applied sciences Article Orthodontic Loads in Teeth after Regenerative Endodontics: A Finite Element Analysis of the Biomechanical Performance of the Periodontal Ligament Cristina Bucchi 1, * , Massimo Del Fabbro 2,3 and Jordi Marcé-Nogué 4,5 1 Department of Integral Adult Dentistry, Oral Biology Research Centre, Universidad de La Frontera, Av. Francisco Salazar 01145, Temuco 4811230, Chile 2 Dipartimento di Scienze Biomediche, Chirurgiche ed Odontoiatriche, Università degli Studi di Milano, Via della Commenda 10, 20122 Milan, Italy; massimo.delfabbro@unimi.it 3 IRCCS Istituto Ortopedico Galeazzi, Via Riccardo Galeazzi, 4, 20161 Milan, Italy 4 Department of Mechanical Engineering, Universitat Rovira i Virgili, Carrer de l’Escorxador, s/n, 43003 Tarragona, Spain; jordi.marce@urv.cat 5 Institut Català de Paleontologia Miquel Crusafont, Universitat Autònoma de Barcelona, Carrer de les Columnes, s/n, 08193 Cerdanyola del Valles, Spain * Correspondence: cristina.bucchi@ufrontera.cl Abstract: The objective of this study was to analyse the stress distribution in the periodontal ligament and tooth structure of a cementum-reinforced tooth, a dentine-reinforced tooth and an immature tooth during orthodontic loads using a finite element analysis. A finite element model of a maxillary incisor and its supporting tissues was developed. The root was segmented into two parts: a part that represented a root in an immature state and an apical part that represented the tissue formed Citation: Bucchi, C.; Del Fabbro, M.; after regenerative endodontics. The apical part was given the mechanical properties of dentine or Marcé-Nogué, J. Orthodontic Loads cementum. The three models underwent simulation of mesial load, palatal inclination and rotation. in Teeth after Regenerative The mean stress values and stress distribution patterns of the periodontal ligament of the dentine- Endodontics: A Finite Element and cementum-reinforced teeth were similar in all scenarios. The maturation of the root, with either Analysis of the Biomechanical dentine or cementum, was beneficial for all scenarios, since the periodontal ligament of the immature Performance of the Periodontal tooth showed the highest mean stress values. Under the condition of this computational study, Ligament. Appl. Sci. 2022, 12, 7063. orthodontic loads can be applied in teeth previously treated with regenerative endodontics, since the https://doi.org/10.3390/ distribution of stress is similar to those of physiologically mature teeth. In vivo studies should be app12147063 performed to validate these results. Academic Editors: Gavriel Chaushu and Joseph Nissan Keywords: orthodontic loads; periodontal ligament; regenerative endodontics; revitalisation; finite element analysis; stress distribution Received: 25 April 2022 Accepted: 29 May 2022 Published: 13 July 2022 Publisher’s Note: MDPI stays neutral 1. Introduction with regard to jurisdictional claims in Regenerative endodontics is a relatively new treatment aimed at regenerating the published maps and institutional affil- pulp-dentine complex after pulp necrosis in immature teeth [1]. Until very recently the iations. standard treatment for such teeth was apexification. However, the latter did not allow further root development or revascularisation of the tooth, which remained with thin and short dentine walls [2]. Regenerative endodontics consists of a new cell-based approach Copyright: © 2022 by the authors. and utilises the stem cells located at the periapical tissues that invade the root canal after Licensee MDPI, Basel, Switzerland. over-instrumentation of the apex [3], and the growth factors present in the dentine are This article is an open access article able to direct stem cell proliferation, migration and differentiation [4], with the final aim distributed under the terms and of revitalising and inducing further root development of the necrotic tooth. Regenerative conditions of the Creative Commons endodontics is considered a successful clinical treatment since most teeth show further root Attribution (CC BY) license (https:// development, healing of periapical lesions and resolution of symptomatology [5]. However, creativecommons.org/licenses/by/ with the exception of occasional studies [6], regeneration of the pulp-dentine complex has 4.0/). Appl. Sci. 2022, 12, 7063. https://doi.org/10.3390/app12147063 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 7063 2 of 11 not been observed in vivo. Thus, the posttreatment development of the root is due to the apposition of cementum-like tissue or osteodentine instead of tubular dentine [7–9]. Dentine is an organised and mineralised tissue with a tubuli-like structure [10] that is able to endure compressive forces, while cementum and other reparative tissues formed af- ter regenerative endodontics have collagen fibres not packed in an orderly fashion [11] and a higher organic composition [12]. As a result, cementum has poorer mechanical properties than dentine [13]. A computational study showed that the mechanical performance of roots reinforced with cementum after regenerative endodontics have been proven to distribute the mechanical stress disadvantageously and unevenly compared to a root reinforced with dentine [14]. In teeth after regenerative endodontics, cementum barely endures stress, which is mainly concentrated in the dentine part of the root [14]. However, it is unknown whether this disadvantageous distribution of stress is transferred and maintained in the periodontal ligament and alveolar bone, producing high peaks of mechanical stress in spe- cific parts of these tissues (i.e., in the neighbouring areas of the dentine with peak stress), or in contrast, whether the periodontal ligament is able to re-distribute the mechanical stress more evenly due to its unique biomechanical and cellular nature [15]. The performance of the periodontal ligament of teeth after regenerative endodontics has not been studied. Periodontal ligament stem cells are sensitive to mechanical loading and play a critical role in periodontal and osseous remodelling during orthodontic movement by regulating the balance between osteoblastic and osteoclastic activity [16]. Orthodontic tooth movement is produced by mechanical forces and promoted by the remodelling of the periodontal ligament and alveolar bone. Under appropriate stimulation, on the compression side, the periodontal ligament becomes compressed, resulting in a focal necrosis [17]. These necrotic sites release chemokines that induce the migration of multinucleated cells that resorb the necrotic periodontal ligament as well as the underlying bone and cementum [17], and this process is stopped only when mechanical stimulation ceases. Thus, appropriate orthodontic forces produce minor reversible injury to the tooth-supporting tissues [17]. However, when these forces are inappropriate, such as in the case of excessive force magnitude or adequate magnitude but with high concentration of the stress in specific areas, they lead to a sustained inflammation process that reabsorbs the root [18]. This has raised the question of whether cementum-reinforced teeth (i.e., teeth that have been previously treated with regenerative endodontics), which seem to distribute mechanical stress disadvantageously compared to physiologically developed roots [14], can endure orthodontic movement or if they are at risk of inflammatory root resorption, as suggested by the European Society of Endodontology Position Statement for Revitalisa- tion [19]. As for today, there is little clinical evidence, limited to case reports, of revitalised teeth treated with orthodontics [20,21]. The objective of this study was to analyse the stress distribution in the periodontal ligament of a cementum-reinforced tooth, a dentine-reinforced tooth and an immature tooth during simulation of orthodontic loads in three different directions using a finite element analysis (FEA) of a maxillary central incisor. 2. Materials and Methods This study was approved by the Ethics Committee of the University of Barcelona and conducted in accordance with the Code of Ethics of the World Medical Association (Decla- ration of Helsinki) for experiments involving humans. The FEA model was constructed based on a µCT scan of the maxilla of a male 59-year-old donor (Quantum FX microCT, PerkinElmer, Waltham, MA, USA). The µCT scan was conducted with 90 kV and 160 mA, FOV 60 mm, exposure time of 4.5 min and voxel size of 118 × 118 × 118 µm. The right maxillary central incisor was selected as the treated tooth and the dental and supporting tissues, including enamel, dentine, cementum, periodontal ligament and alveolar bone, were segmented with Seg3D software (v. 2.4.3, Center for Integrative Biomedical Computing, Salt Lake City, UT, USA). The periodontal ligament was segmented according to its natural shape. It had a maximum Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 11 Appl. Sci. 2022, 12, 7063 3 of 11 software (v. 2.4.3, Center for Integrative Biomedical Computing, Salt Lake City, UT, USA). The periodontal ligament was segmented according to its natural shape. It had a maxi- mum thickness thickness ofmm of 0.38 0.38 in mm theincoronal the coronal part part and aand a minimum minimum thickness thickness of 0.11 of 0.11 mmmm in in the middle the third, middle having third, an hourglass having an hourglassdistribution. As reported distribution. previously As reported [14],[14], previously the root was the root segmented was into two segmented into parts: the coronal two parts: part, part, the coronal representing the immature representing root (stage the immature 4 based root (stage 4 on Cvek’s based classification on Cvek’s [22], [22], classification and and an apical part an apical (the part core (the part core partand andthe thetip tip of root), of the root), representing the tissue formed after regenerative endodontics (Figure 1a). representing Figure 1. Figure Segmentedtissues 1.Segmented tissuesand andmodel modeldevelopment. development. Segmentation Segmentation ofof aa mature mature tooth tooth in inthe thealveolar alveolar bone bone (a). (a). The The apical apical part part (black (black arrow) arrow) of of the the mature mature tooth tooth was was given given thethe mechanical mechanical properties properties ofof dentine dentine or or cementum cementum for for Models Models 22 and and 3, 3, respectively. Segmentation Segmentation of of an an immature immature tooth tooth in in the the alveolar alveolar bone bone (b). (b). Asterisk Asterisk (*): (*): space spaceoccupied occupiedbybythe theapical apicalpapilla. papilla.Image Imageedited editedfrom fromBucchi Bucchietetal., al., 2019 [14]. 2019 [14]. 2.1. 3D 2.1. 3D FEA FEA Models Models and and Tissue Tissue Properties Three different Three different non-linear non-linear models models representing representing three three different different scenarios scenarios were were solved solved using ANSYS using ANSYS Mechanical Mechanical 19.1. 19.1. Each Each model model was was meshed meshed using using tetrahedral tetrahedral elements elements with with aa quadratic quadratic interpolation, interpolation, except except the the elements elements ofof the the mesh mesh of of the the PDL PDL that that were were created created using aa linear using linear interpolation. Three models were Three weredesigned. designed.Model Model1 1represented represented thethe tooth tooth in an immature in an immature statestate (IT) (i.e., coronal part without the apical part) (Figure 1b), Model 2 represented (IT) (i.e., coronal part without the apical part) (Figure 1b), Model 2 represented a physio- a physiologically mature tooth logically mature(DR) (the(DR) tooth coronal (the and apical coronal andpart were apical partgiven werethe mechanical given properties the mechanical prop-of dentine) erties and Model of dentine) and3 Model represented a tooth after 3 represented regenerative a tooth endodontics after regenerative (CR) (coronal endodontics (CR) part waspart (coronal given was thegiven mechanical properties the mechanical of dentine properties and theand of dentine apical thepart thepart apical mechanical the me- properties chanical of cementum) properties (Figure 1a). of cementum) The segmented (Figure alveolar bone 1a). The segmented alveolarwasbone equalwasin the three equal in models, and it was not modified in the IT since the space from the the three models, and it was not modified in the IT since the space from the tip of the roottip of the root to the bottom of the alveolar socket represented the space occupied to the bottom of the alveolar socket represented the space occupied by the apical papillaby the apical papilla (Figure 1b). (Figure 1b). This This made made itit possible possible to to perform perform the the simulations simulations in in standardised standardised models models and and compare the outcomes among compare the outcomes among the models. the models. The material The material properties properties (Young’s (Young’s Modulus Modulus and and the the Poisson Poisson coefficient) coefficient) for for enamel, enamel, dentine, cementum and bone are described in Table 1. Elastic, linear dentine, cementum and bone are described in Table 1. Elastic, linear and homogeneous and homogeneous material properties material propertieswere wereassumed assumed forfor enamel, enamel, dentine, dentine, cementum cementum and and bone,bone, whilewhile a non-a non-linear hyperelastic material was assumed for the periodontal ligament using a nine- linear hyperelastic material was assumed for the periodontal ligament using a nine-pa- parameter Mooney–Rivlin constitutive equation [23], as previously described [14] (Table 2). rameter Mooney–Rivlin constitutive equation [23], as previously described [14] (Table 2). Appl. Sci. 2022, 12, 7063 4 of 11 Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 11 Table 1. Young’s modulus and Poisson coefficient of dental and supporting tissues. Table 1. Young’s modulus and Poisson coefficient of dental and supporting tissues. Tissue Young’s Modulus Poisson Coefficient Tissue Enamel Young’s Modulus 84,100 Poisson0.31 Coefficient Enamel Dentine 84,100 18,600 0.31 0.31 Dentine 18,600 0.31 Cementum 8200 0.3 Cementum 8200 0.3 Bone 14,700 0.31 Bone 14,700 0.31 Table 2. Table 2. Hyperelastic Hyperelasticproperties propertiesof of thethe periodontal ligament. periodontal The Mooney–Rivlin ligament. constitutive The Mooney–Rivlin equa- constitutive tion was used with the listed constants. equation was used with the listed constants. C10 C01 C20 C11 C02 C30 C21 C12 C03 C10 C01 C20 C11 C02 C30 C21 C12 C03 −0.0048 0.00505 0.008 0.0012 0 0.004 0 0 0 −0.0048 0.00505 0.008 0.0012 0 0.004 0 0 0 2.2. Loading Scenario 2.2. Loading Scenario Three typical orthodontic loads (load applied in a mesial direction (mesial load), pal- Three typical atal inclination andorthodontic rotation) wereloads (load applied simulated in a1,mesial for Models 2 and 3.direction Briefly, (mesial load), for the mesial palatal inclination and rotation) were simulated for Models 1, 2 and 3. Briefly, load scenario, a force of 0.6 N was simulated in a mesial direction; for the inclination sce- for the mesial nario, aload loadscenario, of 0.6 Na force of 0.6 N was was simulated insimulated a palatal in a mesial(the direction direction; crownfor the inclination moves from the scenario, vestibularaside loadtoofthe 0.6 palatal N was side); simulated in athe and for palatal direction rotation (theacrown scenario, load of moves 0.6 Nfrom the was ap- vestibular plied at theside to the palatal side); mesiovestibular aspectand for the rotation scenario, a load of 0.6 N was applied [24]. at the mesiovestibular aspect [24]. 2.3. Analysis of Stress 2.3. Analysis of Stress Different metrics of stress distributions were evaluated for the three models (imma- Different metrics of stress tooth ture tooth, dentine-reinforced distributions were evaluated for the and cementum-reinforced three tooth) in models (immature the different load- tooth, dentine-reinforced tooth and cementum-reinforced tooth) in the different loading ing scenarios. Stress values of the tooth and periodontal ligament of each model were an- scenarios. Stress values of the tooth and periodontal ligament of each model were analysed alysed separately to evaluate stress distribution throughout the model. For the tooth tis- separately to evaluate stress distribution throughout the model. For the tooth tissues and sues and the periodontal ligament, we computed the Rankine stress (or maximum stress the periodontal ligament, we computed the Rankine stress (or maximum stress theory) [25] theory) [25] to convert the multiaxial stress state to an equivalent stress state in function to convert the multiaxial stress state to an equivalent stress state in function of the principal of the principal stresses. The plot of maximum principal stresses shows the positive values stresses. The plot of maximum principal stresses shows the positive values of stress pointing of stress pointing the maximum values in tension. the maximum values in tension. Results of the mean stress were summarised. The mean stress for each scenario was Results of the mean stress were summarised. The mean stress for each scenario was reported for the periodontal ligament and teeth in each model situation [26]. reported for the periodontal ligament and teeth in each model situation [26]. 3. Results 3. Results 3.1. Mean Stress Values of the Periodontal Ligament and Tooth Tooth for for Each Each Scenario Scenario The highest mean stress values of the periodontal ligament were observed for mesial rotation (Figure load followed by palatal inclination and rotation (Figure 2). 2). Figure Figure 2. Mean Rankine 2. Mean Rankine stress stress of of the the teeth teeth and and periodontal periodontal ligament ligament of of Models Models 1, 1, 22 and and 33 for for mesial mesial load, palatal inclination and rotation scenarios. Appl. Sci. 2022, 12, 7063 5 of 11 Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of In all scenarios, the mean stress values of the periodontal ligament of the immature In all scenarios, the mean stress values of the periodontal ligament of the immatu tooth were higher than the mean stress values of the periodontal ligament of the dentine- tooth were higher than the mean stress values of the periodontal ligament of the dentin reinforced tooth and the cementum-reinforced tooth (Figure 2). In the mesial load and reinforced tooth and the cementum-reinforced tooth (Figure 2). In the mesial load an palatal inclination scenarios, the mean stress values were slightly higher in the cementum- palatal inclination scenarios, the mean stress values were slightly higher in the cementum reinforced toothreinforced compared to the tooth dentine-reinforced compared tooth (Figuretooth to the dentine-reinforced 2). (Figure 2). In the mesial load scenario, the mean stress value of the dentine-reinforced In the mesial load scenario, the mean stress value of the dentine-reinforced tooth tooth w was higher thanhigher the mean than stress values the mean ofvalues stress both immature and cementum-reinforced of both immature and cementum-reinforced teeth teeth (Fi (Figure 2). In the ure 2). In the palatal inclination scenario, the mean stress of the immaturewas palatal inclination scenario, the mean stress of the immature tooth tooth was sim similar to the dentine-reinforced tooth. The ilar to the dentine-reinforced mean tooth. Thestress meanvalues of the immature stress values tooth tooth we of the immature were higher in the rotation higher in thescenario rotation compared to the mature scenario compared to the teeth (with mature either teeth (withdentine or either dentine or c cementum) (Figure 2). mentum) (Figure 2). Mean stress valuesMean were considerably stress values werelower in the periodontal considerably lower in theligament compared periodontal ligamentto compar the tooth in all models and simulations (Figure 2). to the tooth in all models and simulations (Figure 2). 3.2. Distribution3.2. of the Stress of the Distribution Tooth of the Structure Stress for Mesial of the Tooth Loadfor Mesial Load Structure Figure 3 depictsFigure the distribution 3 depicts theofdistribution the stress inofthe themesial load stress in the scenario forscenario mesial load the threefor the thr models. The distribution models. Theofdistribution the stress inofthethetooth stressstructure andstructure in the tooth periodontal ligament ofligament and periodontal the dentine-reinforced tooth and cementum-reinforced the dentine-reinforced tooth were similar tooth and cementum-reinforced in this tooth were scenario similar in this scenar (Figure 3A–C). (Figure 3A–C). Figure 3. Stress Figure 3. Stress distribution distribution in the in the tooth (A) and tooth in the (A) and inligament periodontal the periodontal (B,C) ofligament (B,C) Models 1, of Models 1 2 and 3 in mesial load scenario. Similar stress patterns are observed between all tissues of Models 2 and 3. of Model and 3 in mesial load scenario. Similar stress patterns are observed between all tissues and 3. Maximum tensile stress of the teeth (A) and ligaments (B,C) are coloured in red. Maximum tensile stress of the teeth (A) and ligaments (B,C) are coloured in red. The highest The highest values values of tensile ofcompromises stress tensile stress compromises a greater areaa of greater area of the the ligament in ligament the in t immature tooth compared to the periodontal ligament of the dentine-reinforced immature tooth compared to the periodontal ligament of the dentine-reinforced tooth and tooth an cementum-reinforced tooth in the mesial load scenario (see amplified cementum-reinforced tooth in the mesial load scenario (see amplified areas of Figure 3B). In areas of Figure 3B the dentine-reinforced tooth, cementum-reinforced tooth and immature tooth, the highest values of tensile stress values in the periodontal ligament were located at the mesial aspect and at the apical third (Figure 3B,C). In the dentine-reinforced tooth, cementum-reinforced tooth and immature tooth, the high- Appl. Sci. 2022, 12, 7063 est values of tensile stress values in the periodontal ligament were located at 6the of 11mesial aspect and at the apical third (Figure 3B,C). 3.3. 3.3.Distribution Distributionof of the Stressofofthe the Stress theTooth Tooth Structure Structure for Palatal for Palatal Inclination Inclination Figure Figure44 depicts thedistribution depicts the distribution of stress of the the stress in palatal in palatal inclination inclination scenario scenario and for and for the the three three models. models. TheThe distribution distribution of the of the in stress stress in the the tooth tooth structure structure and periodontal and periodontal ligament lig- of theof ament dentine-reinforced tooth andtooth the dentine-reinforced cementum-reinforced tooth were similar and cementum-reinforced toothin this scenario were similar in this (Figure 4). scenario (Figure 4). Stressdistribution Figure4.4.Stress Figure distribution in the the tooth tooth(A)(A)and andininthe theperiodontal ligament periodontal (B,C) ligament of Models (B,C) 1, of Models 1, 2 and2 and 3 in3palatal in palatal inclination inclination scenario. scenario. Similar Similar stresspatterns stress patternsare areobserved observed between between all all tissues tissuesofof Mod- elsModels 2 and 23.and 3. Maximum Maximum tensile tensile stress stress of the of the teeth teeth (A)and (A) andligaments ligaments (B,C) (B,C) are arecoloured coloured ininred. red. InInthe the dentine-reinforced tooth and cementum-reinforced tooth, the highest values dentine-reinforced tooth and cementum-reinforced tooth, the highest values of tensile stress in the periodontal ligament were located at the apical third, while for the of tensile stress in the periodontal ligament were located at the apical third, while for the immature tooth, it was located at the apical and coronal third in the palatal inclination immature tooth, 4B,C). scenario (Figure it was located at the apical and coronal third in the palatal inclination scenario (Figure 4B,C). 3.4. Distribution of the Stress of the Tooth Structure for Rotation Scenario 3.4. Distribution of the Stress Figure 5 depicts of the Tooth the distribution of Structure the stress for Rotation in the Scenario rotation scenario for the three models. The Figure 5 distribution depicts the of the stress inof distribution thethe tooth structure stress in theand periodontal rotation ligament scenario of three for the the dentine-reinforced tooth and cementum-reinforced tooth were similar in this scenario models. The distribution of the stress in the tooth structure and periodontal ligament of (Figure 3A–C). the dentine-reinforced tooth and cementum-reinforced tooth were similar in this scenario (Figure 3A–C). Appl. Sci. 2022, 12, 7063 7 of 11 Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 1 Figure 5. Stress Figure 5. Stress distribution distribution in the tooth (A) in andtheintooth (A) and in the the periodontal periodontal ligament ligament (B,C) of Models(B,C) of Models 1, 1, 2 and and 3 in the rotation scenario. Similar stress patterns are observed between all tissues of Models 3 in the rotation scenario. Similar stress patterns are observed between all tissues of Models 2 and 3. and 3. Maximum tensile stress of the teeth (A) and ligaments (B,C) are coloured in red. Maximum tensile stress of the teeth (A) and ligaments (B,C) are coloured in red. In the dentine-reinforced tooth and cementum-reinforced tooth, the highest value In the dentine-reinforced tooth and cementum-reinforced tooth, the highest values of tensile stress in the periodontal ligament were located at the apical third, while for th of tensile stress in the periodontal ligament were located at the apical third, while for immature tooth, it was located at the apical and coronal third in the rotation scenario (Fig the immature tooth, it was located at the apical and coronal third in the rotation scenario ure 5B,C). (Figure 5B,C). 4. Discussion 4. Discussion Regenerative endodontics has several advantages over apexification and root cana Regenerative endodontics has several advantages over apexification and root canal treatment, such as revitalisation and further root development of the immature necroti treatment, such as revitalisation and further root development of the immature necrotic teeth [2]. However, the newly formed tissue lacks the presence of odontoblast-like cel teeth [2]. However, the newly formed tissue lacks the presence of odontoblast-like cells [7] [7] and, as a result, the root maturation is based on reparative tissues [27], such as cemen and, as a result, the root maturation is based on reparative tissues [27], such as cementum- tum-like tissue or osteodentine. Our objective was to determine the biomechanical impl like tissue or osteodentine. Our objective cations over supporting was of tissues to the determine the biomechanical maturation of the root withimplications cementum instead o over supporting dentine, when the tooth is subjected to orthodontic loads. instead tissues of the maturation of the root with cementum of dentine, The results of this FEA stud when the tooth is subjected showed that thetobiomechanical orthodontic loads. The results performance of theof this FEA study periodontal ligamentshowed and tooth struc that the biomechanical ture of the performance dentine- andofcementum-reinforced the periodontal ligament teethand tooth models wasstructure similar ofin the all scenario dentine- and cementum-reinforced and that the periodontal teeth models ligament wasby acted similar reducingin all thescenarios, mean stressand that the values in all model periodontal ligament Theacted by reducing periodontal the acts ligament mean as stress values in a mechanical all models. connection between the tooth and th The periodontal alveolarligament acts as the bone; therefore, a mechanical delivery ofconnection orthodonticbetween the tooth loads to the toothcauses and the stress in th alveolar bone; periodontal therefore, the delivery ligament, of itorthodontic and is related toloads to theoftooth resorption causes stress the periodontal in the tissues [28]. On th periodontal ligament, and it is related to resorption of the periodontal tissues [28]. pressure side, the external mechanical force induces collagenous extracellular matrix re On the pressure side, the external sorption mechanicalligament of the periodontal force induces and bone collagenous [29] due toextracellular matrix clastic activity. The recruite resorption of the periodontal ligament and bone [29] due to clastic activity. The osteoclasts resorb and remodel the alveolar bone during orthodontic movement [29]. A recruited osteoclasts resorb and remodelstress disadvantageous the alveolar bone distribution during with peakorthodontic stress values movement concentrated[29]. A areas o in some disadvantageous stress distribution with peak stress values concentrated in some areas of the periodontal ligament could lead to a pathological process where resorption is not Appl. Sci. 2022, 12, 7063 8 of 11 controlled and affects the tooth’s hard tissues. Root resorption is prone to occur in areas of stress concentration at the compression site during the process of orthodontic tooth movement [30]. Consequently, the question arises as to whether teeth previously treated with regenerative endodontics and that have completed their root development with cementum can endure orthodontic treatment or if they are at risk of root resorption [19]. In our simulation, the mean stress values of the periodontal ligament were similar among the dentine and cementum-reinforced teeth. Moreover, the stress distribution patterns in the periodontal ligament were similar among the dentine- and cementum-reinforced teeth as well. These results suggest that the performance of previously revitalised teeth subjected to orthodontic loads might be comparable to that of physiologically developed teeth, although this needs to be confirmed in in vivo situations. Thus, controlled clinical trials and/or experimental studies are needed. The results of this study also showed that the periodontal ligament acted by dissipating the mechanical loads incurred during simulation of orthodontic loads. Moreover, stress distribution patterns demonstrated that peak stress values were reduced from the teeth to the ligament. Cementum has poorer mechanical properties than dentine [13]. The amount of ce- mentum surrounding the root of physiologically developed teeth (i.e., with dentine) has implications in the mechanical performance of teeth, since mechanical stresses increase at the root apex with increasing thickness of the apical cementum [31]. A previous com- putational study showed a disadvantageous stress distribution along the root in teeth that have completed their root development with cementum compared to physiologically developed teeth [14]. However, it must be noted that these differences in peak stress values are remarkable in high magnitude forces, such as those applied during biting or traumatic dental injuries and are mild with low magnitude forces, such as those applied in orthodon- tics [14]. Regarding stress distribution in the tooth, the present study shows similar mean stress values and stress distribution patterns for cementum- and dentine-reinforced teeth in mesial load, palatal inclination and rotation. In addition, the maturation of the root, with either dentine or cementum, was shown to be beneficial for all scenarios, where the immature tooth and its periodontal ligament showed higher mean stress values. The disadvantageous stress distribution of immature teeth compared to mature teeth has been observed in FEA studies [14,32] and poorer fracture resistance [33] and prognosis [22] have been shown in clinical and animal studies as well [34], and therefore, it is necessary to limit the orthodontic force to prevent root resorption. The finite element model can simulate tooth-periodontal ligament complex under orthodontic loads. It allows the construction of standardised models to simulate the biomechanical performance of teeth and supporting tissues. Remodelling of the periodontal ligament for FEA analyses has been studies extensively, and the large inconsistency in results can be explained by the modelling assumptions (i.e., linear or non-linear, elastic or viscoelastic) given to the models [35]. The periodontal ligament is known to be a non-linear and time-dependent material, as proven by several experimental studies [34–37]. However, finite element model studies simulating orthodontic loads often incorporate homogeneous, linear elastic, isotropic and continuous periodontal ligament properties [17]. Non-linear simulations of the periodontal ligament provide more accurate and reliable calculation for the stresses and strains over several tooth movements than the ones provided by linear simulations [37]. Similarly, viscoelastic time-dependent properties of the ligament are the reason of strain energy dissipation, without which excessive load would cause tooth fracture [38,39]. In this study, we assumed non-linear periodontal ligament properties [17] to capture the elastic nature and deformation capacity of the periodontal ligament. The inclusion of the time-dependent viscoelastic properties was discarded in this study because of the comparative nature of the work. When comparing the same scenario under the three different models we expected the same qualitative results before the relaxation than after the relaxation. Thus, it must be considered that ‘time’ was not a variable included in our analysis, and in consequence this study shows the stress distribution in a specific moment Appl. Sci. 2022, 12, 7063 9 of 11 and not the fatigue caused by the orthodontic treatment. The results of this research must be validated in clinical studies, since no computational study can represent the complex response of the teeth and surrounding tissues under orthodontic loads. To the best of our knowledge, the mechanical properties of the newly formed tissues after regenerative endodontics (periodontal ligament and cementum-like tissue) have not been studied so far. The histological evidence of human teeth treated with regenerative endodontics, limited to case reports and mainly focused on the analysis of the intracanal tissues [6,8,40], do not report differences regarding the shape, thickness nor distribution of the collagen fibres of the newly formed periodontal ligament (the ligament formed between the newly formed hard tissue and the alveolar bone) and the pre-existing one (the periodontal ligament between the immature root and the alveolar bone). Similarly, the tissue responsible for root maturation is histologically similar to the cellular cementum [6,8,40], tissue present in the apical third of mature teeth, or osteocementum. Considering this data, in the present study the shape and mechanical properties given to the periodontal ligament and cementum of the cementum-reinforced tooth were assumed equal to that of a physiologically developed tooth. Thus, the basic data used in this study (i.e., Young’s modulus and Poisson coefficient of the cementum and hyperelastic properties of the periodontal ligament) are not the ones of an immature tooth after regenerative endodontic, but an extrapolation of the mechanical properties of a physiologically developed tooth. Future studies should assess whether the mechanical properties of the immature teeth and of the newly formed tissues after regenerative endodontics are indeed equivalent to those of physiologically developed teeth. 5. Conclusions As far as we know, this is the first study analysing the performance of the periodontal ligament in teeth previously treated with regenerative endodontics. Under the conditions of this finite element analysis study, stress distribution in the tooth and periodontal ligament in a physiologically developed tooth and in a tooth after regenerative endodontics is similar in all three simulations of orthodontic loads. The periodontal ligament acted by dissipating the mechanical loads incurred during orthodontic forces. Maturation of the root with either dentine or cementum is beneficial to the stress distribution in all simulated scenarios. Among the limitations of this computational study, it can be concluded that orthodontic loads can be applied in teeth that have been treated with regenerative endodontics in the past; however, in vivo studies should be performed to validate these results. Future studies should verify the mechanical characteristics of the newly formed tissues after regenerative endodontics to validate these results. Finite element analysis opens new possibilities of research in the field of regenerative endodontics, since the biomechanical performance of teeth and supporting tissues can be simulated in standardised conditions. Future studies can be directed to analyse the mechanical performance of teeth after regenerative endodontics restored with different materials, and its comparison with teeth undergoing other endodontic treatments such as apexification, among others. Author Contributions: Conceptualisation, C.B. and J.M.-N.; methodology, C.B. and J.M.-N.; soft- ware, J.M.-N.; validation, J.M.-N.; formal analysis, C.B., J.M.-N. and M.D.F.; investigation, C.B.; resources, C.B. and J.M.-N.; data curation, C.B. and J.M.-N.; writing—original draft preparation, C.B.; writing—review and editing, C.B., J.M.-N. and M.D.F.; visualisation, C.B.; supervision, M.D.F.; project administration, C.B. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of Universitat de Barcelona. 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