## 3 Results  
### Estimation of Treatment Duration and Real Clinical Time  
The distribution of the normal displacement and mean stress along the PDL profile of the second molar can be seen in Figure 6. The PDL's normal displacement \(U_{N}\) almost followed the same pattern as the corresponding mean stress \(\sigma_{M}\) in the associated PDL element. The ratio of normal displacement to the mean stress \(U_{N}/\sigma_{M}\) along a normalized path characterizing the PDL-root interface, was labeled as \(D_{\sigma_{M}}\) and computed over 180 iterations of molar uprighting. The ratio resulted in \(D_{\sigma_{M}}\approx 0.41\)\(\upmu\)m/KPa, and was almost consistent throughout the treatment simulation; this can be demonstrated in the plotted curves for the first and last iterations. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Young’s Modulus (MPa)** & **Poisson’s Ratio (v)** \\ \hline Bone & 3000 & 0.3 \\ PDL & 0.84 & 0.46 \\ Teeth (Root/Dentin) & 20,000 & 0.3 \\ Teeth (Crown/Enamel) & 70,000 & 0.3 \\ Stainless Steel & 200,000 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the structures involved in both models. Figure 5: Iterative process for simulating long-term orthodontic tooth movement. At each iteration, the applied force ‘F’ produces PDL displacement, which drives bone remodeling and updates the tooth–PDL–bone geometry. The process continues until the corrective forces dissipate and equilibrium is reached. Figure 7 presents the average mean stress in the PDL for each iteration of the molar uprighting simulation (black curve, left y-axis). This stress data was correlated with the bone remodeling velocity (red curve, right y-axis), using the relationship previously established and plotted in Figure 3, as detailed in the methodology section. Using the average mean stress \(\overline{\sigma}_{M}\) and bone remodeling velocity \(R_{v}\) obtained at each iteration in Figure 7, and the corresponding normal displacement in the PDL \(\overline{U}_{N}\) is calculated by multiplying \(\overline{\sigma}_{M}\) by the stress-displacement ratio \(D_{\sigma_{M}}\). The duration of each iteration \(T\) is then determined by dividing \(\overline{U}_{N}\) by the remodeling velocity \(R_{v}\), providing a real-time equivalent for that simulation iteration. This time increment is then added to the total cumulated treatment time \(T_{cum}\), as illustrated in the flowchart in Figure 8. The process continues iteratively until the uprighting objective is achieved. Figure 6: Distribution of normal displacement (\(U_{N}\), black curve) and mean stress (\(\sigma_{M}\), red curve) along the PDL–root interface of the second molar. (**A**) At the beginning of treatment (first iteration). (**B**) At the end of treatment (last iteration). The stress–displacement (\(D_{\sigma_{M}}\)) remains approximately constant at \(\sim\)0.41 \(\upmu\)m/KPa across the interface, indicating a consistent mechanical response during the simulation. Figure 7: Evolution of the average mean stress \(\overline{\sigma}_{M}\) in the PDL (black curve, left y-axis) and the corresponding bone remodeling velocity \(R_{v}\) (red curve, right y-axis) throughout the uprighting procedure, plotted as a function of finite element iterations. The iterative framework outlined in the flowchart (Figure 8) enables the computation of both the duration of each simulation iteration and the corresponding accumulation of treatment time. As shown in the graph below (Figure 9), this process generates the evolution of iteration duration (black curve) and the total cumulated clinical time (red curve) throughout the uprighting simulation. As the uprighting progresses, the iteration duration decreases due to the gradual reduction in mechanical stimulus and remodeling activity, where the first iteration was the longest, lasting approximately 1.17 days, while the final iteration was the shortest, lasting around 0.18 days.