Biomedical Engineering Reference
In-Depth Information
!
j
q
app
/
X
l
m
i
U
i
ð
6
:
13
Þ
i¼1
This expression permits to estimate the apparent density as a function of the strain
energy in an equilibrium state of the bone remodelling process. It can also be used
as an optimization criterion in an iterative optimization procedure. The finite
element method (FEM) was used, combined with this formulation, to predict the
apparent density distribution in the actual bone [
44
]. The proximal femur was the
presented example, and the FEM was used to obtain the stress and strain distri-
butions for each distinct typical loading cases. With the proposed algorithm,
starting from a homogenous density distribution, it was possible to predict density
distributions similar with those found in the real femur within only a few itera-
tions. However, the convergence could not be obtained, and the iteration process
led to non-physiologic states, such as a complete bone with zero density or a
complete bone with cortical density.
Carter assumed that the trabeculae are oriented in the direction of the principal
stresses, respecting the trajectorial hypothesis accepted by many authors. For a
single load case it was shown that an alignment between the material principal
directions and the stress principal directions results in an optimal configuration
with respect to the local stiffness maximization [
47
]. If multiple load cases are
considered, a weighted combination of the normal stress components, with respect
to a normal-vector n, was suggested to serve as a stimulus for trabecular growth in
the corresponding direction. The effective global normal stress r
n
is calculated in
analogy with Eq. (
6.13
).
!
j
r
n
ð
n
Þ
¼
X
l
m
i
P
j¼1
m
j
r
n
i
ð
n
Þ
ð
6
:
14
Þ
i¼1
It was suggested [
46
] that the material stiffness in any direction n was directly
dependent on the magnitude of the corresponding global normal stress r
n
, however
no practical implementation of this trabecular orientation approach was shown.
A modified version of Carter's algorithm was proposed in research works about
the adaptive growth reactions of bone following total hip joint replacement [
48
,
49
].
In order to reduce the number of independent material properties of the orthotropic
case, the bone material was assumed to be transverselly isotropic. This modified
version respect the trajectorial hypotheses of Wolff's law, the directions of the
material axes were aligned with the stress principal direction. The femur density
distribution obtained numerically, in the case of the pre-surgery state, was very
close with the actual density distribution.
Pettermann et al. [
49
] suggested an improved version of Carter's algorithm. It is
based in the assumption that the adaptation of bone tissue can be described
appropriately on the continuum level by using overall tissue material parameters
and stress/strain measures. The specific mechanical stimuli act as driving forces in
