## 2 Materials and Methods  
3D models were constructed from tomography images. We used the cone-beam computed tomography (CBCT) image of a patient with low MSF (Figure 1) from the Affiliation 2. The tomography was scanned using a NewTom 5G (version FP, Imola, Bologna, Italia) under the following settings: mode (81.5 mGy-cm, 110 KVP, and 5 mA); voxel size, 0.25 mm; and slice thickness, 0.25 mm. The scan was originally taken for diagnosis before orthodontic treatment. Three types of teeth were analyzed, including a canine, a double-rooted premolar, and a molar with three roots. The lengths of the teeth (i.e., the long axis) were 23.41 mm, 19.70 mm, and 19.43 mm, respectively. Their shapes and si  
Figure 1: Tomography images of a patient with a low MSF. Penetration depth is the distance between the root apex and the MSF; MS is the maxillary sinus; MSF is the maxillary sinus floor. The tomography images of teeth, PDL, and the alveolar bone were segmented by defining the thresholds of the greyscale values in Mimics (version 19.0; Materialise, Leuven, Belgium). The model comprised teeth, a 0.2 mm PDL, and the manually adjusted alveolar bone. The surface models constructed in Mimics were imported into Geomagic Wrap (version 2017; Geomagic, Raleigh, NC, USA) for constructing the geometry entities. After the laborious work of model processing and repair, which included removing model noise and repairing small holes and smooth and sharp edges, the geometric models were completed, as shown in Figure 3.  
To analyze the influence of the MS anatomy, the penetration depth was individually adjusted for each subject by lowering the MSF. The cavity above the alveolar bone is the MS and there is no material filling in the models. The MS floor is the cortical bone between the cancellous bone and the MS. The roots of teeth penetrated into MS to different depths because of the various heights of the MSF. Penetration depth (\(x\)) is the portion of the root  
Figure 3: Finite element models with 10 different penetration depths for each of the three teeth. (**a**) Finite element models of three types of teeth. (**b**) The manually modified alveolar bone models for the canine. (**c**) The model of the canine and its alveolar bone with 5 mm depth of penetration. FA is the facial-axis point on the crown. (Boundary condition, fixed). Figure 2: Schematic of critical dimension variables (**a**) and the models of the canine (**b**), the premolar (**c**), and the molar (**d**). CR: the center of resistance of the tooth; FA: the facial-axis point on the crown; L: the length of the teeth; h: the bracket height, the distance between FA and the tooth cusp or edge; H: the distance between CR and FA; \(x\): penetration depth of the root; \(y\): the indicator of CR position, the distance from CR to the tooth cusp or edge. penetrating into the MS. It is difficult to obtain real cases involving various depths in clinical practice, which is the advantage of FEA. Various distances between root apices and the MSF were established by modifying the alveolar bone manually (Figure 3b). In the models of different subjects, the depths varied from 0 to 9 mm incrementing 1 mm. Therefore, we had 10 subjects for each of the three types of teeth as shown in Figure 3. The penetration depth was the only difference between subjects with the same tooth. The geometric entities were imported into Abaqus (version 2018; Dassault System, Concord, MA, USA) for FEA. Orthodontists generally stick the bracket to the crown at the facial-axis (FA) point [16]. Therefore, the orthodontic force in this study was applied on the FA point of the crown. The boundary condition was defined by fixing the top and lateral surface of the bone to reflect the natural intraoral environment. The bodily tooth movement in the mesiodistal direction was simulated in this study.