## 2 Materials and Methods
### Geometrical Core Concepts
From Figure 3, the decreasing evolution of the (\(u_{0}(t)\)) signal in relation to (t) can be observed. Additionally, both the increasing evolution of \(y^{\prime}_{00}(t,s)\) signal in relation to (t) and its decreasing evolution in relation to (s) can be remarked. The position (s0) is associated to the visible incisal edge of the tooth and (sf) is associated with the tooth apex. In Equation (6), it can be remarked that for \(t=t_{0}=0\), we obtain \(u_{0}(t)=K_{u}\), and for \(t=t_{tu}\), the plastic residual force (\(\widetilde{u}_{f}\)) is obtained. Usually, it can be considered that the time interval (\(t_{tu}\)) is equal to (or at least comparable to) the duration (\(t_{f}\)) of the orthodontic activation cycle (usually 4 weeks) time between two force applications. Moreover, (T\({}_{1u}\)) and (T\({}_{2u}\)) are the time constants of the elastomeric chain which highlight the (\(u_{0}(t)\)) force dynamics in relation to time. The exponential function (\(F_{OT}(t)\)) which appears in Equation (5) is the component which generates the exponential-damped increase of \(y^{\prime}_{00}(t,s)\) movement in relation to time, having the equation \(F_{OT}(t)=\frac{1}{T_{1}-T_{2}}\cdot e^{-\frac{t}{T_{1}}}+\frac{1}{T_{2}-T_{1} }\cdot e^{-\frac{t}{T_{2}}}\), where (T\({}_{1}\)) and (T\({}_{2}\)) are the time constants of the biomedical (orthodontic) process. In order to choose the time constants (T\({}_{1}\)) and (T\({}_{2}\)), a common variant is proposed, based on the following relations: \(\mu=\frac{t_{f}}{T_{1}+T_{2}}\), \(T_{1}=\frac{t_{f}}{\mu\cdot(1+\alpha)}\), and \(T_{2}=\lambda\cdot T_{1}\). Considering these equations, for the particular value of the coefficients \(\mu=4\) and \(\lambda=1.5\), we have obtained the following forms for the time constants: \(T_{1}=0.1\cdot t_{f}\) and \(T_{2}=0.15\cdot t_{f}\). The (\(\mu\)) and (\(\lambda\)) coefficients represent degrees of freedom in the orthodontic process, their values having the possibility of being modified, taking in consideration the physical particularities of each patient (for example, for the different tissue resistances). It can be easily remarked that for \(\mathrm{t}=\mathrm{t}_{0}=0\) and for the final duration of the orthodontic activation cycle (\(\mathrm{t}=\mathrm{t}_{\mathrm{f}}\)), the exponential function is \(\mathrm{F}_{\mathrm{0T}}(\mathrm{t}_{0})=0\). The (\(\mathrm{K}^{\prime}_{y}\)) coefficient, where \(\mathrm{K}^{\prime}_{y}\)= \(\mathrm{K}_{y}\cdot\widetilde{\mathrm{u}}_{\mathrm{f}}\) represents the steady state value of the \(y^{\prime}_{\alpha}\) response. The function \(\mathrm{F}_{\mathrm{0S}}(\mathrm{s})\) from Equation (5), having the form:  
\[F_{OS}(s)=\gamma_{0}+\gamma_{1}\cdot s \tag{7}\]  
is referring to a straight line equation, located on the tooth's fictional axis enclosed between the following points: \(\alpha=y^{\prime}_{\alpha}=y^{\prime}_{00}(\mathrm{t}_{\mathrm{f}},\mathrm{s }_{0})\) and \(\beta=\pm y_{\beta}=\pm y^{\prime}_{00}(\mathrm{t}_{\mathrm{f}},\mathrm{s}_{ \mathrm{f}})\). As can be observed in Figure 2, the deformation \((-y_{\beta})\) corresponds to a rotation movement and the deformation \((+y_{\beta})\) corresponds to a rototranslational movement. This last movement represents the closest movement to a pure translation, which would be ideal in orthodontic treatment. In Figure 4, the possible evolutions of the \(\mathrm{F}_{\mathrm{0S}}(\mathrm{s})\) function, in relation to the (s) independent variable, are presented. From Figure 4 one can observe that the ramp of the \(\mathrm{F}_{\mathrm{0S}}(\mathrm{s})\) function depends on the position of the point (\(\beta\)). Figure 3: The variation of the applied force in relation to time and the tooth movement in relation both to time (\(t\)) and to position on the fictional shaft (s), both presented in a qualitative manner [22].