## 2 Materials and Methods
### Finite Element Models  
Four finite element models (FEM-FT-01, FEM-FT-02, FEM-PU-01, and FEM-PU-02) were created using the same boundary and loading conditions as the experimental fatigue test setup detailed above (Figure 2a,b). A vertical load was applied to the analytically rigid surface, which was inserted within the horizontal hole of the UHMWPE test block; the lower rigid surface was fixed [16]. These two rigid surfaces were assumed to have a frictionless contact with the test block. The contact interface between the screws and rods was bonded [18,19]. All meshing and simulations were conducted using ANSYS 16.0 (ANSYS Inc., Park City, UT, USA). The pedicle screws, support rods, and UHMWPE test blocks were modeled as linearly elastic materials with the properties detailed in Table 1[16]. The rods were meshed using eight-node hexahedral elements, and the screws used four-node tetrahedral elements. A mesh sensitivity study was performed to ensure the convergence of the mesh solution. The final model had 72,471 elements in each rod, 38,541 elements in each fully threaded polyaxial screw (8582 and 30,059 for the head and body, respectively), and 36,437 elements in each proximally unthreaded polyaxial screw (8582 and 27,855 for the head and body, respectively). The UHMWPE block in the FEM-FT-01 and FEM-PU-01 models had 61,059 elements, and in the FEM-FT-02 and FEM-PU-02 models had 55,832 elements (Table 2). When placed under a 170 N vertical load, the mesh was assumed to converge when the change in von Mises stress on the screws and rods was less than 2%.  
The FEM-FT-01 model was validated by demonstrating that the stiffness of the entire model (43.18 N/mm) was within the range of experimental data (42.78-43.72 N/mm), as shown in Figure 2c.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Modulus (MPa)** & \(\
u\) \\ \hline Ultra-high molecular weight polyethylene (UHMWPE) blocks [16] & 1050 & 0.4 \\ \hline Titanium rods [16] & 110,000 & 0.3 \\ \hline Titanium pedicle screws [16] & 110,000 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of finite element models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Fully Threaded** & \begin{tabular}{c} **Proximally** \\ **Unthreaded** \\ **Polyaxial Screw** \\ **(Head/Body)** \\ \end{tabular} & \begin{tabular}{c} **UHMWPE Block** \\ **of FEM-FT-01 and** \\ **(Head/Body)** \\ \end{tabular} & \begin{tabular}{c} **UHMWPE Block** \\ **of FEM-FT-01 and** \\ **FEM-PU-01** \\ \end{tabular} & **Rod** \\ \hline \hline \begin{tabular}{c} **Type of elements** \\ **elements** \\ \end{tabular} & \multicolumn{4}{c}{4-node tetrahedral} & \multicolumn{4}{c}{8-node hexahedron} \\ \hline \
\begin{tabular}{c} **Number of** \\ **elements** \\ \end{tabular} & 8582/30,059 & 8582/27,855 & 61,059 & 55,832 & 72,471 \\ \hline \hline
\begin{tabular}{c} **Number of nodes** \\ \end{tabular} & 15,448/54,407 & 15,448/49,582 & 109,296 & 99,381 & 289,878 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Type of elements, number of elements, and nodes in each part of the finite element models.  
Figure 2: (**a**) Finite element model in accordance with ASTM F1717 standard configuration. (**b**) Pedicle screw inserted leaving 7.6 mm unsupported length (FEM-FT-02 and FEM-PU-02). (**c**) The axial displacement and load curve of experimental data of EXP-FT-01 and finite element model FEM-FT-01.