## 2 Materials and Methods
### Computing the CR Position Using the Virtual Force Value Method  
When the tooth is in pure translation under a horizontal force, the CR is the acting point through which the force passes [17]. The CR estimation should be based on the morphological variation of teeth and periodontal tissues [21].  
During orthodontics, two principal forces are applied to the tooth: the orthodontic force and the reaction force from the periodontium. They always act in two opposite directions. If the orthodontic force has a common line of action with the reaction force, the tooth generates a bodily movement, and the acting point of the force is the CR of the tooth. If the forces act on the tooth at an offset, they will generate a tipping movement. This is a more common phenomenon in clinical practice since the orthodontic force is applied at the bracket. However, an additional pure moment could move the individual force off its line of action from the bracket to the CR. Therefore, bodily movement can be produced with orthodontic force coupled with a particular moment. The relationship between the value of the moment \(M\) and the offset distance (H) from the bracket to the CR can be directly described with Equation (1) [22].  
\[M=F\times\text{H} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Young’s Modulus (MPa)** & **Poisson’s Ratio** \\ \hline Alveolar Bone & 490 & 0.30 \\ Tooth & 18,600 & 0.31 \\ PDL & Viscoelastic & 0.45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties.  
where \(F\) is the value of the force acting on the facial-axis point (FA) (or the bracket), \(M\) is the value of the additional moment offset the force \(F\), and H is the distance between CR and FA.  
Therefore, the CR position could be indicated by the distance from the CR to the tooth cusp:  
\[y=\mathrm{H}+\mathrm{h}=\frac{M}{F}+\mathrm{h} \tag{2}\]  
where \(y\) is the distance between CR and the cusp, H is the distance between CR and FA point, and h is the offset of FA from the tooth cusp.  
A virtual force value method was used in this study to evaluate the CR position of the specific tooth based on the relationship between the CR position and the M/F for bodily movement in Equation (1). The aim of the method is to calculate the offset of the CR from the cusp (\(y\)) using the FEM. The force system of the tooth in this study was applied on the FA point. The position of the FA point is shown in Figure 2.  
A constant force and a changing moment were applied to the tooth model. The force was set as 1.0 N. The moment was changed over time during the simulation. The moment values ranged from 30% to 70% the length of the tooth. The limits were set as 5.0\(\sim\)13.5 Nmm, 4.0\(\sim\)11.5 Nmm, and 4.0\(\sim\)11.0 Nmm for the canine, the premolar, and the molar teeth, respectively. For each subject, as the magnitude of the moment increased, the rotation center of the tooth gradually shifted from the root apex to the crown. The tooth progressively changed from a distal tipping to a pure translation, and then to a distal root movement. To detect the CR position of a tooth with different penetration depths, a critical frame was selected for when the tooth was pure translated along the distal direction without tipping during the process. The exact distance between the CR and the FA point could be obtained from the value of the moment in this frame, according to Equation (2). The positioning accuracy depends on the increment of the moment value in adjacent frames. In this study, the increment was set as 0.1 Nmm to provide a precision of 0.1 mm for the CR position.