## Introduction

Orthodontically induced inflammatory root resorption (OIIRR) constitutes an undesirable risk connected to orthodontic treatment. Finite element analysis (FEA) is a powerful tool to study the risk of OIIRR. However, its efficiency in predicting OIIRR depends on the insertion of the correct inputs and the selection of an output coherent with the clinical failure mechanism.

## Methods

By combining a systematic review with a 3-dimensional FEA, this article discusses which are the implications of using certain periodontal ligament (PDL) properties (linear and nonlinear models) and failure criteria. Six orthodontic loading regimes were simulated in a maxillary premolar: pure intrusion, buccal tipping, and their combination applied with either a *light* (25 cN) or a *heavy* (225 cN) force. Three stress parameters in the PDL were compared: von Mises stress, minimum principal stress _{,} and hydrostatic stress (σ _{H} ).

## Results

The comparison between linear and nonlinear models showed notable differences in stress distribution patterns and magnitudes. For the nonlinear PDL, none of the light-force models reached the critical compressive hydrostatic stress of 4.7 kPa, whereas all the heavy-force models reached it. In addition, the regions of critical compressive σ _{H} matched with the regions with resorption craters in clinical studies. In linear models, the σ _{H} critical value of 4.7 kPa was reached even in the light-force scenario.

## Conclusions

Only compressive hydrostatic stress in PDL satisfied the requirements to be used as an FEA indicator of OIIRR. However, the requirements were satisfied only when a nonlinear PDL model was considered.

## Highlights

- •
A nonlinear PDL model should be considered to study the risk of root resorption.

- •
Hydrostatic stress is the most appropriate criterion to study root resorption by FEA.

- •
In FEA studies of root resorption, the area of interest should be the PDL and not the root.

Orthodontically induced inflammatory root resorption (OIIRR) constitutes an undesirable risk connected to orthodontic treatment. Its etiology is multifactorial and still not fully understood. Several factors are believed to influence its occurrence, including genetic factors, orthodontic force magnitude, treatment duration, and type of loading regime (eg, continuous vs intermittent forces). ^{,}

Already in 1932, Schwarz associated the occurrence of root resorption with a pressure in the periodontal ligament (PDL) that exceeds capillary blood pressure. Overcompression of the PDL generates a hyalinized zone because of ischemia and necrosis of this tissue and the adjacent cement and alveolar bone. Root resorption can be activated by cells close to the necrotic zone, where tissue response is still viable. Therefore, the positive correlation between increased orthodontic force level and increased root resorption incidence has been associated with the degree of PDL blood obstruction. ^{,}

Nevertheless, the root resorption risk evaluation based only on the intensity of the applied orthodontic force is doubtful because the critical mechanical stimulus to the OIIRR biologic reactions is not the force per se but the stress levels in the PDL. ^{,} The PDL stress depends not only on the applied orthodontic force but also on the anatomic particularities, the displacement restrictions of the dentoalveolar system, and the mechanical behavior of the PDL. Although the local PDL stress cannot be predicted clinically or experimentally, finite element analysis (FEA) can accurately assess it.

The advances in the computer field and medical imaging technology, associated with the current incentive of translational research in health science, explain a large number of OIIRR FEA studies in the last decade. Although the FEA is a valuable tool to study biomechanical issues, its accuracy in predicting clinical outcomes depends on the ability of the user to insert the correct inputs and to select an output coherent with the failure mechanism of interest for the analysis. The present study aimed to critically analyze some input and output parameters previously used to study orthodontically induced root resorption in silico, particularly in relation to the PDL representation and the failure criteria selection. The importance of these parameters is sometimes neglected by the orthodontists who are not familiar with the stress complex concepts that support a precise mechanical analysis or misjudged by an engineer who is not familiar with the biological mechanism of the clinical failure.

In addition to the mechanical consistency of the PDL representation and the coherency between the proposed failure criteria and the biological root resorption mechanism, it is desirable that the regions with the highest stress level match the areas with greater signs of resorption in clinical studies when comparable force systems are applied. From the strong evidence of clinical studies ^{,} ^{,} in a previous systematic review, it could be observed that a low risk of OIIRR is expected for a premolar under a 25 cN orthodontic loading, whereas when a 225 cN force is applied, specific resorption areas could be anticipated depending on tooth movement. Specifically, buccal tipping was associated with resorption mostly on the buccocervical and linguoapical regions, ^{,} whereas intrusion was associated mostly at the apical third. ^{,} It is worth noticing that some applied force systems fail to control the desired movement. When intrusive forces are applied away from the center of resistance of the tooth, the resulting movement is a combination of intrusion and tipping.

Because of some inconsistencies in finite element (FE) models, divergences between FEA and clinical outcome appear and hamper the acceptance of FE studies by orthodontics. Therefore, the current recommendation of orthodontic biomechanical prescriptions is still based only on clinical studies and limits to suggest the use of *light forces* instead of *heavy ones* . By highlighting the importance of some FEA input and output parameters, this article discusses the implications of using certain PDL models and failure criteria, as well as which parameters are more appropriated in OIIRR FE studies. Moreover, this article encourages the development of accurate and evidence-based biomechanical prescriptions, with the aim of optimizing the speed of human tooth movement while preventing root resorption.

## Material and methods

A systematic search was performed to select the published articles that evaluated the risk of OIIRR by using clinical FE models. Three electronic databases were assessed: PubMed, Web of Science, and Scopus ( Fig 1 ). The inclusion and exclusion criteria were applied in 2 different steps (screening and eligibility) by 2 of the authors (J.B.C.M and O.M.U). The selected studies underwent data extraction for identification purposes and FEA inputs and outputs assessment. The flow diagram of the systematic search was plotted in Figure 1 , and Table I summarizes the extraction data of the included articles.

Study | Input – PDL representation | Failure criteria interpretation | ||||
---|---|---|---|---|---|---|

Constitutive model | E (MPa) | ν | Chosen output | Critical value (kPa) | Considered the vessel obstruction | |

Jeon et al (1999) | Linear-elastic | 0.68 | 0.49 | σ _{3 } |
No | |

Jeon et al (2001) | Linear-elastic | 0.68 | 0.49 | σ _{3 } |
No | |

Rudolph (2001) | Linear-elastic | 0.69 | 0.45 | σ _{VM }in the tooth |
No | |

Dorow and Sander (2005) | Linear-elastic | 0.01 0.1 1 |
0.45 | Compressive σ _{H } |
4.7 | Yes |

Sander et al (2006) | Linear-elastic | 0.1 | 0.45 | σ _{H }and principal stresses |
4.7 | Yes |

Hohmann et al (2007) | Linear-elastic | 0.1 | 0.45 | σ _{H }pressure |
4.7 | Yes |

Oyama et al (2007) | Linear-elastic | 0.75 | 0.45 | σ _{VM }in the root |
No | |

Viecilli et al (2008) | Linear-elastic | 0.05 | 0.3 | σ _{1 }, σ _{2 }, σ _{3, }σ _{x }, σ _{y }, σ _{z }, σ _{H } |
No | |

Field et al (2009) | Linear-elastic | 1.18 | 0.45 | σ _{VM }and σ _{1 } |
16 | Yes |

Hohmann et al (2009) | Linear-elastic | 0.1 | 0.45 | σ _{H } |
4.7 | Yes |

Çifter and Saraç (2011) | Linear-elastic | 0.6668 | 0.49 | σ _{VM }in the tooth |
No | |

Kanjanaouthai (2012) | Linear-elastic | 0.68 | 0.49 | σ _{1 }and σ _{3 }in the PDL |
Yes | |

Jing et al (2013) | Linear-elastic | 0.68 | 0.49 | σ _{VM }in the root and PDL |
No | |

Hemanth et al (2015) | Linear-elastic | 0.69 | 0.45 | Principal stress | No | |

Hemanth et al (2015) | Nonlinear | Principal stress | No | |||

Salehi et al (2015) | Linear-elastic | 0.68 | 0.49 | σ _{VM }in the root |
No | |

Jiang et al (2015) | Linear-elastic | 0.5 | 0.45 | σ _{1 }, σ _{3 }, σ _{H }and σ _{VM }in the PDL, root, and alveolar bone |
No | |

Choi et al (2016) | Linear-elastic | 0.05 | 0.49 | σ _{1 }and σ _{3 }in the PDL |
No | |

Liao et al (2016) | Hyperelastic | σ _{H } |
16 | Yes | ||

Papageorgiou et al (2017) ^{, } |
Bilinear | 0.05 0.20 |
0.3 | Von Mises strain in the PDL | No | |

Sugii et al (2018) | Linear-elastic | 50 | 0.49 | σ _{3 } |
No | |

Wu et al (2018) | Linear-elastic | 0.68 | 0.49 | σ _{H } |
Yes | |

Wu et al (2018) | Hyperelastic | σ _{H } |
12.8 | Yes | ||

Long et al (2018) | Linear-elastic | 50 | 0.45 | σ _{VM }in PDL and root |
26 | No |

Zhong et al (2019) | Hyperelastic | σ _{H }in the PDL |
4.7 | Yes |

A 3-dimensional FE model was constructed with axisymmetric geometry to simulate a premolar with its surrounding tissues, including enamel, dentin, cementum, PDL, cortical, and trabecular bone ( Fig 2 ). Material properties were obtained from data previously published. ^{,} All materials were considered elastic, isotropic, linear, and homogeneous, except for the PDL that was modeled as a nonlinear material ( Fig 2 ). The nonlinear model used in the present study was based on Cattaneo et al. In compression, the PDL was represented with a very low elastic modulus (0.005 MPa) up to the 93% strain level, after which an elastic modulus of 8.5 MPa was used to simulate precontact between the roots and the surrounding bone. In tension, the elastic modulus gradually increased from 0.044 MPa at zero strain level to 0.44 MPa at about 50% strain, after which a smaller elastic modulus of 0.032 MPa was used to simulate fiber disruption. The stress-strain curve of nonlinear PDL was plotted.