## 2 The Orthodontic Mathematical Model
### Instantaneous Behavior of the PDL  
The extracted tooth geometry is provided as a set of faces \(\mathcal{F}\) (commonly in STL file format). The PDL is modeled as a set of springs, each of which is connected to the center of a tooth face, while its other end is anchored to the surface of the alveolar bone. The system energy relaxes by changing the tooth position and orientation towards equilibrium under the orthodontic force and PDL stress. Therefore, to calculate the PDL springs' elastic energy, one should be provided with the tooth shape. Each spring \(i\in I\), positioned on the \(i^{th}\) face \(F_{i}\in\mathcal{F}\), is given two mechanical coefficients: a linear stiffness coefficient \(k_{\ell_{i}}\) and a shear stiffness \(k_{S_{i}}\); both are normalized to the relative area \(A_{i}\) they occupy. The latter is introduced to define the relaxed spring configuration as normal to its associated face. Following the conclusions of [9], one sets the \(i^{th}\) spring to be normal to the \(i^{th}\) face area. Recalling that the PDL is assumed to be isotropic tissue having a Poisson ratio of \(\
u=0.45\) (Table 3), the isotropic assumption also implies the connection \(G=\mathrm{E}/(2+2\
u)\)[44], from which one can extract a connection between the \(k_{S_{i}}\) and \(k_{\ell_{i}}\) values. Moreover, as the PDL tissue is thin (\(\sim 0.3\) mm), a single layer of springs will suffice to model it, as depicted in Figure 2. As expected, the shear strain in all our simulation studies did not exceed 4\({}^{o}\), so the small-angle assumption holds. The orthodontic appliances are also modeled as springs (as in most cases, they are indeed springs). To account for the six-dimensional tooth movements (three-dimensional translations and three-dimensional rotations), all calculations were conducted in the configuration space. To do so, define the elastic energy of a tooth configuration \(c\in\mathcal{C}\) as the sum of all (PDL) springs' energies and those of the external orthodontic springs:  
\[U^{tot}=\sum U_{i}^{PDL}+\sum U_{j}^{ext} \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline
**Citation** & **PDL Young Modulus [MPa]** & **PDL Poisson Ratio** \\ \hline
[2, 9– 11, 13– 15] & 0.66–0.69 & 0.45–0.49 \\ \hline
[12] & 1.18 & 0.45 \\ \hline
[39] & 44 & 0.49 \\ \hline
[42] & - & 0.45 \\ \hline
[8] & 0.5 & - \\ \hline
[35] & 0.71 & 0.4 \\ \hline
[3] & 0.17 & 0.4 \\ \hline
[43] & 0.1–0.3 (shear modulus) & - \\ \hline \end{tabular}
\end{table} 
Table 3: Acceptable PDL elasticity modulus and Poisson ratio values. \begin{table}
\begin{tabular}{c c c} \hline
**Citation** & \(\epsilon_{M}\)**or \(\sigma_{M}\)** & \(\epsilon_{m}\) \\ \hline
[17] & \(3000\mathrm{\upmu_{strain}}\) & 1000–1500\(\mathrm{\upmu_{strain}}\) \\ \hline
[7] & \(3000\mathrm{\upmu_{strain}}\) & \(1500\mathrm{\upmu_{strain}}\) \\ \hline
[41] & - & \(1500\mathrm{\upmu_{strain}}\) \\ \hline
[12] & 16 [KPa] & - \\ \hline \end{tabular}
\end{table} 
Table 2: Acceptable PDL range of strains or stresses in OTM. The maximal strains or stresses in the remodeling stage are denoted by \(\epsilon_{M}\) or \(\sigma_{M}\). Beyond these values, significant pathological biological processes may occur. These may require adjustments in the model, which is beyond the scope of this research. The minimal strain \(\epsilon_{m}\) required for the remodeling process to take place is considered in the proposed model. where the energy of the \(i^{th}\) PDL spring is given by:  
\[U_{i}^{PDL}=\frac{k_{\ell_{i}}\Delta\ell_{i}^{2}}{2}+\int_{0}^{\theta_{i}}M_{i}d\theta \tag{2}\]  
The tensile stiffness coefficient \(k_{\ell_{i}}\) is \(A_{i}E/l_{0}\). In the second term, the integrand \(M\) is the torque given by \(M=Fl\cos\theta\). Since \(F_{i}=\tau_{i}A_{i}\) and \(\tau_{i}=Gtan\theta_{i}\), where \(G\) is the shear modulus of the PDL, this implies that \(M_{i}=GA_{i}l_{i}\sin\theta_{i}\). The change in length of the linear spring is marked by \(\Delta\ell_{i}=\ell-\ell_{0}\), and \(\theta_{i}\) denotes the angle between the spring and the tooth face normal (see Figure 2).