## 3 Results and Discussion
### Parametric Study for Material Parameters  
In this parametric study, the material constants, \(C_{10}\), \(k_{1}\), and \(k_{2}\), which are directly related to the young's modulus, stress, and strain of the material, respectively, were investigated. Figure 8 shows the variation of the material parameters according to age in the simulation results for the maximum and minimum trend line range. Consequently, the material constants on the maximum and minimum trend lines continuously increased overall as the age of the specimen increased. According to the line graph, the values of the material constant, \(C_{10}\), increased sharply from 0.055 to 0.01 and from 0.353 to 0.02 on maximum and minimum trend lines, respectively, as age increased from 58 to 89 years. In addition, the values of the material constant, \(k_{1}\), were nonlinearly related with age and increased from 6.5 to 580 on the minimum trend line, whereas the values of the material constant linearly increased from 0.87 to 1.84 on the maximum trend line. Finally, the values for the material constant, \(k_{2}\), are also nonlinearly related with age and increased from 6.0 to 17.9 on the minimum trend line, whereas they are linearly related with age and increased from 63 to 355 on the maximum trend line. In addition, the material constants in the minimum trend line were higher than in the maximum trend line. This may have been due to the large deformation that occurs in the lower failure stress and stretch values, which sharply decreased as the age increased.  
Figure 7: Simulation results under tensile loading for (**a**) axial and (**b**) circumferential directions.  
Moreover, the formulas in this study which can estimate the values of material constant associated with age (\(N\)) shown in Figure 9, could be proposed as follows:  
\[C_{10,max}\ =\ \alpha_{1,max}\times\exp(\alpha_{2,max}\times N)C_{10,min}\ =\ \alpha_{1,min}\times\exp(\alpha_{2,min}\times N) \tag{16}\]  
\[k_{1,max}\ =\ \beta_{1,max}\times N-\beta_{2,max}k_{1,min}\ =\ \beta_{1,min} \times\exp(\beta_{2,min}\times N) \tag{17}\]  
\[k_{2,max}\ =\ \gamma_{1,max}\times\exp(\gamma_{2,max}\times N)k_{2,min}\ =\ \gamma_{1,min}\times N-\gamma_{2,min} \tag{18}\]  
Figure 9 and Table 1 show the coefficients in Equations (16)-(18), and the comparison between material constants and formulae according to age. The R-squared is the proportion of variance in the dependent variable that can be distinguished from the independent variable. As a result, the R-squared ranges from 0.9152 to 0.9979 in the correlation between the values of the material constant and formulae. Therefore, the coefficients according to age can be calculated and predicted, and the simulation can be conducted in patients aged \(>\)78 years.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Constant** & \(\alpha_{1}\), \(\beta_{1}\), \(\gamma_{1}\) & \(\alpha_{2}\), \(\beta_{2}\), \(\gamma_{2}\) & \(R^{2}\) \\ \hline \(C_{10,max}\) & 8.4618E-05 & 0.0818 & 0.9895 \\ \(C_{10,min}\) & 5.4244E-06 & 0.1348 & 0.9152 \\ \(k_{1,max}\) & 0.0485 & 2.0140 & 0.9863 \\ \(k_{1,min}\) & 7.9912E-05 & 0.2278 & 0.9760 \\ \(k_{2,max}\) & 0.2200 & 0.0567 & 0.9979 \\ \(k_{2,min}\) & 15.2 & 844.9 & 0.9868 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coefficients and R-squared in the correlation between values of material constants and formulae.  
Figure 8: Variation of the material parameters according to age in the trend line range ((**a**) maximum and (**b**) minimum).