## 3 Results
### Octahedral Shear Strain in Cancellous Bone  
In the context of mechanostat theory and bone remodeling, the octahedral shear strain has been used as a relevant strain measure to describe the mechanical state of bone. It provides a measure of the deformation experienced by bone tissue, taking into account both axial and shear strains. Researchers have studied the relationship between octahedral shear strain and bone adaptation or remodeling to understand the mechanical stimuli that influence bone behavior. The octahedral shear strain was determined by [PERSON] et al. (2014) as the most significant strain for the mechanostat hypothesis. The mechanostat distinguishes five major strain levels ([PERSON] et al., 2015), as shown in Figure 7. Strain below 1000 \(\upmu\)leads to bone atrophy, and the strain between 1000 \(\upmu\) and 1500 \(\upmu\) is the adapted state for which bone is maintained. The physiological overload level is between 1500 \(\upmu\)c and 3000 \(\upmu\)c, where bone modeling takes place. The pathologic overload above 3000 \(\upmu\)c leads to bone damage and absorption. For strain levels over 25,000 \(\upmu\)c, resulting in spontaneous fracture.  
Figure 5: Graphical representation of stress (MPa) in cancellous bone for; (a) full osseointegration and (b) partial osseointegration.  
Figure 6: Comparison of von Mises stresses in normal and weak bone; (a) full osseointegration and (b) partial osseointegration.  
The octahedral shear strain was calculated by Equation (1).  
\[\varepsilon_{\text{oct}}=2/3\ [(\varepsilon_{1}-\varepsilon_{2})^{2}+(\varepsilon_{2} -\varepsilon_{3})^{2}+(\varepsilon_{3}-\varepsilon_{1})^{2}]^{1/2} \tag{1}\]  
where \(\varepsilon_{\text{oct}}\) is the octahedral shear strain (mm/mm) and \(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\) are the principal strains in directions x, y, and z, respectively.  
Figure 8 shows the maximum octahedral shear strain in cancellous bone. It can be observed that higher strain was in normal bone compared to weak bone in all cases. In the case of full osseointegration for normal bone, the minimum octahedral shear strain was observed in bulk Ti implants, compared to the porous Ti implants. However, for the weak bone, the octahedral shear strain decreased with increasing the porosity of the implant. In the case of partial osseointegration, the octahedral shear strains were higher compared to the full osseointegration. In all cases of weak bone (see Figure 8), where the octahedral shear strains were above 3000 \(\mu\)e, it is expected that bone damage may occur. Such high levels of strain put significant stress on the weakened bone, making it prone to damage according to the mechanostat in Figure 7. When considering full osseointegration with P76, P53, and P31 implants in normal bone, where the maximum octahedral shear strains ranged between 700 \(\mu\)e and 1500 \(\mu\)e, the bone experiences an adaptive strain state. This range of strain stimulates bone adaptation to the implant, enhancing stability and integration which does not change the mass or density of the bone. In the case of full osseointegration in normal bone with bulk Ti, P76, and P53 implants, where the strain was less than 1000 \(\mu\)e, the bone may experience some degree of atrophy. Strain levels within this range indicate that the mechanical stimulus provided by the implant is relatively low, which can result in bone resorption. However, in the case of partial osseointegration with the same set of implants of normal bone, where the strain ranged between 1500 and 3000 \(\mu\)e, the bone faces physiological overload.  
Figure 7: The mechanostat hypothesis shows different levels of strains and corresponding bone remodeling [46].