## 4 Discussion
As shown in the result of displacements, application of the counter load avoids palatal tipping movement during intrusion with MMEA. However, since excessive counter load causes buccal tipping contrarily, the optimum counter load must be calculated. In the results of the optimal counter load, the required load was the largest in _Co_, followed by _Mi_ and _Ap_ because the vector of the intrusive load had the largest horizontal component in _Co_ (Table 3). It indicates that heavy counter load should be applied to avoid tipping movements if the hook is placed near the cervical region owing to anatomical problems, such as palatal shape. To obtain the optimal counter load, the selection of arch wire size and required amount of expansion should be investigated. To reveal them, we performed another analysis using cantilever wire models. For the cross-sectional shape of the wire, three types of rectangular arch wires utilized in clinical practice were prepared. The height and width of the cross sections were 0.016\({}^{\prime\prime}\)\(\times\) 0.022\({}^{\prime\prime}\) (0.406 \(\times\) 0.559 mm), 0.019\({}^{\prime\prime}\)\(\times\) 0.025\({}^{\prime\prime}\) (0.483 \(\times\) 0.635 mm) and 0.021\({}^{\prime\prime}\)\(\times\) 0.025\({}^{\prime\prime}\) (0.533 \(\times\) 0.635 mm), respectively. The length of the wire was set to 6.0 mm; that is, an inter-bracket span between second premolar bracket and first molar buccal tube in maxillary arch [33]. The material was stainless steel and its mechanical properties including Young's modulus and Poisson's ratio were 200.0 GPa and 0.3 [25]. The material was assumed to be homogenous, isotropic, and linearly elastic. The model consisted of tetrahedral solid elements with a side of 0.2 mm. The elements on one end of the wire were fully constraint, and optimal counter load of each hook was applied to the vertical edge in the other end, perpendicularly to the long axis of the model (Appendix A, Figure A1). The displacement in the loading part was measured and defined as the amount of required expansion to obtain each counter load. The results were shown in Appendix A, Table A1. Since the optimal counter load is the value for intrusive load of a unit load, the result increases linearly as the intrusive load increases. If the intrusive load reaches 4.0 N, which is the largest value in previous studies [34], the amount of required expansion is calculated to be only 0.276-0.312 mm even in the 0.016\({}^{\prime\prime}\)\(\times\) 0.022\({}^{\prime\prime}\) wire that is the narrowest of the three. It indicates that the optimal counter load can be obtained with a small amount of expansion regardless of the wire size. However, this expansion load is not fully demonstrated in clinical practice when there is a large bracket/wire play, and it causes improper teeth movement [35]. Therefore, in molar intrusion with MMEA, arch wire that minimizes the bracket/wire play should be selected to achieve molar intrusion without tipping movement. In the case that 0.021\({}^{\prime\prime}\)\(\times\) 0.025\({}^{\prime\prime}\) stainless steel wire is set on 0.022\({}^{\prime\prime}\)\(\times\) 0.028\({}^{\prime\prime}\) slots, the wire should be expanded at first molar region in a ratio of 0.036-0.041 mm to intrusive load of 1.0 N.  
In order to evaluate the effectiveness of intrusion with MMEA, we carried out additional analysis. Intrusion with miniscrews in the palatal and buccal region was analyzed using the same model. In addition to the intrusive load in _Mi_, symmetrical load with the XZ plane was applied in the buccal side. Appendix B, Figure A2 shows the result of the analysis. Comparing with the result of MMEA with optimal counter load in _Mi_,molar intrusion without tipping movement was also observed, but vertical displacement and stress on PDL were approximately one and a half times larger than those of MMEA. However, in clinical practice, bucco-palatal miniscrews cannot necessarily provide the intrusive loads symmetrically owing to the alveolar bone morphology and the location of the tooth root.