## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**t [Weeks]** & **u\({}_{0}\) [grf]** & **s\({}_{c}\) [mm]** & **y\({}_{B}\) [mm]** & **M\({}_{E}\) [grf \(\times\) mm]** & **M\({}_{E}\) [mNm]** & \(\mathrm{y^{\prime\prime}_{\alpha}(s\widehat{s}^{\prime})}\) [mm]** & **L\({}_{E}\) [grf \(\times\) mm]** & **L\({}_{E}\) [fmJ]** & **\({}_{E}\)**L\({}_{d}\)[grf \(\cdot\)mm]** [\(\mu\)J]** \\ \hline
0 & 100 & - & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
0.1 & 75.77 & 19,933 & 0.0189 & 1,510,300 & 154 & 0.0191 & 1.656 & 0.169 & 1.656 \\ \hline
0.2 & 46.22 & 3538 & 0.0648 & 163,530 & 16.675 & 0.0652 & 2.811 & 0.286 & 4.467 \\ \hline
0.3 & 27.92 & 1120.2 & 0.1215 & 31,276 & 3.189 & 0.124 & 2.179 & 0.222 & 6.646 \\ \hline
0.4 & 18.41 & 488.342 & 0.1779 & 8990.4 & 0.916 & 0.1865 & 1.447 & 0.147 & 8.093 \\ \hline
0.5 & 13.83 & 273.258 & 0.228 & 3779.2 & 0.385 & 0.2485 & 0.999 & 0.101 & 9.092 \\ \hline
0.6 & 11.71 & 185.283 & 0.2707 & 2169.7 & 0.221 & 0.3081 & 0.761 & 0.077 & 9.853 \\ \hline
0.7 & 10.76 & 143.038 & 0.3074 & 1539.1 & 0.156 & 0.3648 & 0.637 & 0.065 & 10.49 \\ \hline
0.8 & 10.33 & 119.34 & 0.3392 & 1232.8 & 0.125 & 0.418 & 0.561 & 0.057 & 11.051 \\ \hline
0.9 & 10.14 & 104.363 & 0.367 & 1058.2 & 0.107 & 0.4679 & 0.51 & 0.052 & 11.561 \\ \hline
1 & 10.06 & 93.959 & 0.3914 & 945.227 & 0.0964 & 0.5146 & 0.471 & 0.048 & 12.032 \\ \hline
1.1 & 10.02 & 86.173 & 0.4125 & 863.453 & 0.088 & 0.5583 & 0.438 & 0.044 & 12.470 \\ \hline
1.2 & 10.01 & 80.155 & 0.431 & 802.351 & 0.0818 & 0.5992 & 0.409 & 0.041 & 12.879 \\ \hline
1.3 & 10.005 & 75.276 & 0.4469 & 753.136 & 76.798 & 0.6374 & 0.382 & 0.039 & 13.261 \\ \hline
1.4 & 10.002 & 71.264 & 0.4608 & 712.782 & 72.683 & 0.6734 & 0.36 & 0.036 & 13.621 \\ \hline
1.5 & 10.001 & 67.898 & 0.4727 & 679.047 & 69.243 & 0.707 & 0.336 & 0.034 & 13.957 \\ \hline
1.6 & 10 & 65.042 & 0.483 & 650.42 & 66.324 & 0.7385 & 0.315 & 0.032 & 14.272 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The simulations’ results for the biomedical process. Table 4 contains the simulations in relation to time of the following signals: the applied force (u\({}_{0}\)(t)) (which, in this application, has the same values and the same dynamics as in Figure 5); the position of the rotation center (s\({}_{\rm c}\)(t)); the movement of the tooth's section on which the force is applied (\(y_{\rm x}^{{}^{\prime\prime}}\)(t)); the movement of the tooth's apex (y\({}_{\rm\beta}\)(t)); the elastic moment (M\({}_{\rm E}\)(t)) expressed both in [grf\(\cdot\)mm] and in [mN\(\cdot\) m]; the mechanical work (L\({}_{\rm E}\)(t)) on each considered time interval expressed both in [grf\(\cdot\)mm] and in [mJ]; the cumulated (the total) mechanical work (\(\Sigma\)LE(t)) during the orthodontic treatment expressed both in [grf\(\cdot\)mm] and in [mJ]. From Table 4, the translation regime can be directly identified due to the following results: the tooth's movement has an increasing evolution in relation to time, for any value of (s) independent variable (both the apex movement (y\({}_{\rm\beta}\)(t)) and the  
part of the tooth represented through (y\({}_{\alpha}^{{}^{\prime\prime}}\)(t)) movement are increasing functions in relation to time (t)); the position of the rotation center has a decreasing evolution in relation to time (t), but, for the entire time domain, it is outside the tooth's volume (the minimum value of (s\({}_{\rm c}\)) is 45.045 mm, which is much higher than the tooth's length--27 mm). The simulations' results are presented in Table 4 only for the first four weeks of the treatment due to the same explanations as in the case of the simulations' results presented in Figure 6.  
The evolution in relation to time (t) of the elastic moment (M\({}_{\rm E}\)) (expressed in mN\(\cdot\)m), using the data from Table 4, is presented in Figure 15.  
Both from Table 4 (lines 5 and 6) and from Figure 15, the decreasing evolution of the elastic moment (M\({}_{\rm E}\)(t)) in relation to time, can be observed.