# Using FEM to Assess the Effect of Orthodontic Forces on Affected Periodontium
## 3 Results  
The results obtained in the previously described clinical scenarios and translated into a mathematical model are summarized in the tables below for the lower central incisor (Table 3) and for the lower lateral incisor (Table 4). Representative iconographic elements in various scenarios of force application and lingualization of the lower central incisor (LCI) using a 1 and 3 N force are shown in Figures 2  
stress are mainly concentrated on the buccal apical third of the root and to a lesser degree in the lingual area (Figures 5 and 6). In Figure 7, we observe a severe lingualization of the LCI, the stress being concentrated only in the apical vestibular area. The LLI has similar patterns of stress distribution and movement to the LCI and, as the periodontal tissue loss aggravates, so does the lingualization of the teeth. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Applied Force**} & \multicolumn{3}{c}{**Extent of Periodontal Damage**} \\ \cline{2-4}  & **No HBL** & **33\% HBL** & **66\% HBL** \\ \hline \multirow{4}{*}{0.25 N} & \(\sigma\) ech = 1.03 MPa & \(\sigma\) ech = 1.4 MPa & \(\sigma\) ech = 7.67 MPa \\ \cline{2-4}  & \(\sigma\) c = 0.455 MPa & \(\sigma\) c = 0.697 MPa & \(\sigma\) c = 3.79 MPa \\ \cline{2-4}  & f = 0.065 mm & f = 0.198 mm & f = 2.35 mm \\ \hline \multirow{4}{*}{1 N} & \(\sigma\) ech = 4.16 MPa & \(\sigma\) ech = 5.59 MPa & \(\sigma\) ech = 30.7 MPa \\ \cline{2-4}  & \(\sigma\) c = 1.86 MPa & \(\sigma\) c = 2.79 MPa & \(\sigma\) c = 15.2 MPa \\ \cline{2-4}  & f = 0.26 mm & f = 0.794 mm & f = 9.41 mm \\ \hline \multirow{4}{*}{3 N} & \(\sigma\) ech = 12.3 MPa & \(\sigma\) ech = 16.8 MPa & \(\sigma\) ech = 92.1 MPa \\ \cline{2-4}  & \(\sigma\) c = 5.39 MPa & \(\sigma\) c = 8.37 MPa & \(\sigma\) c = 45.5 MPa \\ \cline{2-4}  & f = 0.782 mm & f = 2.38 mm & f = 28.3 mm \\ \hline \multirow{4}{*}{5 N} & \(\sigma\) ech = 20.5 MPa & \(\sigma\) ech = 28 MPa & \(\sigma\) ech = 154 MPa \\ \cline{2-4}  & \(\sigma\) c = 8.92 MPa & \(\sigma\) c = 14 MPa & \(\sigma\) c = 75.9 MPa \\ \cline{2-4}  & f = 1.3 mm & f = 3.97 mm & f = 47.1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation of lingualization in the LLI. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Applied Force**} & \multicolumn{3}{c}{**Extent of Periodontal Damage**} \\ \cline{2-4}  & **No HBL** & **33\% HBL** & **66\% HBL** \\ \hline \multirow{4}{*}{0.25 N} & \(\sigma\) ech = 0.0271 MPa & \(\sigma\) ech = 0.73 MPa & \(\sigma\) ech = 2.1 MPa \\ \cline{2-4}  & \(\sigma\) c = 0.0272 MPa & \(\sigma\) c = 0.751 MPa & \(\sigma\) c = 2.16 MPa \\ \cline{2-4}  & f = 0.00468 mm & f = 0.124 mm & f = 1.12 mm \\ \hline \multirow{4}{*}{1 N} & \(\sigma\) ech = 1.37 MPa & \(\sigma\) ech = 2.92 MPa & \(\sigma\) ech = 8.43 MPa \\ \cline{2-4}  & \(\sigma\) c = 1.38 MPa & \(\sigma\) c = 2.98 MPa & \(\sigma\) c = 8.67 MPa \\ \cline{2-4}  & f = 0.208 mm & f = 0.497 mm & f = 4.48 mm \\ \hline \multirow{4}{*}{3 N} & \(\sigma\) ech = 4.12 MPa & \(\sigma\) ech = 8.74 MPa & \(\sigma\) ech = 25.3 MPa \\ \cline{2-4}  & \(\sigma\) c = 4.15 MPa & \(\sigma\) c = 8.91 MPa & \(\sigma\) c = 26 MPa \\ \cline{2-4}  & f = 0.626 mm & f = 1.49 mm & f = 13.4 mm \\ \hline \multirow{4}{*}{5 N} & \(\sigma\) ech = 6.86 MPa & \(\sigma\) ech = 14.6 MPa & \(\sigma\) ech = 42.2 MPa \\ \cline{2-4}  & \(\sigma\) c = 6.94 MPa & \(\sigma\) c = 14.9 MPa & \(\sigma\) c = 43.3 MPa \\ \cline{2-4}  & f = 1.04 mm & f = 2.49 mm & f = 22.4 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation of lingualization in the lower central incisor according to the applied force and extent of periodontal damage. Figure 4: LCI using a 1 N force: (**a**) [PERSON] tensions, 66% HBL; (**b**) tooth movement, 66% HBL. Figure 5: Lingualization of the LCI using a 3 N force: (**a**) [PERSON] tensions, no HBL; (**b**) tooth movement obtained, no HBL. Figure 3: Lingualization of the LCI using a 1 N force: (**a**) [PERSON] tensions, 33% HBL; (**b**) tooth movement, 33% HBL.