# FEM-Based Compression Fracture Risk Assessment in Osteoporotic Lumbar Vertebra L1
## 3 Numerical Results and Discussion
### Estimation of the Fracture Risk
#### 3.3.2 Stochastic Models and for the Risk Modelling  
It is well-known that osteoporosis of the trabecular bone also entails decreasing of the thickness of the cortical shell of the lumbar verteb \(\delta\). Therefore, BV/TV and the cortical shell thickness are dependent random variables. In addition, decreasing BV/TV also entails decrease of the cortical shell thickness \(\delta\)[33; 34]. Therefore, the correlation coefficient of these r.vs. is positive. For example in Reference [33], it is obtained that the statistically significant coefficient of correlation between the cortical shell thickness and BV/TV is 0.5, when the significance level is 0.05. According to Reference [34], the coefficient of correlation between BV/TV and the cortical shell thickness is 0.29; however, it is statistically insignificant. To generate the load-bearing capacity \(X_{p,R}\) random numbers, it is assumed that the BV/TV ratio and the cortical shell thickness form a bivariate correlated truncated Gaussian random variable \((X_{\beta},X_{\delta})^{T}\in[M(G_{\beta})-3\sqrt{D(G_{\beta})},M(G_{\beta})+3 \sqrt{D(G_{\beta})}]\times[M(G_{\delta})-3\sqrt{D(G_{\beta})},M(G_{\beta})+3 \sqrt{D(G_{\beta})}]\); where \(\times\) is the Cartesian product of two sets; \(M(G_{\beta})\), \(D(G_{\beta})\), \(M(G_{\delta})\), and \(D(G_{\delta})\) are the means and variances of the correlated untruncated normal random variables \(G_{\beta}\) and \(G_{\delta}\), respectively, that also form a bivariate Gaussian random variable \((G_{\beta},G_{\delta})^{T}\sim N(\vec{\mu},\Sigma)\) of which the mean vector is \(\vec{\mu}=(M(G_{\beta}),M(G_{\delta}))^{T}\) and the covariance matrix is  
\[\Sigma= \begin{bmatrix}D(G_{\beta})&Cor(G_{\beta},G_{\delta})\sqrt{D(G_{\beta })D(G_{\delta})}\\ Cor(G_{\delta},G_{\beta})\sqrt{D(G_{\delta})D(G_{\beta})}&D(G_{\delta})\end{bmatrix}  \tag{11}\]  
where \(Cor(G_{\beta},G_{\delta})\) and \(Cor(G_{\delta},G_{\beta})\) are the correlation coefficients between r.vs. \(G_{\beta}\) and \(G_{\delta}\), \(Cor(G_{\beta},G_{\delta})=Cor(G_{\delta},G_{\beta})\). R.v. \(Y_{p,R}\) can be estimated by the empirical cumulative distribution function (ECDF) and by the empirical quantile function (EQF). Usually, ECDF is defined as follows: \(Pr(Y_{p,R}\leq y)\approx\hat{F}_{Y,p,R}(y)=\frac{1}{n}\sum_{i=1}^{n}I(y_{i} \leq y)\), where \(I\) is the indicator function, \(n\) is the sample size, \(y_{i}\) is the realization \(i\) of r.v. \(Y_{p,R}\); while EQF of r.v. \(Y_{p,R}\) is usually defined as \(Q(p)=inf\{y:Y_{Y,p,R}(y)\geq p\}\), where \(p\) is the required probability. For the evaluation of the fracture risk as the probability \(Pr_{f}=Pr(Y_{p,R}-X_{p,E}\leq 0)\), the external pressure independent r.v. \(X_{p,E}\) can distributed according to various laws. In the present article, it is assumed that the external load r.v. \(X_{p,E}\) is also truncated normal r.v. attaining values from the interval \(X_{p,E}\in[M(G_{p,E})-3\sqrt{D(G_{p,E})},M(G_{p,E})+3\sqrt{D(G_{p,E})}]\), where \(M(G_{p,E})\) and \(D(G_{p,E})\) are the mean and the variances of the untruncated external pressure normal r.v. \(G_{p,E}\). The fracture risk of the lumbar verteb can be expressed as \(Pr_{f}=Pr(Y_{p,R}-X_{p,E}\leq 0)=Pr(Z\leq 0)\approx\hat{F}_{Z}(0)\), where \(\hat{F}_{Z}\) is the ECDF of r.v. \(Z=Y_{p,R}-X_{p,E}\): \(\hat{F}_{Z}(z)=\frac{1}{n}\sum_{i=1}^{n}I(y_{p,R}-x_{p,E}\leq z)\), where \(n\) is the sample size and \(y_{p,R}\) and \(x_{p,E,i}\) are realization of r.vs. \(Y_{p,R}\) and \(X_{p,E}\), respectively. It should be noted that the fracture risk can be also evaluated according to the following known integral [29]:  
\[Pr_{f}=P(Z\leq 0)=\int_{-\infty}^{0}\int_{-\infty}^{+\infty}f_{Y,p,R}(p+z)f_{X,p, E}(p)dpdz \tag{12}\]  
where \(f_{Y,p,R}\) and \(f_{X,p,E}\) are probability density functions (PDFs) of the load-bearing capacity r.v. \(Y_{p,R}\) and the external presure r.v. \(X_{p,E}\), respectively. For the following analysis of the reliability of the vertebra, the concrete parameters of the r.v.