## 2 Materials and Methods
### Geometrical Core Concepts
The values of the time constants (T\({}_{1\rm u}\)) and (T\({}_{2\rm u}\)) (presented between the initial data) were appropriately determined in order for the (u\({}_{0}\)(t)) signal to respect the settling time equal to (t\({}_{\rm fu}\)). The validity of the data presented in Table 1 was documented in the orthodontic literature. Since the error between the data in the literature and the data presented in Table 1 (obtained through simulation) is insignificant, the validities, both of Equation (6) and of the structure parameters (T\({}_{1\rm u}\)) and (T\({}_{2\rm u}\)) values, are proven. Considering a treatment procedure in which the apex movement \(y_{\beta}=0.5\) mm is obtained, the simulation results obtained through the simulation of the model from Equation (5), using the scaled value K\({}_{\rm y}=0.1\) mm/grf of the proportionality constant, are presented in Table 2 (under the numerical form) and in Figure 6 (under graphical form). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & \\ \hline
**t [Weeks]** & \(y_{\rm u}^{{}^{\prime}}\)(\(s\)=\(s_{0}\)=0)[mm] & \(s\)=\(s\)=\(s\)= [mm] & \(y_{\rm u}\) (\(s\)=\(s\)= \(s\)= 0) [mm] & \(s\)=12 [mm] & \(s\)=16 [mm] & \(s\)=20 [mm] & \(y_{\rm f}\)(\(s\)=24) [mm] \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
0.5 & 0.24 & 0.22 & 0.2 & 0.18 & 0.16 & 0.14 & 0.12 \\ \hline
1 & 0.49 & 0.45 & 0.41 & 0.37 & 0.33 & 0.29 & 0.249 \\ \hline
1.5 & 0.669 & 0.61 & 0.55 & 0.5 & 0.44 & 0.39 & 0.33 \\ \hline
2 & 0.78 & 0.71 & 0.65 & 0.58 & 0.52 & 0.45 & 0.39 \\ \hline
2.5 & 0.85 & 0.78 & 0.71 & 0.64 & 0.57 & 0.499 & 0.42 \\ \hline
3 & 0.9 & 0.83 & 0.75 & 0.679 & 0.6 & 0.52 & 0.45 \\ \hline
3.5 & 0.93 & 0.86 & 0.78 & 0.7 & 0.62 & 0.54 & 0.469 \\ \hline
4 & 0.96 & 0.88 & 0.8 & 0.72 & 0.64 & 0.56 & 0.48 \\ \hline
4.5 & 0.97 & 0.89 & 0.81 & 0.73 & 0.649 & 0.56 & 0.48 \\ \hline
5 & 0.98 & 0.9 & 0.819 & 0.73 & 0.655 & 0.57 & 0.49 \\ \hline
5.5 & 0.989 & 0.907 & 0.824 & 0.742 & 0.699 & 0.57 & 0.494 \\ \hline
6 & 0.993 & 0.91 & 0.828 & 0.745 & 0.662 & 0.579 & 0.496 \\ \hline
6.5 & 0.996 & 0.913 & 0.83 & 0.747 & 0.664 & 0.581 & 0.498 \\ \hline
7 & 0.997 & 0.914 & 0.831 & 0.748 & 0.665 & 0.582 & 0.4989 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of the biomedical process model simulation in relation to both independent variables (t) and (s). Figure 5: The evolution in relation to time (t) of the \(u_{0}\)(t) input signal. The interpretation of the simulation results presented in Tables 1 and 2 and in Figures 5 and 6, respectively, for the scaled value \(K_{\rm y}=0.1\) mm/grf of the model proportionality coefficient, can be synthetized as follows:  
1. The orthodontic activation cycle duration is \(t_{\rm f}^{\prime}=7\) weeks (the period in which the signals from Figure 6 accurately reach the new steady state regime implied by the application of the \(u_{0}(t)\) input signal), but from Figure 6 and Table 2, it results that only after four weeks, the output signal \(y_{00}^{\prime}(t,s)\) (for all values of (s) independent variable) reaches with more than 96% from the total increase due to the current activation;
2. The evolution of the force delivered by the elastomeric chain (\(u_{0}(t)\)) [grams force] is an exponentially decreasing one, from the initial value \(K_{\rm u}=100\) [grf] to the residual plastic strength \(\widetilde{u}_{\rm f}=10\) [grf];
3. The incisal edge of the tooth \(y_{\alpha}^{\prime}(t,s_{0})\) [mm] follows an exponentially increasing movement evolution, which superposes, in steady state regime, to the \(y_{\alpha}^{\prime}=1\) [mm] asymptote;
4. All the evolutions of the output signal \(y_{00}^{\prime}(t,s)\) presented in Figure 6 follow increasing exponential evolutions, their dynamics being given by (T\({}_{1}\)) and (T\({}_{2}\)) time constants;
5.