## 4 Discussion
### Congruency/Fitting of the Artificial Root Canals to Instruments
Additionally, the angle of curvature decreases when testing instruments in larger canals. When determining true curvatures in the present study using [PERSON]'s method, the angles were 41.2\({}^{\circ}\), 40.6\({}^{\circ}\), 39.2\({}^{\circ}\), and 36.0\({}^{\circ}\), respectively (Figure 7). The decreasing length of the fractured fragments with increasing diameter of the artificial canals provides further evidence of the changing stress trajectories that result in maximum stress closer to the center of curvature. Using matching artificial canals did not cause any torsional load (Figure 6). Fractographic examination of the fractured surfaces revealed failure due to cyclic fatigue. Fracture schemes were in accordance with those reported in previous studies [44; 45]. Fatigue failure is characterized by macroscopical 'brittleness'. Whereas, in lateral views, some minor plastic deformation may be visible on both sides of the fractures [19], the cross-sectional views usually show a crack propagation perpendicular to the direction of maximum stress, successively reducing the intact mass of the cross-section (Figure 6A-D). Finally, the fracture results in catastrophic failure due to a 'final' load cycle that induces a fast and unstable growth of the crack [37]. Hence, the cross-sectional analysis is crucial to determine the fracture mode (fatigue, torsional, or mixed mode) [19; 38]. Nonetheless, fracture of NiTi endodontic instruments under clinical conditions appears to be mainly caused by a single overload event (causing ductile fracture) rather than solely by alloy fatigue due to the multiple loading cycles [46]. Overload may be attributed to many factors (i.e., atrogenic factors) caused by the active preparation of the root canal. Hence, the longevity of endodontic instruments is highly variable, as the instruments are subjected to multiple dynamic forces due to the cutting/shaping action and axial load in the presence of intracanal debris combined with engagement into the dentinal walls during root canal treatment. Therefore, despite the knowledge of fatigue resistance of NiTi instruments from cyclic fatigue testing [22], direct extrapolation of laboratory parameters to the clinical situation should be done with caution. However, all these considerations suggest that, in further studies, it is reasonable to test instruments with other sizes, tapers, and metallurgies in artificial canals with a diameter in the range of the instrument. This unique setup may help overcoming the controversial discussions about cyclic fatigue testing of endodontic instruments by guaranteeing well \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Percentage Increase in Diameter between Instrument Tip and Artificial Canal Sizes in Relation to Manufacturing Tolerances**} \\ \hline \multirow{3}{*}{**Instrument (Size \# 25)**} & **Matching Tube** & **Instrument Size +0.05** & **Instrument Size +0.10** & **Parallel Tube** \\  & **Instrument Size +0.02** & **mm** & **mm** & **1.2 mm** \\  & **\# 27/100 mm** & **\# 35/100 mm** & **\# 43/100 mm** & **\# 120/100 mm** \\ \hline \multirow{2}{*}{Lower tolerance \# 23} & +17\% & +30\% & +52\% & +522\% \\  & +8\% & +20\% & +40\% & +480\% \\ \multirow{2}{*}{Upper tolerance \# 27} & +0\% & +11\% & +30\% & +444\% \\ \hline \multirow{2}{*}{**Instrument (Size \# 15)**} & **Matching Tube** & **Instrument Size +0.05** & **Instrument Size +0.10** & **Parallel Tube** \\  & **Instrument Size +0.02** & **mm** & **mm** & **1.2 mm** \\  & **\# 17/100 mm** & **\# 25/100 mm** & **\# 35/100 mm** & **\# 120/100 mm** \\ \hline \multirow{2}{*}{Lower tolerance \# 13} & +31\% & +92\% & +269\% & +923\% \\  & +15\% & +67\% & +233\% & +800\% \\ \multirow{2}{*}{Exact size \# 15} & +0\% & +47\% & +206\% & +706\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Percentage increase in diameter between instrument tip and artificial canal sizes in relation to different instruments according to manufacturing tolerances.