# Toward patient-specific prediction of orthodontic tooth movement (OTM): A Maxwell-based phenomenological approach

Yeaseul Lee, Luca Placidi, Michele Tepedino, Yves Bolender, Daniel George

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Yeaseul Lee, Luca Placidi, Michele Tepedino, Yves Bolender, Daniel George. Toward patient-specific prediction of orthodontic tooth movement (OTM): A Maxwell-based phenomenological approach. Mechanics Research Communications, 2026, 153, pp.104641. 10.1016/j.mechrescom.2026.104641. hal-05496982

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# Toward patient-specific prediction of orthodontic tooth movement (OTM): A Maxwell-based phenomenological approach

Yeaseul Lee<sup>a</sup>, Luca Placidi<sup>b</sup>, Michele Tepedino<sup>c</sup>, Yves Bolender<sup>d</sup>, Daniel George<sup>a,∗</sup>

## ARTICLE INFO

### Keywords:

Orthodontic tooth movement (OTM)

Patient-specific

Maxwell model

Canine retraction

Wire relaxation

## ABSTRACT

Tooth displacement in the mesiodistal direction over a 12-week period is predicted using a two degrees of freedom Maxwell model under constant load ranging from 18g to 240g. The viscoelastic behavior of the alveolar bone is represented by capturing both its instantaneous elastic response and its time dependent remodeling behavior. The bone stiffness is derived from patient-specific medical imaging, while the tooth kinetic displacement, represented by the viscosity parameter, was calibrated using experimental literature data reported for canine retraction. The predicted numerical results provide interpatient variability boundaries of tooth displacement, within which the experimental and approximate tooth displacements are located. Next, a three-degrees of freedom Maxwell model accounting for the orthodontic wire was employed to estimate the force relaxation occurring in the wire during treatment. This model serves as the starting point for development of patient-specific forecasts of orthodontic tooth movement.

## Introduction

The application of orthodontic force to the tooth's crown results in the transmission of this force to the alveolar bone through the periodontal ligament (PDL). It is widely accepted that the PDL deformation modulates the process of orthodontic bone remodeling through the deposition of bone on the tension side and the resorption of bone on the unloaded side of the alveolar bone surface [1-3]. Such deformation leads to alterations in the interstitial fluid flow within the bone canaliculi, which, in turn, activates osteocytes that regulate bone turnover [4-8]. Moreover, fluid flow between the bone internal porosities and the PDL facilitates intercellular exchange of nutrients and oxygen [9-11]. Together, this mechanobiological cascade influences osteogenic cells, contributing to the regulation of orthodontic bone remodeling, and consequently, OTM [12,13].

From a continuum modeling perspective, the time-dependent response of bone adaptation due to the interstitial fluid flow under mechanical force can be described using a viscoelastic or more sophisticated, poroviscoelastic model [14,15]. A wide range of studies have incorporated this behavior into their theoretical frameworks to predict bone remodeling under mechanical loads [16-19]. However, theoretical

models of orthodontic bone remodeling are still limited [20], therefore, it remains a challenge to predict the long-term tooth displacement under a wide range of orthodontic loading, particularly due to interpatient variability.

This study proposes a prediction of the long-term tooth displacement under varying magnitudes of constant orthodontic force using a simplified Maxwel based theoretical approach [21]. Under orthodontic loading, the mechanical strain acts primarily on the PDL rather than on the alveolar bone, where the bone remodeling occurs [22]. Therefore, the alveolar bone is assumed to exhibit linear elastic behavior, and its immediate remodeling response is represented by the equivalent spring parameter of the Maxwell model. However, orthodontic bone remodeling is also a time-dependent process driven by biological processes, in which the orthodontic force initiates the mechanobiological cascade that results in bone remodeling. Thus, this delayed alveolar bone adaptation can be described by the equivalent viscosity parameter of Maxwell model. Moreover, force relaxation of the orthodontic equipment through wires is inevitable during OTM as a result of tooth displacement, which necessitates reactivation of the appliance approximately every three to four weeks [23-25]. Consequently, the predicted total tooth displacement obtained under a constant applied load tend to

<footnote><sup>a</sup> University of Strasbourg, CNRS, ICube, Strasbourg, France</footnote>

<footnote><sup>b</sup> Unipegaso of Napoli & MEMOCS Laboratory, L'Aquila, Italy</footnote>

<footnote><sup>c</sup> University of L'Aquila, Department of Biotechnological and Applied Clinical Sciences, L'Aquila, Italy</footnote>

<footnote><sup>d</sup> University of Strasbourg, Department of Dento-Facial Orthopedics, Faculty of Dental Surgery, Strasbourg, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: george@unistra.fr (D. George).</footnote>
be overestimated, as they do not account for the wire force relaxation which directly decreases OTM [26,27]. Therefore, an inverse analysis was performed using a three-degrees of freedom (3DOF) Maxwell model, which extends the 2DOF model by incorporating the orthodontic wire [21].

In the first section, a preliminary estimation of tooth displacement under constant orthodontic force over 12 weeks was performed using the 2DOF model and compared with experimental data. In the second section, force relaxation of the wire was calculated using the 3DOF model under constant displacement conditions, providing an approximate quantification of orthodontic force relaxation over a given period. Together, this framework serves as the foundation for a patient-specific model that may assist practitioners in optimizing clinical strategies.

## Methods

### Modeling tooth displacement with the 2DOF Maxwell model

We propose the use of the 2DOF Maxwell model, which is assumed to be the most straightforward method for describing the bone remodeling process within various ranges of orthodontic loading. In this framework, tooth displacement is described as incorporating both instantaneous and delayed bone remodeling process.

### Definition of the model

The model consists of a spring with stiffness coefficient (k), and a dashpot with viscosity coefficient (c) arranged in series (Fig. 1). The immediate bone elastic response is represented by the spring component derived from patient medical imaging. This component incorporates individual alveolar physiology for the accurate forecast of patient-specific OTM. The time-dependent behavior of bone adaptation under orthodontic loading is captured by the dashpot component with a viscosity parameter connected with a characteristic time (τ), where c = kτ.

Available clinical experimental data are limited to measurements of either tooth displacement (in mm) or the tooth movement rate (in mm/week). By identifying the characteristic time derived from experimental tooth displacement kinetics, the delayed bone remodeling process can be described. We note specifically that the characteristic time is a key indicator, which is inversely proportional to the tooth displacement (u<sub>tot</sub>) versus time graph and under a constant applied force (F) the tooth displacement kinetic (u<sub>tot</sub>) is given by u<sub>tot</sub> = F/kτ. Consequently, a patient with a larger characteristic time is expected to exhibit slower OTM than a patient with a smaller characteristic time. In this framework, the applied orthodontic force (F<sub>ext</sub>) derived from the orthodontic appliance is assumed to be constant (F<sub>ext</sub> = F), as it is the experimental assumption in most of clinical studies [28–33]. Furthermore, the total tooth displacement (u<sub>tot</sub>) under constant applied force (F) can be additively decomposed into two parts: the relative elastic displacement (u<sub>1</sub>) in the spring (bone elasticity) and the remodeling displacement (u<sub>2</sub>) in the dashpot. Therefore, the total tooth displacement (u<sub>tot</sub>) is obtained as u<sub>tot</sub> = u<sub>1</sub> + u<sub>2</sub>, with the initial conditions u<sub>1</sub>(t<sub>0</sub>) = u<sub>2</sub>(t<sub>0</sub>) = 0.

### Theoretical model development

The elastic displacement (u<sub>1</sub>) of the spring under orthodontic force (F<sub>ext</sub>) is derived from the traditional Euler-Lagrange equation (Eq. (1)).

<figure><img src="image_2.png" /><figcaption>Fig. 1. The 2DOF Maxwell model includes a spring representing bone stiffness (k), and a dashpot characterizing bone remodeling (c). The orthodontic force is illustrated by F<sub>ext</sub> generating total tooth displacement (u<sub>tot</sub>), which is the sum of the relative elastic displacement (u<sub>1</sub>) and the remodeling displacement (u<sub>2</sub>).</figcaption></figure>

Therefore, the energy function (U) is defined by the difference between the internal elastic energy (U<sub>int</sub>), due to the spring, and the external energy (U<sub>ext</sub>), due to the orthodontic loading (Eq. (2)). The rate of remodeling displacement (u<sub>2</sub>) is described using the Rayleigh dissipation function (R), which is the result of the damping process (Eq. (3)). In this formulation, the characteristic time (τ) of the remodeling process is expressed through the relation c = kτ.

$$F_{\text{ext}}\& = ku_1 = k(u_{\text{tot}} - u_2) \quad (1)$$

$$U\& = U_{\text{int}} - U_{\text{ext}} = \frac{1}{2}ku_1^2 - F_{\text{ext}}u_{\text{tot}} \quad (2)$$

$$R\& = \frac{1}{2}c\dot{u}_2^2 = \frac{1}{2}k\dot{u}_2^2 \quad (3)$$

The orthodontic force (F<sub>ext</sub>) is presumed to be the constant and produces tooth translation (bodily movement) (Eq. (4)). This force is commonly expressed as an orthodontic load in grams, representing the gravitational force applied. Assuming the aforementioned conditions, the remodeling displacement (u<sub>2</sub>) is obtained using the Rayleigh-Hamilton variational principle, which allows the evolution of the bone remodeling displacement to be described (Eq. (5)). Once the elastic (u<sub>1</sub>) and remodeling (u<sub>2</sub>) displacements are determined, the total tooth displacement (u<sub>tot</sub>) can be calculated (Eq. (6)). By setting the stiffness and viscosity parameters, through the characteristic time, the total tooth displacement (u<sub>tot</sub>) can be obtained numerically under a given applied load.

$$F_{\text{ext}}(t)\& = ku_1(t) = c\dot{u}_2(t) \quad (4)$$

$$u_2(t)\& = u_{\text{tot}}(t) - u_1(t) = \frac{1}{k\tau} \int_0^t F_{\text{ext}}(a) da \quad (5)$$

$$u_{\text{tot}}(t)\& = u_2(t) + \frac{F_{\text{ext}}(t)}{k} \quad (6)$$

### Bone stiffness coefficient

The stiffness coefficient (k) reflects patient-specific bone physiology through bone density, which is a necessary identifier for predicting individual bone remodeling [34]. The most common way to access this information in orthodontic treatment is to use cone beam computed tomography (CBCT), which is widely used and highly regarded for its low X-ray dosage and accessibility [35]. Therefore, the patient's CBCT is collected, comprising 480 slices with an isotropic voxel size of 0.22mm and a 16-bit depth in DICOM format. Data collection was performed according to the Declaration of Helsinki of 1975 and following revisions, and was approved by the local Ethical Committee (Approval no 92/CE/2025). The preliminary 3D segmentation of the scan was then performed using the Dental Segmentator module in 3D Slicer, an open-source medical image processing software (Fig. 2a) [36,37].

The posterior mandibular profile section (Fig. 2a) was designated as the region of interest (ROI). This ROI was subsequently segmented into 2D section as the binary label map in 3D Slicer and exported in nearly raw raster data (NRRD) format (Fig. 2b). Using ImageJ, an open-source image processing software, the approximate mean alveolar bone density (ρ<sub>b</sub>) was calculated based on the mean pixel gray value (pGV) of the corresponding region in the CBCT scan (Eq. (7), Fig. 2c) [38]. The mean pixel gray value of the alveolar bone was determined to be 670.239, which was then converted to the Young's modulus (E) (Eq. (8)) [39,40].

Finally, the Young's modulus of the alveolar bone was multiplied by the CBCT pixel length (l) to obtain the patient-dependent bone stiffness coefficient (k) (Eq. (9)). The resulting parameters are presented in Table 1 (Young modulus, pixel length, stiffness) and bone stiffness (k) is used as the input bone elastic stiffness for the subsequent analyses.

$$\rho_b = 1.9 \cdot 10^{-3} \text{pGV} + 0.105 \ [\text{g/cm}^3] \quad (7)$$
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The patient-dependent stiffness coefficient (k) is obtained using CBCT scan. (a) The initial 3D Segmentation is performed using Dental Segmentator in 3D Slicer, and the region of interest (ROI) is subsequently transferred into 2D section. (b) Binary segmentation (binary label map) was imported from 3D Slicer into ImageJ, with the gray region indicating alveolar bone and the white region denoting the teeth. (c) The calculation of the mean pixel gray value (pGV) of alveolar bone is achieved by selecting the gray region in the binary label map, as defined by the yellow contour in the corresponding CBCT scan (yellow bar = 12 mm).</figcaption></figure>

<caption>**Table 1**</caption>

The input bone elastic parameter (k) is calculated by multiplying the Young's modulus (E), derived from the mean pGV of CBCT scans, by the pixel length (l) of CBCT scans.

<table><thead><tr><td>E [GPa]</td><td>l [mm]</td><td>k [N/mm]</td></tr></thead><tbody><tr><td>1.77</td><td>0.22</td><td>389.4</td></tr></tbody></table>

$$E\& = 0.06 + 0.9\rho_b^2 \ [GPa] \qquad (8)$$

$$k\& = E \cdot l \ [N/mm] \qquad (9)$$

### Characteristic time for bone remodeling kinetic

The characteristic time (τ) is the pivotal parameter, which is connected to the viscosity coefficient (c = kτ), and is used to describe the

delayed bone remodeling process under orthodontic loading. However, clinical data is limited to the measurement of tooth displacement rate or tooth movement. Therefore, the characteristic time (τ) is derived from experimental data on tooth displacement kinetics (û<sub>tot</sub>) as a function of the applied load (F<sub>ext</sub>), either directly from the rate of tooth movement (in mm/week), or from tooth displacement (in mm) subsequently converted into velocity. It should be noted that the slope of the linear fit (due to constant viscosity) of the kinetic-load graph is given by 1/kτ (in mm/(g·week)), where the characteristic time (τ) is inversely proportional to the bone remodeling kinetics (û<sub>tot</sub>) and independent of the applied load (F<sub>ext</sub>).

In the current study, experimental data from a canine retraction in the mesiodistal direction were considered using a friction-free appliance (loop), in which a constant load was applied by a coil spring of sufficient magnitude to induce bodily tooth movement [28–33]. During these experiments, an initial phase was observed that lasted between one and

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Experimental tooth movement rates for canine retraction, averaged across datasets as a function of applied load. Resulting and experimental tooth displacements are expected to lie within the maximum-minimum bounds defined by two fitted lines (red and blue line respectively).</figcaption></figure>
two weeks, during which the kinetics were unstable between the initiation and stabilization rate. As the Maxwell model describes the linear progression of tooth displacement (thus constant kinetics) under constant force, only the stabilized kinetic values recorded after two weeks were considered and subsequently averaged for each dataset at a given applied load.

At first, two linear fits were plotted onto the experimental data to define the boundaries of interpatient variability (minimum and maximum). Therefore, the resulting tooth displacements are expected to lie within these boundaries defined by the two fitted lines (Fig. 3). Next, an ordinary least squares (OLS) fit curve was plotted across all experimental data to provide estimation of average tooth displacement over a long treatment period.

The error of the OLS fit slope (ERR<sub>slope</sub>) is given by its standard deviation in (Eq. (10)), where N is the number of data points, x the corresponding load (in g), y the corresponding tooth displacement kinetic (in mm/week), and m the OLS slope (in mm/(g·week)) (Fig. 4).

$$ERR_{slope} = \frac{1}{N} \sqrt{\sum_{i=1}^{N} \left( \frac{y_i}{x_i} - m \right)^2} \quad (10)$$

### Numerical simulation

The total tooth displacement over 12-weeks period was predicted with orthodontic loads of 18g, 60g, 120g, and 240g (equivalent to 18cN, 60cN, 120cN, and 240cN, respectively). The 2DOF Maxwell model, incorporating the two input parameters (τ and k) derived from the patient's medical image and experimental measurements of the tooth displacement rate, was implemented in Python using the weak formulation (Eqs. (1)–6). Initial conditions were defined with elastic (u₁) and remodeling (u₂) displacements set to zero. The orthodontic loads were modeled as constant (F<sub>ext</sub> = F̄) throughout the simulation. Additionally, with bone stiffness (k) defined as constant, the instantaneous elastic

displacement (u₁) is calculated according to (Eq. (1)) by dividing the applied force (F<sub>ext</sub>) by the stiffness parameter (k). The remodeling displacement (u₂) was calculated by integrating the applied force (F<sub>ext</sub>), followed by multiplication with the inverse of both elastic coefficient (k) and the characteristic time (τ) (Eq. (5)). The total tooth displacement (u<sub>tot</sub>) over the 12-weeks period was then obtained by summing the elastic and remodeling displacements (u<sub>tot</sub> = u₁ + u₂) (Eq. (6)). This leads to a constant linear increase in tooth displacement after brief elastic displacement, where the magnitude of this growth is proportional to the applied load and depends only on the bone stiffness (k) and the characteristic time (τ) (see results and discussion in Section 3). Moreover, with the characteristic time derived from the maximum and minimum fits of the overall experimental data, both experimental and predicted tooth displacements are expected to lie within these boundaries.

### Data validation

A comparison was made between the numerical results and the experimental data concerning average canine displacement over 12 weeks (0, 3, 7, 14, 28, 42, 56, 70, and 84 days) under constant applied loads, as reported by Iwasaki and Nickel [28–30] where the respective subjects comprised 17 and 41 patients, respectively.

### Wire relaxation quantification with the 3DOF Maxwell model

The literature experimental input data for bone remodeling kinetics were collected under the assumption of a constant applied force during canine retraction [28–30,32]. In practice, however, force relaxation of the spring is inevitable as the tooth is moving, necessitating reactivation of the appliance approximately every three to four weeks [23–25]. Hence, the predicted numerical results obtained under constant force conditions are overestimated. To approximately quantify this wire relaxation, a 3DOF Maxwell model that integrates wire stiffness was

<figure><img src="image_5.png" /><figcaption>Fig. 4. The ordinary least squares fit was performed across the experimental data to provide an estimation of average tooth displacement over a long treatment period.</figcaption></figure>
used, enabling of force relaxation via the new characteristic time under constant displacement conditions.

## Definition of the model

The 3DOF Maxwell model is an extension of previously defined 2DOF model (see Fig. 1) by adding a spring with wire stiffness ($k_w$) in series (Fig. 5). In this framework, a constant applied force ($F_{ext} = \bar{F}$) results in relative remodeling displacement ($u_2$), bone displacement ($u_1$), and wire displacement ($u_w$) with the initial conditions $u_1(t_0) = u_2(t_0) = u_w(t_0) = 0$. As the wire is attached to the tooth, a displacement control analysis leads to force relaxation, which provides an approximation of the wire relaxation.

## Theoretical model development

The energy function ($U'$) is defined as the sum of the internal elastic energy ($U'_{int}$) from two springs representing bone and wire stiffness, and the external energy ($U'_{ext}$) associated with the orthodontic loading (Eq. (11)). Building on the aforementioned Euler-Lagrange principle (see Section 2.1.2), three fundamental equations are established from the partial derivatives of the energy function ($U'$) with respect to the displacement (Eq. (12)–14). The remodeling displacement kinetic ($\dot{u}_2$) is described using the Rayleigh dissipation function ($R'$), where the viscosity coefficient (c) is defined by the new characteristic time ($\tau'$) and the equivalent stiffness ($k_{eq} = k \cdot k_w/(k + k_w)$) (Eq. (15)). Finally, the Rayleigh-Hamilton variational principle yields the expression for bone remodeling displacement ($u_2$) (Eq. (16)), which is related to the applied force ($F_{ext}$) by substituting (Eq. (14)) into (Eq. (12)).

$$U' = U'_{int} - U'_{ext} = \frac{1}{2}k(u_1 - u_2)^2 + \frac{1}{2}k_w(u_w - u_1)^2 - F_{ext}u_w \quad (11)$$

$$\frac{\partial U}{\partial u_1} = k(u_1 - u_2) + k_w(u_1 - u_w) = 0 \quad (12)$$

$$\frac{\partial U}{\partial u_2} = k(u_2 - u_1) \quad (13)$$

$$\frac{\partial U'}{\partial u_w} = k_w(u_w - u_1) = F_{ext} \quad (14)$$

$$R' = \frac{1}{2} c \dot{u}_2^2 = \frac{1}{2} k_{eq} \tau' \dot{u}_2^2 \quad (15)$$

$$k(u_2 - u_1) = -c\dot{u}_2 = -F_{ext} \quad (16)$$

To calculate the force relaxation in the wire during OTM due to the tooth displacement, an inverse analysis was performed by applying a constant displacement ($u_w = \bar{u}_w$) to the 3DOF Maxwell model issued from the previously predicted tooth displacement using 2DOF Maxwell framework. By substituting (Eq. (12)) into (Eq. (16)), the remodeling displacement under constant applied displacement is expressed as (Eq. (17)). Accordingly, the wire force relaxation ($F_w$) is defined by substituting (Eq. (17)) into (Eq. (16)). The total tooth displacement at a given time from section 1 under constant force conditions is then assigned as the input constant displacement ($\bar{u}_w$) to observe the corresponding wire relaxation ($F_w$) (Eq. (18)).

$$u_2(t) = \bar{u}_w \left(1 - e^{-\frac{t}{\tau}}\right) \quad (17)$$

<figure><img src="image_6.png" /><figcaption>Fig. 5. The three-degree of freedom model (3DOF) is an extension of the 2DOF Maxwell model (see Fig. 1), incorporating a spring with wire stiffness ($k_w$) in series.</figcaption></figure>

and

$$F_w = c \dot{u}_2 = k_{eq} \tau' \bar{u}_w e^{-\frac{t}{\tau}} \quad (18)$$

## Wire property

In the standard approach to canine retraction, the orthodontic force is applied by a friction free wire referred to as a "loop" attached to the canine and the posterior segment which consists of second premolar and molars, also known as the "anchorage unit" [41]. The activation of this loop is dependent upon the amount of applied force desired by the practitioner, which is achieved through deflection to induce bodily movement of the canine towards the anchorage unit. In many cases, this activation is accomplished using a coil spring, which exerts a constant moment to force ratio on the loop to induce translation of the tooth. Among the materials employed for coil springs, nickel-titanium (NiTi) is most frequently utilized due to its large spring back behavior and relatively high superelasticity [42]. These properties are essential in orthodontic wire, as they enable the loop to revert to its original shape after deformation, thereby allowing practitioners to achieve large deflection control while maintaining a relatively constant force throughout its extended elastic intermediate phase [43]. In this study, the orthodontic coil spring and loop are simplified as a single unit, denoted the single "wire", which is attached to the tooth and delivers the force. Since the force is primarily generated and maintained by the coil spring, this unit is characterized by the mechanical properties of NiTi, with an initial length of 14.25 mm and a diameter 0.356 mm. Moreover, it is presumed that the wire deformation remains within the superelastic phase with strain ranging from 0.01 to 0.04 during the treatment, as indicated by material properties reported in the literature (see Table 2 [44]). Therefore, under a constant displacement ($\bar{u}_w$), the orthodontic wire undergoes simple relaxation ($F_w$).

## New characteristic time for wire force relaxation

In this 3DOF Maxwell model, force relaxation is governed by the new characteristic time ($\tau'$) where $\tau' = c/k_{eq} = \tau k/k_{eq}$ (Eq. (3),15) and the original characteristic time ($\tau$) is derived from the slope of fitted curve to the experimental data (see Fig. 3–4). To account for the force relaxation induced by the wire, its mechanical property is incorporated into the model as wire stiffness ($k_w$). When combined with the bone stiffness ($k$), this results in the equivalent stiffness ($k_{eq}$), which is given by $k_{eq} = k \cdot k_w/(k + k_w)$. It is important to note here that the characteristic time ($\tau$) is the constitutive quantity which depends on the material (bone) not on the external parameter (applied force ($F_{ext}$)). Therefore, under constant displacement conditions ($u_w = \bar{u}_w$), the wire force relaxation ($F_w$) depends only on the new characteristic time ($\tau'$) and the equivalent stiffness ($k_{eq}$) (see Eq. (18)).

## Numerical simulation

In order to approximate the amount of wire force relaxation over a given time period, an inverse analysis was used. This means that the input parameter in this second section using 3DOF Maxwell model under constant displacement conditions, is derived from the results of the first section using 2DOF Maxwell model under constant force conditions. As an example, the numerical result of tooth displacement under a 60 g load at 1 week ($\bar{u}_w = u_{tot}^{60g}(t = 1w)$) was used as the input constant displacement ($\bar{u}_w$), in order to calculate the corresponding wire force

## Table 2

The wire elastic modulus ($E_w$) was derived from the constitutive law for NiTi wire within the elastic intermediate phase [44]. The input wire elastic stiffness ($k_w$) was then calculated by multiplying the wire elastic modulus by the pixel length (l) obtained from the CBCT scan (see Table 1).

<table><thead><tr><td>Strain</td><td>E<sub>w</sub> [GPa]</td><td>k<sub>w</sub> [N/mm]</td></tr></thead><tbody><tr><td>0.01 - 0.04</td><td>1.25</td><td>8.727</td></tr></tbody></table>
relaxation ($F_w$).

### Data validation

As the maximum and minimum boundaries are established through the fitted slopes from which the new characteristic times ($\tau'$) are derived (see Fig. 3), both the clinical wire relaxation time (3–4 weeks) [23–25] and the predicted relaxation time are expected to fall within the relaxation boundaries defined by these new characteristic times ($\tau'$).

## Results and discussion

### Characteristic time

The input characteristic times ($\tau$) used to describe the time dependent bone remodeling process were derived from a linear fit of the experimental data on the bone remodeling kinetics-load graph (linear fit slope = $1/k\tau$). Two linear fits were plotted to determine maximum and minimum interpatient variability boundaries (see Fig. 3), along with the least squares fit for the estimation of average tooth displacement (see Fig. 4). The evaluated corresponding characteristic times are summarized in Table 3.

### Tooth displacement prediction

### Interpatient tooth displacement boundaries

By identifying stiffness and viscosity parameters derived from the characteristic time based on the maximum and minimum slopes of the experimental data ($\tau = 0.0021$ and $0.0270$ week, respectively (Table 3)), total tooth displacement over 12 weeks was predicted using 2DOF Maxwell model under constant applied loads of 18g, 60g, 120g, and 240g. The numerical results were then compared with the experimental data for the canine retraction case [28–30], as shown in Fig. 6.

As expected, the overall experimental data lie within the boundaries of maximum and minimum tooth displacement derived from the corresponding fit slopes (see Fig. 3). However, particularly under the 18 g of load, the tooth displacement during the initial phase (< 2 weeks) falls outside these boundaries. This occurs because experimental tooth displacement kinetics are often unstable during the initial phase, thus only the data collected after 2 weeks were averaged for each dataset to determine the fitted slopes.

In the study of Boester et al. [45] the maximum tooth displacement kinetics are observed within the first week of loading, followed by stabilization after approximately 2–3 weeks. Similarly, Iwasaki et al. [29] report that the highest stiffness in the tooth displacement-time curves occurs during the first week, with stabilization reached after about 2 weeks. These findings indicate that tooth movement kinetics are unstable during the initial phase of treatment, thus this early phase should be modeled separately. However, the proposed Maxwell framework requires constant displacement kinetics from the onset of loading, preventing accurate representation of this transient behavior. Consequently, finer temporal resolutions (hours to days) during the early stage, combined with a modeling approach capable of capturing the nonlinear kinetics from the initial to stabilized phase, is needed to accurately predict the tooth displacement.

After two weeks, the experimental results fall inside these

<caption>**Table 3**</caption>

The slope of the linear fit on the experimental data on the kinetic-load graph [Figs. 3-4] is inversely proportional to the characteristic time ($\tau$). The input characteristic time ($\tau$) was determined from the maximum, minimum, and least squares fits (in bold) to the experimental data.

<table><thead><tr><td></td><td>Fit slope [mm/(g · week)]</td><td>τ [week]</td></tr></thead><tbody><tr><td>Max</td><td>0.0120</td><td>0.0021</td></tr><tr><td>Average</td><td>0.0017</td><td>0.0148</td></tr><tr><td>Min</td><td>0.0009</td><td>0.0270</td></tr></tbody></table>

boundaries. For low applied load (18 g), the experimental data tend toward the maximum boundary (red line in Fig. 6(a)), whereas for high applied load (240g), they tend toward the minimum boundary (blue line in Fig. 6(d)). This is consistent with the interpretation of the kinetics presented in Fig. 3. The results from experimental tooth displacement kinetic weigh more toward the maximum slope (red line in Fig. 3) under low applied loads (< 50g), whereas under high applied loads (> 150 g), they weigh more toward the minimum slope (blue line in Fig. 3).

Moreover, the experimental data used in this study represent average patient value, with no individual level information available. Therefore, at the current stage, we define an interpatient variability range within which individual cases are expected to fall. To enable patient-specific forecasting of OTM, a personalized characteristic time is required, and this can be derived from each patient's tooth displacement kinetics using CBCT. By comparing scans taken at two different time points, the tooth displacement over time can be quantified. Knowing the force applied by the wire at the initial time point (measured using an orthodontic force gauge), a linear curve with slope $1/k\tau$ can be plotted on the tooth displacement kinetic-load graph, starting from the initial kinetic point and expected to lie within the interpatient variability. This enables patient-specific OTM prediction across a wide range of applied loads over a long treatment period, since the characteristic time is a constitutive parameter determined by the material properties rather than by the external load.

### Average tooth displacement

The characteristic time derived from the least square fit over all experimental dataset ($\tau = 0.0148$ week, (Table 3)) was used. The total tooth displacement over 12 weeks was predicted using 2DOF Maxwell model under constant load conditions and subsequently compared with the experimental data of Iwasaki [28,29] et Nickel [30] (Fig. 7) corresponding to the same application. Although a reasonably large percentage error was observed (53 %), this corresponds to the wide variability among the experimental data. At 12 weeks, the predicted total tooth displacement reached 0.38, 1.25, 2.50, and 5.00 mm for orthodontic loads of 18 g, 60 g, 120 g, and 240 g, respectively. We observe as expected that the simulated displacements were consistently lower than the experimental values reported by Iwasaki and Nickel since the measurements of the kinetic tooth displacement were consistently above the fitted regression line (see Fig. 4). Moreover, under the 120 g load, the experimental total tooth displacement reported by Iwasaki et al. [29] is greater than the one observed for the 240 g load, while the values are very close to each other in Nickel et al. [30], indicating that bone remodeling kinetics are not directly related solely to the proportional increase of applied load, but on some other biological factors being patient dependent.

Due to the limited number of studies that met the inclusion criteria, long-term tooth displacement comparisons were restricted to the studies reported by Iwasaki and Nickel [28–30]. In these investigations, the average subject age was 15 years old for a cohort of 17 individuals and 14.8 years old for a cohort of 41 individuals, respectively. Under a 240g load, the reported average velocities of tooth movement were 0.56 mm/week and 0.35 mm/week, respectively. These values are considerably higher than those reported in other studies [32,33,46]. For example, Ren et al. [46] observed a maximum velocity of 0.29 mm/week under a 272g load. Similarly, Yee et al. [32], reported an average velocity of 0.34 mm/week under a 300 g load. The elevated tooth movement kinetics reported by Iwasaki and Nickel [28–30] led to an overall increase in tooth displacement, which may be attributed to the younger age of the subjects, particularly those who had not yet completed craniofacial growth [47–49]. Indeed, a recent study by Wang et al. [50] emphasized that the non-negligible deceleration of OTM observed in adults is due to slower biological responses, resulting in an overall reduction of tooth movement during treatment. Therefore, additional experimental data across a broader range of age groups are needed to enhance validation and improve the fit curve.
<figure><img src="image_7.png" /><figcaption>Fig. 6. Comparison between the predicted total tooth displacement using 2DOF Maxwell model under constant orthodontic loading and experimental data [28–30] over a 12-week period. The red line represents the maximum tooth displacement derived from the maximum fit slope, while the blue line represents the minimum tooth displacement derived from the minimum fit slope (fitted slopes shown by red and blue line in Fig. 3, respectively).</figcaption></figure>

Moreover, it is well characterized that OTM progresses through three distinct phases: an initial phase, during which orthodontic force induces immediate displacement of the tooth within the socket; a lag phase, characterized by PDL repair of necrotic areas and minimal displacement; and, finally, a remodeling phase, initiated once the damaged tissue has been removed [51–53]. Therefore, it can be presumed that the duration of the lag phase depends on the magnitude of the applied load. In other words, an increase in the applied load results in a greater damage to the tissue, which consequently slows down the initiation of the remodeling process due to the process of tissue repairing necrotic areas. This constitutes one of the four hypotheses proposed by Quinn et al. [54].

The duration of the lag phase has been reported to range from 3 to 30 days [52,55]. Given that the optimal load for canine retraction ranges from 50g to 200g [56], the lower or very close experimental tooth displacement observed at 240g compared with 120 g in the studies by Iwasaki [29] and Nickel [30] may be attributed to tissue damage caused by excessive orthodontic force, thereby prolonging the lag phase which slows down OTM. Furthermore, Ren et al. [46], investigated the optimum force: sufficiently high to enhance OTM, yet low enough to minimize the tissue damage [54,57]. Their study concluded that beyond the optimum force threshold, the velocity of tooth movement decreases gradually. This may also explain that optimal force may be found around the 120g load in the studies by Iwasaki [29] and Nickel [30]. Therefore, exceeding this threshold would lead to reduced OTM.

In the proposed Maxwell model framework, the biologically induced tooth displacement kinetics including lag phase are implicitly incorporated through the characteristic time, as these characteristic times are derived from literature based experimental data that already integrates the underlying biological processes. However, these biological

influences remain unattainable as the current framework relies on a phenomenological linear fit (see Fig. 4). To further address this, the early lag phase, caused by tissue damage, could be incorporated into future modeling efforts advancing toward a multiphysics framework where inflammatory cytokines such as TNF-alpha and interleukins-1/6 within the PDL [58] may increase in a force dependent manner, thereby restricting tooth displacement when forces exceed the optimal biological range. The integration of these mechanically driven biological responses could provide the optimal force that can be defined as a balance between minimizing tissue damage and maximizing the applied force to enhance tooth displacement.

### Wire force relaxation quantification

#### New characteristic time

In practice, wire relaxation is inevitable as the tooth is moving, which results in an overestimation of tooth displacement under constant force conditions. To address this, a new characteristic time ($\tau$) was derived from the original characteristic time ($\tau$) as presented above. This newly derived characteristic time ($\tau'$), presented in Table 4, is employed as an input parameter in the 3DOF Maxwell model to estimate wire force relaxation ($F_w$).

#### Average wire force relaxation quantification

Under constant displacement, derived from the total tooth displacement calculated previously at 1 week under a 60 g of load ($\bar{u}_w = u_{tot}^{60g}(t = 1w)$), the wire relaxation was calculated using the 3DOF Maxwell model, as illustrated in Fig. 8. The figure was normalized as the force relaxation evolution will be the same for all load cases since it
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Comparison between the predicted total tooth displacement using 2DOF Maxwell model under constant orthodontic loading and experimental data [28–30] over a 12-week period. The characteristic time derived from the least squares fit slope (green line in Fig. 4) was used as the input characteristic time.</figcaption></figure>

**Table 4**

The new characteristic times ($\tau'$) is defined for inverse analysis under constant displacement control as $\tau' = c/k_{eq} = \tau k/k_{eq}$. The average new characteristic time, derived from the slope of the least squares fit, is shown in bold and used for the average wire force relaxation quantification.

<table><thead><tr><th></th><th>Fit slope [mm/(g - week)]</th><th>τ [week]</th><th>τ' [week]</th></tr></thead><tbody><tr><td>Max</td><td>0.0120</td><td>0.0021</td><td>0.0976</td></tr><tr><td>Average</td><td>0.0017</td><td>0.0148</td><td>0.6759</td></tr><tr><td>Min</td><td>0.0009</td><td>0.0270</td><td>1.2332</td></tr></tbody></table>

depends only on the characteristic time and not on the intensity of the force. As expected, with the average new characteristic time ($\tau' = 0.6759$ week (Table 4)) derived from the original characteristic time ($\tau$), obtained from the least squares fit curve (see Fig. 4), the force relaxation curve lies between those obtained from the maximum and minimum fit slopes (see Fig. 3). The approximate amount of wire force relaxation for one day and one week of constant displacement was also calculated and is presented in (Table 5), showing that the larger the new characteristic time, the slower the force relaxation becomes. The average wire force relaxation was estimated to be approximately 19% after one day and 77% after one week under constant displacement conditions.

In Fig. 8, the total wire force relaxation occurs within a range of 3 days to 6 weeks (blue and red lines, respectively), with the new characteristic time derived from the maximum and minimum fit slope (see Fig. 3). In clinical practice, patients typically visit the orthodontist for appliance reactivation every 3 to 6 weeks [23–25], which fall within this range. The new characteristic time ($\tau'$) being proportional to the original characteristic time ($\tau$) is derived from the experimental data and is therefore a patient dependent parameter. As the new characteristic time ($\tau'$) increases along with slower force relaxation, patients with a larger characteristic time do not need to visit the practitioner for wire reactivation as soon as those with a smaller characteristic time. In other words, patients with a small characteristic time must attend more

frequent appointments, since the orthodontic force relaxes more rapidly. Since our theoretical framework strongly depends on the orthodontic mechanical load, which also governs the activation of mechanosensitive cells. Future extensions of the phenomenological multiphysics model should account for this effect.

**Conclusion**

The present phenomenological model has been developed as a bridge toward patient-specific prediction of tooth displacement across a wide range of orthodontic forces during long-term treatment. The model requires two input parameters: stiffness and viscosity. The stiffness parameter can be directly derived from patient-specific medical imaging, incorporating each patient's unique alveolar physiology. The viscosity parameter is obtained by the characteristic time, which is derived from the linear fit curve of the experimental tooth displacement rate-load graph including the underlying biological phenomena. Together with the patient-specific stiffness parameter, the 2DOF Maxwell framework can provide a robust approximation of tooth displacement over extended time periods and across varying force magnitudes.

The experimental data exhibit considerable variability in the literature, even within a single treatment case (canine retraction). However, this framework can provide the interpatient-variability boundaries through the maximum and minimum slopes across the experimental data, within which the patient-specific tooth displacement is expected to lie. As more reliable experimental data become available, particularly across different age groups and treatment phases, the interpatient variability boundaries can be further refined. Moreover, by analyzing a patient's CBCT scans acquired at two different time points, the patient-specific evolution can be specifically identified.

The quantification of average wire force relaxation was conducted using 3DOF Maxwell model, in which an average relaxation of 77% was observed over a one-week period under constant displacement conditions, a magnitude that cannot be considered negligible. However, the
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Normalized wire relaxation ($F_w$) under constant displacement conditions using 3DOF Maxwell model. Three new characteristic times ($\tau'$) were used as input parameters, upon which the force relaxation depends (see Eq. (18)).</figcaption></figure>

**Table 5**

The approximate wire relaxation was calculated for three new characteristic times ($\tau'$) where $\tau' = c/k_{eq} = \tau k/k_{eq}$. The average force relaxation, shown in bold, was derived from the original characteristic time ($\tau$), obtained from the least squares fit slope to the overall experimental data (see Fig. 4).

<table><thead><tr><th>τ' [week]</th><th>Relaxation (1 day) [%]</th><th>Relaxation (1 week) [%]</th></tr></thead><tbody><tr><td>0.0976</td><td>77</td><td>99</td></tr><tr><td>0.6759</td><td>19</td><td>77</td></tr><tr><td>1.2332</td><td>11</td><td>55</td></tr></tbody></table>

current 2DOF Maxwell model calculates tooth displacement under constant force conditions, and without accounting for force relaxation, the predicted displacement tends to overestimate. As the 2DOF Maxwell model strongly depends on the applied force, future modeling effort should incorporate this effect and by identifying patient-specific characteristic time, this wire force relaxation can also be predicted, providing useful information for practitioners in determining treatment frequency.

Finally, due to the initial tissue damage (lag phase) and the notion of optimal force, tooth displacement kinetics do not always appear proportional to the applied load. The kinetic evolves from zero, leading to an increase in tooth displacement before reaching a constant rate. Furthermore, if the applied force exceeds the optimal force threshold, the rate of tooth movement decreases. This nonlinear behavior of tooth displacement cannot be modeled within this Maxwell based simplified approach where tooth displacement kinetics are constant from the onset. Consequently, a more advanced modeling approach is needed, accounting for the nonlinear kinetics evolution driven by mechanobiological responses, advancing toward a multiphysics framework.

The outlook of this work could be extended to a fully 3D model, as already described in Placidi et al. [21] although not yet implemented numerically. The present formulation can be extended in several directions. For a full 3D setting, one could (i) incorporate regularization via higher gradient theories such as Abali et al. [59], to enhance and improve numerical robustness; (ii) devote specific attention to numerical implementation strategies such as Romano et al. [60], to ensure stable and efficient computation; and (iii) investigate how the predicted 3D remodeling and displacement patterns could be used in the design and optimization of bioinspired metamaterials [61,62]. Moreover, since

advanced manufacturing and multifunctional performance are increasingly relevant in this context, exploring synergies with recent developments in biobased functional composites and dental materials would be particularly valuable [63].

**CRediT authorship contribution statement**

**Yeaseul Lee:** Writing – original draft, Visualization, Validation, Software, Investigation, Formal analysis, Data curation. **Luca Placidi:** Writing – review & editing, Supervision, Methodology, Formal analysis, Conceptualization. **Michele Tepedino:** Writing – review & editing, Supervision, Methodology, Investigation. **Yves Bolender:** Writing – review & editing, Supervision, Methodology, Investigation. **Daniel George:** Writing – review & editing, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization.

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Data availability**

Data will be made available on request.

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