## 3 Results  
FEM models for OTM typically consist of \(>1\) M degrees of freedom (for example, [3]). Here, since our entire model was specially designed for OTM analysis, we can make do with much less. Specifically, since the actual PDL is comprised of short fibers, it suffices to introduce only two springs normal to each tooth facet, as depicted in Figure 2, to model its behavior. However, to accommodate clinical accuracy needs, a series of tests was conducted. For this purpose, an upper canine was subjected to a 300 gf (2.94 N) load with 26,608 facets, 6652 facets, 1662 facets and 830 facets calculated in the initial loading stage before the remodeling process commences (see Figure 4). The model was then tested under the same conditions where remodeling takes place until a 0.5 mm movement of the coil anchorage point was obtained (Figure 6). Our results show that, qualitatively, these choices exhibit very similar behavior. To quantify the differences, 32 experiments were conducted, where the direction of the orthodontic load was varied from distal to mesial on the vestibular plane, with an increment of \(6^{\circ}\) in each experiment. To solidify our conclusion,we followed the Bland-Altman method [48] to validate the results. The Bland-Altman method is typically used to assess the agreement between two measurements that measure the same quantity. It involves plotting the differences between the two measurements against the average of the two measurements. Next, the mean difference and the limits of agreement are calculated. The mean difference indicates the systematic bias between the two measurements, while the limits of agreement provide information about the magnitude of the random error. To implement this, a 100-bin histogram of the stress levels in each face resolution was generated. The area of each face was then normalized, such that the total area of the PDL faces was 1 in order to form an area distribution. Next, the histograms of the different resolutions were compared to that of the 26,608-facet resolution case (the maximal one). We also conducted a similar scenario with a 0.5 mm movement of the load anchorage on the tooth, where the remodeling process was ongoing (Figure 6). This experiment was also repeated 32 times for loads with the same orientations as described above. The same process was repeated for a mandibular molar tooth that also includes a concave surface (Figure 7). A load of 300 gf (2.94 N) was positioned at the mid-buccal surface of models having 29,103 facets, 6834 facets, 1708 facets and 854 facets. The model was simulated during the instantaneous movement, until the remodeling process takes place (Figure 5). The experiments of the canine and the molar tooth were repeated for a 0.5 mm anchorage movement, where the remodeling process was ongoing (see Figures 6 and 7). This experiment was also repeated 32 times. One can conclude that 6652 facets suffices for the estimated requirement of 0.86% (1.4 KPa) accuracy compared with the maximal stress values in the 26,608-facet resolution for the canine case. For the molar case, 6834 facets suffice for a 2.1% (0.7 KPa) accuracy compared to the 29,103-facet resolution in the molar case. Figure 4: Upper right canine facet number optimization as OTM initiated before the remodeling process was ongoing. The red spiral represents the external force. (**a**) 26,608 facets, (**b**) 6652 facets, (**c**) 1662 facets and (**d**) 830 facets. Note that Figures 6 and 7 demonstrate significant differences between the stress levels in the molar and canine teeth under the same load; this is due to the difference in the surface area size of the teeth. Similarly, to compare pairs of resolutions, for example, (a) and (b) in Figure 7, a Bland-Altman analysis was applied. Typical diagrams are provided in Figure 8 for the cases of the 6652- and 6834-facet resolution for the canine and molar cases, respectively. Figure 5: Facet number optimization as OTM initiated (before remodeling started) over a molar tooth. The red spiral represents the external force.