## 2 Materials and Methodology
### Patient Data and Finite Element Model Set-Up  
The patient data from the Knee Grand Challenge [28] was acquired and used in this present study for making 3D tibia bone modeling. Computed tomography (CT) scan data were acquired for a patient (male, 88 years old, 168 cm body height, 66.7 kg body mass,  
Figure 1: Overview of the developed modeling and simulation workflow for analyzing the tibia crack propagation.  
and 23.6 kg/m\({}^{2}\) Body Mass Index (BMI)) with a knee implant instrumented on his right knee side (Figure 2a).  
Semi-automatic segmentation was applied on the CT scan data to segment different knee structures (femur and tibia bones) and TKR implant, using the 3D Slicer software. A thresholding process was performed to segment the bone tissues and the associated implant from surrounding tissues. A threshold value of 810 was used for cortical bone and implant segmentation and a threshold value of 560 was used for cancellous bone segmentation. 3D models of these structures were saved in STL format for further processing in numerical simulation. Tibia bone with and without the implant were considered in the present study. FEM simulations were performed using the Salome-Meca software (2019.03 version). The solver used in this platform was code_aster, an open-source finite element solver developed by EDF R&D in France since 1989, and dedicated to thermo-mechanical analyses of structures. All models were meshed with linear tetrahedral elements (see Figure 2b). Since the crack path was not known a priori, a finer mesh was generated around the crack tip during crack propagation, using automatically adaptive mesh refinement, in order to obtain a good accuracy.  
All bony solids were modeled using homogeneous, isotropic, and linearly elastic laws. Their elastic properties are reported in Table 1, which were used for modeling the tibia bone with and without the implant or total knee replacement (TKR). As listed in the table, elastic modulus (16.6 GPa and 2.4 GPa for cortical and cancellous bones, respectively) and Poisson ratio (0.3) of the bone tissue were taken from the 24-year-old male volunteer [29]. Moreover, elastic properties of the bones associated with two different ages (27-year-old female, 56-year-old female) were also extracted from [20], in which the elastic modulus of the cortical bone was set up as 15.3 GPa and 13.4 GPa for the 27- and 56-years old, respectively. A linear elastic law was also applied for the tibia implant with an elastic modulus of 117 GPa and a Poisson ratio of 0.3 [29]. Regarding the applied boundary conditions, the bottom part of the tibia model was clamped as a physiological boundary condition in standing posture. A displacement of 0.0001 m was prescribed, following the axial direction at the top surface [19]. This displacement value was under normal physiological conditions and could be predicted in in vivo conditions. Furthermore, the interfaces between the cancellous bone and the implant were assumed to be perfectly bonded.  
Figure 2: CT images of the human knee with total knee replacement (TKR) implant of the patient (**a**) and associated meshed models of the tibia bone (cortical and cancellous bones corresponding to the green and yellow parts) and implant (the red part) (**b**).