Biomedical Engineering Reference
In-Depth Information
The results of this benchmark example have shown the ability of the proposed
remodelling algorithm, combined with the proposed new material law and the
NNRPIM accuracy, to predict the principal and secondary trabecular structures for
the two-dimensional analysis.
7.1.2 3D Bone Patch
It is possible to find in the literature three-dimensional benchmark examples
developed to validate bone tissue remodelling algorithms [
6
]. Generally, those
benchmark examples consist on cubic bone patches submitted to localized loads,
which originate the formation of well-known trabeculae structures. In this topic
similar hexahedron bone patches are studied. Firstly consider the 3D patch pre-
sented in Fig.
7.10
a, with a volume 2
1
2mm
3
. A surface load
F ¼ 1
:
0 N/mm
2
, with the direction indicated in Fig.
7.10
a, is applied in two square
areas on the top of the hexahedron patch. On the patch bottom another two square
areas locally constrain the patch movement in all directions. The problem is
analysed considering the regular mesh, with 2,681 nodes, presented in Fig.
7.10
c.
In order to present the results the cubic patch is sectioned by the section presented
in Fig.
7.10
d.
In this example only the proposed material law is considered in the bone
remodelling algorithm. For all studied examples, an uniformly initial density
distribution q
max
app
¼ 2
:
1 g/cm
3
is assumed for the 3D patch, with a Poisson ratio
t = 0.3, regardless the material direction. In the remodelling algorithm it is
assumed
a = b = 0.01
and
a
control
medium
apparent
density
q
control
app
¼ 0
:
4 g/cm
3
.
In Fig.
7.11
is presented the evolution of the trabecular bone remodelling
process until the control apparent density q
control
app
¼ 0
:
4 g/cm
3
is reached. As
expected, the applied loads lead the bone to build vertical trabeculae.
In order to verify the influence of the nodal discretization, a single diagonal
load is considered, as Fig.
7.10
b illustrates. This example is capable to analyse the
influence of the nodal discretization because now, in opposition with the previous
example, the load path is unable to travel from the application point to the con-
strain point following a trivial linear string of field nodes.
The same essential boundary conditions and material properties are assumed.
Again, for the remodelling algorithm, it is assumed a = b = 0.01 and a control
medium apparent density q
control
app
¼ 0
:
4 g/cm
3
. The obtained results are presented
in Fig.
7.12
. It is visible that evolution of the trabecular bone remodelling process
leads to a single diagonal trabecula.
In Fig.
7.13
are presented the three-dimensional section views for both analyses
when the apparent medium density q
app
¼ 0
:
4 g/cm
3
is achieved. Notice that, as
