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Figure 5. Parametric study of dependence of bone mass on a-and load factor.

-k

= 0.01

— ■ -

k

= 0.05

—'

-k

= 0.1

- -X—

- k

= 0.25

*

k

load factor

Figure 5. Parametric study of dependence of bone mass on a-and load factor.

The results obtained for multiple loads are presented in figures 6 and 7. After the optimal density distribution had been obtained, the orientation was computed for each load case independently applied to the final material distribution and superimposed in figure 8.

For each load case the obtained optimal orientations follow the principal strain directions and correspond well in regions of highly oriented trabecular bone in the saggital plane. In particular, there is evidence of the arching arcuate system of trabecular bone originating from the proximal ends of the lateral and medial cortices. Also they agree in general with those observed previously in two-dimensional models (e.g. Carter et al. [17] and Jacobs [9]).

Now lets analyse all the orientations concurrently. From the overlaid figure it is observed that in the region between the epiphyses and the diaphyses the orientations are roughly coincident. This means that the material has a preferential orientation. Therefore in this zone trabecular bone can not be isotropic while maintaining optimality. But, in the epiphyses the three orientations are significantly different. This is problematic, since it indicates that in this region the optimal trabecular bone microstructure should behave as an effective isotropic material to optimally support stresses from all directions and this cannot be achieved by a strictly orthotropic material.

Figure 6. Bone density distribution for multiple load cases and K=0.01. a) Whole femur, b) Elements with /i > 0.2 . c) Elements with pi > 0.4 . d) Femur cross sections.

Figure 7. a) Real femur, b) Density distribution in femur obtained by the model.

Figure 7. a) Real femur, b) Density distribution in femur obtained by the model.

Figure 8. Orientation for multiple loads. 4.2 Implantedfemur

The global model for bone remodelling has also been applied to study the bone adaptation after a total hip arthroplasty. Figure 9 shows the finite element model for the implanted femur. The mesh was built using the femur geometry of the Standardized Femur (Viceconti et al. [29]) while the stem was modelled based on a Tri-Lock prosthesis from DePuy. The problem solved considers a total coated stem, resulting in a perfect adhesion of bone to metal. Mechanically the problem is modelled considering interface between bone and stem fully bonded. Thus there is continuity on the interface of two different materials. A more accurate model for bone/metal interface is possible, if one considers two bodies in contact. However the model should be extended to incorporate contact conditions. In Fernandes et al. [32, 33] a model for global bone remodelling with contact and bone ingrowth is described in detail.

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