# 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  
On the basis of the above given review, for the probabilistic estimations of the load-bearing capacity of the lumbar vertebra \(Pr(Y_{p,R}\leq y_{p})=\hat{F}_{Y,p,R}(y_{p})\) and the failure risk \(Pr_{f}=Pr(Z\leq 0)=F_{Z}(0)\), the following two cases were considered. For the first case (Case I), the following values are adopted: the mean of BV/TV \(M(G_{\beta})=0.1\) and the mean of the cortical shell thickness \(M(G_{\delta})=0.2\) mm; the coefficients of variation \(Cvar(G_{\beta})=\sqrt{D(G_{\beta})}/M(G_{\beta})=0.2\) and \(Cvar(G_{\delta})=0.2\); the correlation coefficient \(Cor(G_{\beta},G_{\delta})=0.35\); the mean of the external load \(G_{p,E}\) attains three values \(M(G_{p,E})\in\{p_{min},1.5p_{min},2.5p_{min}\}\), where \(p_{min}=0.372\)  MPa; and the coefficient of variation for all external pressures is constant: \(Cvar(G_{p,E})=0.20\). For the second case (Case II), the following values are adopted: means \(M(G_{\beta})=0.25\) and \(M(G_{\delta})=0.2\) mm; the coefficients of variation \(Cvar(G_{\beta})=0.2\) and \(Cvar(G_{\beta})=0.2\); the correlation coefficient \(Cor(G_{\beta},G_{\delta})=0.35\); mean of the external load \(G_{p,E}\) attains three values \(M(G_{p,E})\in\{p_{min},1.5p_{min},2.5p_{min}\}\), where \(p_{min}=0.372\)  MPa; and the coefficient of variation for all external pressures is constant: \(Cvar(G_{p,E})=0.2\). Thus, for both Cases I and II, the external load \(G_{p,E}\) attains the same three different means and the coefficient of variation is constant, and for both cases, the coefficient of the correlation between the cortical shell thickness and BV/TV is the same, 0.35. It should be noted that for Cases I and II the mean of the external pressure \(M(G_{p,E})=0.372\) MPa approximately corresponds to the external load of the lumbar vertebra of a person of mass 75 kg in rest, while \(M(G_{p,E})=2.5p_{min}=0.93\) MPa is close to the maximum load during the squat jump of the person. Totally, \(10^{6}\) random values of the r.vs. \(G_{\beta}\) and \(G_{\delta}\) were generated. However, due to the truncation the samples of the realizations of the r.vs. \(X_{\beta}\), \(X_{\delta}\), \(X_{p,E}\), and \(Y_{p,R}\) were smaller than \(10^{6}\) but bigger than \(9\times 10^{5}\). All analyses were performed by the program \"R studio\" [31]. The bivariate Gaussian random numbers were generated using the package \"MASS\" [36]. The realizations of the histograms of the r.vs. \(X_{\beta}\), \(X_{\delta}\) and \(Y_{p,R}\) are shown in Figure 8. The realizations of the histograms of the truncated BV/TV ratio r.v. \(X_{\beta}\) are depicted in Figure 8a,d and the realizations of the truncated cortical shell thickness r.v. \(X_{\delta}\) are depicted in Figure 8b,e while realizations of the histograms of the load-bearing capacity r.v. \(Y_{p,R}\) as well as the 0.05 and 0.95 empirical quantiles of r.v. \(Y_{p,R}\) are depicted in Figure 8c,f. It should be noticed that the histograms depicted in Figure 8a,b,d,e show entire ranges of the random numbers of the r.vs. \(X_{\beta}\) and \(X_{\delta}\), while Figure 8c,d show truncated ones from the right of the intervals of r.vs. \(Y_{p,R}\). As we can see from Figure 8c,f, the probability density function of the load-bearing capacity r.v. \(Y_{p,R}\) can have a positive skew. However, there are such cases of \(M(X_{\beta})\), \(M(X_{\delta})\), \(D(X_{\beta})\), and \(D(X_{\delta})\) in which the histograms of r.v. \(Y_{p,R}\) have a negative skew. It was also observed that at the small values of variances \(D(X_{\beta})\) and \(D(X_{\delta})\), the histograms of the r.v. \(Y_{p,R}\) are symmetrical and the histograms are very similar to the normal distribution probability density function with mean \(M(Y_{p,R})\) and variance \(D(Y_{p,R})\). However, the given Figure 8c,f shows a counterexample, and in general, we cannot state that r.v. \(Y_{p,R}\) is normal provided that r.vs.