## 4 Discussion  
In orthodontic treatment, decision of loading condition may primarily depend on the clinical experience or subjective judgment of orthodontists [22]. Biomechanical evidence for various loading conditions would provide indispensable information to establish a treatment plan; however, it is not justified from an ethical aspect to repeat clinical trials for examining numerous loading conditions. Among various attempts to overcome this difficulty, FE analysis has been commonly utilized. FE analysis is a numerical method that calculates the stresses and determines the mechanical behavior of complex structures [26]. Recently, the predictability of FE analysis in orthodontic treatment has been validated by evaluation with clinical feedbacks [27]. Thus, we evaluated the loading condition for maxillary molar intrusion with MMEA through FE analysis. In malocclusion with an anterior open bite, position of the first molar is strongly associated with vertical occlusal relationship [28]. Therefore, we focused on the maxillary first molar as a target to evaluate the loading condition during intrusion. Regarding the intrusive load, the magnitude of the load varies depending on clinical procedure of the orthodontist and the treatment objective. Even in previous FE studies performing molar intrusion, a large variety of the load was adopted [25; 29; 30]. Hence, the biomechanical evidence with the smallest unit of load was necessary to compare with other loads, and we set 1.0 N as an intrusive load. Concerning the modeling procedure, we utilized simplified average model. Creating the simplified average model does not need much labor and time that are required in modeling based on CT images [31]. In addition, the simplified average model can generalize and evaluate the loading condition without considering numerous varieties of the teeth owing to tooth wear [28] or occlusal trauma [32] that were caused by anterior open bite. However, an estimation error due to simplified modeling should be considered. To evaluate the estimation error, the analysis results of molar intrusion without tipping movement were compared with a previous FE study using CT images. [PERSON] et al. constructed FE models of posterior dental segments based on CT images and an intrusive load of 3.0 N from buccal side was applied [25]. Regarding the ratio of the vertical displacement to the stress on the root surface of the first molar, the previous study demonstrated the ratio in a range from 1.3 to 2.1 \(\upmu\)m/MPa. On the other hand, our analysis with a simplified FE model demonstrated the ratio in a range from 1.5 to 2.0 \(\upmu\)m/MPa. Although these analysis  
Figure 6: The result of maximum von Mises stress on PDL-P and PDL-B. The stress values for various counter loads were expressed by line graph. methods were different in terms of the number of teeth, the magnitude of the load, and modeling procedure, these ratios were almost in same ranges. It indicated that our analysis method using simplified model could produce the result almost in the same range as the FE method using CT images. In displacement results of the crown and root, horizontal displacement had no significant difference in hook positions, on the other hand, vertical displacement showed significant difference between _Co_ and the others (Table 3). Considering with the result of the line graph (Figure 4), in coronal hook position, the amount of intrusion was significantly smaller than the other hook positions. The reason for these results is explained as follows: owing to the sloped shape of the palatal mucosa, the increment from _Co_ to _Ap_ was 2.4 mm in the horizontal direction, whereas 12.0 mm was shown in vertical direction (Figure 2C). Therefore, the vector of the intrusive load had much larger difference in vertical component among three hooks than that of horizontal component. In addition, the vector direction in _Mi_ and _Ap_ were similar compared with _Co_.