Biomedical Engineering Reference
In-Depth Information
acknowledges the orthopedic principle that prolonged straining of a bone
tends to make the bone stiffer and denser, whereas prolonged bed rest or
inactivity will tend to make the bone less stiff and less dense. That theory
postulates a causal relationship between the rate of deposition or resorption
of the bone matrix at any point and the strain at that point in the bone matrix.
(The theory of internal remodeling is discussed in detail in Section 2.4.)
2.2.2 Surface Bone Remodeling
The model of surface bone remodeling presented in Cowin [11] is described
here. In that model, a bone is assumed to be a linear elastic body whose
free surfaces move according to an additional specific constitutive relation.
The additional constitutive relation for the movement of the free surface is
the result of a postulate that the rate of surface deposition or resorption is
proportional to the change in the strain in the surface from a reference value
of strain. At the reference value of strain there is no movement of the surface.
To express the constitutive equation for the surface movement in equation
form, Cowin [11] introduced the following notations:
Q
is a surface point
on the bone (see Figure 2.1),
n
is an outward unit normal vector of the
tangent plane to the surface of the bone at
Q,
and
U
is the velocity of the
remodeling surface in the
n
direction. The velocity of the surface in any
direction in the tangent plane is zero because the surface is not moving
tangentially with respect to the bone. The hypothesis for surface remodel-
ing is that the speed of the remodeling surface is linearly proportional to
the strain tensor ε
ij
:
0
UCQQ Q
=
()[()
ε
−ε
(
)]
(2.1)
ij
ij
ij
where
ε
0
are the reference values of strain at point
Q,
where no remodeling
occurs, and
C
ij
(
n
, Q) are the coefficients of the surface remodeling rate that
are, in general, dependent on the point
Q
and the normal
n
to the surface at
Q.
()
ij
b
(
t
)
a
(
t
)
n
Q
FIGURE 2.1
Geometry of the diaphysis of the bone.
