## 2 Materials and Methods
### Geometrical Core Concepts
Therefore, the values of the \((\gamma_{0})\) and \((\gamma_{1})\) coefficients also result from Figure 4 and are presented in Equations (8) and (9):  
\[V=\frac{1}{2}\cdot\pi\cdot P\cdot Q\cdot r_{f}, \tag{2}\]0 \[V=\frac{1}{2}\cdot\pi\cdot P\cdot Q\cdot r_{f}, \tag{2}\]1  
having to mention the fact that \(\gamma_{0}=y^{\prime}_{\alpha}\) and \(\gamma_{1}=\frac{\gamma_{\beta}-y^{\prime}_{\alpha}}{s_{\rm f}}<0\) when we consider the case where \(y_{\beta}>0\). Going back to the \(y^{\prime}_{00}(t,s_{0})\) evolution from Figure 3, we must mention that this deformation is easily observable by the orthodontist in the oral cavity during an orthodontic activation cycle (from \(t_{0}\) to \(t_{\rm f}\)). The evolution \(y^{\prime}_{00}(t,s_{0}<s<s_{\rm f})\) can be only estimated, since the deformation \((+y_{\beta})\) of the tooth apex cannot be precisely known, due to its positioning in the alveolar bone. The median between the \(y^{\prime}_{00}(t,s_{0})\) and \(y^{\prime}_{00}(t,s_{\rm f})\) deformations corresponds to the fictional axis of the tooth for which the equation \(F_{\rm OS}(s)=\gamma_{0}+\gamma_{1}\cdot s\) was presented in Equation (7). This median slides its ends between the points \((t_{0},s_{0})\) and \((t_{0},s_{\rm f})\) and the points \((t_{\rm f},s_{0})\) and \((t_{\rm f},s_{\rm f})\) from \[V=\frac{1}{2}\cdot\pi\cdot P\cdot Q\cdot r_{f}, \tag{2}\]2 Consequently, this right line is the geometric locus of the points disposed along the entire length of the fictional axis of the tooth during the entire orthodontic activation cycle. Only the observable relative deformation \(y^{\prime}_{00}(t,s_{0})\) presented in Equation (5) is considered to be rigorous and, therefore, can be taken into consideration. General Example for Scaling \(K_{y}\) Coefficient Based on Main Initial Data  
Orthodontic teeth movement occurs only after force application. One way of applying force is using elastomeric chains, especially in situations that imply closing spaces. Usually, the force decreases in time. The literature offers multiple and various studies regarding the force degradation of these elastomeric chains. Depending on the fabrication materials, the manufacturers, and the different conditions in the oral cavity, studies show that various levels of force decrease in time. Some state that the force of the plastic chains decreases by half in only one week, compared to the memory chains which lose only 20% of their initial value in the same amount of time [23]. Others proved that the force of the elastomeric chains decreases by 20% after only 24 h [24]. An in vitro study, conducted by [PERSON] et al., demonstrated that the AO-Memory and the Ormco chains keep most of their initial force at the end of a four-week time interval [25]. [PERSON] et al. state that, in different conditions in the oral cavity, the elastomeric chains have a substantial force decay in the first seven days [26]. Another in vitro study shows that force degradation of the elastomeric chains reaches levels of 10-40% after the first 4 h; 20-50% alter 24 h, 30-60% after four weeks [27]. [PERSON] et al. showed that after four weeks, only 30-40% of the force of  
\[V=\frac{1}{2}\cdot\pi\cdot P\cdot Q\cdot r_{f}, \tag{2}\]3 The possible evolutions of the \(F_{0S}(s)\) function. the traditional orthodontic chains was retained [28]. After careful analysis of the literature, the conclusion is that every study offers different results regarding the force degradation of the orthodontic elastomeric chains during an orthodontic activation cycle-period between two force applications. In this context, further research is imperatively needed. The initial data used in the simulation example are:  
1. Tooth's parameters: \(s_{\mathrm{f}}^{\prime}=24\) mm; \(\widetilde{\mathrm{s}}=4\) mm; \(\widetilde{\mathrm{s}}^{\prime}=4\) mm; \(\mathrm{s}_{\mathrm{f}}=s_{\mathrm{f}}^{\prime}-\left(\widetilde{\mathrm{s}}+ \widetilde{\mathrm{s}}^{\prime}\right)=16\) mm
2. Duration of active force: \(\mathrm{t}_{\mathrm{fu}}=1.2\) weeks
3.