Biomedical Engineering Reference
In-Depth Information
where
1
2
+
q
N
( )
233
QRQs
NAQ
+
++
q
N
( )
233
QRQs
NAQ
+
++
B
=
+
C
+
+
(
CC
+
),
R
R
θ
Z
2
1
2
+
q
N
( )
233
QRQs
NAQ
+
++
q
N
( )
233
QRQs
NAQ
+
++
B
=
+
C
+
+
(
CC
+
),
θ
θ
RZ
2
1
2
+
q
N
( )
233
QRQs
NAQ
+
++
q
N
( )
233
QRQs
NAQ
+
++
B
=
+
C
+
+
(
CC
+
),
(2.56)
Z
Z
R
θ
2
C
N
C
N
C
N
R
θ
RZ
θ
Z
B
=
,
B
=
,
B
=
,
R
θ
RZ
θ
Z
2
Qq
NR
(3
+
1)
( )
233
QRQs
NAQR
+
++
+
1
3
sQ
R
B
=−
+
++
(
CC C
)
θ
ZR
2
2.4 A Simple Theory of Internal Bone Remodeling
Based on Equations (2.4)-(2.7), Cowin and Van Buskirk [27] presented a
theoretical solution for internal bone remodeling induced by a medullary
pin; Tsili [28] derived a solution for the case of bone remodeling induced by
casting a broken femur. Papathanasopoulou et al. [29] developed a model for
internal remodeling of poroelastic bone filled with fluid. In this section, the
developments in Tsili and Papathanasopoulou et al. are briefly summarized
for reference in subsequent chapters.
2.4.1 Internal Remodeling Induced by Casting a Broken Femur
Tsili [28] considered a femur in a steady state subjected to a constant ten-
sile axial load
P
0
(
t
) due to the weight of the lower leg. The bone is assumed
to have a uniform reference volume fraction ξ
0
and
e
is constant in the whole
solution domain. At
t
= 0, the bone was cast. Then this bone was under
the tensile load
P
0
(
t
) and under a constant external pressure
P
1
(
t
) due to the
cast. Tsili's purpose with this work was to predict
e
(
t
0
), where
t
0
is the time at
which the plaster cast is removed. The bone (femur) is modeled as a hollow
circular cylinder with constant inner and outer radii
a
and
b,
respectively.
The boundary conditions of this problem are
σ
rr
= σ
r
θ
= σ
rz
= 0 at
r
=
a
(2.57)
σ
rr
= −
P
1
σ
r
θ
= σ
rz
= 0
at
r
=
b
(2.58)
