**3.1. Voxel FE model of human proximal femur**

30 Apoptosis and Medicine

**remodeling** 

procedures.

*AF*

probability *f*\* of bone resorption/formation.

 

\*

boundary condition.

on *Pf*\* and *Pr*\*.

where *AFf* reaches the maximum *AFfmax* at *εc* = *εmax*. It is noted that the pathological overuse is

The mathematical models of bone remodeling established in low, physiological, and high strain ranges were integrated by introducing transition regions. In the transition regions between low and physiological range (*εdu* < *εc < εpl*) or between physiological and high range (*εpu* < *εc < εol*), the probabilities of bone resorption or formation in each window were combined as a linearly weighed sum as given by Eqs. (10)-(12) resulting the resultant

c du c du

pl du pl du

εε εε f Γ ,ε Γ ε εε εε εε

c pu c pu

ol du ol pu

\*\* \* ( , ) ( ( , ) 0) *f cc cc Pf f* 

\*\* \* ( , ) ( ( , ) 0) *r cc cc Pf f* 

Using this integrated model, trabecular bone structure was simulated by the following

1. The initial shape of the trabecular bone is discretized by voxel finite elements and

2. The equivalent stress and strain distributions are determined under a given mechanical

3. The probabilities *Pf*\* and *Pr*\* of bone formation or resorption are calculated for every

4. The activation frequency *AFf* and *AFr* of bone formation or resorption are calculated for every voxel on bone surface by Eqs. (7) and (9). These are rounded to the nearest integer

5. Bone surface movement (voxel removal or addition) is determined stochastically based

εε εε

*f*

HS

P

*f*

Young's modulus and Poisson's ratio are given.

voxel on bone surface by Eqs. (10) - (12).

number and truncated by *AFfmax* or *AFrmax*.

c c c pl c pu

 

*fmax c max*

<sup>1</sup> <sup>1</sup> <sup>1</sup>

2 2

not considered here and the upper limit *εmax* is enforced for *εc* in high strain range.

**2.4. Integrated mathematical model and simulation procedures for bone** 

 

*fmax c ol*

*sin*

*AF*

*AF*

*max ol f c*

ol c ε ε

 

 

LS c du

εε εε 1 P <sup>Γ</sup> ε εε

1 Γ P ε εε

P ε ε

c HS pu c ol

LS c du c pl

*f*

 

 

*ol c max*

(9)

(10)

(11)

(12)

The three-dimensional profile of human proximal femur is reconstructed using computed tomography (female, 70 years, hip osteoarthritis), the resolution of the images was 0.7 mm so that the entire volume could be captured with slice image of 512512 cubic voxels. As the random initial trabecular structure in proximal femur, an artificial trabecular structure was given by the random-placement of hollow spheres with inside and outside diameters of 2100 and 4200 μm, respectively (Kwon et al., 2010a). One voxel was transferred to one finite element of eight node brick element. The isotropic elastic body with Young's modulus of 20.0 GPa and Poisson's ratio of 0.3 was assumed for trabecular bone material. The cortical outer surface was fixed throughout the remodeling simulation, and the intra-structure, from the head to mid-shaft, was remodeled in accordance with the same remodeling procedure. The fixed boundary conditions are imposed on the distal end. In the daily loading condition, we considered the one-legged stance, abduction, and adduction, which accounted for 50, 25, and 25% of this condition as illustrated in Fig. 3. The loading magnitudes and frequencies of these three stances were shown in Table 1 (Beaupré et al., 1990; Adachi et al., 1997). The extreme ranges of motion of abduction and adduction were assumed. The trabecular structure obtained under this daily loading condition was referred to as the normal structure and also used for the initial structure for the remodeling simulation under reduced weight-bearing conditions (Kwon et al., 2010b).

**Figure 3.** Standard weight-bearing cases at each stance (daily-loading condition).
