## 4 Discussion
In other words, molar intrusion with MMEA can be expected similar tooth movement to bucco-palatal miniscrews, and the risk of root resorption can be reduced although the amount of vertical displacement is smaller. Regarding the load direction, MMEA can determine the load direction freely by adjusting the shape of the arm. In addition, to perform maxillary molar intrusion in both sides, MMEA requires only two miniscrews, whereas intrusion with bucco-palatal miniscrews requires four miniscrews. It contributes to reduction in invasive and economic burdens on the patient. From these perspectives, we consider that MMEA would have some advantages for molar intrusion. In orthodontic treatment, root resorption is a frequent consequence of tooth movement, especially intrusion and tipping movement [36; 37]. As shown in Figure 5, the optimal counter load brought no tipping movement, and the stress value on the PDL became lower. Although it remains a matter of speculation because stress distribution would not be accurate owing to the simplified root shape, the risk of root resorption may be reduced under optimal load conditions [38]. Regarding the maximum von Mises stress values on the PDL, although ANCOVA cannot be performed since the results draw curves, the results appeared to be divided into two groups, i.e., \(Co\) and the others (Figure 6). This characteristic was similar to the result of vertical displacement. [PERSON] et al. [39] described that intrusion caused larger stress on PDL than tipping movement when the load magnitude was same. Thus, the stress on the PDL might be affected by vertical displacement. A possible explanation of a parabolic-like change on PDL-B is that the palatal tipping movement disappeared, and buccal tipping movement occurred instead as the counter load increased. Since the stress changed along with the magnitude of tipping movement [39], it showed a parabolic-like change. In our study, there were at least four limitations. The first is related to the mechanical property. A linear elastic analysis was performed for simplicity; however, the bone tissues and PDL were anisotropic and inhomogeneous. In fact, the mechanical properties of the PDL have not been fully clarified. PDL possesses anisotropic and non-linear characteristics owing to the orientation of collagen fibers [40]. However, tooth movement can be approximated to clinical results, even if the PDL was assumed to be a linear, isotropic material [41]. Furthermore, the difference between the linear and non-linear values can be ignored in the FE analysis for tooth intrusion within a load of 2.5 N [42]. Therefore, the PDL was assumed to be a linear, isotropic, and homogeneous material in the study. Secondly, due to the simplified geometries of the tooth, stress distribution of PDL could not be evaluated accurately. Stress distribution determines biological response of PDL that initiates tooth movement or PDL necrosis [43]. Hence, in this study, the biological response to stress was also uncertain. Thirdly, our FE model was simplified compared with clinical condition. Because we applied a single tooth model, transmission of intrusive load to adjacent teeth is not calculated. Therefore, the tipping movement and the required counter load in our model are larger than in a non-simplified model. However, if this model is interpreted as the condition with maximum consideration for possibility of tipping movement, our result can potentially provide the safest clinical index. The final imitation was that the counter load was considered in only one direction, but the load could be applied in various directions in clinical practice. Various directions of the counter load should be considered in further studies. As an outlook for the FE study in orthodontics, we consider that it is necessary to clarify the relationship between the calculated stress and the biological response such as alveolar bone remodeling and tooth movement in detail.