Biomedical Engineering Reference
In-Depth Information
and
ec ce
2
0
1
=+++ε+ ε
ce
AeA
ij
(2.9)
0
1
2
ij
ij
ij
0 1
are material constants. Equations
(2.1)-(2.9) consist of the basic formulation of surface and internal bone remod-
eling. More details as to the solution procedure and application of bone
remodeling theory are described in the remaining sections of this chapter.
0
1
where
CCc
,
,
,
ccA
,
,
, and
A
ijkm
ijkm
012
ij
ij
2.3 A Simple Theory of Surface Bone Remodeling
In the previous section a simple constitutive equation (2.1) for surface bone
remodeling was presented. Based on Equation (2.1) and the theory of linear
elastic solid material, Cowin and Van Buskirk [23] developed a simple theory of
surface bone remodeling and applied it to the problem of predicting the surface
remodeling that would occur in the diaphysial region of a long bone as a result
of superposed compressive load and as a result of a force-fitted medullary pin.
Papathanasopoulou, Fotiadis, and Massalas [24] extended it to the case of poro-
elastic material with fluid. A brief review of the development in Cowin and
Van Buskirk and Papathanasopoulou et al. is presented in this section.
2.3.1 Basic Equations of the Theory
As mentioned in Subsection 2.2.2 and also indicated in Cowin and Van
Buskirk [23], Equation (2.1) by itself does not constitute the complete theory.
The theory of surface bone remodeling is completed by considering a bone
as a linear elastic solid and implementing the related elastic equations. For
simplicity, Cowin and Van Buskirk assumed the bone to be transversely iso-
tropic with the axis of a long bone as the axis of symmetry and considered
to have only cylindrical boundaries (see Figure 2.1). Therefore, the basic
equations can be written within the framework of a cylindrical coordinate
system. Based on that assumption, the strain-stress relations for the bone
are written as
ε= σ−µσ −
µ
σε = σ
1
1
A
(
)
,
,
rr
rr
T
θθ
zz
r
θ
r
θ
E
E
2
G
T
A
T
ε= σ−µσ −
µ
σε = σ
1
1
A
(
)
,
,
(2.10)
θθ
θθ
T r
zz
θ
z
θ
z
E
E
2
G
T
A
A
ε=
µ
σ+σ+ σ
1
1
A
A
(
)
,
ε = σ
zz
θθ
rr
zz
rz
rz
E
E
2
G
A
A
