## 2 Materials and Methods
### Model 1: Canine Retraction Simulation  
The first computer-assisted design (CAD) model was developed based on previous studies that assessed canine translation velocity under continuous, constant forces [24,25]. These studies, which bridge clinical observations with experimental data, are essential for establishing a novel FE modeling technique capable of estimating real clinical time for a single simulation iteration, and thus the total treatment duration.  
For that purpose, a two-dimensional (2D) CAD representation of the maxillary canine was reconstructed, with 0.2 mm of PDL thickness and alveolar bone surrounding the tooth using SolidWorks(r) software (Version 2022, Dassault Systemes, Velizy-Villacoublay, France).  
The incorporated appliances included a standard edgewise bracket of 0.022-inch slot size bonded on the maxillary canine and a segmental 0.019 \(\times\) 0.025-inch stainless steel wire shaped to resemble a vertical open loop with a height of 13 mm which engaged the canine on the distal side (Figure 1). Using Abaqus/Standard software (Version 2022, Dassault Systemes, Velizy-Villacoublay, France), constant forces ranging from 0 to 800 cN with an increment of 10 cN, were applied to translate the canine distally and carry out the simulation process.  
Experimental studies [24; 25] provided clinical data on canine translation velocity under constant forces of up to 1600 cN. In the current FE simulation, constant forces ranging from 50 to 800 cN were applied, as this range was sufficient to establish the correlation. The simulation outcomes included tooth displacements and PDL deformations, which were used to calculate the real-time duration of each numerical step. The overall workflow for obtaining this calibration is outlined in Figure 2. By calculating the bone remodeling velocity and average mean stress in the PDL, a correlation was established and can be seen in Figure 3. This relationship enabled the translation of stress levels into remodeling velocities and served as the basis for the time calibration applied in the molar uprighting model. Additionally, Figure 3 illustrates the original relationship between tooth displacement velocity and applied force [24].  
The process summarized above presents a framework for relating the bone remodeling velocity to average mean stress in the PDL, allowing its application across a range of orthodontic loading conditions. This relationship can be leveraged to estimate the equivalent real clinical time for any applied force at any simulation iteration, ultimately enabling the prediction of both total orthodontic treatment duration and tooth movement rate using FE modeling. To authors' knowledge, this study is the first to introduce such an approach.  
Figure 1: Finite element (FE) representation of Model 1 for the canine retraction using vertical loop mechanics. The mesh distribution illustrates local refinement around the periodontal ligament (PDL) to ensure the accuracy and convergence of the results. A distal force ‘F’ is applied through a looped spring to translate the maxillary canine, simulating experimental retraction conditions for time calibration.  
Figure 3: Tooth displacement velocity and bone remodeling velocity plotted as functions of the applied retraction force and the corresponding computed average mean stress in the PDL. The black curve represents tooth displacement velocity during canine retraction based on experimental data from previous studies [24]. The red curve shows the bone remodeling velocity derived from the current study using Model 1.  
Figure 2: Flowchart illustrating the calibration process used to convert FE simulation iterations into real clinical treatment time. The left pathway represents canine retraction using vertical loop mechanics, combining experimental data and simulations to derive a relationship between bone remodeling velocity and average mean PDL stress. This relationship was then applied to the molar uprighting simulation (right pathway) to compute the clinical time duration of each iteration.