## 3 Results
### CR Position and the M/F of teeth with Different Penetration Depths  
The virtual force value method was carried out by applying a constant force and a gradually increasing moment. The tooth first tipped distally, then translated, and then gradually tipped mesially. The value of the moment in the critical increment was obtained when the tooth was purely translated without tipping. Then, the CR positions were evaluated based on Equation (2). The CR positions for teeth with different penetration depths (\(x\)) are presented in Figure 4.  
As the sinus floor sinks, the CR tends to move toward the cervical margin. The CR for an ordinary periodontal condition (\(x\) = 0 mm) was located approximately 69.4%, 63.4%, and 63.8% of the tooth length from crown to root apex for the canine, premolar, and molar, respectively. With the increase in the penetration depth from 0 mm to 9 mm, the location of the CR dropped linearly to 53.9%, 43.4%, and 39.9%, respectively. In the posterior teeth, the position of the impedance center even dropped to below half the length of the tooth. The CR of the molar and premolar dropped faster than that of the canine. The relationship between the CR position and penetration depth for the three types of teeth were described well using a linear model, as follows:  
\[y/\mathrm{L}=(\mathrm{m}x+\mathrm{n}) \tag{6}\]  
where \(y\) is the distance between CR and tooth cusp, L is the length of the tooth, \(x\) is the penetration depth of the root, the slop m and intercept n are the parameters. The values of the parameters for each tooth are shown in Table 2.  
Figure 4b and Table 2 show the simulation results of the CR positions against the height of the MSF. The blue lines in the graphs are the fitting curves for each tooth. The slope (m) is the descending length of the CR for every millimeter of root penetration. The descending speed of the posterior teeth was much higher than that of the canines. Anatomical structure changes have a greater impact on posterior teeth. The coefficient of determination (r-squared) values was 0.9965 for the canine, 0.9961 for the first premolar, and 0.9796 for the molar. The high r-squared fitting values indicate a good fit for the linear models. Moreover, the orthodontic M/F for bodily movement should be equal to the distance between the bracket and the CR (H). By substituting Equation (6) into Equation (2), the relation between the M/F and the penetration depth \(x\) could be expressed:  
\[\frac{M}{F}=(\mathrm{m}x+\mathrm{n})\mathrm{L}-\mathrm{h} \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Tooth** & **m** & **n** & **r-squared** \\ \hline The Canine & \(-\)0.017 & 0.696 & 0.9965 \\ The Premolar & \(-\)0.023 & 0.655 & 0.9961 \\ The Molar & \(-\)0.024 & 0.610 & 0.9796 \\ \hline \end{tabular} r-squared is the coefficient of determination.  
\end{table} 
Table 2: Parameters of the linear model between the CP position and penetration depth for different teeth. Figure 4: Changes of CR positions of three teeth against the penetration depths. (**a**) Schematic of the CR positions for different depths of root penetration. The blue horizontal lines present the different heights of MSFs. Colored circles with black strokes indicate the changes of CR positions because of the MSF sinking. Different transparency of the graphics represents different depths and corresponding CR positions. (**b**) The ratio of \(y\) to L against the penetration depth based on the simulation results. Experimental data and the linear model fitting. where h is the bracket height, the distance between the cusp and the bracket, which is a constant during the treatment. The CR position and the M/F in bodily movement for each type of the teeth were presented in Table 3.  
On the basis of the simulation results, an M/F of approximately 11.27 mm, 8.00 mm, and 8.40 mm can generate the bodily movement of the canine, the first premolar, and the molar with a normal periodontal anatomy, respectively. As the depth of root penetration increased, the M/F required to maintain tooth translation decreased rapidly.