Simulation of a three-dimensional craniofacial structure under the application of orthodontic loads

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Simulation of a three-dimensional

craniofacial structure under the

application of orthodontic loads

Joa˜o Correia

, Rui F Martins

and Pedro F Santos

Abstract

The aim of this study was to develop a three-dimensional model of a patient’s craniofacial structure to be analysed using the finite element method in order to estimate the forces required to carry out a dental positioning’s correction. The three-dimensional model was composed by several anatomical structures, namely the teeth, the periodontal ligaments, and the trabecular and cortical bones, which were modelled with the aid of computed tomography cone beam images. The tomographic images were analysed and reconstructed using the 3D Slicer software, while the assembly of the anato-mical structures modelled, as well as the contact surfaces between contiguous parts, was defined in computer-aided design software. Bone remodelling and the occurrence of tissue’s injuries were considered during the numerical simula-tions carried out. By imposing displacements to each tooth, it was possible to calculate the orthodontic loads needed to carry out dental correction (reaction forces), as well as the distributions of stresses and deformations inherent to the clinical treatment, allowing to obtain a craniofacial structure capable of simulating the dental movements of upper and lower arches with anatomical realism. In addition, this methodology constitutes a personalised dental medicine that could lead to the development of highly customised orthodontic appliances where different mechanical loads could be applied individually to each tooth to achieve the foreseen dental correction.

Keywords

Craniofacial modelling, orthodontic forces, finite element method, stresses and strains, customised orthodontic appliances

Date received: 17 February 2018; accepted: 15 April 2018

Introduction

The periodontal ligament (PDL) performs an important role in the orthodontic treatment process, more specifi-cally in dental migration.1–3In fact, appropriate stress levels applied on this structure promote a local inflam-mation, which favours bone remodelling by absorption and deposition of bone tissue in the compressed and distended regions of the PDL, respectively.4–9

Nevertheless, it is important to differentiate ortho-dontic and orthopaedic forces. For that purpose, Silva10quantified the limit for the magnitude’s force in 400 gf (approximately 3.92 N); so, only forces under this value are considered orthodontics. In addition, Lee11 proposed that the optimum stress range to be applied to PDL’s should be comprehended between 165 and 185 gf/cm2 (0.0165–0.0185 N/mm2) to obtain an adequate dental movement rate. An additional cri-terion proposed by Ferreira3 stated that the optimal orthodontic forces to be applied to the PDL should

produce a stress condition slightly higher than the pres-sure exerted by the blood in the capillary vessels, which is about 25 gf/cm2(0.0025 N/mm2), in order to have an efficient orthodontic movement, which means the exis-tence of a minimum stress capable of causing bone remodelling.

In the work herein presented, and having into account the information referred previously, the use of a three-dimensional (3D) tomographic reconstruction

1_{UNIDEMI, Department of Mechanical and Industrial Engineering, Faculty}

of Sciences and Technology, Universidade Nova de Lisboa, Monte de Caparica, Portugal

2_{Egas Moniz Clinic of Orthodontics, Universidade Egas Moniz, Monte de}

Caparica, Portugal Corresponding author:

Rui F Martins, UNIDEMI, Department of Mechanical and Industrial Engineering, Faculty of Sciences and Technology, Universidade Nova de Lisboa, Campus de Caparica, 2829-516 Monte de Caparica, Portugal. Email: [email protected]

technology has allowed modelling the craniofacial ana-tomical structures of a patient using computer-aided design (CAD) software. These biomechanical models were then used as functional prototypes and were stud-ied through computational simulations based on the finite element method (FEM). Hence, the 3D solid models were discretized and evaluated when subjected to the application of specific orthodontic forces. This procedure allowed to quantify the dental movement rates and their centres of rotation, as well as the stresses induced at the PDLs (distribution, maximum and aver-age values), and thus to predict the areas where prefer-ential bone remodelling will take place in the alveolar cavity, as well as the number of load reactivations to be carried out (iterations) on each tooth for its correct repositioning while avoiding tissue’s lesions. In addi-tion, two virtual models developed were combined with computer-aided manufacturing (CAM) technology, more specifically rapid prototyping equipment, using stereolithography apparatus (SLA) technology, for its physical construction.

Materials and methods

Modelling the craniofacial models

To obtain the geometry of the patient’s craniofacial structures, a 3D reconstruction of the images obtained on computed tomography cone beam (CTCB) was used. A total of 598 original axial images were captured in CTCB, each having a thickness of 0.3 mm, which allowed a scanning with a total height of 179.4 mm. In addition, each original axial image was represented by a square matrix having 663 3 663 pixels (663 columns per 663 lines) with a distance between the pixels’ centres equal to 0.3 mm. Consequently, the tomographic exam allowed to obtain a 3D representation with a volume of 198.9 3 198.9 3 179.4 mm3, space enough to obtain a representation of the anatomical areas required for this research. The tomographic images were processed and reconstructed using the software 3D Slicer. With the objective of guaranteeing that all models had all their surfaces free from any flaw, such as open spaces on the surface, disconnected edges and vertices, regions with no thickness, and inter-penetration between adjacent triangular elements, repairing tools from two software were used, namely Blender (for repairing simpler defects) and Meshmixer (for those defects with greater complexity). Meshmixer was also used to smooth the rough appearance of certain surface regions, preserving the congenital geometries, but causing an increase in the number of triangular elements that defined the stereolithographic (STL) models, which could limit the computational processing capacity. To avoid this, the Quadratic Edge Collapse Decimation feature, available in the MeshLab software, was used, which allowed to reconstruct the model, maintaining its original topol-ogy, while choosing the desired number of triangular elements.

The anatomical structures referred previously were then imported from a STL format into a CAD software (SolidWorks), being individually converted into the native file type, and were assembled in its anatomical position. Hence, using the Cavity tool, a highly accu-rate Boolean subtraction operation, which allowed to remove overlapping regions between the models, it was possible to create contact surfaces between contiguous models using Contact Sets defined as bonded and assuming that all nodes, edges and faces in contact were rigidly connected. Two assemblies were created namely the maxillary/cranial and the mandibular struc-tures (Figures 1 and 2, respectively).

Figure 1. Exploded perspective of the components of the skull/maxilla, and their representation in the form of the complete structures (a) and their cross-sectional view (b).

Figure 2. Exploded perspective of the mandibular components, and their representation in the form of the complete structures (a) and their cross-sectional view (b).

Simulation of tooth correction/repositioning

The clinical study used in this research involved 2D computed cephalometric analyses (Slavicek and Sato analyses) and a 3D one (COMPASS), in which angular and linear measurements of the cephalometric tracing were analysed. Hence, the main orthodontic problems in the dental arrangement of both arches were deter-mined, which were the following:

 Need to carry out a dental alignment to correct den-tal crowding and diastemas;

 Need to promote a mandibular advancement to enhance occlusal contact points, namely, the mesial cusp of the mandibular first molar should advance until occluding between the maxillary first molar and the second premolar;

 The distal side of the maxillary canine should slide in the mesial of the maxillary canine and the distal of the maxillary lateral incisor;

 Guarantee enough space for an adequate dental implant to replace the missing tooth no. 36;

 Establish occlusion between upper and lower inci-sors to correct the open bite problem.

Based on the problems evidenced by the cephalo-metric analyses, the individual repositioning of each tooth was carried out using the Translate and/or Rotate tools available in Blender software (Figures 3 and 4). Then, the measurement of linear displacements and angular rotations was carried out with the 3D CAD software SolidWorks, by comparing the initial and the final position of the longitudinal axis of each tooth defined by the intermediate points of the incisal edge and the apical root or, in the case of the molars, between an intermediate point of the bifurcated region and a central point of the occlusal face. The values of the displacements according to the three main Cartesian coordinates (X, Y, Z), as well the rotations

Figure 3. Representation of the symmetry axes (upper and lower) which are (a) non-coincident before dental correction and (b) coincident after the correction.

Figure 4. Representation of the right sagittal view: (a) before dental correction and (b) after dental correction and resolution of malocclusion class II.

(u) along the longitudinal axis, as shown in Figure 5, were measured and are presented in Tables 1 and 2.

Regarding the finite element analysis (FEA) carried out, the craniofacial models, including both mandibu-lar and maxilmandibu-lary structures, were discretized by 3D second-order finite elements, with tetrahedral shape, and the cortical bone models were fixed in areas located far away from the regions where dental displa-cement occurred, as represented in green in Figure 6.

The mandibular structure (Figure 6(a)) had its vol-ume discretized by a total of 754,252 elements, 1,186,998 nodes and 3,556,032 degrees of freedom, while the maxillary/cranial assembly (Figure 6(b)) was defined by 1,233,126 elements, 1,893,543 nodes and 5,627,169 degrees of freedom.

In a first phase, the simulations were carried out considering the total displacements applied in a one sin-gle step (Du, total; Tables 1 and 2). However, this pro-cedure did not simulate bone remodelling process and the tooth’s movement in the alveolar cavity. Hence, to consider the cellular phenomena of bone remodelling, it was decided to divide the total deformation into an integer value of ‘n’ adjustments (for each tooth), each corresponding to one incremental displacement (Du, n), which was multiple of the total (Du, total). Thus, in practice, each ‘n’ increment corresponds to an initial movement that the dental structure carries out inside the alveolar cavity (before bone remodelling phenom-ena occur) to a new equilibrium position, correspond-ing to the balance of the orthodontic force in defeatcorrespond-ing the PDL constraints. The choice of the number of ‘n’ increments needed for the complete movement of each tooth was made considering the following criteria:  The maximum stress registered in the PDL should

be in the range of 0.0165–0.0185 MPa to promote the optimal tooth movement rate, as suggested by Lee.11

 The mean stress value in the PDL should be higher than the minimum stress capable of causing the inflammatory process resulting from bone remodel-ling ( . 0.0025 MPa, according to Ferreira3).  The magnitude of the resulting force applied to the

tooth structure must be in the orthodontic range ( \ 3.92 N), as proposed by Silva.10

In addition, based on the assumption that the anato-mical structures have linear elastic properties, each iteration provided the same deformation condition (Du, n) for the successive application conditions of the same load (DP, n). The mesh definition parameters (Mesh Size and Aspect Element Ratio) of the various models, as well as the linear elastic properties of the PDLs (Poisson’s coefficient, n, and Young’s modulus, E), were chosen to benefit the accuracy of the numerical results obtained. In this way, four different parameters were considered (Table 3) for the definition of the lin-ear elastic properties of the PDLs,12–14and three differ-ent FE meshes, with differdiffer-ent levels of refinemdiffer-ent, from

T able 1. V alues of the displacements and rotations imposed on the longit udinal axis of maxillar y teeth based on virt ual dental corr ection performed in softwa re Blender . T ooth no . D X1 (mm) D Y1 (mm) D Z1 (mm) D X2 (mm) D Y2 (mm) D Z2 (mm) u (° ) Mov ement type 11 0.3501 –1.8647 –0.0733 –1.5273 –1.2658 –0.0194 –17.31 Tipping + bodily motion + rotation 21 0.7955 –0.9683 0.2254 –0.2479 –1.5124 –0.0402 –7.18 Tipping + bodily motion + rotation 12 1.1249 –2.2545 –1.6585 1.3407 –1.9365 –1.4837 –20.9 Bodily motion + extrusion + rotation 22 0.6076 –1.0529 –0.877 1.7088 0.2534 –0.3147 –3.14 V ertical tooth’ s root + bodily motion + rotation 13 1.4392 –3.4425 –1.3994 1.0205 –2.7611 –1.2577 0 Bodily motion + extrusion 23 0.2356 –0.2287 0.5242 0.7892 –0.4897 0.3857 8.36 Bodily motion + int rusion + rotation 14 3.0587 –2.6652 –0.9099 –1.5192 –2.2225 –1.6502 –4.64 Tipping + extrusion + rotation 24 –0.1505 –0.1069 0.08 0.6776 0.1674 0.08 –10.02 Tipping + rotation 15 2.9116 –2.2251 –0.1447 –0.9514 –2.8981 –0.0637 –19.12 Bodily motion + tip ping + rotation 25 –2.2586 0.1524 0.5741 1.3127 –0.2816 0.4752 0.59 Tipping + rotation 16 0 –4.0309 0.39 0 –4.0309 0.39 0 Bodily motion + int rusion 26 0 –1.0208 1.1415 0 –1.0208 1.1415 0 Bodily motion + extrusion 17 0 –3.0996 0.07 0 –3.0996 0.07 0 Bodily motion + int rusion 27 0 –0.4951 –0.722 0 –0.4951 –0.722 0 Bodily motion + extrusion 18 0 –2.9697 0 0 –2.9697 0 0 Bodily motion 28 0 –0.4551 –2.1918 0 –0.4551 –2.1918 0 Extrusion

coarse to fine, were defined, in order to evaluate the convergence of numerical results.

Therefore, five preliminary structural analyses were carried out, each of which related to a type of basic orthodontic loading, namely tipping, bodily, intrusion, extrusion and rotation; two analyses considered a man-dibular central incisor subjected to inclination and translation (Figure 7(a) and (b), respectively), and the remaining to a central maxillary incisor subjected to intrusion, extrusion and rotation (Figure 7(c) and (d), respectively). The main purpose of these studies was to compare the numerical results with bibliographical data to validate the numerical models.2,10,11,15–17

Results and its discussion

Introduction

According to the study carried out by Bright and Rayfield,18the accuracy of numerical simulations varies according to the level of refinement of the finite element meshes and the results are only considered equivalent if they differ by a relative error lesser than 5%. From the validation analyses carried out, it was possible to con-clude that the convergence of the results is strongly influenced by the refinement of the finite element mesh (Table 4) and by the elastic properties of the PDL (Table 3). In fact, the greater the refinement of the FE meshes, the longer the results take to converge and higher the probability of this process to fail. This is associated with the large dimension of the global stiff-ness matrix and the considerable number of degrees of freedom (DOFs) involved in the numerical calculations. In addition, the use of lower Young’s modulus (E) is associated with longer computing times required for the

T able 2. V alues of the displacements and rotations imposed on the longit udinal axis of mandibular teeth based on virtual dental co rr ection performed in softw ar e Blender . T ooth no . D X1 (mm) D Y1 (mm) D Z1 (mm) D X2 (mm) D Y2 (mm) D Z2 (mm) u (° ) Mov ement type 41 –0.1477 –0.9999 –0.0617 –0.0478 –0.65 24 –0.1762 –5.07 Bodily motion + intrusion + rotation 31 0.0342 –0.1739 –0.4738 0.0342 –0.41 59 –0.3812 0 Tipping + intrusion 42 –0.1335 –0.1647 –0.5835 –0.6355 –1.76 15 –0.1867 2.3 Bodily motion + extrusion + rotation 32 –0.0346 0.0592 0.1306 –0.0025 0 0.2508 0 Extrusion 43 –0.3341 0.3587 –0.764 0.3971 0.453 7 –0.8035 22.29 Bodily motion + intrusion + rotation 33 0.1412 0.6187 –0.1414 0.1412 0.141 2 0.6187 –0.1414 Bodily motion + intrusion 44 –0.6399 1.1565 –1.0711 –1.9394 –1.93 94 3.1748 –1.22 Bodily motion + root verticalization + rotation 34 0.9397 2.0706 –0.2 0.9397 0.939 7 2.0706 –0.2 Bodily motion + intrusion 45 –0.428 1.0364 0.1594 –1.7006 –1.70 06 3.8511 –0.3604 Bodily motion + root verticalization + rotation 35 1.1284 2.0974 0.0774 1.0117 1.011 7 1.7197 0.0431 Bodily motion + rotation 46 –1.2298 0.9333 –0.3421 –1.4289 –1.42 89 3.2483 –0.7976 Root verticalization + bodily motion + rotation 47 –0.9578 –7.9807 1.0245 0.4987 0.498 7 –3.6222 –1.143 Root verticalization + bodily motion + rotation 37 0.0701 1.4785 –1.3609 –0.4182 –0.41 82 3.3884 –2.1471 Root verticalization + bodily motion + rotation 38 –1.4234 –6.4661 –0.476 –1.6302 –1.63 02 –4.4377 –0.9791 Root verticalization + bodily motion + rotation

Figure 5. Representation of the axes and definition of displacements and rotations in (a) teeth located in lower arch and (b) teeth located in upper arch.

convergence of results. Beyond the mechanical proper-ties defined in Table 3, other properproper-ties were defined for the cortical and trabecular bones, as well as for the teeth (Table 5).

Comparing the relative errors obtained for the vali-dation analyses carried out with mesh configurations Mesh2and Mesh3, it was verified that the convergence of results reached an acceptable precision for both refinements (relative error \ 5%); it was also found

that Andersen et al.12elastic configuration for the PDL structures showed adequate results. Accordingly, the following parameters were decided to be used in all numerical analyses carried out:

 Andersen et al.12– elastic properties (E = 0.07 MPa; n= 0.49) for the PDL model;

 Mesh2 configuration for the finite elements meshes.

Figure 6. Complete models of the mandibular (a) and cranium-maxillary structures (b), with the FE meshes defined, as well as the respective boundary conditions (green arrows).

Table 3. Different values for Poisson’s coefficient (n) and Young’s modulus (E) applied to the PDL during the validation phase.12–14

Poisson’s coefficient, n Young’s modulus, E (MPa)

Andersen et al.12 0.49 0.07

Cai et al.14 0.49 0.68

Cattaneo et al.13linear low 0.3 0.044

Cattaneo et al.13linear high 0.3 0.175

PDL: periodontal ligament.

Table 4. Level of refinement for the FE meshes defined in the validation analyses carried out.

Mesh Part Mesh size (mm) No. of finite elements No. of DOF

Mesh 1| Mesh 2| Mesh 3 Tooth 2| 1| 0.06 93,573| 122,944| 267,849 429,123| 581,208| 1,248,903

PDL 1| 0.5| 0.03

Cortical and trabecular bones

4| 2| 0.12

PDL: periodontal ligament; DOF: degree of freedom.Element aspect ratio, a/b, equal to 1.

Table 5. Linear elastic properties used in the definition of the isotropic models of cortical/trabecular bones and teeth.19–22

Young’s modulus, E (MPa)

Coefficient of Poisson, n

Density, r (kg/m3) Shear modulus,

G (MPa) Compressive yield stress (MPa) Cortical bone 13.7 3 103 0.3 1900 6500 120 Trabecular bone 1.37 3 103 0.3 900 260 4 Teeth 19.6 3 103 0.3 2450 – –

Validation analyses

Table 6 summarises the results obtained for the appli-cation of the five types of basic orthodontic movement (Figure 7). It was possible to conclude that:

 The application of a tipping force equal to 0.39 N to the buccal face of tooth’s no. 41 induced a maxi-mum displacement of 0.087 mm in its occlusal face, equalling the average value obtained by Jones.16  The imposition of a tipping force of 0.49 N to the

buccal face of tooth’s no. 41 promoted the rotation around its centre of rotation (CR) by an identical value of what was obtained by Moyers22 and Ferreira3and allowed obtaining a maximum stress value of 0.01771 and 0.01552 MPa in the cervical lingual and labial margin of the PDL, respectively, which fit in the necessary stress range to promote tooth movement according to Lee.11

 The application of 0.25 N together with 2.17 N mm at an intermediate point of the buccal face of

tooth’s no. 41 resulted in a stress value of 0.0012 MPa in the apex region of PDL, which is close to the result obtained by Rudolph et al.15  The movement of extrusion applied to tooth no. 11

through the application of a load value of 0.49 N resulted in 0.0017 MPa at the root apex of the PDL; this value was equal to the result obtained by Rudolph et al.15in a similar study.

 The rotation of tooth no. 11 due to the application of four forces of 0.1225 N resulted in a stress value of 0.00131 MPa at the root apex of PDL; this value was equal to the obtained by Rudolph et al.15

Numerical simulation of teeth correction (total

displacement imposed)

A structural analysis was carried out using the FE soft-ware SolidWorks Simulation, in which each teeth was forced to move to its final position in one step time (Du, total), according to the displacements shown in

Figure 7. (a) 0.49 N tipping force applied to a centre point of the buccal surface of tooth no. 41; (b) 0.25 N force associated with a 2.17 N mm moment of force applied to a centre point of the tooth 41’s crown to promote a bodily movement; (c) promotion of an intrusion movement on tooth no. 11 by the application of four forces of 0.0625 N/each in a central plane of its crown; (d) extrusion movement caused by the application of four forces of 0.1225 N in a central plane of the tooth crown no. 11; (e) the rotational movement of tooth no. 11 by the application of four forces of 0.0625 N in an intermediate plane of its crown, perpendicular to the longitudinal tooth’s axis.

Table 6. Numerical results obtained during the validation analyses carried out under the application of basic orthodontic forces.

Tooth’s movement Force (N) Moment of

force (N mm) Maximum tooth’s displacement (mm) Stress in PDL’s root apex region (MPa) Maximum stress in PDL’s lingual/cervical region (MPa) Maximum stress in PDL’s labial/cervical region (MPa)

Tipping (tooth no. 41) 0.49 – 0.1156 0.001759 0.01771 0.01552

0.39 – 0.087 0.001437 0.01346 0.01151

Bodily motion (tooth no. 41) 0.25 2.17 0.0065 0.001194 0.0023332 0.0021086

Intrusion (tooth no. 11) 0.25 – 0.00636 0.001055 0.0023805 0.0018434

Extrusion (tooth no. 11) 0.49 – 0.0121 0.001702 0.002862 0.003863

Rotation (tooth no. 11) 0.49 – 0.024132 0.001311 0.0079014 0.0048369

T able 7. FEM results for mandi bular structur es subjected to total imposed displacements for dental corr ection (D u, total). T ooth no . F or ce Moment of for ce Relation M/F Maximum str ess in PDL (N/mm 2 ) A verage str ess in PDL (N/mm 2 ) Magnitude (N) V ector co mponents Magnitude (N mm) V ector componen ts 41 12.498 ux = 0 :074 uy = 0 :995 uz =0 :0616 87.748 vx = 0 :996 vy =0 :0734 vz =0 :052 7.03 0.294 0.066 31 4.879 ux =0 :0899 uy = 0 :691 uz = 0 :717 9.67 vx = 0 :993 vy = 0 :118 vz = 0 :003 1.94 0.1181 0.039 42 21.066 ux = 0 :323 uy = 0 :946 uz = 0 :027 187.814 vx = 0 :944 vy =0 :3287 vz =0 :0246 8.94 0.488 0.0852 32 3.012 ux =0 :0245 uy = 0 :173 uz =0 :9846 19.37 vx = 0 :934 vy = 0 :355 vz = 0 :028 6.3 0.0922 0.0176 43 14.596 ux = 0 :145 uy =0 :8413 uz = 0 :521 139.419 vx =0 :9898 vy = 0 :092 vz = 0 :109 9.72 0.5787 0.0852 33 14.275 ux =0 :1992 uy =0 :9597 uz = 0 :198 147.234 vx =0 :969 vy = 0 :129 vz =0 :2107 10.31 0.3057 0.0488 44 42.166 ux = 0 :545 uy =0 :7589 uz = 0 :356 485.755 vx =0 :8198 vy =0 :3804 vz = 0 :428 11.55 0.7202 0.1645 34 43.955 ux =0 :4907 uy =0 :8702 uz = 0 :044 419.01 vx =0 :8245 vy = 0 :43 vz =0 :368 9.55 0.9276 0.1381 45 43.82 ux = 0 :288 uy =0 :9402 uz = 0 :183 467.131 vx =0 :9416 vy =0 :2596 vz = 0 :215 10.64 0.7147 0.1289 35 89.725 ux =0 :5366 uy =0 :8434 uz =0 :029 857.787 vx =0 :7561 vy = 0 :504 vz =0 :418 9.57 1.424 0.2019 46 81.576 ux = 0 :445 uy =0 :8458 uz = 0 :294 1034.363 vx =0 :9336 vy =0 :3502 vz = 0 :075 12.75 0.9687 0.2027 47 93.386 ux =0 :3748 uy = 0 :921 uz = 0 :107 445.89 vx = 0 :708 vy = 0 :679 vz =0 :1953 4.83 1.353 0.3328 37 74.176 ux = 0 :135 uy =0 :8224 uz = 0 :553 694.963 vx =0 :9797 vy =0 :041 vz = 0 :196 9.32 1.1405 0.2341 38 89.805 ux = 0 :237 uy = 0 :969 uz = 0 :072 636.22 vx = 0 :854 vy =0 :2509 vz = 0 :456 7.1 1.7102 0.3128 FEM: fi nite elemen t method; PDL: pe riodon tal lig amen t.The vo n Mises str es ses indu ced in the PDLs w e re obtain ed usin g the tool ‘Pr obe Result On se lect ed en tities’. T able 8. FEM results for maxillar y structur e s subjected to total imposed d isplacements for dental corr ection (D u, total). Tooth Fo rc e Moment of for ce Relation M/F Maximum str ess in PDL (N/mm 2 ) A verage str ess in PDL (N/mm 2 ) M agnitude (N) V ector componen ts Magnitude (N mm) V ector components 11 21.07 9 ux = 0 :086 uy = 0 :939 uz = 0 :332 139.333 vx =0 :991 vy = 0 :13 vz = 0 :031 6.61 0.3484 0.06472 21 32.25 6 ux = 0 :406 uy = 0 :884 uz = 0 :233 344.244 vx =0 :8604 vy = 0 :291 vz = 0 :418 10.67 0.3473 0.0761 12 70.24 2 ux =0 :3872 uy = 0 :83 uz = 0 :401 720.302 vx =0 :886 vy =0 :1912 vz =0 :4224 10.25 1.0312 0.1777 22 31.64 1 ux =0 :9734 uy = 0 :145 uz = 0 :177 414.183 vx =0 :0844 vy =0 :8199 vz =0 :5663 13.09 0.4792 0.0766 13 79.55 8 ux =0 :3723 uy = 0 :876 uz = 0 :307 847.854 vx =0 :8569 vy =0 :1715 vz =0 :486 10.66 0.77 0.1771 23 17.32 3 ux =0 :8647 uy = 0 :416 uz =0 :2811 203.115 vx =0 :114 vy =0 :8702 vz =0 :4793 11.73 0.3931 0.0611 14 77.06 6 ux =0 :0701 uy = 0 :907 uz = 0 :415 710.976 vx =0 :8488 vy = 0 :308 vz =0 :4297 9.23 1.202 0.17796 24 20.42 9 ux =0 :9609 uy = 0 :201 uz = 0 :191 255.821 vx =0 :1981 vy =0 :8629 vz =0 :4648 12.52 0.3129 0.04577 15 54.02 6 ux =0 :1444 uy = 0 :981 uz = 0 :13 582.768 vx =0 :9779 vy =0 :0317 vz =0 :2065 10.79 1.1423 0.16155 25 15.66 1 ux = 0 :798 uy =0 :1807 uz =0 :5747 95.543 vx = 0 :546 vy = 0 :629 vz = 0 :553 6.1 0.43263 0.06442 16 88.38 6 ux = 0 :005 uy = 0 :993 uz =0 :1177 727.89 vx =0 :8555 vy =0 :0985 vz =0 :5084 8.24 1.1192 0.20492 26 44.11 ux =0 :1971 uy = 0 :769 uz =0 :6076 256.994 vx =0 :8053 vy = 0 :368 vz = 0 :465 5.83 0.33966 0.09683 17 106 .521 ux =0 :1589 uy = 0 :987 uz = 0 :035 1038.005 vx =0 :7405 vy =0 :1124 vz =0 :6626 9.74 1.2251 0.17893 27 11.92 6 ux =0 :3675 uy = 0 :927 uz =0 :0803 121.386 vx =0 :9359 vy =0 :0118 vz = 0 :352 10.18 0.28232 0.05054 18 74.8 ux = 0 :06 uy = 0 :985 uz =0 :159 596.012 vx =0 :6656 vy =0 :0725 vz =0 :7428 7.97 1.254 0.1899 28 28.07 3 ux =0 :0602 uy = 0 :446 uz = 0 :893 190.896 vx =0 :2546 vy =0 :8694 vz = 0 :423 6.8 0.59496 0.1451 FEM: finite ele ment method; PDL: periodon tal ligament.The vo n Mise s str esses induce d in the P DLs w e re obt ained using the tool ‘Pr obe Result O n select ed enti ties’.

equivalent stress values (von Mises) in the PDLs were calculated, as well as the load (DP, total) to be applied for each imposed displacement (reaction forces: force and moment of force). In Figure 8, the components of the forces/moment of forces are represented in a sche-matic view and the values of the above-mentioned vari-ables are presented in Tvari-ables 7 and 8, for the mandibular and maxillary structures, respectively; in Figures 9 and 10 are shown the teeth displacement and the equivalent stress distributions in each PDL (man-dibular and maxillary), respectively.

The number of iterations, n, that each tooth must move in each increment (Du, n) in order to reach the total migration (Du, total) is presented in Table 9, and that guarantees the coexistence of the three essential conditions in the PDL, in each iterative process, namely the maximum stress in the range of 0.0165–0.0185 MPa and an average stress greater than 0.0025 MPa, as well as the application of an orthodontic force magnitude lower than 3.92 N. As an example, for the mandibular right central incisor (tooth no. 41, Table 9), a value ‘n’ of 17 iterations was defined, meaning that the total dis-placement (Du, total) for this tooth should be divided

Figure 8. Representation of the resultant force vector (F) and

its components (FX, FY, FZ), as well as the vector of moment of

force (M) and its respective vector components (MX, MY, MZ)

applied at an intermediate point of the vestibular face of the dental crown, which coincides with the Cartesian coordinate system’s origin.

Figure 9. Representation of the total displacements induced in the mandibular (a) and maxillary (b) structures at the end of the FEM simulation process.

Figure 10. Distribution of equivalent stresses (von Mises) in the periodontal ligaments of the mandibular (a) and maxillary (b) structures due to the imposition of total displacements on the respective teeth (Tables 1 and 2, respectively).

T able 9. Estimativ e of the av erage and maximum str esses induced in the PDL, in each iteration, wher e each toot h is for ced to carr y out an im posed displacement (D u, n) under the application of a for ce and moment of for ce (D P, n). T ooth no . Number of iterations needed (n) Maximum stress induced in PDL for each iteration (MPa) A verage str ess induced in PDL for each iteration (MPa ) Magnitude of for ce to be applied in each iteration (N) Moment of for ce to be induced in each iteration (N mm) 11 19 0.01834 0.00341 1.1094 7.3333 21 20 0.01736 0.00381 1.6128 17.2122 12 57 0.01809 0.00312 1.2323 12.6369 22 27 0.01775 0.00284 1.1719 15.3401 13 42 0.01833 0.00422 1.8942 20.187 23 22 0.01787 0.00278 0.7874 9.2325 14 66 0.01821 0.0027 1.1677 10.7724 24 17 0.01841 0.00269 1.2017 15.0483 15 62 0.01842 0.00261 0.8714 9.3995 25 24 0.01803 0.00268 0.6525 3.981 16 61 0.01835 0.00336 1.449 11.9326 26 20 0.01698 0.00484 2.2055 12.85 17 67 0.01829 0.00267 1.5899 15.4926 27 16 0.01765 0.00316 0.7454 7.5866 18 68 0.01844 0.00279 1.1 8.7649 28 33 0.01803 0.0044 0.8507 5.7847 41 17 0.01729 0.00388 0.7352 5.1616 31 7 0.01687 0.00557 0.697 1.3814 42 27 0.01808 0.00316 0.7802 6.9561 32 5 0.01843 0.00352 0.6024 3.874 43 32 0.01808 0.00266 0.4561 4.3568 33 17 0.01799 0.00287 0.8397 8.6608 44 42 0.01715 0.00392 1.004 11.5656 34 51 0.01819 0.00271 0.8619 8.2159 45 40 0.01787 0.00322 1.0955 11.6783 35 77 0.01849 0.00262 1.1653 11.1401 46 53 0.01828 0.00382 1.5392 19.5163 47 80 0.016913 0.00416 1.1673 5.5736 37 68 0.016772 0.00344 1.0908 10.22 38 95 0.018002 0.00329 0.9453 6.6971 PDL: per iodon tal lig ament .

T able 10. Results obtained in each mandibular tooth when subjected to the first iterativ e displacement of dental corr ection (D u, 1). T ooth F o rc e Moment of for ce Relation M/F Maximum str ess in PDL (N/mm 2 ) A verage str ess PDL (N/mm 2 ) Magnitude (N) V ector components Magnitude (N mm) V ector components 41 0.734 (0.16 %) ux = 0 :074 uy = 0 :996 uz =0 :0558 5.158 (0.07%) vx = 0 :996 vy =0 :0759 vz =0 :0519 7.03 0.0172 (0.52%) 0.00389 (0.26%) 31 0.691 (0.86 %) ux =0 :0899 uy = 0 :691 uz = 0 :717 1.34 (3%) vx = 0 :992 vy = 0 :127 vz = 0 :005 1.94 0.0169 (0.18%) 0.0056 (0.54 %) 42 0.781 (0.1% ) ux = 0 :324 uy = 0 :946 uz = 0 :024 6.986 (0.43%) vx = 0 :944 vy =0 :3293 vz =0 :0238 8.94 0.01814 (0.33%) 0.00316 (0%) 32 0.597 (0.9% ) ux = 0 :017 uy = 0 :158 uz =0 :9873 3.76 (2.94%) vx = 0 :962 vy = 0 :262 vz = 0 :07 6.3 0.01817 (1.41%) 0.0035 (0.57 %) 43 0.452 (0.9% ) ux = 0 :148 uy =0 :8401 uz = 0 :522 4.393 (0.83%) vx =0 :9902 vy = 0 :091 vz = 0 :106 9.72 0.01806 (0.11%) 0.00266 (0%) 33 0.844 (0.51 %) ux =0 :1987 uy =0 :9596 uz = 0 :199 8.704 (0.5%) vx =0 :9691 vy = 0 :128 vz =0 :2106 10.31 0.01757 (2.33%) 0.00289 (0.7%) 44 1 (0.39%) ux = 0 :543 uy =0 :7606 uz = 0 :355 11.554 (0.1%) vx =0 :8162 vy =0 :3885 vz = 0 :428 11.55 0.01716 (0.06%) 0.00392 (0%) 34 0.861 (0.1% ) ux =0 :4914 uy =0 :8698 uz = 0 :044 8.221 (0.06%) vx =0 :8247 vy = 0 :43 vz =0 :367 5 9.55 0.01818 (0.05%) 0.00271 (0%) 45 1.106 (0.96 %) ux = 0 :285 uy =0 :9414 uz = 0 :181 11.771 (0.79%) vx =0 :942 vy =0 :258 vz = 0 :215 10.64 0.01788 (0.06%) 0.00323 (0.31%) 35 1.164 (0.11 %) ux =0 :5366 uy =0 :8433 uz =0 :0292 11.14 (0%) vx =0 :7562 vy = 0 :504 vz =0 :4176 9.57 0.01899 (2.7%) 0.00259 (1.15%) 46 1.554 (0.96 %) ux = 0 :44 uy =0 :836 uz = 0 :328 19.81 (1.5%) vx =0 :9362 vy =0 :3437 vz = 0 :074 12.75 0.01828 (0%) 0.00383 (0.26%) 47 1.171 (0.32 %) ux =0 :3716 uy = 0 :923 uz = 0 :103 5.657 (1.5%) vx = 0 :721 vy = 0 :666 vz =0 :1923 4.83 0.01695 (0.22%) 0.00415 (0.24%) 37 1.086 (0.44 %) ux = 0 :134 uy =0 :8288 uz = 0 :543 10.122 (0.96%) vx =0 :9801 vy =0 :0407 vz = 0 :194 9.32 0.01677 (0.01%) 0.00344 (0%) 38 0.946 (0.07 %) ux = 0 :224 uy = 0 :972 uz = 0 :064 6.714 (0.25%) vx = 0 :86 vy =0 :2473 vz = 0 :447 7.1 0.018 (0.01%) 0.00329 (0%) PDL: per iodon tal lig ament .Relativ e err ors betw een the nume rical results and those ideally ca lculated in T able 10 . T able 11. Results obtained in each maxillar y toot h when subjected to the first iterativ e displacement of dental corr ection (D u, 1). T ooth Fo rc e Moment of for ce Relation M/F Maximum str ess in P D L (N/mm 2 ) A verage str ess in PDL (N/mm M agnitude (N) V ector componen ts Magnitude (N mm) V ector componen ts 11 1.101 (0.76%) ux = 0 :075 uy = 0 :936 uz = 0 :345 7.38 (0.64 %) vx =0 :9947 vy = 0 :103 vz = 0 :008 6.7 0.01845 (0.6%) 0.00349 (2.35%) 21 1.565 (2.96%) ux = 0 :402 uy = 0 :894 uz = 0 :195 16.903 (1.8%) vx =0 :8559 vy = 0 :293 vz = 0 :426 10.8 0.01783 (2.71%) 0.00392 (2.89%) 12 1.223 (0.76%) ux =0 :3688 uy = 0 :832 uz = 0 :415 12.62 (0.13%) vx =0 :8844 vy =0 :196 vz =0 :4235 10.32 0.01791 (1%) 0.0031 (0.32 %) 22 1.188 3 (1.4%) ux =0 :9602 uy = 0 :15 uz = 0 :236 15.338 (0.01%) vx =0 :1514 vy =0 :8102 vz =0 :5663 12.91 0.01792 (0.96%) 0.00286 (0.32%) 13 1.894 (0.01%) ux =0 :3732 uy = 0 :876 uz = 0 :306 20.175 (0.06%) vx =0 :8562 vy =0 :173 vz =0 :4868 10.65 0.01842 (0.49%) 0.00423 (0.24%) 23 0.791 5 (0.52%) ux =0 :868 uy = 0 :416 uz =0 :2715 9.151 (0.88%) vx =0 :1242 vy =0 :8693 vz =0 :4784 11.56 0.01777 (0.56%) 0.00278 (0%) 14 1.15 (1.52 %) ux =0 :0713 uy = 0 :925 uz = 0 :374 10.657 (1.07%) vx =0 :8515 vy = 0 :29 vz =0 :4365 9.27 0.01811 (0.55%) 0.00268 (0.74%) 24 1.165 (3.05%) ux =0 :9654 uy = 0 :184 uz = 0 :185 14.969 (0.53%) vx =0 :1808 vy =0 :8598 vz =0 :4776 12.85 0.01822 (1.03%) 0.00269 (0%) 15 0.872 (0.07%) ux =0 :1513 uy = 0 :98 uz = 0 :127 9.354 (0.48%) vx =0 :976 vy =0 :0337 vz =0 :2152 10.73 0.01832 (0.54%) 0.0026 (0.38 %) 25 0.656 (0.54%) ux = 0 :8 uy =0 :1799 uz =0 :5718 3.97 (0.28 %) vx = 0 :539 vy = 0 :635 vz = 0 :553 6.052 0.01794 (0.5%) 0.00268 (0%) 16 1.442 (0.48%) ux =0 :031 uy = 0 :993 uz =0 :113 12.01 (0.65%) vx =0 :8494 vy =0 :091 vz =0 :5198 8.329 0.01833 (0.11%) 0.00336 (0%) 26 2.191 (0.66%) ux =0 :1931 uy = 0 :768 uz =0 :6108 12.765 (0.66%) vx =0 :8021 vy = 0 :372 vz = 0 :467 5.826 0.01711 (0.77%) 0.00485 (0.21%) 17 1.595 (0.32%) ux =0 :1561 uy = 0 :987 uz = 0 :034 15.568 (0.49%) vx =0 :7409 vy =0 :1158 vz =0 :6615 9.761 0.01809 (1.09%) 0.00266 (0.37%) 27 0.75 (0.62 %) ux =0 :3614 uy = 0 :932 uz =0 :0173 7.668 (1.07%) vx =0 :9368 vy =0 :0265 vz = 0 :349 10.224 0.01777 (0.68%) 0.00317 (0.32%) 18 1.09 (0.91 %) ux = 0 :062 uy = 0 :997 uz = 0 :044 8.586 (2.04%) vx =0 :6425 vy =0 :0941 vz =0 :7605 7.877 0.01835 (0.49%) 0.00278 (0.36%) 28 0.853 (0.27%) ux =0 :061 uy = 0 :447 uz = 0 :893 5.798 (0.23%) vx =0 :257 vy =0 :8681 vz = 0 :425 6.797 0.01812 (0.5%) 0.0044 (0%) PDL: pe riodon tal lig amen t.Relativ e err ors betw een the nume rical results and those ideally ca lculated in T able 10.

into 17 sequential displacements of equal value (Du, n), resulting in 17 orthodontic loads of equal value to be applied (DP, n). This model also assumes that the reac-tivation of the orthodontic forces occurs at the end of each iteration and the model is defined in the linear elastic regime. Hence, the maximum and average von Mises equivalent stresses in the PDL, for each tooth and single iteration, is also given in Table 9.

To ascertain the validity of imposing partial displa-cement to each tooth (Tables 1, 2 and 9), several numer-ical simulations were carried out for the first iteration required (Du, 1). For this purpose, the displacements indicated in Tables 1 and 2 were divided by the number of iterations, n, indicated in Table 9 for each maxillary and mandibular tooth. Hence, the loads (force and moment of force) needed to apply to each tooth were obtained, as well as the values of the resulting stresses (maximum and average) in each PDL. These values are shown in Tables 10 and 11 for the mandibular and max-illary structures, respectively.

Comparing the required orthodontic loads and the maximum and average stresses estimated in the PDL (presented in Table 9), with the numerical results included in Tables 10 and 11, it was possible to verify a maximum relative error of 3%. Thus, it can be con-cluded that the division of the total displacements by ‘n’ iterations is valid for the considered hypotheses, making the incremental displacement model an accep-table way to foresee the computational dental correc-tion process, which allows a suitable approximacorrec-tion of bone remodelling phenomena.

Rapid prototyping

The research herein presented can lead to the develop-ment of highly customised orthodontic appliances, where different mechanical loads could be individually applied to each tooth. A customised appliance was 3D printed using a SLA machine in order to impose the displacements to the patients’ teeth (first iteration) (Figure 11).

Conclusion

The aim of this study was to estimate the forces required to carry out dental positioning’s correction using the FEM. By imposing displacements to each tooth, the methodology developed allowed to quantify the allowable orthodontic loads and number of reacti-vations to apply in each dental structure to allow dental migration process. All the anatomical models were free of any geometric error, being able to be used in the numerical simulations and to be processed using rapid prototyping.

The validation of numerical results and the justifica-tion for the procedures were made via reference to some papers, either of clinical or computational type, in order to guarantee the coexistence of three conditions in the PDLs in each iterative process, namely a maximum stress in the range of 0.0165–0.0185 MPa and an aver-age stress greater than 0.0025 MPa, as well as the appli-cation of an orthodontic force lower than 3.92 N. Several iterations were, therefore, foreseen for dental correction and an orthodontic appliance, corresponding to the first iteration calculated, was 3D printed. Hence, the research herein presented can lead to the develop-ment of highly customised orthodontic appliances where different mechanical loads can be individually applied to each tooth without putting in risk the period-ontal tissues. This should allow the overall desired movement of the upper and lower arches and assure proof of concept. In addition, this work can contribute to decrease the duration of the orthodontic treatment and to increase its efficiency. Nevertheless, biological and anatomical confirmation of results will be gathered in the future, in order to validate the numerical work herein presented.

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: All authors affirm that they have no financial affiliation or involvement

Figure 11. Customised orthodontic appliance produced with SLA technology and referred to the first iteration in the patients’ maxillary teeth correction (a) Virtual 3D prototype used to generate the layered model used during 3D printing (b) Manufactured part.

interest in the subject or materials discussed in this arti-cle, nor have any such arrangements existed in the past 3 years. The authors deny any conflicts of interest. Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article. ORCID iD

Rui F Martins https://orcid.org/0000-0001-8155-0079

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