Biomedical Engineering Reference
In-Depth Information
Fig. 7.7 Evolution of the trabecular architecture in the square bone patch. a FEM analysis [
4
].
b FEM analysis [
5
]. c Meshless analysis using the proposed material law. d Meshless analysis
using Lotz material law [
1
]
however if the same interest point shows q
app
[ 1
:
3 g/cm
3
the trabecular material
curve is assumed. It is not a continuous process, as it is in the proposed material
law. In Lotz material law when an interest point leaves the cortical curves and
passes to the trabecular curve a hug leap in the material laws occurs (notice, for the
the material properties of the cortical bone and the trabecular bone). This maybe in
the origin of the difference between the results obtained with the regular and the
irregular nodal distributions.
In Fig.
7.7
the results obtained with the NNRPIM are compared with the results
obtained with the FEM [
4
,
5
]. Being this example a benchmark problem, the nodal
distribution used in both FEM studies respects the same nodal density used in the
meshless analysis. It is visible that the FEM results available in the literature [
4
,
5
],
Fig.
7.7
a, b, resemble the results of a simple topological structural problem, in
opposition the results obtained with the meshless method, Fig.
7.7
c, d, look like
the real trabecular architecture which can be seen in X-ray plates. The results of
the meshless method are obtained considering a = b = 0.01. As it is visible in
Figs.
7.4
,
7.6
, both a = b = 0.01 and a = b = 0.02 lead to good results when
compared with the FEM solution, Fig.
7.7
a, b. Therefore in the following studies
presented in this topic the parameters a and b are assumed as: a = b = 0.01.
In order to validate the remodelling algorithm when distinct mechanical load
cases are consider, the bone patch was subjected to two individual loads; a load
case L1, Fig.
7.1
a, and a load case L2, Fig.
7.1
d. In both cases the applied load has
the same magnitude. In this study only the proposed anisotropic material law was
considered and the problem was analysed with the same nodal distributions pre-
sented in Fig.
7.1
b, c.
In a first approach, both loads are applied with the same number of cycles.
Therefore, load case L1 was applied with 1,000 cycles and load case L2 was also
applied with 1,000 cycles. The results are shown in Fig.
7.8
. As it was expected
the trabecular remodelling resembles in both directions (x and y).
A final test regarding the square bone patch example was conducted. The two
load cases already referred were now applied with the following condition: load
case L1 was applied with 1,000 cycles and load case L2 with 5,000 cycles. All the
