## 2 Materials and Methods  
Five endodontic NiTi instruments, two composed of conventional alloys and three made of heat-treated alloys were considered [10; 11]. Instruments were digitalized in a micro-CT scanner system (BrukermicroCT, Kontich, Belgium) and irradiated with an x-ray beam at 90 kV and the camera resolution was set at 9 \(\upmu\)m\(\cdot\)pixel\({}^{-1}\). Scans were acquired by revolving the specimen at a 1\({}^{\circ}\) rotation step, thus, obtaining 360 projection images. The 3D model of the instruments was reconstructed by assembling (approximately) 700 cross sectional images. Briefly, image datasets were processed via ScanIP(r) (3.2, Simpleware Ltd., Exeter, UK). Image segmentation and filtering procedures were used and a stereolithography model (STL) of the instruments was created. The STL files were exported and a dedicated software (SpaceClaim 2019, ANSYS Company, Canonsburg, PA, USA) was employed (Figure 1). Then, the volume was checked, the gaps between triangular faces were fixed and small faces were merged into a single face. In sequence, 20 perpendicular planes were created along the endodontic instrument length (2 mm each) to split the STL file in segments. These segments were filled with a skin surface and joined to form one volumetric solid and exported in STEP (Standard for the Exchange of Product model data) format. These procedures were repeated for all the endodontic instruments. Using modeling software (Rhinoceros 6.0, McNeel North America, Seattle, WA, USA), an artificial canal was created. Thus, a three-dimensional rectangular solid (32 \(\times\) 20 \(\times\) 6 mm) was created containing a simulated canal with an angle of curvature of 90\({}^{\circ}\), a radius of 5 mm, with the center of the curvature 7 mm from the tip of the instrument. For each instrument, one specific canal was created presenting the same volume, conicity and shape than the instrument. Five sets of canals and instruments were considered, and a computer assisted engineering software (ANSYS 17.2 Company, Commonwealth of Pennsylvania, USA) was used. A mesh convergence test (10%) was carried out to assist with determining the number of nodes and elements for each set [35]. The technical features of the analyzed models are summarized in Table 1. The frictional contact (0.2) between the instrument and canal was assumed, according to the motion of the file into the root canal. The constraints were applied for nodal displacements at the bottom of the artificial curved canal in all directions. The volume of the instrument, reaction force in bending and elastic modulus of each instrument (according the employed NiTi alloy) are summarized in Table 2.  
Figure 1: Stereolithography model (STL) files obtained with MicroCT of analyzed instruments. F360 and F6 Sky Taper are made of conventional NiTi alloy. Hyflex CM, Protaper Next and One Curve are constituted of heat treated NiTi alloys. Mechanical properties (elastic modulus--E, Poisson's ratio--v) of two different dentine types were used: a sound dentine (E = 18 GPa, \(\
u\) = 0.3) and a highly mineralized peritubular dentine (E = 42 GPa, \(\
u\) = 0.3) [14; 35; 36]. For a conventional NiTi alloy, a value of 43.9 GPa was assumed for the elastic modulus, whereas E = 24.6 GPa was considered for the heat treated instruments [37]. Two different loading conditions were considered for each instrument at the same time (Figures 2 and 3). The first condition representing the bending moment during the insertion of the endodontic instrument inside of the curved canal is shown in Figure 2.  
This reaction force during the instrument bending was dependent upon the shape (cross section), elastic modulus and displacement of the instrument. For this reason, this reaction force was assessed using a static structural analysis for each instrument, simulating the bending condition during insertion into the canal. The reaction force reported in N was recorded to be used in the second part of the simulation, with the instrument working in rotation inside the dentine root canal.