## 3 Results  
Figure 3 shows the tooth movement process under the reference condition, that is, an archive of 0.016 \(\times\) 0.022 inch, a helical loop with a height of 8 mm, and a gable bend of 20\({}^{\circ}\). Although brackets were not included in the FEM model, their dummies were drawn to indicate the space occupied by the brackets. The mean stress in the PDL is depicted on the root with color contours.  
When the spring was activated (Figure 3A), a mesial force of 2 N, an intrusive force of 0.3 N, and a moment of 9.4 N-mm were applied to the premolar. A distal force of 2 N and a moment of 5.7 N-mm were applied to the anterior teeth. The maximum equivalent stress of 770 MPa was raised in the spring, which was less than the yield stress of TMA (1010 MPa) [28]. After the activation, the teeth moved with repeated calculations. At \(N\) = 500 (Figure 3B), the canine tipped lingually 5.5\({}^{\circ}\), and its crown moved distally 2.1 mm. The central incisor tipped lingually 7.9\({}^{\circ}\). The premolar tipped mesially 2.7\({}^{\circ}\), and its crown moved mesially 0.4 mm. At \(N\) = 2700 (Figure 3C), the tipping angle of the canine was reduced to 0.1\({}^{\circ}\), at which the anterior teeth were considered to translate or move bodily. The number of repeated calculations required for the bodily movement was denoted by \(N_{b}\). The canine and the premolar moved distally 1.8 mm and mesially 0.6 mm, respectively. The sum of them, 2.4 mm, is the space closing distance, which was denoted by \(C\). At \(N\) = 4000 (Figure 3D), the canine deviated from bodily movement and tipped labially. The tipping angle of \(-\)2.6 means the canine tipped labially 2.6\({}^{\circ}\). On the labial side of the anterior tooth, the stress at the root apex changed from compression to tension, depending on the movement pattern.  
Figure 4 shows the bodily movement of the canine when the gable bend was changed to 10\({}^{\circ}\) and 30\({}^{\circ}\). As the gable bend increased, \(N_{b}\) and \(C\) decreased. The tipping angle of the premolar increased.  
Figure 5 shows the case when the loop height was changed to 10 mm and 6 mm. As the loop height decreased, \(N_{b}\) and \(C\) and the tipping angles of the premolars and central incisor decreased.  
Figure 3: Tooth movement process with the helical loop spring. The anterior teeth first tip, then become upright. At \(N_{b}\) = 2700, the canine achieves bodily movement.  
Figures 6 and 7 show the cases where the wire size was increased to 0.018 \(\times\) 0.025 inch and the helical loop was replaced with the vertical loop, respectively. In both cases, \(N_{b}\), \(C\), and the tipping angle of the central incisor decreased compared to the reference condition (Figure 3C).  
Using the number of repeated calculations required for the bodily movement, \(N_{b}\), and the achieved space closing distance, \(C\), we defined a parameter \(C/N_{b}\) that expressed the space closing speed. The relationship between \(C/N_{b}\) and \(C\) is shown in Figure 8.  
Figure 4: Effect of gable bend. When the gable bend increases, closing distance \(C\) and the number of repeated calculations \(N_{b}\) decreases.  
Figure 5: Effect of loop height. When loop height decreases, closing distance \(C\) and the number of repeated calculations \(N_{b}\) decreases.  
Figure 6: Effect of the wire size. When the wire size increases, closing distance \(C\) and the number of repeated calculations \(N_{b}\) decreases.