## 4 Discussion  
In this study, we attempted to establish a relationship between orthodontic forces and supporting structures influenced by the MS for different types of the teeth in a patient-specific case. The CR positions and stress distributions for teeth with different root penetration depths were evaluated. A canine, a premolar with double roots, and a molar with three roots were analyzed. The relationship between the M/F and orthodontic force magnitude against the penetration depths was established by Equations (7) and (9) based on a special case. The results shown in Tables 3 and 5 reflect the influence of the MS on the orthodontic force for bodily movement. As an example, if there are three millimeters of penetration for a canine, the 2 N force should be modified by a factor of 0.86 to 1.72 N to obtain a similar stress level to that of the 2 N force applied at a tooth that does not penetrate the MSF. Moreover, an M/F of 15.15 is needed for bodily movement of the tooth. A virtual force method was promoted to evaluate the CR position in this study. The center of resistance was defined as the point where the applied force induced parallel movement. Many researchers determine the CR position of teeth using FE analysis. They generally adjust the acting point of the force to find the position of the CR [25]. The virtual \begin{table}
\begin{tabular}{c c c c c} \hline
**Tooth** & **a** & **b** & **c** & **r-squared** \\ \hline The canine & 0.05864 & 0 & 2.300 & 0.9754 \\ The premolar & 0.01931 & 0 & 1.052 & 0.9837 \\ The molar & 0.01411 & 0 & 0.571 & 0.9696 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of the quadratic model between the average stress and the penetration depth for different teeth. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Penetration Depth (mm)**} & \multicolumn{3}{c}{**Average Stresses along the PDL under 1 N**} & \multicolumn{3}{c}{**Compensation Coefficient for Orthodontic Force**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Canine**} & **Premolar** & **Molar** & **Canine** & **Premolar** & **Molar** \\ \hline
0 & 2.4023 & 0.5674 & 0.5423 & 1.00 & 1.00 & 1.00 \\
1 & 2.5042 & 0.6171 & 0.6458 & 0.98 & 0.98 & 0.98 \\
2 & 2.6297 & 0.6356 & 0.6839 & 0.91 & 0.93 & 0.91 \\
3 & 2.8029 &  
force method applies a force and a moment at a fixed FA point. We changed the value of the moment to obtain an equivalent effect of changing the acting point of the force. It is easier to change the value or the number of the force system than to move the force to different points. The distance between the CR and the FA point is accurate to 0.1 mm in this study. Moreover, decreasing the increment of the moment value was able to improve the accuracy of CR position. The relationship between the CR position and the force system for a bodily movement is shown in Equation (2). It is generally accepted that the M/F is crucial for the center of rotation of tooth movement. Small miscalculations in the M/F can move the rotation center and change the type of tooth movement [26; 27]. However, a more appropriate expression would be that the deviation of the M/F and the distance from the applied force to the CR (H) determine the movement pattern of the tooth. For example, fewer degrees of tipping were found for the incisors in the models with bone loss than the models with a normal bone height in response to the same force and moment [28]. This is the result of CR migration caused by bone loss. The distance H increased after the migration of the CR. Therefore, greater torque is required to tip the teeth to the same degree. During orthodontic treatment, the moment value is limited by the anatomy and appliance geometry. Orthodoxtists usually control the moment by means of special bending of the arch wire, thus presetting the inclination direction of the bracket groove. The results in Tables 3 and 5 may help clinicians to calculate the orthodontic force for patients with a low MSF.