# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1
## 3 Numerical Results and Discussion
### Estimation of the Fracture Risk
#### 3.3.3 Fracture Risk Modelling Data, Results, and Discussion
\(X_{\beta}\) and \(X_{\delta}\) are symmetrically truncated dependent normal random variables with nonnegative correlation coefficient. As we can see from Figure 8c,f, the 0.05 quantiles of the load-bearing capacity of the vertebra r.v. \(Y_{p,R}\) increase from \(p_{0.050}=0.355\) MPa for Case I up to \(p_{0.05}=0.638\) MPa for Case II. Analogously, the 0.95 quantiles increase from \(p_{0.95}=0.913\) MPa up to \(p_{0.95}=2.364\) MPa. The quantiles of the load-bearing capacity r.v. \(Y_{p,R}\) can serve as extra information on the fracture risk of the lumbar vertebra. It should be noted that the correlation between r.vs. \(X_{\beta}\) and \(X_{\delta}\) in general is important for r.v. \(Y_{p,E}\). For example, for the considered Case I, the increase of the correlation coefficient \(Cor(G_{\beta},G_{delta})\) from 0.35 up to 0.5 decreases the small quantiles of the load-bearing capacity r.v. \(Y_{p,R}\) from \(p_{0.05}=0.355\) up to \(p_{0.05}=0.347\) and increases the big quantiles from \(p_{0.05}=0.913\) up to \(p_{0.95}=0.929\). For Case II, the increase the correlation coefficient from \(Cvar(G_{\beta},G_{\delta}=0.35)\) up to \(Cvar(G_{\beta},G_{\delta})=0.5\) also entails a decrease of small value quantiles of r.v. \(Y_{p,R}\) from \(p_{0.05}=0.638\) up to \(p_{0.05}=0.620\), but the increase in the correlation coefficient also increases the big quantile values from \(p_{0.95}=2.364\) up to \(p_{0.95}=2.423\). Figure 8: Realizations of the probability density functions of the random variables \(X_{\beta}\) (**a**,**d**); r.v. \(X_{\delta}\) (**b**,**e**); and r.v. \(Y_{p,R}\) (**c**,**f**). The fracture risks of static verteba loads of Case I and Case II as well histograms of r.vs. \(Y_{p,R}\), \(X_{p,E}\), and \(Z=Y_{p,R}-X_{p,E}\) can be seen in Figure 9. It was obtained that with increasing the cortical shell thickness from \(\delta=0.2\) (Case I) to \(\delta=0.25\) (Case II) and the BV/TV ratio from \(\beta=0.1\) (Case I) to \(\beta=0.2\) (Case II), the failure risk \(Pr_{f}\) decreases from 0.0904 to \(2.752\cdot 10^{-3}\) for \(M(G_{p,E})=p_{min}=0.372\) MPa; from 0.417 to 0.0337 for \(M(G_{p,E})=1.5p_{min}=0.558\) MPa; and from 0.897 to 0.260 for \(M(G_{p,E})=2.5p_{min}=0.93\) MPa. The obtained results suggest that, for the considered supposed person, the squat jump inevitably would lead to the fracture of a lumbar verteba, while a relatively small increasing BV/TV and the cortical shell thickness \(\delta\) significantly decrease the fracture risk \(Pr_{f}\). The correlation between r.vs. \(X_{\beta}\) and \(X_{\delta}\) is also important for the fracture risk \(Pr_{f}\). For example, for the considered Case I, the increase of the correlation coefficient \(Cor(G_{\beta},G_{\delta})\) from 0.35 up to 0.5 increases the fracture risk from \(Pr_{f}=0.0904\) up to \(Pr_{f}=0.0963\) and from \(Pr_{f}=0.417\) up to \(Pr_{f}=0.418\) at small values of the loading, i.e., as \(M(G_{p,E})\in\{p_{min},1.5p_{min}\}=\{0.372,0.558\}\) MPa. However, at big load values, the increasing correlation coefficient decreases the fracture risk from \(Pr_{f}=0.897\) up to \(Pr_{f}=0.892\) as \(M(G_{p,E})=2.5p_{min}=0.93\) MPa. For Case II, the increase in the correlation coefficient from \(Cvar(G_{\beta},G_{\delta}=0.35)\) up to \(Cvar(G_{\beta},G_{\delta})=0.5\) increases the fracture risk from \(Pr_{f}=0.002752\) up to \(Pr_{f}=0.00387\) as \(M(G_{p,E})=p_{min}=0.372\) MPa; from \(Pr_{f}=0.0337\) up to \(Pr_{f}=0.0392\) as \(M(G_{p,E})=1.5p_{min}=0.558\) MPa; and from \(Pr_{f}=0.260\) up to \(Pr_{f}=0.270\) as \(M(G_{p,E})=2.5p_{min}=0.93\) MPa. Figure 9: Realizations of the probability density functions of the random variables \(X_{p,E}\) and \(Y_{p,R}\) (**a**) for Case I and (**c**) for Case II; and empirical cumulative distribution functions \(\hat{F}_{Z,i}(z)\) of r.v.