Biomedical Engineering Reference
In-Depth Information
The surface remodeling equation (2.1) can be written in cylindrical coor-
dinates as
U
=
C
R
ε
rr
+
C
Z
ε
zz
+
C
θ
ε
θθ
+
C
RZ
ε
rz
+
C
R
θ
ε
r
θ
+
C
θ
Z
ε
θ
z
−
C
0
(2.15)
where
0
0
0
0
0
0
0
CC
=ε+ε+ε+ε+ε+ε
θθθ
CCC
C
C
(2.16)
R r
Z
zz
RZ
rz
Rr
θ
θ
θ
Zz
θ
with the subscripts
R,
Z,
and θ referring to the radial, axial, and tangential
directions, respectively. Substituting Equation (2.10) into Equation (2.15),
we have
U
=
B
R
σ
rr
+
B
Z
σ
zz
+
B
θ
σ
θθ
+
B
RZ
σ
rz
+
B
R
θ
σ
r
θ
+
B
θ
Z
σ
θ
z
−
C
0
(2.17)
where
=−
µ
C
E
C
E
−
µ
C
C
G
R
ZZ
R
θ
R
θ
B
,
B
=
,
R
R
θ
E
2
R
Z
R
R
=−
µ
C
E
C
E
−
µ
C
E
C
G
θ
ZZ
Z
RR
R
RZ
Z
B
,
B
=
,
(2.18)
θ
RZ
2
R
C
E
µ
C
G
Z
Z
RZ
B
=−+
(
CC
E
)
,
B
=
Z
R
θ
RZ
2
Z
Z
Z
Equations (2.10)-(2.18) are the basic equations for the solution of bone
surface remodeling problems.
2.3.2 Bone Remodeling of Diaphysial Surfaces
To illustrate the application of the theory just described, a simple example of
surface remodeling occurring in a diaphysial region of a long bone under a
constant load [23] is summarized in this subsection.
Cowin and Van Buskirk [23] considered a hollow circular cylinder com-
posed of linearly elastic bone materials subjected to a constant compressive
axial load
P
for an indefinite time period. As a result of the applied load
P,
the hollow cylinder, which initially had an internal radius
a
o
and an exter-
nal radius
b
o
,
will remodel into a cylinder with an internal radius
a
∞
and an
external radius
b
∞
. The objective of the theory is to predict the instantaneous
values
a
(
t
) and
b
(
t
) of these radii. Cowin and Van Buskirk assumed that the
surface remodeling rate coefficients are different on the periosteal and endos-
teal surfaces, but that each is constant on these surfaces throughout the diaph-
ysis. Since σ
zz
is the only nonzero stress, it follows from Equation (2.17) that
0
UB
=σ−
C
(2.19)
p
Zp z
p
