Bone Tissue Remodelling Analysis - Meshless Methods in Biomechanics - page 275

(a) (b) (c)

Fig. 7.35 Combination of all mechanical cases. a Initial von Mises effective stress isomap.
b Final von Mises effective stress isomap. c Final obtained trabecular architecture

the NNRPIM to achieve a femoral internal trabecular bone architecture similar
with the real density distribution found in the femur bone.

7.2.3.1 Femoral Two Dimensional Analysis

The femur is a long bone articulated in the hip-bone. The body weight is directly
applied in the femur head. This long bone is a natural choice to validate an
anisotropic remodelling algorithm, since it is a well-studied bone in biomechanics
and the trabecular structure in the proximal femur is relatively well oriented. An
example of a X-ray plate of the proximal femur is presented in Fig. 7.37 a. From
Fig. 7.37 a it is possible to empirically obtain the compressive and tensile lines
indicated in Fig. 7.37 b. It was used the geometry of a two-dimensional proximal
femur model proposed in the literature [ 26 ]. The domain was discretized with the
nodal distribution presented in Fig. 7.38 .
The femur loading history was approximated by the three-load cases used by
Beaupré et al. [ 14 , 15 ], each consisting of one parabolic distributed load over the
joint surface, nbc 1 , and another parabolic distributed load on the trochanter, nbc 2 ,
representing the abductor muscle attachment. In Fig. 7.39 it is possible to observe
the resultant of each applied parabolic distributed load and the correspondent
direction. For the three considered mechanical cases, all degrees of freedom are
constrained in the basis ebc 1 .
As in previous example, in a first step, each one of the load cases presented in
Fig. 7.39 are separately analysed. In order to determine the initial bone tissue
material properties using the proposed phenomenological material law, in each
analysis it is imposed an initial uniform density distribution q max

app ¼ 2 : 1 g/cm 3 . The
Poisson ratio is assumed as t ¼ 0 : 3 and the apparent density control value is
considered as q control

app ¼ 1 : 2 g/cm 3 . The a and b parameters, governing the growth
and the decay of the bone tissue, are assumed as: a = b = 0.01.
The results obtained regarding the first mechanical case proposed by Beaupré
et al. [ 14 , 15 ] are presented in Fig. 7.40 . In Fig. 7.40 a it is presented the von Mises
effective stress isomap obtained for the first step of the iterative remodelling