Finite Element Analysis Model for Assessing Expansion Patterns from Surgically Assisted Rapid Palatal Expansion

Summary

A set of novel finite element models of surgically assisted rapid palatal expansion (SARPE) that could perform a clinically required amount of expander activation with various angles of buccal osteotomy was created for further analysis of the expansion patterns of the hemimaxillae in all three dimensions.

Abstract

Surgically assisted rapid palatal expansion (SARPE) was introduced to release bony resistance to facilitate skeletal expansion in skeletally mature patients. However, asymmetric expansion between the left and right sides has been reported in 7.52% of all SARPE patients, of which 12.90% had to undergo a second surgery for correction. The etiologies leading to asymmetric expansion remain unclear. Finite element analysis has been used to evaluate the stress associated with SARPE in the maxillofacial structures. However, as a collision of the bone at the LeFort I osteotomy sites occurs only after a certain amount of expansion, most of the existing models do not truly represent the force distribution, given that the expansion amount of these existing models rarely exceeds 1 mm. Therefore, there is a need to create a novel finite element model of SARPE that could perform a clinically required amount of expander activation for further analysis of the expansion patterns of the hemimaxillae in all three dimensions. A three-dimensional (3D) skull model from cone beam computed tomography (CBCT) was imported into Mimics and converted into mathematical entities to segment the maxillary complex, maxillary first premolars, and maxillary first molars. These structures were transferred into Geomagic for surface smoothing and cancellous bone and periodontal ligament creation. The right half of the maxillary complex was then retained and mirrored to create a perfectly symmetrical model in SolidWorks. A Haas expander was constructed and banded to the maxillary first premolars and first molars. Finite element analysis of various combinations of buccal osteotomies at different angles with 1 mm clearance was performed in Ansys. A convergence test was conducted until the desired amount of expansion on both sides (at least 6 mm in total) was achieved. This study lays the foundation for evaluating how buccal osteotomy angulation influences the expansion patterns of SARPE.

Introduction

Surgically assisted rapid palatal expansion (SARPE) is a commonly used technique for transversely expanding the maxillary bony structure and the dental arch in skeletally mature patients¹. The surgery involves a LeFort I osteotomy, a mid-palatal corticotomy, and, optionally, the release of the pterygoid-maxillary fissure². However, undesired expansion patterns from SARPE, such as uneven expansion between left and right hemimaxillae³ and dentoalveolar process buccal tipping/rotation⁴, have been reported, which could lead to failure of SARPE, and sometimes, even requiring additional surgeries for correction⁵. Previous studies have indicated that the variation in circum-maxillary osteotomies may play a significant role in post-SARPE expansion pattern^2,3, as the collisions between the bone blocks at the Le Fort I osteotomy sites can contribute to the uneven resisting force of lateral expansion of the hemimaxillae and to the rotation of the hemimaxillae with the alveolar edges below the cut moving inwards while the dentoalveolar process expands^3,4. Therefore, there is a need to investigate the effects of different osteotomy directions, especially the buccal osteotomy, on post-SARPE expansion patterns.

Several finite element analysis (FEA) models have been set up to evaluate the force distribution during SARPE. However, the amount of expansion set in these models is limited to up to 1 mm, which is far below the required clinical amount^{6,7,8,9,10,11,12}. Inadequate expansion in FEA models can lead to erroneous predictions of post-SARPE outcomes. More specifically, the collision between the bones at the osteotomy site, as reported by Chamberland and Proffit⁴, may not be demonstrated if the expander is not adequately turned, which may not reflect the true clinical reality. With the limited amount of expansion built in the previous models, the outcome evaluations of these models were focused on stress analysis. However, the stress analysis of FEA in dentistry is usually conducted under static loading with the mechanical properties of materials set as isotropic and linearly elastic, which further restricts the clinical relevance of the FEA studies¹³.

Furthermore, most of these studies did not consider the thickness of the surgical instrument at the osteotomy site^{6,7,8,10,11,12}, often setting friction to zero at the cuts as part of the boundary conditions. However, this setting oversimplifies the contacts between the hard and soft tissues. It may significantly impact the distribution of force and the resulting expansion pattern of the hemimaxillae.

Nevertheless, no available literature has investigated the effect of osteotomy on post-SARPE asymmetry using finite element analysis (FEA) models. All the current studies employed models with symmetrical osteotomy patterns^{6,7,8,9,10,11,12,14}, which do not reflect the reality of clinical practice where the osteotomies may differ on each side of the skull. The lack of literature examining the effect of asymmetrical osteotomies on post-SARPE asymmetry represents a significant knowledge gap that must be addressed.

Therefore, the goal of this study is to develop a novel FEA model of SARPE that can truly mimic the clinical conditions, including the expansion amount and osteotomy gap, and investigate the expansion patterns of the hemimaxillae in all three dimensions with various designs of the osteotomy. Such an approach would provide valuable insight into the mechanics underlying post-SARPE expansion patterns and serve as a useful tool for clinicians in the planning and execution of SARPE procedures.

Protocol

This study utilized a pre-existing, de-identified, pre-treatment CBCT image of a patient who had SARPE as part of the treatment plans. The study was conducted in accordance with the Declaration of Helsinki and approved by the Institutional Review Board (protocol #853608).

1. Sample acquisition and tooth segmentation

1. Acquire a human CBCT image of the head in a natural head position that includes the patient's maxillary complex, including the maxillary basal bone, maxillary alveolar bone, and maxillary dentition.
2. Import the CBCT Digital Imaging and Communications in Medicine (DICOM) files into Mimics software.
   1. Create New Project (Ctrl + N), select all the DICOM images, and click Next and Convert.
   2. Define the direction of the model (A: anterior, P: posterior, T: top, B: bottom, L: left, R: right) and click OK.
3. Segment the file into maxillary complex, maxillary first premolars, and maxillary first molars.
   1. Click Thresholding, select an appropriate threshold to segment bones, and click Apply.
   2. Create new masks and click Edit Masks, using Draw and Erase to segment the patient's maxillary complex, maxillary first premolars, and maxillary first molars.
4. Export the targets as stereolithography (STL) files.
   1. Right-click on masks and select Calculate 3D to generate 3D objects.
   2. Right-click on 3D objects, select STL+, choose the demanded objects, and press Add and Finish to create STL files.

2. Surface smoothing and creation of cancellous bone and periodontal ligament space

1. Import the STL files into Geomagic software.
   1. Click File > Open, select the STL files, then press Open.
   2. Choose Millimeters for the data in Units pop-up window and click OK.
2. Smooth the surface of the maxillary complex, maxillary first premolars, and maxillary first molars.
   1. Click Polygons > Remove Spikes, click and drag the smoothness level near Low, click Apply, and OK.
   2. Click Polygons > Relax Polygons, click and drag the smoothness level near Min, click Apply, and OK.
   3. Click Polygons > Repair intersections, choose Relax/Clean in the Mode window, click Apply, and OK.
3. Modify the surface of the model into a continuous and closed region.
   1. Click and drag the sharp surface, and press delete to create a hole.
   2. Click Polygons > Fill Holes, use Fill, Fill Partial, Create Bridges in the Fill Method window to fill up the holes, click Apply, and OK.
4. Convert the 2D surface to a 3D solid model and export it as a computer-aided design (CAD) file.
   1. Click Edit > Phase > Shape Phase, select Edit contours to sketch the contours of the surface, then click OK.
   2. Click Draw Patch Layout and draw quadrilateral meshes to cover all the surfaces, then click OK.
   3. Click Construct Grids, define a proper Resolution, and click OK to generate a finer mesh.
   4. Click Fit Surfaces, click Apply, and OK to construct a 3D solid model.
   5. Click File > Save as to export the 3D model and save it in an IGES file (named Maxilla).
5. Create the cancellous bone by reducing the volume of the maxillary complex by 1 mm from the buccal alveolar surface. Create periodontal ligament space by expanding the contour of the roots by 0.2 mm.
   1. Click Polygon Phase, choose Delete in the Contour Lines window, select Preserve in the Patch Layout window, then press OK to convert the 3D solid model into a 2D surface.
   2. Click Polygons > Offset, enter -1 mm and 0.2 mm in the Distance panel for cancellous bone and periodontal ligament, then click Apply and OK.
   3. Click Edit > Phase > Shape Phase, select Restore Patch Layout and press OK.
   4. Click Construct Grids, define a proper Resolution, and click OK to generate a finer mesh.
   5. Click Fit Surfaces, click Apply, and OK to construct a 3D solid model.
   6. Click File > Save as to export the 3D model and save it in IGES files (named CB and PL).

3. Construct an anatomical symmetric maxilla model

1. Import the CAD files into SolidWorks.
   1. Click File > Open, select the Maxilla file, and press Open to import the CAD file.
   2. Click File > Save to save the file into the Part format.
2. Construct the cancellous bone below the palatal plane (PP).
   1. Click Insert > Part, select the CB file, and press Open to import the CAD file.
   2. Click Insert > Reference Geometry > Plane, choose three feature points on the palatal plane, and click OK to create a cutting plane.
   3. Click Insert > Features > Split, choose the palatal plane in Trim Tools, and click Cut Part to create a cutting preview.
   4. Tick the checkboxes in the Resulting Bodies, and click OK to separate the cancellous bone.
   5. Click the cancellous bone above the palatal plane, right click and press Delete in the Body section.
3. Construct the periodontal ligament of maxillary first premolars and maxillary first molars.
   1. Click Insert > Part, select the PL file, and press Open to import the CAD file.
   2. Click Insert > Features > Intersect, and choose Maxilla and PL in the Selections window.
   3. Select Create both in the Selections window, choose the periodontal ligament part in the Region List, then click OK to generate the ligament.
4. Perform a midpalatal cutting plane from the anterior nasal spine (ANS) to posterior nasal spine (PNS) and retain the right half of the maxillary complex.
   1. Click Insert > Reference Geometry > Plane, choose three feature points on the midpalatal plane, and click OK to create a cutting plane.
   2. Click Insert > Features > Split, choose the palatal plane in Trim Tools, and click Cut Part to create a cutting preview.
   3. Tick the checkboxes in the Resulting Bodies, and click OK to separate the maxillary complex.
   4. Click the left half of the maxillary complex, right-click, and press Delete in the Body section.
5. Mirror the right half of the maxillary complex and create an identical left half.
   1. Click Insert > Pattern/Mirror > Mirror, and choose the midpalatal plane in Mirror Face/Plane.
   2. Choose all the right half maxillary complex in Bodies to Mirror, and click OK to generate the left half of the maxillary complex.

4. Create a Haas expander and band to the maxillary first premolars and first molars

1. Construct the premolar band and molar band.
   1. Click Insert > Part, select the PL file, and press Open to import the CAD file.
   2. Click Insert > Features > Split, choose the teeth in the PL file, and set a Uniform Scaling of 1.05. Click OK to generate bands with 0.5 mm in thickness.
   3. Click Insert > Reference Geometry > Plane, choose three feature points on the occlusal plane, and click OK to create a reference plane.
   4. Click Insert > Reference Geometry > Plane, choose the occlusal plane, and set an offset distance of 1.5 mm. Click OK to create the first cutting plane.
   5. Click Insert > Reference Geometry > Plane, choose the occlusal plane, and set an offset distance of 4.0 mm. Click OK to create the second cutting plane.
   6. Click Insert > Features > Split, and choose the first and second plane in Trim Tools and the teeth in Target Bodies. Click Cut Bodies to create a cutting preview.
   7. Tick the checkboxes in the Resulting Bodies, and click OK to separate the teeth.
   8. Click the band above the first plane and below the second plane, right-click, and press Delete in the Body section.
2. Construct the acrylic plate.
   1. Click Insert > Reference Geometry > Plane, choose three feature points on the hard palate plane, and click OK to create a sketch plane.
   2. Click Insert > Sketch, draw an acrylic plate refer to the Haas expander, and click Exit Sketch.
   3. Click Insert > Boss/Base > Extrude, choose the sketch of the acrylic plate, set 5 mm in Depth, and click OK.
   4. Click Insert > Features > Flex, and bend the acrylic plate to fit the anatomy of the palate.
   5. Click Insert > Features > Fillet/Round, and fillet the sharp edges of the acrylic plate in a radius of 1 mm.
3. Construct the expander arms.
   1. Click Insert > Reference Geometry > Plane, choose three feature points on the band, and click OK to create a sketch plane (named P1).
   2. Click Insert > Sketch, draw a circle 2 mm in diameter, and click Exit Sketch (named C1).
   3. Click Insert > Reference Geometry > Plane, choose three feature points on the acrylic plate, and click OK to create a sketch plane (named P2).
   4. Click Insert > Sketch, draw a circle 2 mm in diameter, and click Exit Sketch (named C2).
   5. Click Insert > Reference Geometry > Plane, choose the P2 plane, and set an offset distance of 6 mm. Click OK to a sketch plane.
   6. Click Insert > Sketch, draw a circle 2 mm in diameter, and click Exit Sketch (named C3).
   7. Click Insert > Boss/Base > Loft, and choose the C1, C2, and C3 sketch in the Profiles window.
   8. Select the band and the acrylic plate in the Feature Scope window, tick Merge Result in the Options window , and click OK.

5. Design the osteotomy

1. Create a 1 mm thick plane, equivalent to the diameter of a bur usually used by the surgeon, from the corner of the piriform aperture (Alar) towards the infra zygomatic crest (IZC) at various degrees from the horizontal plane.
   1. Click Insert > Reference Geometry > Plane, choose three feature points on the osteotomy plane (0°, 10°, 20°, or 30° to the horizontal plane), and click OK to create the plane (named O1).
   2. Click Insert > Reference Geometry > Plane, choose the osteotomy plane, and set an offset distance of 1.0 mm. Click OK to create an inferior cutting plane (named O2).
   3. Click Insert > Features > Split, choose the O1 and O2 plane in Trim Tools, and click Cut Part to create a cutting preview.
   4. Tick the checkboxes in the Resulting Bodies, and click OK to separate the maxillary complex.
   5. Click the body between O1 and O2 planes, right-click, and press Delete in the Body section.
2. Export models with different buccal osteotomy angles in Parasolid Model Part File (X_T) for analysis.
   1. Click File > Save as, and choose Parasolid (x_t) in the File Type list.
   2. Click Save to export the models for finite element analysis software.

6. Finite element analysis

1. Import and set the material parameters of the maxillary complex model into Ansys software.
   1. Click and drag the Static Structural in Toolbox to create an analysis workspace.
   2. Double click the Engineering Data, and set Young's modulus and Poisson's ratio of all the materials in Properties. The material properties of different structures^12,15,16 are listed in Table 1.
   3. Double-click Geometry, click File > Import External Geometry File, then click Generate to import the maxillary complex model.
   4. Click Create > Boolean, and generate the cortical bone and periodontal ligament by Boolean with the cancellous bone and teeth.
2. Set up the finite element analysis model.
   1. Double-click the Model, and click Geometry to select the material properties for each part.
   2. Right-click Mesh and click Generate Mesh to build the elements on the model.
   3. Click Connections, and assign the soft/small part in Contact Bodies and the stiff/large part in Target Bodies.
   4. Assign the contact type and friction coefficient in Definition. The connection properties of different parts¹⁷ are listed in Table 2.
   5. Right-click Connections, click Insert > Spring to connect the upper and lower parts of the osteotomy plane. Set the springs as 1 mm long with spring constant k = 60 N/mm and place one spring at each grid node.
3. Set a clinically acceptable force along the x-axis (perpendicular to the midline) on the acrylic plate on various combinations of osteotomies.
   1. Right-click Static Structural, click Insert > Fixed Support and set the structure on the palatal plane immovable.
   2. Right-click Static Structural, click Insert > Force and set a 150 N force to apply on the acrylic plate with a direction away from the medial line.
   3. Right-click Solution, and click Insert > Deformation > Total to monitor the deformation of the expansion.
4. Conduct a convergence test until expansions on both sides are achieved.
   1. Click Solve on the toolbars, and wait until the Force Convergence level reaches the Force Criterion.
   2. Click Total Deformation to display the expansion results.
5. Measure the displacements of the anatomic landmarks in all three dimensions as the results of expansion. Suggest the following landmarks to be used to evaluate the expansion pattern:
   Mesioincisal line angle of the maxillary central incisor (U1).
   Buccal cusp tip of the maxillary first premolar (U4).
   Mesiobuccal cusp tip of the maxillary first molar (U6).
   Lateroinferior corner of the piriform aperture (Alar).
   Infra-zygomatic crest (IZC).
   Midpoint of the expander.

Results

The demonstration model utilized the CBCT image of a 47-year-old female with maxillary deficiency. In the generated model, the anatomic structure of the nasal cavity, the maxillary sinus, and the periodontal ligament space for the expander anchored teeth (first premolar and first molar) are preserved (Figure 1).

To simulate the surgical procedure accurately, the nasal septum, lateral walls of the nasal cavity, and pterygomaxillary fissure were separated from the m...

Discussion

The direction of the buccal osteotomy in SARPE can be either a horizontal cut from the nasal aperture before stepping down at the maxillary buttress area or a ramped cut from the piriform rim towards the buttress corresponding to the maxillary first molar, as described by Betts². Either way, the osteotomy extends well below the zygomatic process of the maxilla. However, most current FEA studies on SARPE use a horizontal cut extending posteriorly at the same level as the piriform rim

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Ansys	Ansys	Version 2019	Ansys is a software for finite element analysis that can solve complicated models based on differential equations. The expansion results of different buccal osteotomy angles were analyzed through this software.
Geomagic Studio	3D Systems	Version 10	Geomagic Studio is a software for reverse engineering that can generate digital models based on physical scanning points. This study built cancellous bone and periodontal ligaments through this software.
Mimics	Materialise	Version 16	Mimics is a medical 3D image-based engineering software that efficiently converts CT images to a 3D model. This study reconstructed a maxilla complex through the patient's DICOM images.
SolidWorks	Dassault Systèmes	Version 2018	SolidWorks is a computer-aided design software for designers and engineers to create 3D models. A Haas expander was designed and drawn through this software in this study.