## Introduction

As a powerful numerical approximation tool, finite element analysis (FEA) has been widely used to predict stress and strain distributions in facial bones generated by orthodontic appliances. Previous FEA models were constructed on the basis of a linear elastic phase of the bone response (eg, elastic bone strains to loading). However, what is more useful for clinical understanding would be predicting long-term strains and displacements of bone-segments responding to loading, yet tissue responses are (1) not promptly observable and (2) hard to predict in nature.

## Methods

Viscoelastic property of the mandibular bone was incorporated into FEA models to visualize long-term, time-dependent stress and strain patterns in the mandible after being exposed to orthopedic stress. A mandible under loading by a Herbst appliance was modeled, and outcomes of the constructed elastic and viscoelastic models were compared.

## Results

Patterns and magnitudes of the displacement throughout the mandible predicted by the viscoelastic model were exhibited in accordance with previous clinical outcomes of Herbst appliance therapy. The elastic models exhibited similar displacement patterns; however, the magnitude of the displacements in the models was invariably small (approximately 1 per 100) compared with those outputs of corresponding viscoelastic models. The corresponding maximum stress level in our viscoelastic mandible subjected to the Herbst appliance with the same loading was considerably low and relaxed in various regions when compared with the elastic model.

## Conclusions

We suggest that a viscoelastic model of the mandible mimics our general prediction of orthopedic treatment outcomes better than those by elastic models.

## Highlights

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The viscoelastic model may mimic long-term displacements from orthopedic appliances.

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The viscoelastic model showed low, relaxed strains over time that appear clinically relevant.

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The model showed a 3.1 mm forward downward chin displacement after 6 months of Herbst appliance use.

To measure the clinical effectiveness of an orthodontic appliance, stress exerted by the orthodontic appliance to the bone needs to be analyzed because the loading applied to the bone through the corresponding strain in the soft tissue matrix is responsible for bone remodeling over time. ^{,} Throughout the years, many approaches, such as brittle lacquer, photoelasticity, and holography, have been used to study the effects of orthodontic force on bones. Finite element analysis (FEA) simulates complex biologic structures and their biomechanical behaviors under different conditions, and various forces were used in orthodontics for many decades. In 1984, Williams et al first used FEA as a tool to study the center of rotation of maxillary incisors in relation to elastic properties of the periodontal ligament. However, owing to a lack of reports on material properties and oversimplified geometries, most studies using FEA were remote from clinical applications. Using FEA, many researchers attempted to show stress and strain distributions on the maxilla and mandible generated by orthodontics appliances such as expanders to Class II correctors, ^{,} facemasks, and temporary anchorages devices.

As studies using FEA gain popularity in the orthodontic field, we note that the validity of their research relies on the soundness of input data. Therefore, defining proper material properties, accuracy in geometry, applicable forces, and boundary conditions, as well as types of analysis depending on the nature of the problem, are crucial for the soundness of a model. Digital Imaging and Communications in Medicine files converted from 3-dimensional (3D) cone-beam computed tomography (CBCT) images can conveniently be exported to an FEA software package, which enables researchers to build individual models to test.

Despite all major advancements in the field, most previous studies examining the clinical effects of orthodontic appliances employed a set of linear elastic material properties to simulate behaviors of viscoelastic bone tissue showing nonlinear behaviors. ^{,} With elastic models, it is impossible to calculate displacements of the bone over a long period of treatment time, which is crucial for studying the end results of orthodontic appliances. In addition, the propagation of stress and its resultants in the bone during the treatment process cannot be captured in an elastic model because, in general, an elastic model can only express instantaneous behaviors of the bone. In contrast, a viscoelastic model factors a time-effect into account. Thus, a viscoelastic model express changes over time. In this study, we aimed to compare structural behaviors of a viscoelastic model of the mandible with Herbst appliance in action compared with those of a linear elastic model of the mandible. Herbst appliances were chosen because there are ample clinical data which would help examine and understand the results of this study. ^{,}

## Material and methods

Briefly, the captured geometry of a mandible from CBCT in Digital Imaging and Communications in Medicine image was converted to a stereolithography (STL) file. Then, the file was transformed into a 3D computer-aided design model that can be interpreted by Finite element method (FEM) software (Abaqus; Dassault Systemes Simulia Corp, Providence, RI) for analysis. The geometry was then discretized (meshed), and material properties (such as elastic or viscoelastic) were assigned, and then finally, the applied forces from the appliance as well as the boundary conditions were specified.

A full volume CBCT image on a boy aged 10 years with skeletal Class II was used. The CBCT was taken with CareStream CS9300 (Carestream Health, Rochester, New York, NY) at the following settings: 90 kVp; 5 mA; exposure time of 8 seconds; resolution of 180 micro-meter. The Ma 4. Invivo software (Anatomage, San Jose, Calif) was used to derive the file in STL format. The STL files were then transferred to Abaqus, which is a FEA software package with pre- and postprocessing capabilities. Using the 3D image obtained from the CBCT, the geometry was imported and meshed using other modules of Abaqus. Material properties —namely, Young modulus (or modulus of elasticity) and Poisson ratio—were assigned in accordance with the values in the Table .

Modulus of elasticity (E) | Material properties of cortical bone | |
---|---|---|

Poisson ratio (υ) | Retardation period (τ) | |

13,700 MPa | 0.3 | 50 min |

For viscoelastic models, the Prony series parameters were chosen for the study because the material behaves close to the Maxwell model. The retardation period was assumed to be 50 minutes, and the treatment period was assumed to be 4300 hours, which approximates 6 months. Although the Kelvin-Voigt model is usually used in the literature to capture the viscoelastic behavior of the cortical bone, it appears reasonable for clinical orthodontic treatments to assume that the cortical bone shows significant plastic behavior as well. This behavior in the cortical bone is often observed and suggested by researchers in biomechanical engineering.

After defining the material properties, the boundary conditions were imposed as the translational lock in all the global directions for elements on the condylar heads of the mandible, as shown in Figure 1 . To study the deformation of the mandible, degrees of freedom (ie, movement of the node in 1 or more directions x, y, and z) must be restricted to avoid rigid body motion. Such constraints are termed as boundary conditions. In addition, a static force of 40 N in the vertical and 60 N in the horizontal direction was imposed through masticatory muscles to the first molar regions on the mandible to simulate the loading from the Herbst appliance. These forces were adopted from the average bite-force, and the forces from the masticatory muscles reported previously in the literature. ^{,}

A pair of models using 2 separate material properties were constructed and analyzed. For 1 model, the elastic properties of the cortical bone were incorporated for the immediate elastic response. To account for the time parameter, the identical model except for viscoelastic material properties was constructed and analyzed. One of the purposes of this phase was to see the progression of how stresses and strains emerge and propagate through the course of treatment during Herbst appliance treatment. In addition, the magnitude of displacement, as well as the change in stress levels due to creep, were of interest in this study. In the postprocessing phase (which is the last phase of an FEA study), we observed the results of applied forces in the form of displacements and stress distributions. These results were visualized by contour maps in colored magnitudes of the outputs. One should note that the value of the displacement obtained from our elastic model could be very small because of the high elastic modulus of the cortical bone that we adopted. In addition, the elastic response would reflect a transitory spontaneous behavior of the mandible because creep or stress relaxation were not a part of the interpretation.

## Results

In both models, we exhibit principal stresses, von Mises stresses, and the magnitude of each displacement. von Mises stress was calculated to predict yielding of the bone. ^{,} The total number of nodes employed for the model was 26,260, along with the total number of elements of 52,235.

Figure 2 shows the magnitude of displacements when the elastic material properties were incorporated. Various colors in different areas represent the range of their corresponding displacements; red indicates an instant and maximum displacement, and blue indicates minimum displacements. The FEM analysis revealed that the maximum displacement resulted from the Herbst appliance in the elastic model was 0.04 mm at the chin in a forward and downward direction.