Effects of Increasing the Orthodontic Forces over Cortical and Trabecular Bone during Periodontal Breakdown—A Finite Elements Analysis

Background and Objectives: Herein we used numerical analysis to study different biomechanical behaviors of mandibular bone subjected to 0.6 N, 1.2 N, and 2.4 N orthodontic loads during 0–8 mm periodontal breakdown using the Tresca failure criterion. Additionally, correlations with earlier FEA reports found potential ischemic and resorptive risks. Materials and Methods: Eighty-one models (nine patients) and 243 simulations (intrusion, extrusion, rotation, tipping, and translation) were analyzed. Results: Intrusion and extrusion displayed after 4 mm bone loss showed extended stress display in the apical and middle third alveolar sockets, showing higher ischemic and resorptive risks for 0.6 N. Rotation, translation, and tipping displayed the highest stress amounts, and cervical-third stress with higher ischemic and resorptive risks after 4 mm loss for 0.6 N. Conclusions: Quantitatively, rotation, translation, and tipping are the most stressful movements. All three applied forces produced similar stress-display areas for all movements and bone levels. The stress doubled for 1.2 N and quadrupled for 2.4 N when compared with 0.6 N. The differences between the three loads consisted of the stress amounts displayed in color-coded areas, while their location and extension remained constant. Since the MHP was exceeded, a reduction in the applied force to under 0.6 N (after 4 mm of bone loss) is recommended for reducing ischemic and resorptive risks. The stress-display pattern correlated with horizontal periodontal-breakdown simulations.


Introduction
Bone structure (i.e., cortical, and trabecular bone) along with periodontal ligament (PDL) are the surrounding supporting tissues of the tooth and directly influenced by both orthodontic forces and the periodontal breakdown process [1][2][3].The two components of bone should be analyzed together since clinically they behave as a continuum and their biomechanical behavior in both the intact and reduced periodontium is multifactorial, depending on the cortical and trabecular structural continuum, material and structural properties, and internal micro-architectural changes [3][4][5][6][7][8][9][10]. Trabecular bone is better vascularized and innervated (also holding the bone marrow, with a regenerative function) than the cortical component (with more structural support and a protective function) [3,7,8,11].
The effects of orthodontic movements over the stress-extension areas in bone are variable (i.e., some are biomechanically more stressful), with various inconclusive reports of intrusion/extrusion [22], and translation and tipping [1,20,21] to be more stressful for the intact periodontium.No information was found for the reduced periodontium except our previous comparative studies reporting for both the intact and reduced periodontium that rotation and translation [2,10,[12][13][14]17,18] are more stressful than the other movements, with increasing effects correlated with bone-loss progression.
Clinically during orthodontic treatment various levels of bone loss are found, signaling various levels of ischemic and resorptive risks that could alter the prognosis of the treatment if not managed with care.The only available noninvasive method of investigation for their biomechanical behavior is FEA analysis (i.e., a mathematically based algorithm method allowing an individual assessment of each component of a living structure) [2,10,[12][13][14]17,18,23,24,27,29].
The FEA method is widely used in the engineering field with excellent accurate results, since its use is well documented and correlated with the type of analyzed material (i.e., each material has a specific designed failure criteria that better describes its biomechanical behavior, e.g., ductile, brittle, liquid, or gas according to yielding theory [10,[12][13][14]).There are reports that only an FEA numerical simulation enables accurate biomechanical studies assessing and predicting stress distribution in living dental tissues [23,24,27,29].
The well-vascularized dental tissues (PDL, NVB and dental pulp) are more sensitive to ischemia and resorptive process than the less-vascularized ones (dentine, cementum, cortical and trabecular bone), that are capable of withstanding a higher quantitative stress for a longer period.Thus, this is the explanation for the biomechanical behavior of the absorption-dissipation mechanism under light orthodontic forces that determine stresses lower than MHP in PDL, NVB and dental pulp, and higher quantitative values in dentine, cement, and bone [2,10,[12][13][14]17,18].
For enabling the correlations of the results, only four bone-tooth numerical studies [1,[20][21][22] were found, for the intact periodontium and with one or two movements, reporting contradicting results: intrusion/extrusion vs. tipping as more stressful, variable amounts of stress and extension areas for the same movements and applied forces, PDL stresses exceeding the 16 KPa of MHP, no correlations with the MHP and no mention of the failure-criteria suitability issues.
Our step-by-step previous research [2,10,[12][13][14]17,18] investigated the employment of proper failure-criteria material types based on investigating each component of dental tissues, as well as their biomechanical behavior during various levels of bone loss when subjected to orthodontic loads and movements.We [2,10,[12][13][14]17,18] reported the Tresca criterion (designed for ductile non-homogenous materials) to be the suitable criterion for dental tissues.Another assessed issue was related to the maximum amount of orthodontic load that can be applied during orthodontic movements to each component of dental tissues in order to avoid ischemic and resorptive risks, since no studies investigating the gradual horizontal breakdown employing the Tresca criterion were found in the current research flow.The results reported that, in the intact periodontium, 0.6-1.2N is relatively safely applied, while after 4 mm of bone loss forces of 0.2-0.6N are recommended.The previous study [2] regarding mandibular bone reported Tresca as being the most suitable criterion for bone, by comparing five of the most-used failure criteria in various bone loss levels and under 0.5 N.However, no data regarding ischemic and resorptive risks related to the biomechanical behavior of applied loads in periodontal breakdown were found.
The goals of our study were (a) to assess the biomechanical behavior of cortical and trabecular bone as a continuum subjected to increasing orthodontic forces during horizontal periodontal breakdown; (b) to assess the differences between orthodontic movements and loads when subjected to increasing loads; (c) to correlate the results with other FEA-related reports of dental tissues for identifying potential resorptive and ischemic risks.

Materials and Methods
This FEA numerical analysis is part of a stepwise larger research project [2,10,[12][13][14]17,18] (with the clinical protocol no.158, 2 April 2018) that investigated the biomechanical behavior of teeth and surrounding supporting tissues under light orthodontic forces in both intact and reduced periodontium.
The inclusion criteria were intact arches, intact teeth, no malposition, intact or slightly reduced bone loss, non-inflamed periodontium, availability for regular checkups, indication of orthodontic treatments.The contraindications were in opposition to the above.In the research project more patients were initially included; however, at the end only nine qualified (four males/five females, mean age 29.81 ± 1.45).
Each patient received a CBCT examination (ProMax 3DS, Planmeca, Helsinki, Finland, voxel size 0.075 mm) of the mandibular region (with the two molars and premolars).
The grey shades DICOM images were manually segmented using Amira 5.4.0 (Visage Imaging Inc., Andover, MA, USA), allowing the identification of small tissues that were not recognized by the automated algorithm.Thus, our models contained: enamel; dentincementum; dental pulp; NVB; PDL; cortical bone; and trabecular bone.The base of a stainless-steel bracket was manually reconstructed on the vestibular surface of the second premolar.The models guarded only the second lower premolar while the alveolar socket of the other teeth was filled with bone (Figure 1).These nine models had various but limited levels of bone loss (i.e., in the cervical third of the alveolar socket).The bone loss and missing PDL were manually reconstructed, thus obtaining nine models with intact periodontium.
the research project more patients were initially included; however, at the end only nine qualified (four males/five females, mean age 29.81 ± 1.45).
Each patient received a CBCT examination (ProMax 3DS, Planmeca, Helsinki, Finland, voxel size 0.075 mm) of the mandibular region (with the two molars and premolars).
The grey shades DICOM images were manually segmented using Amira 5.4.0 (Visage Imaging Inc., Andover, MA, USA), allowing the identification of small tissues that were not recognized by the automated algorithm.Thus, our models contained: enamel; dentin-cementum; dental pulp; NVB; PDL; cortical bone; and trabecular bone.The base of a stainless-steel bracket was manually reconstructed on the vestibular surface of the second premolar.The models guarded only the second lower premolar while the alveolar socket of the other teeth was filled with bone (Figure 1).These nine models had various but limited levels of bone loss (i.e., in the cervical third of the alveolar socket).The bone loss and missing PDL were manually reconstructed, thus obtaining nine models with intact periodontium.Each of these 9 models was then subjected to a gradual horizontal reduction of bone and PDL of around 1 mm, obtaining a periodontal breakdown of 1-8 mm (for each patient 9 models were obtained with various bone losses, totaling 81 3D models).All these models were anatomically correct (e.g., 5.06-6.05 million C3D4 tetrahedral elements, 0.97-1.07 million nodes, and a global element size of 0.08-0.116mm, for the 9 intact periodontium models).The PDL had a variable thickness of 0.15-0.225mm, including NVB, while cementum was reconstructed as dentine (since they have similar physical properties, and their separation was found to be impossible).
The mesh models displayed various surface irregularities (201 element warnings meaning 0.0039% from a total of 5,117,355 elements of the bone continuum) and no error warnings (Figure 1G, yellow spots).All these surface irregularities were in non-essential areas not interfering with the quasi-continuity of the stress-affected areas.Moreover, each software had internal checking and testing algorithms that did not allow a new step if the earlier one had not been passed.
The numerical simulations were performed using Abaqus 6.13-1 (Dassault Systèmes Simulia Corp., Maastricht, The Netherlands), employing the Tresca (maximum shear stress) failure criterion suited for ductile materials with physical properties [2,10,[12][13][14]17,18] (Table 1) similar to other FEA numerical studies.The simulated orthodontic Each of these 9 models was then subjected to a gradual horizontal reduction of bone and PDL of around 1 mm, obtaining a periodontal breakdown of 1-8 mm (for each patient 9 models were obtained with various bone losses, totaling 81 3D models).All these models were anatomically correct (e.g., 5.06-6.05 million C3D4 tetrahedral elements, 0.97-1.07 million nodes, and a global element size of 0.08-0.116mm, for the 9 intact periodontium models).The PDL had a variable thickness of 0.15-0.225mm, including NVB, while cementum was reconstructed as dentine (since they have similar physical properties, and their separation was found to be impossible).
The mesh models displayed various surface irregularities (201 element warnings meaning 0.0039% from a total of 5,117,355 elements of the bone continuum) and no error warnings (Figure 1G, yellow spots).All these surface irregularities were in non-essential areas not interfering with the quasi-continuity of the stress-affected areas.Moreover, each software had internal checking and testing algorithms that did not allow a new step if the earlier one had not been passed.
The results were color-coded projections of various colors and extension areas of the shear stress (qualitative and quantitative), that were compared and correlated with both previous studies of our team [2,10,[12][13][14]17,18] and with the four identified bone-tooth research papers [1,[20][21][22].

Results
The analysis herein was performed over eighty-one models in a total of 243 FEA simulations, with no influence whatsoever due to gender, age, or periodontal status.
There was a constant and coherent visible pattern of stress increase and extension areas for all movements and loads strictly correlated with the progression of the periodontal breakdown.Rotation seemed to be the most stressful among the five orthodontic finemovements for the mandibular bone.
Qualitatively, FEA simulations displayed similar color-coded projections of stress distribution areas (i.e., Figures 2 and 3) for all three forces independently of their amount (i.e., 0.6 N, 1.2 N and 2.4 N).The three forces were selected based on the fact that they are considered relatively safe to apply in both intact and reduced periodontium and for our being able to corelate these simulations with previous ones investigating other tissular components (PDL, dental pulp, NVB, bone) [2,10,[12][13][14]17,18].
The boundary conditions included homogeneity, linear elasticity, and isotropy similar with the other FEA numerical analyses mentioned above.
The results were color-coded projections of various colors and extension areas of the shear stress (qualitative and quantitative), that were compared and correlated with both previous studies of our team [2,10,[12][13][14]17,18] and with the four identified bone-tooth research papers [1,[20][21][22].

Results
The analysis herein was performed over eighty-one models in a total of 243 FEA simulations, with no influence whatsoever due to gender, age, or periodontal status.
There was a constant and coherent visible pattern of stress increase and extension areas for all movements and loads strictly correlated with the progression of the periodontal breakdown.Rotation seemed to be the most stressful among the five orthodontic finemovements for the mandibular bone.
Qualitatively, FEA simulations displayed similar color-coded projections of stress distribution areas (i.e., Figures 2 and 3) for all three forces independently of their amount (i.e., 0.6 N, 1.2 N and 2.4 N).
Medicina 2023, 59, x FOR PEER REVIEW 5 of 16 movements were intrusion, extrusion, rotation, tipping, and translation under three applied forces (0.6 N approx.60 g, 1.2 N approx.120 g, and 2.4 N approx.240 g) at the bracket base (Figure 1).The base of the model had zero displacements (encastered).The three forces were selected based on the fact that they are considered relatively safe to apply in both intact and reduced periodontium and for our being able to corelate these simulations with previous ones investigating other tissular components (PDL, dental pulp, NVB, bone) [2,10,[12][13][14]17,18].
The boundary conditions included homogeneity, linear elasticity, and isotropy similar with the other FEA numerical analyses mentioned above.
The results were color-coded projections of various colors and extension areas of the shear stress (qualitative and quantitative), that were compared and correlated with both previous studies of our team [2,10,[12][13][14]17,18] and with the four identified bone-tooth research papers [1,[20][21][22].

Results
The analysis herein was performed over eighty-one models in a total of 243 FEA simulations, with no influence whatsoever due to gender, age, or periodontal status.
There was a constant and coherent visible pattern of stress increase and extension areas for all movements and loads strictly correlated with the progression of the periodontal breakdown.Rotation seemed to be the most stressful among the five orthodontic finemovements for the mandibular bone.
Qualitatively, FEA simulations displayed similar color-coded projections of stress distribution areas (i.e., Figures 2 and 3) for all three forces independently of their amount (i.e., 0.6 N, 1.2 N and 2.4 N).Quantitatively, the amount of displayed stress doubled (for 1.2 N) and quadrupled (for 2.4 N) when compared with 0.6 N of force (Table 2).The quantitative increase rate (i.e., a%, m%, c%) was similar for all three forces independently of the levels of bone loss displaying a gradual increase up to a doubling of the amount of stress (at 8 mm of loss) when compared with the intact periodontium.Quantitatively, the amount of displayed stress doubled (for 1.2 N) and quadrupled (for 2.4 N) when compared with 0.6 N of force (Table 2).The quantitative increase rate (i.e., a%, m%, c%) was similar for all three forces independently of the levels of bone loss displaying a gradual increase up to a doubling of the amount of stress (at 8 mm of loss) when compared with the intact periodontium.Both qualitatively and quantitatively, vestibular cervical stress was the highest for all three movements (rotation, translation, and tipping) and forces, seeming to display higher risks of ischemia and further resorptive processes.The extension of vestibular cervical-stress areas progressively increased with bone loss.
Rotation, translation, and tipping movements produced the highest amounts of stress both inside and outside the alveolar socket (after 4 mm of bone loss) seeming to be prone to increased ischemic and resorptive risks when compared with the intrusion and extrusion movements.Outside the alveolar socket, a progressive extension of vestibular stress to the middle-and apical vestibular third was seen for all five movements correlated with bone loss.Inside the alveolar socket, in the intact periodontium, all five movements produced limited vestibular cervical stress with no visible (except rotational) stresses in the apical and middle third.However, with the progression of periodontal breakdown (especially after 4 mm of loss), stress areas were extended to the entire inside surface of the alveolar socket.Thus, this confirms a correlation between bone loss and increased risks of apical and middle third ischemic and resorptive processes for the alveolar socket after 4 mm of loss even for 0.6 N of applied force.
In both the intact and reduced periodontium, the 3D models displayed stress areas found at the base, since it was encastered and with zero displacements (i.e., simulating mandibular stiffness).These areas were more visible after 4 mm of bone loss due to the reduction in the bone height and periodontal ligament (i.e., surrounding supporting tissues) with no influence, however, over the stress distribution in the alveolar socket.These areas of stress near the bases of the models were expected (similar to other numerical analyses).
Extrusion (Figures 2A and 3A) displayed limited vestibular cervical stress (where the bone is thinner) for the intact periodontium, with a progressive extension to the middle and apical third of the alveolar socket from 4 to 8 mm of periodontal breakdown.A bone loss level of 8 mm displayed the highest extension of stress areas to the entire alveolar socket.Quantitatively, for 0.6 N of force the apical third stress in the alveolar socket exceeded the MHP by four times.The cervical third stress had a limited increase interval of 149.83-168.34KPa (for 0.6 N, 0-8 mm bone loss), almost ten times higher than the MHP, signaling potential ischemic and resorptive risks.On the lingual side of the model, the base area displayed limited stress areas (due to boundary conditions with zero displacement) correlated with bone loss, with no significance whatsoever for the alveolar-socket stress display.
Intrusion (Figures 2B and 3B) displayed a similar behavior to the extrusion movement.
Rotation (Figures 2C and 3C) displayed mostly vestibular cervical stress for the intact periodontium, with a progressive but limited extension (much smaller than intrusion/extrusion) to the apical third after 4 mm of loss.It also displayed the highest and most extended stress areas (both vestibular and lingual) at the base of the models due to the assumption of zero displacements (with no influence over the stress distribution in the alveolar socket).Quantitatively, rotation produced the highest amount of stress among the five movements.
Tipping (Figures 2D and 3D) movement showed only vestibular cervical stress in the intact periodontium.However, along with progression of bone loss there was a visible extension of stress to the vestibular and lingual middle and apical third of the socket.On the lingual and vestibular sides near to the base of the model, limited visible stresses were displayed due to assumed boundary conditions.
Translation (Figures 2E and 3E) movement displayed limited cervical-third vestibular stress in the intact periodontium, with a progressive extension to the proximal and lingual cervical-third sides of the socket correlated with bone loss.

Discussion
These 243 FEA simulations performed over eighty-one models aimed to assess the biomechanical behavior of cortical and trabecular bone as a continuum subjected to increasing orthodontic forces during horizontal periodontal breakdown.Additionally, the aim was to assess the differences between orthodontic movements when subjected to different loads, as well as to correlate results with other FEA-related reports of dental tissues for finding potential resorptive and ischemic risks.It must be emphasized that herein being the first investigation into these issues, and since the current research flow is scarce, the only available sources of comparison were our previous research [2,10,[12][13][14]17,18] and the four bone-tooth studies [1,[20][21][22].
This manuscript further develops the previous published research [2] (being a stepby-step developed study).The previous research focused on selecting suitable failure criteria for the numerical study of bone (as a structural continuum) by employing five failure criteria used in dental FEA analysis, 0.5 N applied force, five movements and 0-8 mm periodontal breakdown.The qualitative and quantitative results provided by the simulations prove that only failure criteria for ductile materials are suitable (i.e., giving accurate results when compared with the other three) for bone study, with in particular Tresca being the best option.We must emphasize that only our studies employed Tresca, we being the first to use it in dental field research.
On the other hand, our research took the next logical step in the study of bone, investigating how much the increase in orthodontic force affects both qualitative and quantitative results.Thus, we employed the previous proven Tresca (best suitable criterion) and, under 0.6 N, 1.2 N, and 2.4 N, we analyzed how the changes affect the intact bone and the reduced periodontium, and what the ischemic and resorptive risks are.
In our simulations, the color-coded projections of the stress-distribution areas (qualitative results) in both the intact and reduced periodontium were similar for all three applied forces.However, the amounts of stress displayed by these areas increased along with the applied force (doubled for 1.2 N and quadrupled for 2.4 N when compared with 0.6 N).These above results suggest that the differences between these light orthodontic forces consist of stresses (quantitative) displayed in the color-coded stress areas, while their location and extension remain constant (qualitative).There seems to be a correlation between the progression of bone loss (i.e., surrounding supporting tissues), extension of stress areas to the alveolar socket, and increase in displayed amounts of stress (in agreement with previous data [2,10,[12][13][14]17,18]).
Previous studies investigating PDL, dental pulp and NVB behavior during 0-8 mm bone loss under the same five orthodontic movements and three forces (correlated with the MHP) reported 0.6-1.2N to be safe for the intact periodontium and 0.2-0.6N safe for various levels of bone loss [17,18].The results herein agree with these observations.The biomechanically similar stress display areas in the intrusion and extrusion movements during gradual horizontal periodontal breakdown are due to the periodontal ligament functions (absorption-dissipation stress ability) [2,10,14,17,18].As the surrounding supporting tissues (bone and ligament) are reduced, the stress distribution changes so that the entire tooth alveolar socket absorbs these stresses [2,10,14,17,18].We must emphasize that only intrusion and extrusion, among the five movements, displayed such a rapid extension of the stress areas in the apical and middle third of the alveolar socket after 4 mm of bone loss, in agreement with previous studies [2,10,14,17,18].This behavior seems to lead to the conclusion that with bone loss the risks of apical and middle-third alveolar-socket ischemia and resorptive processes for these two movements increase when compared with the other three movements (due to better vascularization of this area-NVB, dental pulp, perforating vessels) [2,12,13,17,18].Moreover, quantitatively, in the apical third of the alveolar socket the amount of stress displayed for 0.6 N has a range of 61.33-67.3KPa for 4-8 mm of loss, around four times higher than the physiological MHP of 16 KPa.Nevertheless, the cervical-third stress is nine times higher (i.e., 0.6 N and the intact periodontium) than MHP stress, but much lower when compared with the other three movements.This observation seems also to be supported by the clinical and anatomical data that the apical third of the alveolar socket has better vascularization than the cervical third [2,10,[12][13][14]17,18].
Rotation, translation, and tipping displayed mainly vestibular cervical-third stress in the alveolar socket during the entire periodontal-breakdown simulation (due to PDL absorption-dissipation ability), leading to the conclusion that ischemic and resorptive risks are higher in these areas, in agreement with our previous studies [2,10,[12][13][14]17,18].Moreover, the amount of cervical stress for a rotational applied force of 0.6 N reached a maximum of 402.71 KPa for 8 mm bone loss (i.e., twenty-five times higher than the MHP).These findings agree with our earlier results [2].On the other hand, from the clinical data, 0.6 N of force is considered to be light and safe to be used in the intact periodontium [19], but there are no available studies for the reduced periodontium, except our previous ones [2,10,[12][13][14]17,18].Thus, despite 0.6 N being considered safe for the intact periodontium, for the assessment of safety in the reduced periodontium we can only rely on data present in the current research flow (i.e., FEA studies) [2,10,[12][13][14]17,18].
The above findings correlate with previous reports [2,12,13,[15][16][17][18] of 0.6 N effects over PDL (higher risks of ischemia and further loss) and their recommendations [15][16][17][18] of a reduction to 0.2-0.4N in the applied force in cases of periodontal breakdown (4-8 mm of loss).It also confirms a correlation between bone loss and increased risks of apical and middle-third ischemic and resorptive processes for the alveolar socket after 4 mm of loss even for 0.6 N of applied force.However, it must be emphasized that bone as a continuum is less vascularized than PDL and dental pulp-NVB and thus less prone to circulatory disturbances.Therefore, the maximum force safely applied in these tissues (i.e., PDL, pulp, tooth, bone, NVB) should be the one that is safe for the weakest component (i.e., PDL and NVB), and thus, previous recommendations [17,18] should be considered.
Another aspect that should be addressed is related to the fact that in both the intact and reduced periodontium, the models displayed stress areas found at the base, since it was encastered and with zero displacements (i.e., simulating mandibular stiffness).These areas were more visible after 4 mm of bone loss due to the reduction in bone height and periodontal ligament (i.e., surrounding supporting tissues) with no influence however over the stress distribution in the alveolar socket.
No FEA bone-tooth studies employing the Tresca failure criterion were found.However, four FEA intact periodontium tooth-bone studies using the Von Mises criterion and similar boundary conditions were found [1,[20][21][22].
Merdji et al. [20] (lower third molar, intact periodontium, single model, sample size of one, Von Mises criterion, intrusion: 10 N, tipping/translation: 3 N, bone: 142,305 elements, global element size: 0.25-1 mm), qualitatively reported (similar with the results herein) cervical-third stress for all three movements.Qualitatively, Merdji et al. [20] reported that the stress display was both on the vestibular and lingual cervical third of the alveolar socket since both sides of the bone were of equal thickness (probably due to three-rooted anatomy and artificial anatomical positioning), and from this point of view is closer to boneimplant [3,7,9,[23][24][25][26][27][28] stress display.In our models, the correct anatomical reconstruction avoided these shortcomings (the second premolar displayed a vestibular alveolar socket wall thinner than the lingual one); thus, the stress distribution areas were found toward the thinner and less resistant wall.
We assumed that in the appearance of these differences an important role was played both by boundary conditions (global element size 0.25-1 mm and 142,305 elements [20] vs. global element size 0.08-0.116mm and 5,117,355 elements herein) and the anatomy of the model (idealized third molar [20] vs. our anatomically accurate second premolar).
Field et al. [21] (intact periodontium, two models, sample size of two, Von Mises criterion, tipping: 0.5 N, canine model: 23,565 elements, incisor-canine-first premolar model: 32,812 elements, global element size: 1.2 mm) qualitatively reported comparable results (such as extension and topography) to this study, but with different color-coded codification meaning high stress intensity (contradicting the color codification herein).We must emphasize that Field et al. [21] applied 0.35 N/0.5 N of tipping, a force considered light [19], reporting extended red high stresses [21] vs. limited-extension blue-green lower stresses in our simulations (much closer to the clinical behavior of a reduced force).
We assumed that these differences were due to modeling issues and boundary conditions (i.e., global element size 1.2 mm and 23,565-32,812 elements [21] vs. global element size 0.08-0.116mm, 5.06-6.05 million elements, and 0.97-1.07 million nodes in our simulations).In support of these assumptions, we consider relevant the reported [21] amount of 32 KPa of hydrostatic pressure in the apical third of PDL and Von Mises stresses of 235.5-324.5 KPa in the entire PDL, exceeding by far the reported MHP of 16 KPa, suggesting high ischemia and resorptive risks for a light force of 0.35 N/0.5 N, contradicting both clinical data [19] and other FEA studies (i.e., 0.5 N proven to be safely applied in both PDL and pulp-NVB up to 8 mm of loss with amounts of stress lower than MHP) [17,18].Field et al. [21] did not perform correlations with physiological MHP and reported significant qualitative and quantitative differences between the biomechanical behavior of the two models (i.e., higher extension of the stress areas for the multi-teeth model when compared with the single-tooth model).In our simulations both qualitative and quantitative results were lower than those Field et al. [21] reported, and we expect that the same pattern should be displayed by a multi-teeth model.
Shaw et al. [22] (upper incisor, intact periodontium, one model, sample size of one, Von Mises criterion, intrusion, extrusion, tipping, translation and rotations, model: 11,924 elements and 20,852 nodes) reported lower amounts of cervical stress (i.e., intrusion/extrusion 1.664 KPa, translation 0.6 KPa, tipping 0.54 KPa and rotation 0.015 KPa), intrusion and extrusion as the most stressful movements, and comparable stress-display areas.We assumed that these differences were due to modeling issues and boundary conditions.Shetty et al. [1] (upper 1st molar, intact periodontium, one model, sample size of one, Von Mises criterion, intrusion and tipping: 150 N, model: 30,838 nodes and 167,089 elements) quantitatively reported 1.33-1.95MPa for intrusion and 2.16-8.15MPa for tipping, with tipping as more stressful than intrusion (in agreement with the results herein) and qualitatively displaying extended stress areas in the entire alveolar socket for both movements (and contradicting our results herein).It must be emphasized that these models [1] are not anatomically accurate and were simplified (i.e., lower numbers of nodes and elements) along with other boundary conditions; thus, the differences (i.e., high amounts of stress and extension areas) could be explained.In the simulations here, there were no qualitative differences between 0.6 and 2.4 N of force, so we expect that this pattern should be kept for higher forces.Nonetheless, the quantitative results here rapidly increased so higher amounts of stress could be possible under higher forces.However, the fundamental issues reside in the appliance time and the integrity of supporting tissues (more than 0.6 N is prone to ischemic and resorptive risks in the reduced periodontium if kept for longer periods of time).
The limits of FEA studies are related to the anatomical accuracy of the model and boundary conditions, as well as to the selection of the proper failure criteria.It must be emphasized that another limit is that FEA studies cannot entirely reproduce clinical situations and that clinically pure movements rarely happen (mostly, a combination of them is met).
The proper failure criteria are related to the type of analyzed material (ductile or brittle, homogenous-nonhomogeneous), since each criterion was specially designed for a certain type of biomechanical behavior.The Tresca criterion is the single criterion suited for all components of teeth and the surrounding supporting tissues (proven by previous FEA studies) [2,10,[12][13][14]17,18].
Few FEA studies assessed the non-linear vs. linear elasticity issues, reporting quantitative differences for light orthodontic forces (1 N).Thus, Hemanth et al. [15,16] reported 20-50% less force needed for non-linear elasticity when compared to linear for obtaining the same results.Nevertheless, Hemanth et al. [15,16] employed a brittle material criterion (maximum and minimum principal stresses) for biomechanically analyzing PDL (which is ductile) under extremely small movements (intrusion and tipping), using a model with an idealized anatomy, all with a high potential of altering the accuracy of the results.More studies are needed for assessing this issue.
The anatomical accuracy of the analyzed model also has a potential for altering the accuracy of results (as previously shown).An FEA model should be based entirely on CBCT data, with a reduced voxel size (not artificially reconstructed based on a simplified idealized anatomy), and with a mesh displaying a large number of elements and nodes (e.g., the 5.06-6.05 million elements and 0.97-1.07 million nodes used herein vs. 142,305 elements [20]; 23,565-32,812 elements [21]; 30,838 nodes and 167,089 elements [1]; 148,097 elements and 239,666 nodes [15,16]; 11,924 elements and 20,852 nodes [22]); and a small global element size (e.g., 1.2 mm [21]; 0.25-1 mm [20] vs. the 0.08-0.116mm used herein).
FEA studies (i.e., numerical simulations) are the only available method to individually study such small and complex structures as dental living tissues.However, despite the individual study of each component, clinically, all these components work and function as a continuum; thus, this shows the importance of correlations with MHP and between available studies.Both our results herein and our previous [2,10,[12][13][14]17,18] studies were focused on addressing these issues for matching the numerical results with clinical data (since, due to misunderstanding of the FEA methodology [1,3,7,9,15,16,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], most of the dental FEA studies are not as accurate as those from the engineering field).Our research is the first in the above-mentioned direction; thus, we tried to explain the main issues regarding the use of FEA in dental studies for obtaining more accurate results and identifying and using one general single failure criterion (found to be Tresca [2,10,[12][13][14]17,18]) for the study of all dental structures.

1.
Both intrusion and extrusion after 4 mm of bone loss displayed extended stress display in the apical and middle third of the alveolar socket, seeming to show higher ischemic and resorptive risks for these areas even for 0.6 N.

2.
Rotation, translation, and tipping displayed the highest amounts of stress and cervicalthird stress with higher ischemic and resorptive risks after 4 mm of loss for 0.6 N.

3.
Based on quantitative results, rotation, translation, and tipping seem to be more stressful when compared with intrusion and extrusion.4.
All three applied forces produced similar stress display areas for all movements and bone levels.5.
The amount of stress doubled for 1.2 N and quadrupled for 2.4 N when compared with 0.6 N. 6.
The differences between the three orthodontic forces consist of stress (quantitative) displayed in the color-coded stress areas, while their location and extension (qualitative) remained constant in both the intact and reduced periodontium.7.
Since the MHP was exceeded in all simulations, a reduction in the applied force to under 0.6 N (after 4 mm of bone loss) is recommended for reducing the ischemic and resorptive risks.8.
The stress display pattern correlated with horizontal periodontal-breakdown simulations.

Practical Implications
Little information is available about the biomechanical behavior of bone as a continuum during periodontal breakdown.Moreover, the effects (distribution areas and amounts of stress) of various applied orthodontic forces are unknown.Thus, this research studied the effects of three orthodontic forces (0.6 N, 1.2 N and 2.4 N) over 0-8 mm bone loss during five orthodontic movements.It also correlated the results with other studies with the same boundary conditions but investigating dental pulp, the periodontal ligament, the neuro-vascular bundle and the surrounding bone, thus supplying a clearer and more comprehensive image of the entire biomechanical behavior of these tissues.The results are extremely important for both clinical practitioners and researchers.The stress distribution under intrusion and extrusion seems to favorize the ischemic and resorptive risks in the apical and middle third of the alveolar socket while rotation, translation, and tipping have a similar effect localized to the cervical third of the alveolar socket.Moreover, after 4 mm of bone loss a reduction in the amount of force to under 0.6 N is recommended.Both researchers and clinicians can benefit from the comprehensive image created by the FEA studies and better understand the advantages but also the limits of this method of investigation of small anatomical tissues.

Table 1 .
Elastic properties of materials.

Table 1 .
Elastic properties of materials.

Table 1 .
Elastic properties of materials.