## 2 Materials and Methods
### The Stress Distribution along PDL and the Compensation Coefficient of Orthodontic Force  
Orthodontic force was applied to the tooth and transmitted to the PDL, where most OTM mechanotransduction occurred. Stresses in the PDL, rather than forces applied to the crown, were considered to be the critical regulator of bone remodeling [23], which is the biological process of OTM. The hydrostatic pressure was assumed to be the indicator of the ideal force for moving the tooth [24]. Therefore, we hypothesize that similar hydrostatic stress along the PDL generates similar bone remodeling and the tooth movement. A coefficient compensation method was used in this study to evaluate the forces for a low MSF based on this hypothesis.  
It is difficult to determine the exact value of an ideal force for orthodontic tooth movement because of the individual differences in the periodontal materials and the biochemical reaction. The FEA in this study focuses on the influence of the anatomical structure of the alveolar bone. On the based of the hypothesis that a similar stress level generates similar tooth movement, we calculated the compensation coefficient of the force magnitude for the tooth penetrating into the MSF (\(x>0\) mm) so that the tooth produces a similar periodontal stress level as models with a normal periodontal anatomy (\(x=0\) mm).  
The distal translation of the tooth through the MS was simulated for each subject. A force of 1 N and the corresponding moment calculated based on the M/F obtained in Section 2.1 were applied to each subject to generate a bodily movement. Models with a normal periodontal anatomy (\(x=0\)) were the control group. Models with a tooth penetrating into the MSF (\(x>0\) mm) were the test groups.  
The hydrostatic stress distribution along the PDL of different subjects was evaluated using FEA when the tooth moved through the sinus to different penetration depths. The finite element model is composed of elements and nodes. The hydrostatic stress was calculated in each element.  
\[\sigma_{i}=\frac{1}{3}(\sigma_{11}+\sigma_{22}+\sigma_{33}) \tag{3}\]  
where \(\sigma_{11}\), \(\sigma_{22}\), and \(\sigma_{33}\) are three principle stresses; \(\sigma_{i}\) is the hydrostatic stress on the element \(i\).  
Therefore, the average stress on the PDL is defined as the sum of the hydrostatic stress at each element on the PDL surface divided by the number of elements as follows:  
\[\sigma_{\rm{ave}}=\frac{1}{N}\sum_{i=1}^{N}\sigma_{i} \tag{4}\]  
where \(\sigma_{\rm{ave}}\) is the average stress on the PDL under a force of 1 N, \(N\) is the number of the elements in the PDL model. It was assumed as an indicator to quantify the stress levels of each model.  
The average stress of the models in the control group, expressed as \(\sigma_{\rm{ave}}(0)\), is the target. The average stress of the models in the test group was expressed as \(\sigma_{\rm{ave}}(x)\). \(x\) is the penetration depth. Stress is defined as the internal force in the unit area, therefore the magnitude of orthodontic force is generally proportional to the stress. In order to produce the similar hydrostatic stress level to \(\sigma_{\rm{ave}}(0)\) along the PDL in the test group, the compensation coefficient of force magnitude can be expressed as:  
\[R(x)=\frac{\sigma_{\rm{ave}}(0)}{\sigma_{\rm{ave}}(x)} \tag{5}\]  
where \(R(x)\) is the compensation coefficient of force magnitude for the model with \(x\) mm penetration depth; \(\sigma_{\rm{ave}}(x)\) is the average stress along the PDL for the model with \(x\) mm penetration depth under 1 N force. The compensation coefficient was used to evaluate the changes in the orthodontic forces when the teeth penetrating into the MSF.