# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1
## 3 Numerical Results and Discussion
### Estimation of the Fracture Risk
#### 3.3.1 Approximation of the Maximum von Mises Stress  
The approximation of the calculated maximum von Mises stresses of the lumbar vertebra was conducted by using the program \"R studio\" [31] and the package \"nlme\" [32] over the set \(\mathbb{D}\times\mathbb{B}\times\mathbb{P}\), where \(\mathbb{D}\), \(\mathbb{B}\), and \(\mathbb{P}\) are sets of values of the cortical shell thickness \(\delta\), BV/TV ratio \(\beta\), and the external load \(p\), respectively: \(\delta\in\mathbb{D}=\{0.5,0.4,0.2\}\) mm, \(\beta\in\mathbb{B}=\{0.1,0.2,0.35\}\), and \(p\in\mathbb{P}=\{0.15,0.30,0.45,0.60,0.75\}\) MPa. The set of 45 points of the maximum von Mises stresses  
Figure 6: Results of the dynamic analysis. \(\mathbb{D}\times\mathbb{B}\times\mathbb{P}\) of the static analysis obtained by the finite element method that were used to obtain the approximating polynomial is given in Table 4.  
The approximating polynomial of the maximum von Mises stresses of the static analysis is as follows:  
\[ \begin{split}\sigma\approx\hat{g}(p,\beta,\delta)=& a_{0}+a_{1}p+a_{2}\beta+a_{3}\delta+a_{11}p^{2}+a_{33}\delta^{2}+a_{12}p \beta+a_{13}p\delta+a_{123}p\beta\delta+\\ & a_{112}p^{2}\beta+a_{113}p^{2}\delta+a_{221}\beta^{2}p+a_{223} \beta^{2}\delta+a_{331}\delta^{2}p+a_{332}\delta^{2}\beta+a_{12233}p\beta^{2} \delta^{2}+\\ & a_{11233}p^{2}\beta\delta^{2},\delta\in[0.2,0.5]\text{ mm},\,\beta\in[0.1,0.35],\,p\in[0.15,0.75]\text{ MPa}\end{split}  \tag{9}\]  
where the coefficients are as follows: \(a_{0}=40.095\), \(a_{1}=109.545\), \(a_{2}=-33.064\), \(a_{3}=-153.799\), \(a_{11}=56.976\), \(a_{33}=152.808\), \(a_{12}=-410.969\), \(a_{13}=-348.377\), \(a_{123}=704.586\), \(a_{112}=-153.953\), \(a_{113}=-117.969\), \(a_{221}=737.456\), \(a_{223}=97.428\), \(a_{331}=361.940\), \(a_{332}=56.473\), \(a_{12233}=-2913.652\), and \(a_{11233}=702.954\). It should be noted that only the coefficients \(a_{i}\) for which the statistical hypothesis \(H_{0}:a_{i}=0\) is rejected and the alternative \(H_{1}:a_{i}\
eq 0\) is accepted are present in the given approximating polynomial Equation (9). The hypotheses were tested using the Student \(t\)-test. The maximum and minimum residuals are as follows: \(max\{g_{FEM}(p_{i},\beta_{i},\delta_{i})-\hat{g}(p_{i},\beta_{i},\delta_{i}) \}=0.883\) MPa and \(min\{g_{FEM}(p_{i},\beta_{i},\delta_{i})-\hat{g}(p_{i},\beta_{i},\delta_{i}) \}=-0.838\) MPa, where \((p_{i},\beta_{i},\delta_{i})\in\mathbb{D}\times\mathbb{B}\times\mathbb{P}\) and \(g_{FEM}(p_{i},\beta_{i},\delta_{i})\) stand for the stress values calculated by the FEM. The relative residuals \((\hat{g}(p_{i},\beta_{i},\delta_{i})-g_{FEM}(p_{i},\beta_{i},\delta_{i}))/g_ {FEM}(p_{i},\beta_{i},\delta_{i})\) versus the stresses calculated by FEM are depicted in Figure 7. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Analysis**} & \multirow{2}{*}{**Thickness, mm, \(\delta\)**} & \multirow{2}{*}{**BV/TV, \(\beta\)**} & \multicolumn{5}{c}{**External Load, MPa, \(p\)**} \\ \cline{5-8}  & & & **0.15** & **0.30** & **0.45** & **0.60** & **0.75** \\ \hline \multirow{8}{*}{Static} & \multirow{8}{*}{0.5} & 0.10 & 2.9 & 6.1 & 9.3 & 12.3 & 15.2 \\  & & 0.20 & 2.0 & 4.1 & 6.1 & 8.2 & 8.5 \\  & & 0.35 & 1.6 & 3.4 & 5.0 & 6.7 & 8.5 \\ \cline{2-8}  & & 0.10 & 3.4 & 6.7 & 10.2 & 14.1 & 17.0 \\  & 0.4 & 0.20 & 2.1 & 4.2 & 6.3 & 8.4 & 10.5 \\  & & 0.35 & 1.8 & 3.5 & 5.4 & 7.3 & 9.1 \\ \cline{2-8}  & & 0.10 & 18.9 & 24.1 & 31.3 & 40.8 & 49.4 \\  & 0.2 & 0.20 & 13.3 & 18.4 & 23.2 & 28.1 & 33.5 \\  & & 0.35 & 12.1 & 17.4 & 21.0 & 24.5 & 28.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The maximum von Mises stresses of the static analysis calculated by finite element method (FEM) at different \(\delta\), \(\beta\), and \(p\)  
Figure 7: Relative residuals versus calculated values of the maximum von Mises stress.