## Introduction

The center of resistance (C _{Res} ) is regarded as the fundamental reference point for predictable tooth movement. Accurate estimation can greatly enhance the efficiency of orthodontic tooth movement. Only a handful of studies have evaluated the C _{Res} of a maxillary first molar; however, most had a low sample size (in single digits), used idealized models, or involved 2-dimensional analysis. The objectives of this study were to: (1) determine the 3-dimensional (3D) location of the C _{Res} of maxillary first molars, (2) evaluate its variability in a large sample, and (3) investigate the effects of applying orthodontic load from 2 directions on the location of the C _{Res} .

## Methods

Cone-beam computed tomography scans of 50 maxillary molars from 25 patients (mean age, 20.8 ± 8.7 years) were used. The cone-beam computed tomography volume images were manipulated to extract 3D biological structures via segmentation. The segmented structures were cleaned and converted into virtual mesh models made of tetrahedral triangles having a maximum edge length of 1 mm. The block, which included the molars and periodontal ligament, consisted of a mean of 7753 ± 2748 nodes and 38,355 ± 14,910 tetrahedral elements. Specialized software was used to preprocess the models to create an assembly and assign material properties, interaction conditions, boundary conditions, and load applications. Specific loads were applied, and custom-designed algorithms were used to analyze the stress and strain to locate the C _{Res} . The C _{Res} was measured in relation to the geometric center of the buccal surface of the molar and the trifurcation of the molar roots.

## Results

The average location of the C _{Res} for the maxillary first molar was 4.94 ± 1.39 mm lingual, 2.54 ± 2.7 mm distal, and 7.86 ± 1.66 mm gingival relative to the geometric center of the buccal surface of the molar and 0.136 ± 1.51 mm lingual ( *P* <0.01), 1.48 ± 2.26 mm distal ( *P* <0.01), and 0.188 ± 1.75 mm gingival ( *P* >0.01) relative to the trifurcation of the molar roots. In the anteroposterior (y-axis) and the vertical (z-axis) planes, the C _{Res} showed significant association with root divergence ( *P* <0.01).

## Conclusions

The C _{Res} of the maxillary first molar was located apical and distal to the trifurcation area. It showed significant variation in its location. The 3D location of and also varied with the force direction. In some samples, this deviation was large. For accurate and predictable movement, tooth-specific C _{Res} need to be calculated.

## Highlights

- •
Accurate estimation of the center of resistance (C

_{RES }) is critical for efficient tooth movement. - •
CBCT scans of 50 maxillary first molars were converted to finite element models.

- •
The C

_{RES }was located apical and distal to the trifurcation area. - •
Significant amounts of individual variation were noted.

- •
The 3-dimensional location of C

_{RES }changes with force direction.

Efficient tooth movement depends on the relationship between the line of action of force and the center of resistance (C _{Res} ) of a tooth. Therefore, knowledge of the location of C _{Res} is critical in predicting tooth movement and minimizing unwanted side effects. C _{Res} is defined as the point of greatest resistance to tooth movement or a point that allows a tooth to display pure translation without exhibiting a rotational tendency (a moment) when a force vector, regardless of its direction, passes through it. ^{,} With exact knowledge of the location of C _{Res} , the force system (force and moment) that must be applied to the crown of a tooth to achieve the desired movement can be determined.

Historically, the C _{Res} of a maxillary first molar has been approximated at the trifurcation of the roots or the midheight of the cervical third of the root. Although initial reports were anecdotal, ^{,} subsequent experimental analyses confirmed the theoretical approximation of the C _{Res} . Worms et al clinically determined the location of the C _{Res} for the maxillary molars in the area of the root trifurcation. Dermaut et al, using laser speckle interferometry, identified its location slightly occlusal to the trifurcation of the roots. However, these early studies located the C _{Res} by projecting the complicated 3-dimensional (3D) structure of a maxillary first molar as a 2-dimensional (2D) entity in 1 direction. As a result, at the clinical level, determining the C _{Res} and predicting maxillary first molar movement for a given application of force has proved to be difficult. Recently, Viecilli et al reported that geometric asymmetry of a tooth and periodontal ligament (PDL) implies the existence of axes of resistance (A _{Res} ) in 3 directions (anteroposterior, vertical, and buccolingual) that do not intersect at a single point. In other words, the axes do not define a 3D C _{Res} as a singular point but rather as a volume. Dathe et al used vector and tensor calculus to show that C _{Res} of a tooth only exist in 2-dimensions or very special symmetrical spatial configurations. Experimental studies using finite element analysis (FEA) have also verified the absence of a definite localized center (point) of resistance.

The emergence of numeric methods of evaluating tooth displacement, like FEA, have provided more flexibility on modeling conditions. FEA is a computational method used to analyze the effects of external forces and other physical properties on the deformation and stress distribution over objects such as artificial hard tissues and human anatomic structures. Its usefulness in designing, analyzing, and finding answers to dental biomechanical problems has been proven. In addition, the introduction of computed tomography scanning and cone-beam computed tomography (CBCT) scanning has thrust models and calculations from the 2D world into 3-dimensions. In spite of these technological advancements, most evaluations of C _{Res} have been carried out on idealized models. The few studies that did evaluate actual teeth focused specifically on maxillary anterior teeth. However, of all the major drawbacks discussed above, the one that stands out is the small sample size, quite often in single digits. Adequate sample size will ensure better representation of the C _{Res} for a particular tooth type and would limit the influence of outliers or extreme observations. However, FEA requires special training and is time-consuming; therefore, building large sample sizes is not easy. However, with technological advancements, computational power and modeling capabilities have increased. The ability to create and study complex models has become easier too. An increase in computer processing power and software complexity has also allowed researchers to use 3D radiographs to extract accurate anatomic models for use in advanced software by segmenting the teeth, bone, PDL, and various other structures instead of relying on idealized models. ^{,}

The objectives of this study were to: (1) determine the 3D location of the C _{Res} of the maxillary first molar in anatomically correct teeth, (2) evaluate its variability in a large sample, (3) compare it to the available 2D data, and (4) investigate the effects of applying orthodontic load from 2 directions on the position of the C _{Res} .

## Material and methods

This was a descriptive, cross-sectional study involving the analysis of previously acquired radiographic images of patients. CBCT scans of 25 patients were randomly selected from a pool of images at the University of Connecticut Orthodontics and Radiology clinics. The sample was obtained from CBCTs of 14 male and 11 female patients whose ages ranged from 11.60 to 61.20 years. The average age was 20.80 ± 8.70 years. A total of 50 maxillary first molars were identified. The inclusion criteria were: (1) radiographically identifiable dental anatomy and healthy periodontal tissues, (2) no radiographic evidence of periodontal bone loss or root resorption, (3) voxel size of no larger than 350 μm, and (4) no or little scatter or degrading attributes of the image quality.

The research was approved by the institutional review board (no. 17-071S-2). CBCT scans had been performed on an i-CAT Imaging System (Imaging Sciences International, Hatfield, Pa). The image voxel size for the 50 molars ranged between 0.20-0.33 mm. Each selected volume was examined for missing teeth, voxel size, field of view, and overall quality of the image. Once selected, each volume was completely deidentified of any patient information and cropped to include just the maxilla and teeth.

Selected CBCT digital imaging and communications in medicine files were exported into 3D image processing and editing software Mimics (version 19.0; Materialise, Leuven, Belgium) for segmentation. Threshold segmentation was used within Mimics to separate desired anatomic structures from the rest of the tissue. To achieve accurate tooth, root, and bone anatomy, manual manipulation of the segments was employed as necessary to remove extraneous tissue incorporated in the segments after using the software’s preprogrammed threshold levels.

Before any meshing could be performed, the segmented models from the CBCT were cleaned and optimized in a preprocessing step using 3-Matic software (version 11.0; Materialise). Sharp edges, holes, imperfections, and bumps were removed so that the models were smooth; however, care was taken to maintain their anatomic integrity. During the meshing process, the surface models were meshed using the adaptive remesh feature to provide as many near equilateral triangles (1 mm side length for the maxilla and teeth and 0.50 mm for periodontal ligament) as possible on the model surfaces while still preserving the overall anatomic details.

The models were imported as “orphan” meshes into Abaqus (CAE 2017; Dassault Systèmes, Vélizy-Villacoublay, France) for processing in preparation for the analysis. Material properties for bone and teeth were assigned a Young’s modulus of 17,000 MPa and Poisson’s ratio of 0.30 ( Table I ). The PDL was assigned nonlinear, hyperelastic properties according to the Ogden model proposed by Huang et al (μ1 = 0.07277; α1 = 16.95703; D1 = 3 × 10 ^{−7} ). Specific densities were also assigned to these 3 properties: 1.85 g/cm ^{3} for bone, 2.02 g/cm ^{3} for teeth, and 1.00 g/cm ^{3} for PDL (density of water, as PDL is mostly composed of water). ^{,} The thickness of human PDL has been reported to be around 0.10 to 0.30 mm (average ∼0.20 mm). ^{,} Because of a lower CBCT resolution (0.25-mm voxel size), it was not clearly shown in the images. Thus, the root was identified first, and the PDL was grown from its surface. The PDL was modeled as a fiber-reinforced matrix with a uniform thickness of 0.20 mm.

Structure | Elastic modulus (MPa) | Poisson’s ratio | Specific density (g/cm ^{3 }) |
---|---|---|---|

Teeth | 17,000 | 0.30 | 2.02 |

Bone | 17,000 | 0.30 | 1.85 |

PDL | 0.05 | See text | 1.00 |

Two-node link elements were created to connect the nodes on the root and cortical bone surfaces to simulate the fibers. A 4-node tetrahedral element was used to model the bone and molars. Details of modeling have been reported previously. The block, which included the molars and periodontal ligament, consisted of a mean of 7753 ± 2748 nodes and 38,355 ± 14,910 tetrahedral elements ( Table II ). Nodes lining the socket surfaces of the molar were constrained in 3 degrees of freedom for displacement to most closely model the force system associated with instantaneous movement. Tie constraints were used between PDL and bone and between PDL and tooth to connect the nodes at the material interfaces. Bone rigidity was modeled by simply assigning zero displacements in all directions as boundary conditions for all nodes.

Feature | Mean | Standard deviation | Range |
---|---|---|---|

Tooth length (mm) | 19.30 | 1.47 | 6.54 |

Root length (mm) | 11.10 | 1.55 | 6.23 |

Nodes | 7753 | 2748 | 12,287 |

Elements | 38,355 | 14,910 | 61,206 |

Volume | 1177 | 207 | 838 |

To standardize the force vectors and locate the position of the C _{Res} , a cartesian coordinate system (x-y-z) was constructed and defined by the following orientations: x-axis (buccolingual axis) in the transverse direction with the buccal portion in the positive direction, y-axis (anteroposterior or labio-lingual axis) oriented along the midpalatal suture with the posterior portion in the positive (+) direction, and z-axis (superoinferior or occlusogingival axis) in the vertical direction with the superior or gingival portion of the model in the positive direction. The coordinate system was applied in 3 different ways: (1) a global coordinate system was established with its origin (O) between the facial surfaces of the central incisors occlusal to the incisive papilla located on a line bisecting the interincisor and intermolar widths in the x-y-plane ( Fig 1 ); (2) two reference points were constructed: R and T for each maxillary first molar. The R point/node was defined as the geometric center on the buccal surface of the crown. This site was chosen to approximate the location where an operator typically places a bracket to apply orthodontic forces. The T point was defined as a point/node representing the trifurcation point of the 3 roots. This has historically been used as a reference point for locating C _{Res} .

Steps were assigned to the model for each simulation for the molar. In each step, a 1 Newton (N) concentrated static load (F _{n} ) was applied on a randomly selected node (x) in either the positive y-direction (simulating a distalization force) or the positive z-direction (simulating an intrusive force). After each step, a user-defined algorithm created in Python, was run within the Abaqus interface to analyze the reaction force system and subsequent moments created as a result of load application. The algorithm automatically suggested a new node to apply the load such that a moment of near-zero magnitude is created within the force system. This proceeded in an iterative process until the node location was found to be closest to zero when a force is applied through it ( Fig 2 ). Because of the randomly generated location of the nodes and the inherent distance between any 2 nodes (for this study, approximately 0.5 mm), it was almost impossible to find a location where exactly a zero moment is generated. This algorithm has been previously described in detail.

The location of the C _{Res} is reported as distances from the defined reference points: O, R, and T. Descriptive statistics have been used to analyze the findings. For measurements relative to the T node, *P* values were also calculated. The total tooth length, root length, nodal location of C _{Res} in the buccolingual (x), mesiodistal (y), and occlusogingival (z) directions, and the differences between the calculated A _{Res} in the y and z directions using the finite element method are shown in Tables III-V .

Variables | C _{Res }coordinates (mm) |
||
---|---|---|---|

x | y | z | |

O | |||

Mean | 23.80 | 33.80 | 8.75 |

SD | 3.06 | 3.33 | 2.48 |

SE | 0.433 | 0.471 | 0.35 |

95% CI | 22.90-24.60 | 32.80-34.70 | 8.05-9.46 |

Range | 12.00 | 11.60 | 10.50 |

R | |||

Mean | 4.94 | 2.54 | 7.86 |

SD | 1.39 | 2.70 | 1.66 |

SE | 0.20 | 0.38 | 0.24 |

95% CI | 4.55-5.34 | 1.78-3.31 | 7.39-8.33 |

Range | 6.52 | 11.60 | 7.82 |

T | |||

Mean | 0.14 | 1.48 | 1.48 |

SD | 1.51 | 2.26 | 1.75 |

SE | 0.21 | 0.32 | 0.25 |

95% CI | −0.29 to 0.57 | 0.84-2.13 | −0.31 to 0.69 |

Range | 5.95 | 9.37 | 11.00 |

P |
0.5243 | <0.0001 | 0.4496 |

Significance | NS | ∗ | NS |