Biomedical Engineering Reference
In-Depth Information
1
are material constants. Using these approximations, the remodeling rate
equation (3.37) can be simplified as
ε ε
AAAAccee
E
0
E
1
0
1
0101 0
1
0
1
0
,
where
C
0
,
C
1
,
C
2
,
,
,
,
,
,
,
,
,
λλκκχ
,
,
,
,
and
χ
i
i
ij
ij
ij
ij
ij
ij
i
i
i
i
3
3
ė
= α(
e
2
−2β
e
+ γ)
(3.40)
by neglecting terms of
e
3
and the higher orders of
e,
where α, β, and γ are
constants. The solution to Equation (3.40) is straightforward and has been
discussed by Hegedus and Cowin [34]. For the reader's convenience, the
solution process is briefly described here. Let
e
1
and
e
2
denote solutions to
e
2
−2β
e
+ γ = 0 that is,
e
1, 2
= β ± (β
2
− γ)/
1/2
(3.41)
When β
2
< γ,
e
1
and
e
2
are a pair of complex conjugates and the solution of
Equation (3.40) is
2
(
γ−β
β−
)
2
2
et
()
=β+γ−β αγ−β +
(
) tan(
t
)arctan
(3.42)
e
0
where
e
=
e
0
is the initial condition. When β
2
= γ, the solution is
=−
−
+α −
ee
eet
1
0
et
()
e
(3.43)
1
1(
)
1
0
Finally, when β
2
> γ, we have
ee
(
−+ −
e
)
e
(
ee
)exp(
α
(
eet
−
) )
10
2
2
1
0
1
2
et
()
=
(3.44)
(
ee
−+−
)
(
ee
)exp(
α
(
eet
−
) )
0
2
1
0
1
2
Since it has been proved that both solutions (3.42) and (3.43) are physically
unlikely [4], we will use solution (3.44) in our numerical analysis.
3.4 Semianalytical Solution for Inhomogeneous
Cylindrical Bone Layers
The solution obtained in the previous section is suitable for analyzing
bone cylinders if they are assumed to be homogeneous [4]. It can be useful
if explicit expressions and a simple analysis are required. In fact, however,
all bone materials exhibit inhomogeneity. In particular, for a hollow bone
