## 3 Stochastic Approach  
The probabilistic calculation was performed in the Anthill software, in which the analysis was performed using the SBRA (simulation-based reliability assessment) method. The SBRA method applies the probability theory and statistics to stochastic inputs. The output variables affecting the safety of a structure or machined part are usually expressed as truncated histograms. Due to the random nature of the input variables, the resulting output variables also possess a random character. This allows a probabilistic assessment of the engineering problem [18; 19]. The method is based on the direct Monte Carlo approach, which analyzes the reliability function using sequences of pseudo-random simulations (realizations). The simulation works with a large number of pseudo-random samples (in our case, \(10^{6}\) random simulations), and the input and output values are defined using bounded histograms [18; 19; 20; 21]. In our calculation, input stochastic variables used screw and bone dimensions and their mechanical properties. Thanks to this, possible inaccuracies in production, different bone anatomy, and their mechanical properties could be included in the calculation. Output \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Quantity** & **Definition of Quantity** & **Quantity** & **Definition of Quantity** \\ \hline \multirow{2}{*}{L\({}_{\text{k}1}\), L\({}_{\text{k}2}\) fragments} & Lengths of bone & \multirow{2}{*}{\(\Delta\)} & Mutual displacement of the bone \\  & & fragments & fragments (drawn to each other \\  & & & by the screw tightening) \\ \hline \multirow{2}{*}{\(A_{\text{k}1}\), \(A_{\text{k}2}\), \(A_{\text{k}3}\)} & Bone areas in the & \multirow{2}{*}{\(E_{\text{k}}\)} & \multirow{2}{*}{Modulus of elasticity of the bone} \\  & individual parts of the & & \\  & screw & & \\ \hline \multirow{2}{*}{\(A_{a}\), \(A_{\text{k}p}\), \(A_{c}\)} & Screw areas in individual & \multirow{2}{*}{\(E_{\text{1}}\)} & Modulus of elasticity of the \\  & parts & & screw \\ \hline \multirow{2}{*}{L\({}_{\text{1a}}\), L\({}_{\text{l}b}\), L\({}_{\text{1c}}\)} & Screw lengths in individual parts & \multirow{2}{*}{\(a_{S}\)} & Measurement—effect of the force \\  & & & sensor (correction to the real \\  & individual parts & & osteosynthesis not containing \\  & & & any sensor) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definition of deterministic input variables. variables were axial normal force, maximal stress in the screw, and reliability function. Figure 8 shows histogram of the axial normal force. Reliability function \(\mathrm{Fs}\) is defined as  
\[\mathrm{Fs}=\mathrm{Re}-\sigma\mathrm{max}, \tag{4}\]  
where \(\mathrm{Re}\) is the yield stress of a screw and \(\sigma\mathrm{max}\) is the maximal stress in the screw. The situation where \(\mathrm{Fs}\leq 0\) expresses the probability of a condition where the thread cut in the bone fragment is sheared. The calculated reliability function \(\mathrm{Fs}\) is shown in Figure 9. Thread shearing and thus, plastic deformation, occurred in the bone with a probability of only 0.0493 (i.e., of 4.93%). If thread is sheared during surgery, the headless screw can be replaced with a larger diameter headless screw, and the surgical procedure can subsequently be completed, which further reduces the risk to the patient. Figure 10 shows the 2D distribution of the reliability function \(\mathrm{Fs}\). Towards the center, the frequency of the reliability function values increased, with higher frequencies of occurrence shown in red. The green color indicates the limit line, which divides the reliability function into negative and positive parts. If the reliability function is negative (i.e., \(\mathrm{Fs}<0\)), the headless screw is in the region of permanent plastic deformation and thread shearing  
Figure 8: Histogram for axial normal force \(\mathrm{F}\) in the headless screw (minimum = 62.7 N, median = 162.6 N, mean = 164.3 N, maximum = 357.9 N)—Anthill software. Figure 9: Distribution of the reliability function \(\mathrm{Fs}\)—Anthill software.