## 3 Results  
The FEA results were analyzed for each configuration by comparing the moment along the \(z\)-axis and the resulting moment-to-force-ratio (M:F) delivered to the tooth on the XY plane. The M:F values describe the quality of the force system [35], while the absolute values of M and F define the amount of orthodontic movement. Figure 9 reports the displacement occurring for each simulation. Table 2 shows how the combination of pressure point and attachment produced the maximum tooth displacement (0.085 mm), which is justified by the higher forces and moments measured at the tooth CRES compared with the other configurations. Table 2 also shows high force and moment values for the standard aligner; however, the loads are not in the desired directions, as demonstrated by Figure 8, in which one can notice that with the standard aligner the resulting movement is approximately a mesiodistal tipping of the maxillary canine. Figure 10 shows the patient's documentation at the end of the orthodontic treatment carried out by using AirNivol (AirNivol S.p.A., 56023, Cascina, (PI), Italy) [36] aligners. Table 3 shows the cephalometric data of the patient at the end of treatment. The stability of the obtained results were analyzed after four years (Figure 11). Figure 8: Intraoral photographs at the beginning of the aligner treatment. \begin{table}
\begin{tabular}{c c c c} \hline  & **Standard Aligner** & \
\begin{tabular}{c} **Double** \\ **Attachment** \\ \end{tabular} &
\begin{tabular}{c} **Attachment and** \\ **Activation Point** \\ \end{tabular} \\ \hline Maximum displacement (mm) & \(0.075\) & \(0.058\) & \(0.085\) \\ \hline Fx (N) & \(-0.36\) & \(-0.26\) & \(-0.64\) \\ \hline Fy (N) & \(-0.93\) & \(-1.53\) & \(-3.62\) \\ \hline Fz (N) & \(-1.5\) & \(-1.09\) & \(-3.75\) \\ \hline Mx (Nmm) & \(4.23\) & \(0.79\) & \(-4.3\) \\ \hline My (Nmm) & \(1.1\) & \(-1.23\) & \(11.5\) \\ \hline Mz (Nmm) & \(7.6\) & \(5.77\) & \(12.3\) \\ \hline Mz/Fx (mm) & \(-19.5\) & \(-22.2\) & \(-19.21\) \\ \hline Mz/Fy (mm) & \(-8.2\) & \(-3.8\) & \(-3.4\) \\ \hline My/Fx (mm) & \(6.7\) & \(4.73\) & \(-17.8\) \\ \hline My/Fz (mm) & \(5\) & \(1.13\) & \(-3.1\) \\ \hline Mx/Fy (mm) & \(-11.7\) & \(-0.51\) & \(1.2\) \\ \hline Mx/Fz (mm) & \(-3.7\) & \(-0.72\) & \(1.2\) \\ \hline \end{tabular}
\end{table} 
Table 2: Maximum displacement and loads delivered to the tooth by the different aligners. The force systems are measured at the tooth’s center of resistance. Figure 9: The displacement occurring for each simulation is shown by the different views. It can be seen how the activation point with attachment generated the maximum displacement (red). Moreover, the approximate center of rotation can be retrieved for each simulation as it is located in the blue region. Figure 10: Cont. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Sagittal Jaw Relationship**} \\ \hline  & **Patient Value** & **Normal Value** \\ \hline Maximillary position SNA (\({}^{\circ}\)) & 80.7 & 82 \(\pm\) 3.5 \\ Mandibular position SNB (\({}^{\circ}\)) & 81.0 & 80 \(\pm\) 2.5 \\ Sagittal jaw relation ANPg (\({}^{\circ}\)) & \(-\)0.3 & 2 \(\pm\) 2.5 \\ \hline \multicolumn{3}{c}{**Vertical jaw relationship**} \\ \hline Maximillary inclination SN’ANS-PNS (\({}^{\circ}\)) & 8.3 & 8 \(\pm\) 3 \\ Mandibular inclination SN’GoGn (\({}^{\circ}\)) & 30.1 & 33 \(\pm\) 2.5 \\ Vertical jaw relation ANS/PNS’GoGn (\({}^{\circ}\)) & 21.8 & 25 \(\pm\) 6 \\ \hline \multicolumn{3}{c}{**Dento-basal relationship**} \\ \hline Maximillary incisor inclination U1’ANS-PNS (\({}^{\circ}\)) & 114.2 & 110 \(\pm\) 6 \\ Mandibular incisor inclination L1’GoGn (\({}^{\circ}\)) & 85.7 & 94 \(\pm\) 7 \\ Mandibular incisor compensation (L1-Apo) (mm) & 2.1 & 2 \(\pm\) 2 \\ \hline \multicolumn{3}{c}{**Dental relationship**} \\ \hline Overjet (mm) & 2.8 & 3.5 \(\pm\) 2.5 \\ Overbite (mm) & 2.0 & 2.5 \(\pm\) 2.5 \\ Interincisal angle (\({}^{\circ}\)) & 138.2 & 132 \(\pm\) 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cephalometric data at the end of treatment.