## 2 Materials and Methods
### Scenarios of Spinal Rod for Finite Element Analysis  
The spinal FE model was developed from lumbar 3 to sacrum 1, and the surgical part of the lumbar spine was set from lumbar 3-4. The porosity values of the spinal implants were selected to be 0%, 34%, 50%, 73%, and 81%, based on the results of the diamond porous structure tensile test by [PERSON] et al. Additionally, the FE analysis for porosity values of 75%, 60%, 40%, and 15% was performed by estimating the mechanical properties according to the Ramberg-Osgood formula. The true stress-true strain curve was used to improve the simulation precision of the vertebral body and spinal implant behavior. ANSYS workbench 2019 R1 (ANSYS Inc., Pittsburgh, PA, USA) was used for the FE analysis [36; 39; 40].  
The material properties at different porosities, which were not presented in the study by [PERSON] et al., were estimated according to the following Ramberg-Osgood formula:  
\[\widetilde{\epsilon}=\frac{\widetilde{\sigma}}{E}+\left(\frac{\widetilde{ \sigma}}{H}\right)^{1/n}, \tag{1}\]  
To estimate the material properties, the true stress-true strain behavior of the Ti-6 Al-4V porous structure was obtained from the tensile test results presented by [PERSON] et al., and, subsequently, the variables of the Ramberg-Osgood formula (strain hardening exponent (\(n\)) and strength coefficient (\(H\))) were obtained [36; 39]. Table 2 lists the details of the variables of the Ramberg-Osgood formula from the tensile test result presented by  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p\) (\%) & \(n\) & \(H\) & \(E\) (GPa) & \(\
u\) \\ \hline
0 & 0.0605 & 1236.232 & 118.7 & 0.31 \\
34 & 0.0815 & 559.114 & 41.8 & 0.31 \\
50 & 0.0897 & 339.078 & 20.4 & 0.32 \\
73 & 0.0963 & 169.512 & 7.4 & 0.35 \\
81 & 0.1102 & 101.977 & 3.7 & 0.42 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Strain hardening exponent (\(n\)), strength coefficient (\(H\)), elastic modulus (\(E\)), and Poisson’s ratio (\(\
u\)) of Ti–6 Al–4V at different porosities (\(p\)).  
Figure 3: Comparison of range of motion (ROM) of intact FE model between the current study and conventional studies of [PERSON] et al. and [PERSON] et al.: (**a**) flexion-extension; (**b**) lateral bending; (**c**) rotation.  
The formulae for estimating the material properties of Ti-6 Al-4V at different porosities were to be expressed by the following Equations (2)-(5), and the corresponding porosity, that is, 75%, 60%, 40%, and 15%, was estimated using these same formulae. Figures 4 and 5 show the variable estimation of Ramberg-Osgood formula and the estimated stress-strain curves, respectively. A total of nine scenarios were selected for FE analysis from 0% to 81%, and the porosity and material properties of each scenario are summarized in Table 3.  
\[n=0.056p+0.061, \tag{2}\]  
\[H=1280.925p^{2}-2420.704p+1235.172, \tag{3}\]  
\[E=179,288p^{2}-285,485p+118,546, \tag{4}\]  
\[\
u=0.584p^{2}-0.466p+0.400. \tag{5}\]  
Figure 4: Estimation formula of material properties of Ti–6 Al–4V at different porosities (\(p\)): (**a**) strain hardening exponent (n); (**b**) strength coefficient (\(H\)); (**c**) elastic modulus (\(E\)); and (**d**) Poisson’s ratio (\(\
u\)).  
Figure 5: Stress (\(\sigma\))-strain (\(\varepsilon\)) behavior of Ti–6 Al–4V at different porosities: (**a**) porosity 15%; (**b**) porosity 40%; (**c**) porosity 60%; (**d**) porosity 75%.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scenario** & _p (\%)_ & _E (GPa)_ & \(v\) & \(\sigma_{0.2}\) (MPa)_ \\ \hline
1 & 0 & 118.7 & 0.31 & 898.8 \\
2 & 15 & 79.8 & 0.31 & 585.3 \\
3 & 32 & 41.8 & 0.31 & 331.7 \\
4 & 40 & 33.0 & 0.31 & 281.0 \\
5 & 50 & 20.4 & 0.32 & 193.5 \\
6 & 60 & 11.8 & 0.33 & 135.5 \\
#7 & 73 & 7.4 & 0.35 & 90.8 \\
#8 & 75 & 5.3 & 0.38 & 73.9 \\
#9 & 81 & 3.7 & 0.41 & 50.3 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Scenarios for spinal motion simulation using spinal implants at different porosities (_p_): elastic modulus (_E_), Poisson ratio (_v_), and 0.2% yield strength (\(\sigma_{0.2}\)) at each scenario.