## 2 The Orthodontic Mathematical Model
### Remodeling Process  
The remodeling process is modeled as the anchorage displacement of the springs. For example, the \(i^{th}\) pair of springs depicted in Figure 2, which is subjected to a strain, will eventually result in a movement of the anchorage point when the remodeling process initiates. The rate at which the repositioning of the \(i^{th}\)-associated springs takes place depends on the exerted strains \(\varepsilon_{i}=\frac{\Delta\ell_{i}}{L}\) and \(\varepsilon_{si}=tan(\theta_{i})\), yielding the stresses \(\sigma_{i}=E\varepsilon=\frac{k_{\ell_{i}}\Delta\ell_{i}}{A_{i}}\) and \(\tau_{i}=Gtan(\theta_{i})=\frac{k_{\xi_{i}}tan(\theta_{i})}{\ell_{i}A_{i}}\). Note that our model also applies when the tooth undergoes pure rotation. For clarity, recall that the remodeling stage initiates when no instantaneous movement can be applied. Resorption refers to the case where the PDL springs' length changes towards their neutral length, removing bone tissue, while formation refers to the case where the PDL springs shorten. Moreover, the remodeling process is not uniformly distributed. In every instant, some PDL springs may undergo remodeling, while others will not. Explicitly, remodeling occurs under the following stress criterion, which is directly derived from the strains criterion:  
\[\sigma_{m}<\sigma_{i}<\sigma_{M} \tag{5}\]  
For values above \(\sigma_{M}\), the behavior of the PDL significantly differs, and thus, it is beyond the scope of this study. \(\sigma_{m}\) is the corresponding stress that realizes the minimal \(1500\mu_{strain}\) requirement (Table 1). Equation (6) describes the repositioning \(\Delta s_{i}\) of the spring anchorage in the alveolar bone due to the tensile stress.  
\[\Delta s_{i}=\Delta\ell_{i}k_{\ell_{i}}\beta_{\ell}\tilde{s}_{i}+tan(\theta_ {i})\beta_{\frac{G(\ell_{0}\hat{n}_{i}-\ell_{i}\hat{s}_{i})}{\ell_{i}\parallel \ell_{0}\hat{n}_{i}-\ell_{i}\hat{s}_{i}\parallel}} \tag{6}\]  
This encapsulates the fact that the spring anchoring point \(s_{i}\) aims to align back (Figure 2). \(k_{\ell_{i}}\) and \(k_{\xi_{i}}\) are the normal and shear coefficients of spring \(i\), and \(\theta_{i}\) is the angle between the spring and the normal to \(i\)-th facet. The force-to-distance conversion coefficient is denoted \(\beta_{\ell}\) and is set to \(1\) for formation and to \(4\) for resorption, reflecting the acceptable resorption and formation rates' ratio. \(\beta_{s}\) is \(1\) since tangent movement is symmetric in all directions in the isotropic assumption. Equation (6) implies that the anchorage movement rate is proportional to the stress applied upon the spring.  
Apart from straining the PDL fibers in their longitudinal axes, the PDL has incompressible qualities that should be accounted for in the model. The magnitude of the tangent stress is calculated using the properties of the shear modulus for isotropic materials:  
\[\frac{E}{2(1+\
u)}=G=\frac{F_{s}}{A}cot(\theta_{i}) \tag{7}\]  
Here, \(E\), \(G\) and \(\
u\) are the PDL modulus, shear modulus and Poisson ratio, respectively, and recall [PERSON]'s law for shear stress \(\tau=G\tan\theta\). Therefore, the new position of the spring's anchorage is the accumulation of the movements due to strains in the PDL fibers' directions and shear strains.