bone layer is sufficiently thin, we can replace
r
with its
mean value
R
and let
r
=
a
+
s,
where 0 ≤
s
≤
h,
and
a
and
h
are the inner radius
and the thickness of the thin bone layer, respectively. Thus, Equation (3.45)
is reduced to
c
c
/
c
11
12
c
cR
1
12
11
−
u
u
∂
∂
c
r
11
r
=
s
σ
σ
ψ
(/
cc
R
−
1)
r
r
12
11
2
R
(3.46)
c
c
λ
13
11
1
−
c
cc
R
11
+
ε+
T
z
c
(1
−
c
/)
c
(/
−λ
1)
13
12
11
12
11
1
R
This equation can be written symbolically as
∂
∂
{} {}{}{}
[]
(3.47)
FGFH H
L
=
+
+
T
s
