## 4 Discussion
In the present study, linear tetrahedral elements were used. This choice was based on two main reasons. First, it was recommended in the used FE solver (Code_Aster) to use linear elements for the XFEM methods. This element type facilitated the crack identification. Second, during the crack proposition study, for each step of crack advancement, the model needed to rerun to compute the fracture mechanics parameters. To assure a balance between result accuracy and efficient computing time, it was plausible to select the model with linear elementsafter performing a convergence study on the mesh size. Note that we also performed simulations with the quadratic tetrahedral elements. The obtained outcomes showed that the values of stress intensity factor and the energy release rates were about 10% higher than those obtained with linear elements in the same mesh size. Additionally, both types of elements provided the same trends and converged quite well. However, the quadratic model required about 10 times of memory and computing time (37.10 s and 338.40 s for medium linear and quadratic mesh configurations) than those of linear elements. Moreover, 10% of the difference between the linear and quadratic elements of the same mesh sizes was not significant, due to the uncertainty of the boundary conditions and material properties. Another effecting factor of our simulation outcomes might link to the segmentation error. In the present study, a thresholding process was performed. Due to the high contrast level of the bone tissue from CT scan, different bone structures (cortical and cancellous bone tissues and implant in titanium) were extracted in a straightforward manner. However, further investigation should take the sensitivity of the threshold values into consideration. Moreover, it is practically impossible to validate the simulation outcomes with in vivo patient data in a non-invasive manner. Thus, we adopted the strategy to compare the simulation outcomes with experimental observations. First, the stress intensity factor and the energy release rate from our simulations were closely in agreement with literature data. Moreover, in comparison to cracked ex vivo bone samples (pig femur in transverse and longitudinal planes with high-speed X-ray imaging technique [37], human proximal femoral diaphysis specimens with backscattered electron microscopic imaging technique [38]), the crack pattern was quite stable with no large deflection. The direction of crack was also in agreement with experimental observations. However, our simulations could not reproduce the micro-cracks around the crack path as commonly observed in the experimental cracked specimens [39; 40]. One of the limitations of the present study relates to the use of only tibia structure and associated implant part. Thus, a full knee model including femur, tibia, cartilage, and ligament structures and the associated full TKR implant needs to be investigated in the future, to have a full view on the crack propagation behavior within the human knee. Another limitation dealt with the use of linear elastic law and generic material properties for modeling bone tissue. It is important to note that the material properties play an important role in the crack propagation analysis, especially when the ageing effect is considered. Bone tissue commonly exhibits an anisotropic, inhomogeneous, nonlinear, thermo-rheologically viscoelastic material. Thus, further studies on the integration of more complex behavior law and associated patient-specific properties of the bone tissue from medical imaging data [41; 42] would be performed. In particular, more mechanical failure testing data on older bone specimens, especially tibia bones, need to be acquired to reconstruct the more appropriate S-N curves for patients with total knee replacement, who are generally older than the age range (27-year-old and 56-year-old) used in this present study.