Biomechanical considerations on tooth-implant supported fixed partial dentures
Identifieur interne : 003097 ( Pmc/Corpus ); précédent : 003096; suivant : 003098Biomechanical considerations on tooth-implant supported fixed partial dentures
Auteurs : Konstantinos X. Michalakis ; Pasquale Calvani ; Hiroshi HirayamaSource :
- Journal of Dental Biomechanics [ 1758-7360 ] ; 2012.
Abstract
This article discusses the connection of teeth to implants, in order to restore partial edentulism. The main problem arising from this connection is tooth intrusion, which can occur in up to 7.3% of the cases. The justification of this complication is being attempted through the perspective of biomechanics of the involved anatomical structures, that is, the periodontal ligament and the bone, as well as that of the teeth- and implant-supported fixed partial dentures.
Url:
DOI: 10.1177/1758736012462025
PubMed: 23255882
PubMed Central: 3487629
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PMC:3487629Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Biomechanical considerations on tooth-implant supported fixed partial dentures</title><author><name sortKey="Michalakis, Konstantinos X" sort="Michalakis, Konstantinos X" uniqKey="Michalakis K" first="Konstantinos X" last="Michalakis">Konstantinos X. Michalakis</name><affiliation><nlm:aff id="aff1-1758736012462025">Division of Removable Prosthodontics, Department of Prosthodontics, School of Dentistry, Aristotle University of Thessaloniki, Thessaloniki, Greece</nlm:aff></affiliation><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author><author><name sortKey="Calvani, Pasquale" sort="Calvani, Pasquale" uniqKey="Calvani P" first="Pasquale" last="Calvani">Pasquale Calvani</name><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author><author><name sortKey="Hirayama, Hiroshi" sort="Hirayama, Hiroshi" uniqKey="Hirayama H" first="Hiroshi" last="Hirayama">Hiroshi Hirayama</name><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">23255882</idno><idno type="pmc">3487629</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3487629</idno><idno type="RBID">PMC:3487629</idno><idno type="doi">10.1177/1758736012462025</idno><date when="2012">2012</date><idno type="wicri:Area/Pmc/Corpus">003097</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">003097</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Biomechanical considerations on tooth-implant supported fixed partial dentures</title><author><name sortKey="Michalakis, Konstantinos X" sort="Michalakis, Konstantinos X" uniqKey="Michalakis K" first="Konstantinos X" last="Michalakis">Konstantinos X. Michalakis</name><affiliation><nlm:aff id="aff1-1758736012462025">Division of Removable Prosthodontics, Department of Prosthodontics, School of Dentistry, Aristotle University of Thessaloniki, Thessaloniki, Greece</nlm:aff></affiliation><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author><author><name sortKey="Calvani, Pasquale" sort="Calvani, Pasquale" uniqKey="Calvani P" first="Pasquale" last="Calvani">Pasquale Calvani</name><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author><author><name sortKey="Hirayama, Hiroshi" sort="Hirayama, Hiroshi" uniqKey="Hirayama H" first="Hiroshi" last="Hirayama">Hiroshi Hirayama</name><affiliation><nlm:aff id="aff2-1758736012462025">Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</nlm:aff></affiliation></author></analytic><series><title level="j">Journal of Dental Biomechanics</title><idno type="eISSN">1758-7360</idno><imprint><date when="2012">2012</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>This article discusses the connection of teeth to implants, in order to restore partial edentulism. The main problem arising from this connection is tooth intrusion, which can occur in up to 7.3% of the cases. The justification of this complication is being attempted through the perspective of biomechanics of the involved anatomical structures, that is, the periodontal ligament and the bone, as well as that of the teeth- and implant-supported fixed partial dentures.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Br Nemark, P I" uniqKey="Br Nemark P">P-I Brånemark</name></author><author><name sortKey="Zarb, Ga" uniqKey="Zarb G">GA Zarb</name></author><author><name sortKey="Albrekstsson, T" uniqKey="Albrekstsson T">T Albrekstsson</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Adell, R" uniqKey="Adell R">R Adell</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Albrektsson, T" uniqKey="Albrektsson T">T Albrektsson</name></author><author><name sortKey="Zarb, Ga" uniqKey="Zarb G">GA Zarb</name></author><author><name sortKey="Worthington, P" uniqKey="Worthington P">P 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<front><journal-meta><journal-id journal-id-type="nlm-ta">J Dent Biomech</journal-id><journal-id journal-id-type="iso-abbrev">J Dent Biomech</journal-id><journal-id journal-id-type="publisher-id">DBM</journal-id><journal-id journal-id-type="hwp">spdbm</journal-id><journal-title-group><journal-title>Journal of Dental Biomechanics</journal-title></journal-title-group><issn pub-type="epub">1758-7360</issn><publisher><publisher-name>SAGE Publications</publisher-name><publisher-loc>Sage UK: London, England</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">23255882</article-id><article-id pub-id-type="pmc">3487629</article-id><article-id pub-id-type="doi">10.1177/1758736012462025</article-id><article-id pub-id-type="publisher-id">10.1177_1758736012462025</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Biomechanical considerations on tooth-implant supported fixed partial dentures</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name><surname>Michalakis</surname><given-names>Konstantinos X</given-names></name><xref ref-type="aff" rid="aff1-1758736012462025">1</xref><xref ref-type="aff" rid="aff2-1758736012462025">2</xref></contrib><contrib contrib-type="author"><name><surname>Calvani</surname><given-names>Pasquale</given-names></name><xref ref-type="aff" rid="aff2-1758736012462025">2</xref></contrib><contrib contrib-type="author"><name><surname>Hirayama</surname><given-names>Hiroshi</given-names></name><xref ref-type="aff" rid="aff2-1758736012462025">2</xref></contrib></contrib-group><aff id="aff1-1758736012462025"><label>1</label>Division of Removable Prosthodontics, Department of Prosthodontics, School of Dentistry, Aristotle University of Thessaloniki, Thessaloniki, Greece</aff><aff id="aff2-1758736012462025"><label>2</label>Division of Graduate and Postgraduate Prosthodontics, Department of Prosthodontics and Operative Dentistry, Tufts University School of Dental Medicine, Boston, MA, USA</aff><author-notes><corresp id="corresp1-1758736012462025">Konstantinos X Michalakis, Division of Removable Prosthodontics, Department of Prosthodontics, School of Dentistry, Aristotle University of Thessaloniki, 3, Gregoriou Palama str., Thessaloniki 54622, Greece. Email: <email>kmichalakis@hotmail.com</email></corresp></author-notes><pub-date pub-type="collection"><year>2012</year></pub-date><pub-date pub-type="epub"><day>29</day><month>10</month><year>2012</year></pub-date><volume>3</volume><elocation-id>1758736012462025</elocation-id><permissions><copyright-statement>© The Author(s) 2012</copyright-statement><copyright-year>2012</copyright-year><copyright-holder content-type="sage">SAGE Publications</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/2.5/"><license-p>This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</license-p></license></permissions><abstract><p>This article discusses the connection of teeth to implants, in order to restore partial edentulism. The main problem arising from this connection is tooth intrusion, which can occur in up to 7.3% of the cases. The justification of this complication is being attempted through the perspective of biomechanics of the involved anatomical structures, that is, the periodontal ligament and the bone, as well as that of the teeth- and implant-supported fixed partial dentures.</p></abstract><kwd-group><kwd>Biomechanics</kwd><kwd>periodontal ligament</kwd><kwd>bone</kwd><kwd>tooth to implant connection</kwd></kwd-group></article-meta></front><body><p>Several changes have been made since the introduction of implants for dental prosthetic rehabilitation. The initial concept introduced by Professor Per-Ingvar Brånemark and his associates recommended placement of six implants in the anterior mandibular or maxillary area and construction of a fixed detachable hybrid prosthesis, made of a metal substructure, denture teeth, and heat polymerized acrylic resin material.<sup><xref ref-type="bibr" rid="bibr1-1758736012462025">1</xref><xref ref-type="bibr" rid="bibr2-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr3-1758736012462025">3</xref></sup></p><p>Since then, researchers and clinicians modified the initial implant treatment options. Implants are now being used for the restoration of maxillary complete edentulism, as well as for the rehabilitation of partial edentulism.<sup><xref ref-type="bibr" rid="bibr4-1758736012462025">4</xref><xref ref-type="bibr" rid="bibr5-1758736012462025"></xref><xref ref-type="bibr" rid="bibr6-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr7-1758736012462025">7</xref></sup> Furthermore, connection between teeth and implant fixtures has been advocated as a feasible way to provide prosthetic reconstructions, when anatomical limitations—for example, sinus or mental nerve proximity and lack of sufficient bone quantity—or financial restrictions are present<sup><xref ref-type="bibr" rid="bibr8-1758736012462025">8</xref>,<xref ref-type="bibr" rid="bibr9-1758736012462025">9</xref></sup> (<xref ref-type="fig" rid="fig1-1758736012462025">Figure 1</xref>).</p><fig id="fig1-1758736012462025" position="float"><label>Figure 1.</label><caption><p>Implant-to-tooth connection with no apparent tooth intrusion.</p></caption><graphic xlink:href="10.1177_1758736012462025-fig1"></graphic></fig><p>It should be mentioned that initially separation of the implants from abutment teeth has been recommended, because successfully osseointegrated implant fixtures are practically ankylosed in the bone, while teeth present some mobility due to the presence of the periodontal ligament (PDL). The efficacy of the implant-to-tooth connection has been discussed in the dental literature.<sup><xref ref-type="bibr" rid="bibr10-1758736012462025">10</xref>,<xref ref-type="bibr" rid="bibr11-1758736012462025">11</xref></sup> Due to the complications (<xref ref-type="fig" rid="fig2-1758736012462025">Figure 2</xref>) that have been observed, there is an argument among the clinicians regarding the long-term prognoses of these restorations.<sup><xref ref-type="bibr" rid="bibr12-1758736012462025">12</xref>,<xref ref-type="bibr" rid="bibr13-1758736012462025">13</xref></sup></p><fig id="fig2-1758736012462025" position="float"><label>Figure 2.</label><caption><p>Clinical evidence of evident tooth intrusion in an implant-to-tooth connection. Note the discrepancy at the attachment and at the occlusal surfaces of the restorations supported by the tooth and the implant.</p></caption><graphic xlink:href="10.1177_1758736012462025-fig2"></graphic></fig><p>The purpose of this article is to review the existing literature and discuss the biomechanics of the tooth’s supporting tissue (PDL), the biomechanics of implants’ supporting tissue (bone), as well as the biomechanics of the tooth–implant connection, in an attempt to focus on the tooth intrusion phenomenon that is sometimes clinically and radiographically observed.</p><sec id="section1-1758736012462025"><title>Biomechanics of the PDL</title><p>The PDL is the connective tissue structure surrounding the root of the tooth and connecting it with the adjacent bone. Its thickness is about 0.2 mm. Histochemical and morphometric analyses of the PDL have demonstrated that it is composed of collagen, oxytalan and elaunin fibers, a ground substance made of glycosaminoglycans, and blood vessels that occupy 4%–47% of the tissue’s volume.<sup><xref ref-type="bibr" rid="bibr14-1758736012462025">14</xref><xref ref-type="bibr" rid="bibr15-1758736012462025"></xref><xref ref-type="bibr" rid="bibr16-1758736012462025"></xref><xref ref-type="bibr" rid="bibr17-1758736012462025"></xref><xref ref-type="bibr" rid="bibr18-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr19-1758736012462025">19</xref></sup> Microscopically, the PDL is constructed from fibers with high spatial organization, which can be categorized into six groups: (a) transeptal, (b) alveolar crest, (c) horizontal, (d) oblique, (e) apical, and (f) interradicular (only between the roots of the multirooted teeth).<sup><xref ref-type="bibr" rid="bibr20-1758736012462025">20</xref>,<xref ref-type="bibr" rid="bibr21-1758736012462025">21</xref></sup> These specialized connective tissue fibers are not unidirectionally distributed. They connect the tooth to the alveolar bone and absorb functional and parafunctional forces originated both from the teeth and other adjacent anatomical structures (i.e. lips, cheeks, and tongue). Another feature that affects the biomechanical response of the PDL is the presence of numerous vessels that exert a hemodynamic pressure.<sup><xref ref-type="bibr" rid="bibr22-1758736012462025">22</xref><xref ref-type="bibr" rid="bibr23-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr24-1758736012462025">24</xref></sup> The exact mechanism with which the PDL supports the tooth is not clear.<sup><xref ref-type="bibr" rid="bibr25-1758736012462025">25</xref></sup> Three main hypotheses have been made in order to explain the action of the PDL: (a) the <italic>tensional mechanism of tooth support</italic>,<sup><xref ref-type="bibr" rid="bibr26-1758736012462025">26</xref><xref ref-type="bibr" rid="bibr27-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr28-1758736012462025">28</xref></sup> which supports the notion that a gradual unfolding of the fibers takes place in order to transmit the loads to the surrounding alveolar bone, (b) the <italic>viscoelastic model</italic>,<sup><xref ref-type="bibr" rid="bibr29-1758736012462025">29</xref>,<xref ref-type="bibr" rid="bibr30-1758736012462025">30</xref></sup> which considers that tooth displacement is actually controlled by vascular elements and the fibers have a less important function, (c) the <italic>collagenous thixotropic system</italic>, which supports the idea that the PDL possesses thixotropic gel properties.<sup><xref ref-type="bibr" rid="bibr31-1758736012462025">31</xref>,<xref ref-type="bibr" rid="bibr32-1758736012462025">32</xref></sup></p><p>The PDL seems to play an important role in the function of the stomatognathic system by (a) its mechanoreceptors, which transmit to the central nervous system essential information regarding the occlusal forces,<sup><xref ref-type="bibr" rid="bibr33-1758736012462025">33</xref>,<xref ref-type="bibr" rid="bibr34-1758736012462025">34</xref></sup> (b) furnishing of stimulating tensile loads to the alveolar bone, and (c) the provision of physiologic tooth mobility. This mobility differs among different groups of teeth. Teeth with one root present higher mobility than teeth with multiple roots.<sup><xref ref-type="bibr" rid="bibr35-1758736012462025">35</xref></sup> It has also been demonstrated that mobility also varies from day to day or even from hour to hour.<sup><xref ref-type="bibr" rid="bibr36-1758736012462025">36</xref></sup> Physiologic teeth mobility can be classified as horizontal and vertical. The upper limit of horizontal tooth mobility has been found to be T<sub>500</sub> = 0.15 mm for single-rooted teeth and T<sub>500</sub> = 0.10 mm for multirooted teeth, where T<sub>500</sub> is the range of tooth movement under an occlusal force of 0.5 kg.<sup><xref ref-type="bibr" rid="bibr37-1758736012462025">37</xref>,<xref ref-type="bibr" rid="bibr38-1758736012462025">38</xref></sup> Vertical tooth mobility occurs in two stages: (a) the initial or intrasocket stage, in which the tooth moves within the boundaries of the PDL;<sup><xref ref-type="bibr" rid="bibr39-1758736012462025">39</xref></sup> this movement occurs with forces of about 100 lb and it ranges between 50 and 100 µm.<sup><xref ref-type="bibr" rid="bibr40-1758736012462025">40</xref></sup> (b) The secondary stage, which occurs gradually and entails elastic deformation of the alveolar bone in response to increased horizontal forces.<sup><xref ref-type="bibr" rid="bibr40-1758736012462025">40</xref></sup></p><p>It has been observed that there is a nonlinear force–displacement relationship for a tooth, regardless of the direction of occlusal loading. Logarithmic relationships between applied forces and teeth displacement have been demonstrated after application of both horizontal and vertical forces.<sup><xref ref-type="bibr" rid="bibr41-1758736012462025">41</xref>,<xref ref-type="bibr" rid="bibr42-1758736012462025">42</xref></sup> Under the same loading–unloading rates, a hysteresis loop was obtained for axial and horizontal loadings, as well as for rotational movements.<sup><xref ref-type="bibr" rid="bibr43-1758736012462025">43</xref>,<xref ref-type="bibr" rid="bibr44-1758736012462025">44</xref></sup> However, more recent animal studies have indicated that for occlusal loads between 0.1 and 4.0 N, there is a two-phase parabolic relationship with a discontinuity occurring within the force range of 0.5–0.8 N.<sup><xref ref-type="bibr" rid="bibr45-1758736012462025">45</xref><xref ref-type="bibr" rid="bibr46-1758736012462025"></xref><xref ref-type="bibr" rid="bibr47-1758736012462025"></xref><xref ref-type="bibr" rid="bibr48-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr49-1758736012462025">49</xref></sup> These parabolic relationships present the following forms:</p><p><disp-formula id="disp-formula1-1758736012462025"><mml:math id="math1-1758736012462025"><mml:mtable><mml:mtr><mml:mtd><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>F</mml:mi><mml:mn>1</mml:mn><mml:mi>M</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width="0.25em"></mml:mspace><mml:mtext>for forces between</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.25em"></mml:mspace><mml:mtext>and</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn><mml:mspace width="0.25em"></mml:mspace><mml:mtext>N</mml:mtext></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>F</mml:mi><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mtext>for forces between</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>8</mml:mn><mml:mspace width="0.25em"></mml:mspace><mml:mtext>and</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mn>4</mml:mn><mml:mspace width="0.25em"></mml:mspace><mml:mtext>N</mml:mtext></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p><p>where <italic>I</italic> is the intrusion, <italic>F</italic> is the force, and <italic>M</italic><sub>1</sub> and <italic>M</italic><sub>2</sub> are the slopes of the log displacement versus log force curves. It has also been demonstrated that <italic>M</italic><sub>1</sub> is sensitive to the rate of application of occlusal forces, while <italic>M</italic><sub>2</sub> is sensitive to periodontal fluid exchanges.</p><p>Horizontal forces create greater tooth displacements than vertical loads. It has been shown that a 1-N horizontal force applied for a period of 2 s causes a movement of 150 µm, while the same force applied vertically produces a 15–20 µm displacement. It should be mentioned, however, that tooth movement produced by horizontal loads is also dependent on the duration, rate, and exact point of force application. These horizontal forces can cause buccolingual and mesiodistal movements, as well as extrusion.<sup><xref ref-type="bibr" rid="bibr50-1758736012462025">50</xref></sup></p><p>A biomechanical analysis of the PDL has been attempted in the past, based on numerical techniques. The formulation of models for the explanation of living material behavior involves the development of relations between stress, strain, and their derivatives with respect to time.<sup><xref ref-type="bibr" rid="bibr51-1758736012462025">51</xref></sup> The construction of these constitutive models many times adopted simplifications and assumptions in order to overcome inherent difficulties in obtaining experimental data. In that manner, isotropy and linear elasticity have been incorporated in the mathematical modeling.<sup><xref ref-type="bibr" rid="bibr52-1758736012462025">52</xref></sup> However, this simplification may only be used for fairly small stretches, where the real performance of many biological fibers may be sufficiently approached by a linear relationship between force and stretch. In those cases, the stress–strain relationship is expressed by<sup><xref ref-type="bibr" rid="bibr53-1758736012462025">53</xref></sup></p><p><disp-formula id="disp-formula2-1758736012462025"><label>(1)</label><mml:math id="math2-1758736012462025"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mi>c</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.25em"></mml:mspace></mml:mrow></mml:mfrac></mml:mrow></mml:mfrac><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p>where <italic>c</italic> is an intrinsic property of the fiber, which is independent of the unloaded length of the fiber, <italic>λ</italic> is the stretch parameter, and <italic>λ<sub>c</sub></italic> is the critical value after which there is a sharp increase of the force in the fiber. For small range extensions, <italic>λ</italic> ≈ 1. Therefore</p><p><disp-formula id="disp-formula3-1758736012462025"><mml:math id="math3-1758736012462025"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>λ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mspace width="0.25em"></mml:mspace></mml:mrow></mml:mfrac><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></disp-formula></p><p>and equation (1) becomes</p><p><disp-formula id="disp-formula4-1758736012462025"><label>(2)</label><mml:math id="math4-1758736012462025"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>λ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula> (2)</p><p>The quantity <italic>λ-</italic>1, in equations (1) and (2) is referred to as strain, which can also be expressed as <italic>ϵ</italic>. Thus, equation (2) becomes</p><p><disp-formula id="disp-formula5-1758736012462025"><label>(3)</label><mml:math id="math5-1758736012462025"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mi>ε</mml:mi></mml:mrow></mml:math></disp-formula></p><p>It should be mentioned, however, that the majority of biologic materials do not demonstrate an elastic behavior, but rather a viscoelastic one. They present the phenomena of both the relaxation and creep. When they are subjected to a cyclic loading, the loading and the unloading processes follow somewhat different paths, which delineate an area representing the amount of energy dissipated as heat during the process (<xref ref-type="fig" rid="fig3-1758736012462025">Figure 3</xref>). This area is equal to <inline-formula id="inline-formula1-1758736012462025"><mml:math id="math6-1758736012462025"><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:mo>∮</mml:mo><mml:mrow><mml:mi>σ</mml:mi><mml:mi>d</mml:mi><mml:mi>ε</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math></inline-formula>, where <italic>σ</italic> is the stress and <italic>ϵ</italic> is the strain.</p><fig id="fig3-1758736012462025" position="float"><label>Figure 3.</label><caption><p>Different paths of loading and unloading of a viscoelastic material.</p></caption><graphic xlink:href="10.1177_1758736012462025-fig3"></graphic></fig><p>The big percentage of water (and other biologic fluids) in the tissues is responsible for their viscoelastic behavior, which in a one-dimensional model is expressed as</p><p><disp-formula id="disp-formula6-1758736012462025"><label>(4)</label><mml:math id="math7-1758736012462025"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mfrac><mml:mn>1</mml:mn><mml:mi>l</mml:mi></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <inline-formula id="inline-formula2-1758736012462025"><mml:math id="math8-1758736012462025"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the damping coefficient and <inline-formula id="inline-formula3-1758736012462025"><mml:math id="math9-1758736012462025"><mml:mrow><mml:mi>d</mml:mi><mml:mi>l</mml:mi><mml:mo>/</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math></inline-formula> is the time derivative that measures the rate of change of the fiber’s length.</p><p>Moreover, the wavy configuration of the PDL’s fibers and the change in their conformation under loading assist the description of the nonlinear stress–strain curve observed, which displays a small stiffness to external loads for moderately big strains and higher and quasi-constant stiffness for bigger strains.<sup><xref ref-type="bibr" rid="bibr54-1758736012462025">54</xref><xref ref-type="bibr" rid="bibr55-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr56-1758736012462025">56</xref></sup></p><p>There is a question, however, whether one model is sufficient to fully describe the behavior of the PDL, since when a force is exerted on the tooth, some regions of the PDL will always be under compression while others will be under tension.</p><p>Experimental data by Bergomi et al.<sup><xref ref-type="bibr" rid="bibr57-1758736012462025">57</xref></sup> have demonstrated that the compressive load response of the PDL is viscous, and it may be described by the following pseudoplastic, time-dependent constitutive equation</p><p><disp-formula id="disp-formula7-1758736012462025"><label>(5)</label><mml:math id="math10-1758736012462025"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>λ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo>|</mml:mo><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mrow><mml:mo>|</mml:mo><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>where <italic>T</italic> is the stress and <italic>λ</italic> is the stretch ratio.</p><p>However, this viscous model cannot be assumed for the fibers under tension, since the role of the fluid phase was not found to be equally distributed between the tensile and the compressive parts of the load cycles.<sup><xref ref-type="bibr" rid="bibr39-1758736012462025">39</xref></sup> While the results of the compressive phase can be expected to be an interaction amid the porous matrix (fibers, vessels, and ground substance) and the fluid phase (blood and interstitial fluid), the same concept does not seem to apply for the tensile phase. The latter is presumed to be a tissue restructuring, with gradual unfolding, reorientation, and stretching of the collagen fibers.<sup><xref ref-type="bibr" rid="bibr55-1758736012462025">55</xref>,<xref ref-type="bibr" rid="bibr58-1758736012462025">58</xref>,<xref ref-type="bibr" rid="bibr59-1758736012462025">59</xref></sup> Other facts that need to be considered include the movement of the interstitial fluid from compressive to tensile areas, the porosity of the surrounding bone that receives blood and tissue fluids, and the elasticity of the bone that is usually assumed to be negligible due to the big difference with that of the PDL.<sup><xref ref-type="bibr" rid="bibr60-1758736012462025">60</xref>,<xref ref-type="bibr" rid="bibr61-1758736012462025">61</xref></sup> It has been also postulated that the PDL can be treated as a hyperelastic matrix including numerous collapsible voids. Under compression, these pores can be filled with fluids that in this way can move to nearby areas of the PDL or forced in the adjacent alveolar bone.<sup><xref ref-type="bibr" rid="bibr62-1758736012462025">62</xref></sup> This movement of the fluid may also be accompanied by electrokinetic effects, for example, streaming potentials and currents, as the ions circulate through the charged solid medium.<sup><xref ref-type="bibr" rid="bibr63-1758736012462025">63</xref>,<xref ref-type="bibr" rid="bibr64-1758736012462025">64</xref></sup> The permeability of both the neighboring bone and cementum should be taken into account and can be obtained using Darcy’s relation<sup><xref ref-type="bibr" rid="bibr65-1758736012462025">65</xref><xref ref-type="bibr" rid="bibr66-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr67-1758736012462025">67</xref></sup></p><p><disp-formula id="disp-formula8-1758736012462025"><label>(6)</label><mml:math id="math11-1758736012462025"><mml:mrow><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>μ</mml:mi></mml:mfrac><mml:mi>κ</mml:mi><mml:mfrac><mml:mrow><mml:mi>A</mml:mi><mml:mi>Δ</mml:mi><mml:mi>P</mml:mi></mml:mrow><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <italic>Q</italic> is the flow rate, <italic>µ</italic> is the dynamic viscosity of the fluid, <italic>κ</italic> is the permeability of the medium, <italic>A</italic> is the cross-sectional area to flow, Δ<italic>P</italic> is the pressure drop, and <italic>L</italic> is the length over which the pressure drop takes place.</p><p>In that sense, PDL’s response can be based on Biot’s concept of dynamic poroelasticity and poromechanics, that is, the study of fluid-saturated porous media whose performance on external actions is influenced by the fluid occupying the pores.<sup><xref ref-type="bibr" rid="bibr62-1758736012462025">62</xref>,<xref ref-type="bibr" rid="bibr68-1758736012462025">68</xref>,<xref ref-type="bibr" rid="bibr69-1758736012462025">69</xref></sup> Therefore, the modeling requires a pore-permeated matrix that is filled with a fluid. The solution to this problem involves equations of linear elasticity for the solid medium, Navier–Stokes equations for the fluid, and as mentioned before, Darcy’s equation for the fluid flow. In order to model the solid matrix, Bergomi et al.<sup><xref ref-type="bibr" rid="bibr62-1758736012462025">62</xref></sup> have adopted a modification of Ogden’s<sup><xref ref-type="bibr" rid="bibr70-1758736012462025">70</xref>,<xref ref-type="bibr" rid="bibr71-1758736012462025">71</xref></sup> material model strain energy density</p><p><disp-formula id="disp-formula9-1758736012462025"><label>(7)</label><mml:math id="math12-1758736012462025"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>W</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>Ν</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi>μ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>λ</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>λ</mml:mi><mml:mn>3</mml:mn><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>β</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula></p><p>where <italic>λ<sub>j</sub></italic> are the principal stretch ratios; <italic>N, µ<sub>i</sub></italic>, and <italic>α<sub>i</sub></italic> are material constants; and <italic>J</italic> is the Jacobian determinant.</p><p>It can be understood that this presents a highly complicated problem with blood and lymph vessels, as well as the ground substance forming a continuum arrangement with viscoelastic properties. On the other hand, the collagen fibrils that are embedded in the ground substance alter the symmetry characteristics of the material.<sup><xref ref-type="bibr" rid="bibr56-1758736012462025">56</xref></sup> As a result, an anisotropic mechanical response and nonlinear behavior are expected, due to the multidirectional arrangement of the collagen fibrils. The crimped conformation of the fibrils and the anisotropy of the material have led some research groups to adopt a hyperelastic model in order to describe the response of the PDL.<sup><xref ref-type="bibr" rid="bibr56-1758736012462025">56</xref>,<xref ref-type="bibr" rid="bibr72-1758736012462025">72</xref></sup> In brief, a material can be called hyperelastic or Green elastic when the material’s response is such that the stress power and the stress work may be obtained from a scalar valued potential <inline-formula id="inline-formula4-1758736012462025"><mml:math id="math13-1758736012462025"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula> such that<sup><xref ref-type="bibr" rid="bibr73-1758736012462025">73</xref></sup></p><p><disp-formula id="disp-formula10-1758736012462025"><label>(8)</label><mml:math id="math14-1758736012462025"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>ϕ</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mstyle mathvariant="bold"><mml:mi>E</mml:mi></mml:mstyle></mml:mrow></mml:mfrac><mml:mo>:</mml:mo><mml:mstyle mathvariant="bold"><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mstyle><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mstyle mathvariant="bold"><mml:mi>E</mml:mi></mml:mstyle><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:msub><mml:mstyle mathvariant="bold"><mml:mover accent="true"><mml:mi>E</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mstyle><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p><disp-formula id="disp-formula11-1758736012462025"><label>(9)</label><mml:math id="math15-1758736012462025"><mml:mrow><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:mrow><mml:mi>t</mml:mi></mml:munderover><mml:mi>ω</mml:mi><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mspace width="0.25em"></mml:mspace><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mstyle mathvariant="bold"><mml:mi>E</mml:mi></mml:mstyle><mml:mo>]</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mspace width="0.25em"></mml:mspace><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mstyle mathvariant="bold"><mml:mi>E</mml:mi></mml:mstyle><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mi>Δ</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></disp-formula></p><p>where <italic>ω</italic> is the stress power per unit volume, <bold>E</bold> is the state of strain, <inline-formula id="inline-formula5-1758736012462025"><mml:math id="math16-1758736012462025"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula> is the elastic energy or strain energy per unit volume, and <italic>w</italic> is the stress work per unit volume, when the material is deformed.</p><p>The following equations also hold</p><p><disp-formula id="disp-formula12-1758736012462025"><label>(10)</label><mml:math id="math17-1758736012462025"><mml:mrow><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula></p><p><disp-formula id="disp-formula13-1758736012462025"><label>(11)</label><mml:math id="math18-1758736012462025"><mml:mrow><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:mrow><mml:mi>t</mml:mi></mml:munderover><mml:mi>ω</mml:mi><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:mrow><mml:mi>t</mml:mi></mml:munderover><mml:mi>σ</mml:mi><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:mrow><mml:mi>ε</mml:mi></mml:munderover><mml:mi>σ</mml:mi><mml:mi>d</mml:mi><mml:mi>ε</mml:mi></mml:mrow></mml:math></disp-formula></p><p>where <italic>σ</italic> is the stress and <italic>ϵ</italic> is the strain.</p><p>The deformation of the PDL under functional and parafunctional loads will as a result have the accumulation of some energy, which may be expressed as<sup><xref ref-type="bibr" rid="bibr56-1758736012462025">56</xref></sup></p><p><disp-formula id="disp-formula14-1758736012462025"><label>(12)</label><mml:math id="math19-1758736012462025"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p>where <italic>F</italic> is the deformation gradient and <italic>W</italic>(<italic>F</italic>) is the strain energy density function. This equation may also be expressed in terms of the principal invariants of the Cauchy–Green tensor as</p><p><disp-formula id="disp-formula15-1758736012462025"><label>(13)</label><mml:math id="math20-1758736012462025"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Taking into account the isotropic contribution of the matrix (<italic>W<sub>m</sub></italic>), the anisotropic contribution of the fibrils (<italic>W<sub>f</sub></italic>), and the interaction of these two media (<italic>W<sub>mf</sub></italic>), the above-described equation may be expressed as</p><p><disp-formula id="disp-formula16-1758736012462025"><label>(14)</label><mml:math id="math21-1758736012462025"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>I</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>It should be mentioned, however, that <italic>W<sub>mf</sub></italic> is usually omitted. Consequently, the Cauchy stress tensor is given by</p><p><disp-formula id="disp-formula17-1758736012462025"><label>(15)</label><mml:math id="math22-1758736012462025"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:munderover><mml:msub><mml:mi>σ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfrac><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mover accent="true"><mml:mi>W</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:msub><mml:mi>n</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <inline-formula id="inline-formula6-1758736012462025"><mml:math id="math23-1758736012462025"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are the principal stresses and <italic>n<sub>i</sub></italic> the related principal directions of stresses.</p><p>The possibility of a progressive failure of the collagen fibrils and the surrounding matrix has also been evaluated in the past and elasto-damage models have been developed to accommodate the behavior of the PDL under large strains.<sup><xref ref-type="bibr" rid="bibr54-1758736012462025">54</xref>,<xref ref-type="bibr" rid="bibr72-1758736012462025">72</xref></sup> This model considers the failure as irreversible, and it neglects the repairing cellular activity that is overcome by the speed of the mechanical damage process. The deformation of the PDL collagen fibers, until total failure, follows the hierarchical structure: (a) collagen molecular deformation, (b) fibril deformation, and (c) fiber deformation.<sup><xref ref-type="bibr" rid="bibr74-1758736012462025">74</xref>,<xref ref-type="bibr" rid="bibr75-1758736012462025">75</xref></sup> This implies that the molecular elongation is the first incident in the deformation process, with changes in the <italic>D</italic>-period structure. On a molecular level, a type I collagen molecule is made of three polypeptide chains twisted together to form a helix structure. Studies of Cowan et al.<sup><xref ref-type="bibr" rid="bibr76-1758736012462025">76</xref></sup> and Sasaki and Odajima<sup><xref ref-type="bibr" rid="bibr77-1758736012462025">77</xref></sup> have demonstrated that the distance <italic>d</italic> between adjacent amino acids can be a measure for the molecular strain <italic>ϵ<sub>d</sub></italic></p><p><disp-formula id="disp-formula18-1758736012462025"><label>(16)</label><mml:math id="math24-1758736012462025"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <italic>d</italic><sub>0</sub> is the distance between the neighboring amino acids before the force application.</p><p>At the fibril level, the distance <italic>D</italic> (~67 nm) between two neighboring molecules can also be a measure for the strain<sup><xref ref-type="bibr" rid="bibr74-1758736012462025">74</xref>,<xref ref-type="bibr" rid="bibr78-1758736012462025">78</xref></sup></p><p><disp-formula id="disp-formula19-1758736012462025"><label>(17)</label><mml:math id="math25-1758736012462025"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>D</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <italic>D</italic><sub>0</sub> is the distance between the neighboring amino acids before the force application.</p><p>It should be mentioned that the strain observed for a collagen molecule is the strain in the helix pitch. On the other hand, the strain for the <italic>D</italic>-period occurs by a rearrangement of the molecules.<sup><xref ref-type="bibr" rid="bibr75-1758736012462025">75</xref></sup> At the higher level of the hierarchy, the collagen fibers are responsible for the tensile response of the PDL. The wavy configuration that the fibers have in the relaxed state makes them unable to resist elongation. Therefore, they cannot offer any mechanical contribution until they are stretched. The number <italic>n</italic> of fibers participating, when a tensile force is exerted and causes a stretch <italic>x</italic>, is<sup><xref ref-type="bibr" rid="bibr79-1758736012462025">79</xref></sup></p><p><disp-formula id="disp-formula20-1758736012462025"><label>(18)</label><mml:math id="math26-1758736012462025"><mml:mrow><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>n</mml:mi><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math></disp-formula></p><p>where <italic>µ</italic> is the mean value of the length of the collagen fibers and <italic>s</italic> is the standard deviation.</p><p>When some tensile force is exerted on a tooth, we have to assume that not all PDL fibers will participate immediately. Since all fibers have the same modulus of elasticity and limit strain, the fibers will start failing in the same sequence in which they started participating. According to Haut,<sup><xref ref-type="bibr" rid="bibr79-1758736012462025"></xref><xref ref-type="bibr" rid="bibr80-1758736012462025">80</xref></sup> when a fiber is stretched beyond its limit strain (<italic>ϵ<sub>l</sub></italic>), it will finally rupture, and the number of fibers that participate at a stretch <inline-formula id="inline-formula7-1758736012462025"><mml:math id="math27-1758736012462025"><mml:mrow><mml:mi>x</mml:mi><mml:mo>/</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is</p><p><disp-formula id="disp-formula21-1758736012462025"><label>(19)</label><mml:math id="math28-1758736012462025"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign="center"><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>n</mml:mi><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>−</mml:mo><mml:mi>μ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="center"><mml:mtext>with</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mi>x</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula></p><p>Therefore, the damage function <italic>D<sub>f</sub></italic> of the PDL fibers is given by<sup><xref ref-type="bibr" rid="bibr56-1758736012462025">56</xref></sup></p><p><disp-formula id="disp-formula22-1758736012462025"><label>(20)</label><mml:math id="math29-1758736012462025"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="0.25em"></mml:mspace><mml:mi>w</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mspace width="0.25em"></mml:mspace><mml:mi>x</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ε</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><p>In the same manner, a damage of the matrix may also occur due to a combination of creep and cyclic loading. The damage function <italic>D<sub>m</sub></italic>(<italic>t</italic>) can be found through a continuum damage mechanics approach, where the damage parameter <italic>D<sub>m</sub></italic> is defined through the use of effective stress <inline-formula id="inline-formula8-1758736012462025"><mml:math id="math30-1758736012462025"><mml:msup><mml:mi>σ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula><sup><xref ref-type="bibr" rid="bibr81-1758736012462025">81</xref>,<xref ref-type="bibr" rid="bibr82-1758736012462025">82</xref></sup></p><p><disp-formula id="disp-formula23-1758736012462025"><label>(21)</label><mml:math id="math31-1758736012462025"><mml:mrow><mml:msup><mml:mi>σ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>σ</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>D</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <italic>σ</italic> is the applied stress. It should be mentioned that the net stress is the true stress due to the damage growth, which results in the reduction of the load bearing area (<italic>A<sub>l</sub></italic>). Consequently, the net stress is based on the net area. For the undamaged matrix <italic>D<sub>m</sub></italic> = 0 and <inline-formula id="inline-formula9-1758736012462025"><mml:math id="math32-1758736012462025"><mml:mrow><mml:msup><mml:mi>σ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>σ</mml:mi></mml:mrow></mml:math></inline-formula>. However, as <italic>D<sub>m</sub></italic> → 1, <italic>A<sub>l</sub></italic> → 0 and <inline-formula id="inline-formula10-1758736012462025"><mml:math id="math33-1758736012462025"><mml:mrow><mml:msup><mml:mi>σ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></inline-formula></p><p>The development of damage within the matrix can be found using the Kachanov’s<sup><xref ref-type="bibr" rid="bibr83-1758736012462025">83</xref></sup> damage growth law</p><p><disp-formula id="disp-formula24-1758736012462025"><label>(22)</label><mml:math id="math34-1758736012462025"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>η</mml:mi></mml:msup></mml:mrow></mml:math></disp-formula></p><p>where <italic>C</italic> and <italic>η</italic> are the material constants.</p><p>The damage function of the matrix is</p><p><disp-formula id="disp-formula25-1758736012462025"><label>(23)</label><mml:math id="math35-1758736012462025"><mml:mrow><mml:mi>D</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:munderover><mml:mo stretchy="false">[</mml:mo><mml:mi>σ</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>η</mml:mi></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>η</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:math></disp-formula></p><p>The damage of the matrix and fibers results in a degenerative condition of the PDL, which may be a key factor in the intrusion phenomenon sometimes observed when teeth are connected with implants.</p></sec><sec id="section2-1758736012462025"><title>Biomechanics of the bone</title><p>The bone is composed of mineral (65%), organic matrix (35%), cells, and water. The organic matrix is basically made of collagen creating a meshwork with spaces filled with the bone mineral that is in the form of crystals. These crystals come in different shapes, such as plates, rods, and needles, and are basically impure hydroxyapatite.<sup><xref ref-type="bibr" rid="bibr84-1758736012462025">84</xref></sup></p><p>The residual bone of the maxilla and the mandible is the bone of the alveolar process that remains after the extractions of the teeth. It consists of cortical and trabecular (cancellous) bone. These two different types of bone are different in many aspects, including development, architecture, function, blood supply, rate of turnover time, as well as the degree of age-dependent changes.<sup><xref ref-type="bibr" rid="bibr84-1758736012462025">84</xref><xref ref-type="bibr" rid="bibr85-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr86-1758736012462025">86</xref></sup> The cortical bone is very dense, with a porosity percentage of 3%–5%. It has microscopic channels and it can be further classified as primary or secondary. The primary cortical bone consists of highly organized lamellar layers, which have a thickness of 3–7 µm containing fine fibers that run approximately in the same direction. It should be mentioned, however, that in neighboring lamellae, these collagen fibers may have different orientations.<sup><xref ref-type="bibr" rid="bibr87-1758736012462025">87</xref>,<xref ref-type="bibr" rid="bibr88-1758736012462025">88</xref></sup> On the other hand, the secondary cortical bone consists of layers that are disrupted by tunnels of osteons. The trabecular (cancellous) bone is formed by intersecting calcified tissue (trabeculae) and presents a porous structure, with porosity up to 90%.<sup><xref ref-type="bibr" rid="bibr89-1758736012462025">89</xref></sup> It can be understood that bone presents an anatomical structure that is hierarchically organized with an irregular arrangement and orientation of its components. As a result, bone presents heterogeneity and anisotropy.<sup><xref ref-type="bibr" rid="bibr90-1758736012462025">90</xref></sup> It has been demonstrated that the mechanical properties of bone vary at different levels. However, it is not known whether these differences are due to the influence of the microstructure or due to the application of different research methods.</p><p>The modulus of elasticity of the bone has been shown to be related to its density<sup><xref ref-type="bibr" rid="bibr91-1758736012462025">91</xref><xref ref-type="bibr" rid="bibr92-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr93-1758736012462025">93</xref></sup></p><p><disp-formula id="disp-formula26-1758736012462025"><mml:math id="math36-1758736012462025"><mml:mrow><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mn>3790</mml:mn><mml:msup><mml:mi>έ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>06</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></disp-formula></p><p>where <italic>E</italic> is the modulus of elasticity (in MPa), is the strain rate (in s<sup>−1</sup>), and <italic>ρ</italic> is the density of the bone (in g cm<sup>−3</sup>). This equation applies for samples including both cortical and trabecular bones. Research on trabecular bone specimens revealed that the modulus of elasticity and the strength are proportional to the square of apparent density of the tissue and are therefore proportional to one another.<sup><xref ref-type="bibr" rid="bibr94-1758736012462025">94</xref></sup></p><p>Although the exact properties of the cancellous bone have not been clearly identified, there is strong evidence that Young’s modulus and the response of the trabecular bone to compressive and tensile loadings are quite different from those of the cortical bone. A correlation between the density and the compressive strength in trabecular bone has been established<sup><xref ref-type="bibr" rid="bibr93-1758736012462025">93</xref></sup></p><p><disp-formula id="disp-formula27-1758736012462025"><mml:math id="math37-1758736012462025"><mml:mrow><mml:msup><mml:mi>σ</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:msup><mml:mi>έ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>06</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></disp-formula></p><p>where <italic>σ<sup>−</sup></italic> represents the compressive strength, is the strain rate (in s<sup>−1</sup>), and <italic>ρ</italic> is the density of the bone (g cm<sup>−3</sup>).</p><p>Therefore, the suggestion of Wolff<sup><xref ref-type="bibr" rid="bibr95-1758736012462025">95</xref></sup> that compact bone is simply more dense cancellous bone tissue is not an accurate statement when only the mechanical properties of these two tissues are considered. In fact, it seems that the proximity as well as the orientation of the trabeculae regulates the mechanical properties of cancellous bone.<sup><xref ref-type="bibr" rid="bibr96-1758736012462025">96</xref></sup> Other factors influencing the properties of the bones include the age, sex, metabolic and hormonal functions, and location (i.e. femur, jaw, etc).</p><p>The cortical–trabecular bone proportion can vary greatly between individuals. A jaw bone quality classification has been proposed by Lekholm and Zarb:<sup><xref ref-type="bibr" rid="bibr97-1758736012462025">97</xref></sup> type 1: almost the whole bone consists of homogenous cortical bone, type 2: a small core of dense trabecular bone is surrounded by thick cortical stratum, type 3: a big core of dense trabecular bone is surrounded by a thin layer of cortical bone, and type 4: a big part of trabecular bone of low density is surrounded by thin cortical stratum.</p><p>Cortical bone is considered an anisotropic material, due to the presence of rigid hydroxyapatite-like mineral particles that are embedded in a collagen fiber matrix. According to Fratzl et al.,<sup><xref ref-type="bibr" rid="bibr98-1758736012462025">98</xref></sup> the anisotropic shapes of the mineral particles are responsible for the anisotropy presented in the mechanical properties of the cortical bone. In general, cortical bone presents a viscoelastic behavior.<sup><xref ref-type="bibr" rid="bibr99-1758736012462025">99</xref></sup> Initially, under a load application to the bone, an elastic reaction is observed. Subsequently, a strain that develops and increases over time is seen. The rate of the strain depends on the magnitude of the force. It has been demonstrated that the strain decreases towards zero, after the removal of the load. This indicates that the bone demonstrates a passive creep behavior. Tennyson et al.<sup><xref ref-type="bibr" rid="bibr100-1758736012462025">100</xref></sup> found that the stress–strain relationship of post-mortem bovine femoral cortical bone obeys to the Kelvin–Voigt model equation of the form:</p><p><disp-formula id="disp-formula28-1758736012462025"><label>(24)</label><mml:math id="math38-1758736012462025"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mi>ε</mml:mi><mml:mo>+</mml:mo><mml:mi>η</mml:mi><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula></p><p>where <italic>E</italic> is a long-term elastic constant and <italic>η</italic> is the viscosity.</p><p>The viscoelastic behavior of the cortical bone was shown to be related to its hydration state. Therefore, a reduction in the water content has been proven to result to a significant decrease of the amount of viscoelasticity of the human femoral cortical bone.<sup><xref ref-type="bibr" rid="bibr101-1758736012462025">101</xref></sup> Another study by Adharapurapu et al.<sup><xref ref-type="bibr" rid="bibr102-1758736012462025">102</xref></sup> has demonstrated that the modulus of elasticity of dry cortical bone specimens is lower than that of rehydrated specimens.</p><p>The Maxwell–Wiechert model (<xref ref-type="fig" rid="fig4-1758736012462025">Figure 4</xref>) can also be used in order to accommodate a distribution of relaxation times due to different molecular segment lengths, which as a result have a faster relaxation time for shorter segments than for the longer ones.<sup><xref ref-type="bibr" rid="bibr103-1758736012462025">103</xref></sup> In that model, the total stress equals the stress of the isolated spring plus the sum of the stresses of all the Maxwell spring–dashpot arms</p><p><disp-formula id="disp-formula29-1758736012462025"><label>(25)</label><mml:math id="math39-1758736012462025"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:munder><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∑</mml:mo></mml:mstyle><mml:mi>j</mml:mi></mml:munder><mml:msub><mml:mi>σ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><fig id="fig4-1758736012462025" position="float"><label>Figure 4.</label><caption><p>Illustration of a general form of the Maxwell–Wiechert model.</p></caption><graphic xlink:href="10.1177_1758736012462025-fig4"></graphic></fig><p>It should be mentioned, however, that previous research has shown that a linear viscoelastic theory with a single relaxation mechanism does not have an application to bone.<sup><xref ref-type="bibr" rid="bibr101-1758736012462025">101</xref>,<xref ref-type="bibr" rid="bibr104-1758736012462025">104</xref></sup> Therefore, constitutive models with both viscoelastic and viscoplastic components have been proposed in order to explain the behavior of the cortical bone. In that model, the viscoelastic component can be described by the following equation, which contains two linear viscoelastic mechanisms, namely 1 and 2<sup>105</sup></p><p><disp-formula id="disp-formula30-1758736012462025"><label>(26)</label><mml:math id="math40-1758736012462025"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>V</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>η</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula></p><p>where <italic>VE</italic> is the viscoelastic strain and 0 is used for the long-term equilibrium behavior.</p><p>The viscoplastic behavior of the cortical bone may be obtained from the dependence of yield stress on strain rate, from the work of McElhaney,<sup><xref ref-type="bibr" rid="bibr106-1758736012462025">106</xref></sup> as well as that of Crowninshield and Pope,<sup><xref ref-type="bibr" rid="bibr107-1758736012462025">107</xref></sup> and is given by the following equation</p><p><disp-formula id="disp-formula31-1758736012462025"><label>(27)</label><mml:math id="math41-1758736012462025"><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>V</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>σ</mml:mi><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>σ</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mo>|</mml:mo><mml:mi>σ</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></disp-formula></p><p>where <italic>k</italic> is calculated from the inverse of the slope of the trend of the stress–strain curve, while <italic>S</italic><sub>0</sub> is the natural logarithm given by the stress axis intercept.</p><p>The final viscoelastic–viscoplastic model may be obtained from equations (26) and (27), where the total strain rate is given by</p><p><disp-formula id="disp-formula32-1758736012462025"><label>(28)</label><mml:math id="math42-1758736012462025"><mml:mrow><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>V</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mover accent="true"><mml:mi>ε</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>V</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></disp-formula></p><p>The porous structure of the cancellous bone as a result has a different mechanical behavior than that of the cortical bone. The existence of trabecular rods–plates and the pores that are filled with a viscous fluid form a matter with a complex mechanical response during loading.<sup><xref ref-type="bibr" rid="bibr108-1758736012462025">108</xref><xref ref-type="bibr" rid="bibr109-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr110-1758736012462025">110</xref></sup> The big difference between the bulk modulus of the solid and the fluid phases, as well as the minute local deformations, as a result has a biphasic mechanical response to external loads, which may be expressed by a poroelastic model.<sup><xref ref-type="bibr" rid="bibr61-1758736012462025">61</xref>,<xref ref-type="bibr" rid="bibr110-1758736012462025">110</xref></sup> Trabecular bone is an anisotropic material and a reformation of the equations of anisotropic poroelasticity has been presented by Thompson and Willis.<sup><xref ref-type="bibr" rid="bibr111-1758736012462025">111</xref></sup> However, for simplicity reasons, the equations presented by Cowin<sup><xref ref-type="bibr" rid="bibr61-1758736012462025">61</xref></sup> were for an isotropy case, and the isotropic stress–strain relationship is expressed as</p><p><disp-formula id="disp-formula33-1758736012462025"><label>(29)</label><mml:math id="math43-1758736012462025"><mml:mrow><mml:mn>2</mml:mn><mml:mi>G</mml:mi><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>ν</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>α</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>ν</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p>while the stress–strain relations are</p><p><disp-formula id="disp-formula34-1758736012462025"><label>(30)</label><mml:math id="math44-1758736012462025"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>α</mml:mi><mml:mi>p</mml:mi><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>G</mml:mi><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>G</mml:mi><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>ν</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <italic>σ<sub>ij</sub></italic> represents the components of the total stress tensor, <italic>ϵ<sub>ij</sub></italic> represents the components of the strain tensor in the solid phase, <italic>p</italic> is the pore pressure, <italic>G</italic> and <italic>v</italic> are the shear modulus and Poisson’s ratio of the trabecular bone under the drained condition, respectively, and <italic>α</italic> represents the ratio of the fluid ratio changes due to the volume discrepancies of the bone when it is loaded under drained conditions.</p><p>It has been demonstrated that bone under the application of a cyclic load can develop microcracks that can ultimately cause fatigue failure. It should be mentioned, however, that these phenomena have been observed in vitro.<sup><xref ref-type="bibr" rid="bibr112-1758736012462025">112</xref>,<xref ref-type="bibr" rid="bibr113-1758736012462025">113</xref></sup> There is data indicating that under normal conditions, these microcracks do not necessarily lead to failure. On the contrary, a remodeling process can be initiated in order to overcome the potential destructive effects of the cycling loading. It should be mentioned that strain seems to be the key factor between loading forces and bone remodeling. It has been found that the mass of bone is controlled by an equilibrium between resorption and bone formation, which are controlled by the endocrine—and the mechanically produced—drive, respectively. There is a requirement of a few cycles of dynamic loading for an osteogenic response.<sup><xref ref-type="bibr" rid="bibr114-1758736012462025">114</xref></sup></p><p>Besides the mechanical damage accumulation, a biological one may also take place. According to Nash,<sup><xref ref-type="bibr" rid="bibr115-1758736012462025">115</xref></sup> this is expressed as</p><p><disp-formula id="disp-formula35-1758736012462025"><label>(31)</label><mml:math id="math45-1758736012462025"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>S</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>H</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>D</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <italic>D<sub>T</sub></italic> is the total damage at any time, <italic>D<sub>S</sub></italic> represents the damage due to a general stress, <italic>H</italic> is the damage that is repaired by the healing process, <italic>D<sub>D</sub></italic> is the damage due to disease, and finally <italic>D<sub>A</sub></italic> is the damage caused by the aging process. This model applies to cyclic loading cases, and <italic>D<sub>S</sub></italic> is calculated from the following equation<sup><xref ref-type="bibr" rid="bibr116-1758736012462025">116</xref></sup></p><p><disp-formula id="disp-formula36-1758736012462025"><label>(32)</label><mml:math id="math46-1758736012462025"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>S</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∑</mml:mo></mml:mstyle><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>n</mml:mi><mml:mi>N</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <italic>N<sub>i</sub></italic> represents the number of loading cycles that is required to cause failure at the <italic>i</italic>th stress level, <italic>n<sub>i</sub></italic> is the number of cycles executed at the <italic>i</italic>th stress level, and <italic>m</italic> represents the number of significant stress levels.</p><p>Carter and Caler<sup><xref ref-type="bibr" rid="bibr116-1758736012462025">116</xref></sup> have presented a model of stress-related bone damage in order to provide a comprehensive model of the damage–repair process in living bone. This model assumes that within the bone, there is a progressive accumulation of cracks that is related to a damage function <italic>D<sub>c</sub></italic>, which takes values between 0 and 1. When <italic>D<sub>c</sub></italic> has a value of 0, there is no creep damage, and when <italic>D<sub>c</sub></italic> = 1, a creep fracture occurs. It can be understood that damage accumulation is a time-dependent process, and the cumulative creep damage due to cycle loading is</p><p><disp-formula id="disp-formula37-1758736012462025"><label>(33)</label><mml:math id="math47-1758736012462025"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mstyle mathsize="140%" displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:munderover><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>k</mml:mi><mml:mi>σ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:math></disp-formula></p><p>where <italic>k</italic> and <italic>c</italic> are the constants.</p><p>It has been demonstrated that during fatigue loading, an internal damage takes place that results in a progressive loss of bone stiffness and strength that eventually leads to failure.<sup><xref ref-type="bibr" rid="bibr117-1758736012462025">117</xref></sup> Another study by the same group has concluded that the type of load, that is, tensile or compressive, has a different effect on the type of damage observed. Tensile fatigue load creates a widespread failure at the cement lines, as well as at the interlamellar cement bands. On the other hand, a compressive fatigue load produces several oblique microcracks that are influenced by stress concentrations created by lacunae and vascular canals.<sup><xref ref-type="bibr" rid="bibr118-1758736012462025">118</xref></sup> The properties of the cortical and the trabecular bones determine their behavior to the application of mechanical loads. Application of a compressive cyclic load as a result has a greater bone deposition in cortical bone than when a tensile cyclic load is applied.<sup><xref ref-type="bibr" rid="bibr113-1758736012462025">113</xref></sup> This phenomenon is due to different generated electric potentials in bone. Bone compression as a result has generation of a negative electric potential, while a tensile load produces positive potentials.<sup><xref ref-type="bibr" rid="bibr119-1758736012462025">119</xref></sup></p><p>An insight into the complicated processes of bone adaptation has been attempted by McNamara et al.,<sup><xref ref-type="bibr" rid="bibr120-1758736012462025">120</xref></sup> who have claimed that bone remodeling is initiated by accumulative damage. An assumption has been made that even at the remodeling equilibrium, some damage exists, and the repair rate is related to the damage rate</p><p><disp-formula id="disp-formula38-1758736012462025"><label>(34)</label><mml:math id="math48-1758736012462025"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mo>⋅</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>f</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <italic>dX</italic> is the repair rate, <italic>dt</italic> is the damage rate, <italic>C</italic> represents the remodeling coefficient, and <italic>ω<sub>eff</sub></italic> represents the effective damage of the bone.</p><p>A model that includes the cellular activity within the bone has also been described:<sup><xref ref-type="bibr" rid="bibr121-1758736012462025">121</xref>,<xref ref-type="bibr" rid="bibr122-1758736012462025">122</xref></sup></p><p><disp-formula id="disp-formula39-1758736012462025"><label>(35)</label><mml:math id="math49-1758736012462025"><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:msub><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:msub><mml:mi>n</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi>n</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula></p><p>where <italic>λ</italic> is the surface area fraction available, <italic>a</italic> represents cellular activity, <italic>n</italic> represents cellular number, while <italic>b</italic> and <italic>c</italic> represent osteoblastic and osteoclastic activity, respectively. This theory was recently tested by a research group and gave encouraging results.<sup><xref ref-type="bibr" rid="bibr123-1758736012462025">123</xref></sup></p></sec><sec id="section3-1758736012462025"><title>Biomechanics of tooth-implant-supported fixed partial dentures</title><p>It can be seen that bone exhibits a completely different biomechanical behavior than the PDL. Connection of teeth, which present some mobility, with implants, which are practically ankylosed in the bone, presents some risks that should be taken into account before making such a decision.</p><p>A careful treatment planning should be made by assessing the following biomechanical factors:</p><list list-type="bullet"><list-item><p>Mobility of the teeth to be connected with implants.</p></list-item><list-item><p>Number of teeth and implants to be connected.</p></list-item><list-item><p>Occlusal forces including</p><list list-type="simple"><list-item><p>(a) magnitude</p></list-item><list-item><p>(b) duration</p></list-item><list-item><p>(c) distribution</p></list-item><list-item><p>(d) direction</p></list-item></list></list-item></list><list list-type="bullet"><list-item><p>Rigidity of the prosthesis.</p></list-item><list-item><p>Type of connection (rigid or nonrigid).</p></list-item><list-item><p>Type of the bone.</p></list-item></list><sec id="section4-1758736012462025"><title>Mobility of teeth to be connected with implants</title><p>As mentioned before, periodontally healthy teeth have a physiologic mobility due to the PDL. Different groups of teeth present different degrees of mobility. Posterior teeth usually present less mobility than anterior teeth. Canines display less mobility than incisors. On the other hand, implants present no mobility. Clinical research has indicated that the Periotest<sup>®</sup> values<sup><xref ref-type="bibr" rid="bibr124-1758736012462025">124</xref>,<xref ref-type="bibr" rid="bibr125-1758736012462025">125</xref></sup> of successfully osseointegrated fixtures in the mandible range from −4 to −2. Mandibular canines present values between −1 and +4.<sup><xref ref-type="bibr" rid="bibr126-1758736012462025">126</xref></sup> The higher damping values found at implants have been attributed to the absence of the PDL. Another reason accounting for these values is that titanium’s elastic modulus is 110–117 GPa<sup><xref ref-type="bibr" rid="bibr127-1758736012462025">127</xref></sup> as compared to dentin’s, which is 12–14.7 GPa.<sup><xref ref-type="bibr" rid="bibr128-1758736012462025">128</xref></sup> Young’s modulus of adjacent anatomical structures, such as cortical and trabecular bones, as well as that of the PDL, is important, but it will be discussed later on.</p><p>There is controversial information in the dental literature regarding the periodontal status of the teeth to be connected to implants. Kindberg et al.<sup><xref ref-type="bibr" rid="bibr13-1758736012462025">13</xref></sup> have reported that treatments with periodontally sound teeth and implants splinted together in one-piece prostheses with rigid connections show excellent long-term results. Nevertheless, Cordaro et al.<sup><xref ref-type="bibr" rid="bibr129-1758736012462025">129</xref></sup> have shown that when teeth are connected to implants with a rigid connection, the intrusion phenomenon is more likely to happen to teeth with an intact periodontal support than to teeth with a reduced periodontal support.</p><p>The position of the teeth—to be connected to implants—in the arch does not seem to be associated with the intrusion phenomenon.<sup><xref ref-type="bibr" rid="bibr8-1758736012462025">8</xref></sup> However, the proximity of the tooth to the implant is probably important and it will be discussed in the next section.</p></sec><sec id="section5-1758736012462025"><title>Number of teeth and implants to be connected</title><p>Information on how the number of teeth connected with implants influences tooth intrusion is scarce. A retrospective multicenter study, which investigated the complications arising from the connection of teeth with implants, has reported that the intrusion of the supporting teeth was most common in the prosthesis design with one implant connected to one tooth.<sup><xref ref-type="bibr" rid="bibr12-1758736012462025">12</xref></sup> This finding was based on an up to 3-year follow-up of 220 abutment teeth connected with 185 implants in 111 patients. The authors of the aforementioned study<sup><xref ref-type="bibr" rid="bibr12-1758736012462025">12</xref></sup> found that the intrusion of natural teeth retainers was 5%, while 72.72% of this was attributed to a one-to-one implant-to-tooth connection. However, another clinical study<sup><xref ref-type="bibr" rid="bibr8-1758736012462025">8</xref></sup> that evaluated 339 implants connected to 313 teeth in 123 patients for up to 15 years reported an incidence of tooth intrusion in 7.31% of the cases. Prostheses supported by one tooth and one implant accounted for 44.44% of the total intrusion cases. An interesting point of this study, which is not mentioned by the authors, regards cases in which multiple natural teeth are connected to implants. The data support the notion that in the majority of the cases that developed an intrusion, the intrusion was found in natural teeth abutments that were adjacent to the implants.</p></sec><sec id="section6-1758736012462025"><title>Occlusal forces</title><sec id="section7-1758736012462025"><title>Magnitude</title><p>The magnitude of chewing and maximum biting forces varies considerably among the individuals.<sup><xref ref-type="bibr" rid="bibr130-1758736012462025">130</xref><xref ref-type="bibr" rid="bibr131-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr132-1758736012462025">132</xref></sup> It should be mentioned, however, that in the majority of the patients, the chewing forces are smaller than the maximum biting forces. Normally, when the individual has a fixed complete denture, chewing forces range between 55 and 165 N, while maximum biting forces can usually be between 264 and 336 N. The latter can very rarely reach values of 4340 N.<sup><xref ref-type="bibr" rid="bibr133-1758736012462025">133</xref></sup> Biting forces are influenced by many factors such as type of restorations, premature contacts, skeletal and anatomical factors and parafunctional habits. It has been proved by clinical studies that the forces are noticeably different whether the fixed partial dentures are supported by natural teeth or implants. The presence of bilateral terminal abutments and bilateral or unilateral cantilevers is also important for the biting force magnitude. Existence of natural teeth or a complete denture on the opposing arch also plays a role.<sup><xref ref-type="bibr" rid="bibr134-1758736012462025">134</xref></sup> The presence of premature contacts, even if these are as small as 100 µm, can increase the biting forces substantially.<sup><xref ref-type="bibr" rid="bibr135-1758736012462025">135</xref>,<xref ref-type="bibr" rid="bibr136-1758736012462025">136</xref></sup></p><p>Skeletal factors may influence the magnitude of chewing and maximum biting forces. It has been demonstrated that subjects with low Frankfort mandibular plane angle (FMA) can produce almost twice as much occlusal forces, in the molar region, when compared to individuals with high FMA.<sup><xref ref-type="bibr" rid="bibr137-1758736012462025">137</xref>,<xref ref-type="bibr" rid="bibr138-1758736012462025">138</xref></sup> The anatomy of the masseter muscle, especially its cross-sectional area, can also influence the biting force exerted in the molar area.<sup><xref ref-type="bibr" rid="bibr139-1758736012462025">139</xref></sup> Parafunctional habits such as clenching and bruxism may affect both the magnitude and the duration of the forces.</p></sec><sec id="section8-1758736012462025"><title>Duration</title><p>Occlusal forces are exerted on the teeth and/or on the restorations, both while swallowing and chewing. During the waking state, occlusal contacts from swallowing occur about every 2 min.<sup><xref ref-type="bibr" rid="bibr140-1758736012462025">140</xref></sup> During sleep, the occlusal contacts due to swallowing are irregular and much less frequent.<sup><xref ref-type="bibr" rid="bibr141-1758736012462025">141</xref></sup> It has been estimated that from swallowing alone, opposing teeth touch about 1500 times each day.<sup><xref ref-type="bibr" rid="bibr142-1758736012462025">142</xref></sup></p><p>Prolonged duration of occlusal contacts is observed in subjects with parafunctional habits such as bruxism and clenching. A study has shown that subjects with parafunctional activities occlude their teeth seven times longer than those with no such habits (38.7 min vs 5.4 min).<sup><xref ref-type="bibr" rid="bibr143-1758736012462025">143</xref></sup></p></sec><sec id="section9-1758736012462025"><title>Distribution</title><p>Distribution of forces on occluding units has an impact on both muscular contraction and the mechanical effect on each tooth.<sup><xref ref-type="bibr" rid="bibr144-1758736012462025">144</xref></sup> However, the outcome of uneven force distribution is different for anterior and posterior teeth. For anterior teeth, the greatest proprioceptive inhibition results if only one tooth is in contact with its antagonist.<sup><xref ref-type="bibr" rid="bibr145-1758736012462025">145</xref></sup> The problem arises from the fact that in that scenario, all the muscular contractions and the resultant forces will be placed on that tooth, with detrimental effects. Posterior teeth are quite different from anterior ones in the sense that an occlusal contact on a tooth stimulates the muscles to their greatest level of contractibility. Therefore, existence of a single posterior occlusal contact does not trigger the proprioception to stop muscle contraction. As a result, the posterior tooth will take all the potential force that the elevator muscles can generate. The aim of the clinicians should be an even distribution of biting forces in maximum intercuspation to as many posterior teeth as possible.<sup><xref ref-type="bibr" rid="bibr146-1758736012462025">146</xref></sup> Nevertheless, definite occlusal concepts have not been developed regarding implant prostheses.<sup><xref ref-type="bibr" rid="bibr132-1758736012462025">132</xref></sup> Even worse, when the treatment involves both natural teeth and implants, furnishing of a specific occlusal scheme is empirical, since the literature on this subject is scarce.</p><p>Whether or not equal distribution of occlusal forces between natural teeth and implants is mandatory is not known yet. The dissimilar physical characteristics of the PDL and the bone as a result have a different behavior of teeth and implants when subjected to forces. Under a 20-N force, teeth will usually intrude 50 µm,<sup><xref ref-type="bibr" rid="bibr28-1758736012462025">28</xref></sup> while implants will intrude about 2 µm under the application of the same force.<sup><xref ref-type="bibr" rid="bibr147-1758736012462025">147</xref></sup> As a result, if occlusal equilibration is performed in such a way as to have an equal distribution of forces under light contacts, the implants will be subjected to a force overload when heavier occlusal contacts will be exerted, since the natural teeth will intrude into their sockets while implants will not. Consequently, in a tooth-implant fixed partial denture, as the tooth will intrude, the prosthesis will act as a cantilever to the implant. On the other hand, if the equilibration is performed under heavy occlusal contacts, the forces will be evenly distributed between teeth and implants.<sup><xref ref-type="bibr" rid="bibr148-1758736012462025">148</xref></sup> The number of splinted teeth and implants, mobility of periodontally involved abutment teeth, crown to implant ratio, quality of the bone where the implant was placed, location of the fixed tooth-to-implant prosthesis in relation to the elevator muscles and parafunctional activities are important factors that have to be evaluated when the occlusal equilibration is performed.</p></sec><sec id="section10-1758736012462025"><title>Direction</title><p>Vertically directed occlusal loads can be beneficiary to the teeth, since they do not induce excessive mobility that lateral loads can create.<sup><xref ref-type="bibr" rid="bibr149-1758736012462025">149</xref>,<xref ref-type="bibr" rid="bibr150-1758736012462025">150</xref></sup> One of the goals of occlusal adjustment is the redirection of forces along the long axes of the teeth. According to the experimental data and biomechanical calculations, osseointegrated implants should ideally be loaded axially.<sup><xref ref-type="bibr" rid="bibr151-1758736012462025">151</xref></sup> Off-axial forces due to erroneous implant placement, wrong prosthesis design, or mediotrusive (nonworking) contacts will create bending moments given by the following equation<sup><xref ref-type="bibr" rid="bibr152-1758736012462025">152</xref></sup></p><p><disp-formula id="disp-formula40-1758736012462025"><label>(36)</label><mml:math id="math50-1758736012462025"><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mo>⋅</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:math></disp-formula></p><p>where <italic>M</italic> is the bending moment (in N m), <italic>F</italic> is the force (in N), and <italic>L</italic> represents the lever arm (in m). These bending moments will be transferred from the prosthesis to the implant and will ultimately end in the bone.</p><p>It should be mentioned that in tooth-to-implant fixed prosthesis, even axial directed forces can create bending moments to the implant, if the natural tooth has excessive mobility or if it intrudes. These forces will be applied to the prosthesis, the implant prosthetic components (i.e. fastening screw and abutment), the implant itself, and the bone that is in close contact to the fixture.</p></sec></sec><sec id="section11-1758736012462025"><title>Rigidity of the prosthesis</title><p>Fixed partial or complete dentures should be made with certain dimensions to ensure minimal flexure, which is essential for the longevity of the restoration. A thickness of 0.3 mm for base metal alloys and 0.5 mm for high noble alloys, as well as 3 × 3 mm<sup>2</sup> sized connectors are essential to obtain a quite rigid prosthesis.<sup><xref ref-type="bibr" rid="bibr153-1758736012462025">153</xref></sup> This concept has been used successfully in traditional prosthodontics for many years. Rigidity of the metal substructure guarantees that there will be no fracture of the overlying esthetic material (acrylic or porcelain) and no cement wash out at the abutments. However, whether the connection between implants and teeth should be rigid or not is not resolved yet. This clinical approach is quite new and is still under investigation, because of frequent complications. These include fracture of implant components, damage, and/or intrusion of the abutment teeth.<sup><xref ref-type="bibr" rid="bibr154-1758736012462025">154</xref></sup></p><p>Several theories have been developed in order to explain the tooth intrusion phenomenon:</p><list list-type="order"><list-item><p><italic>Disuse atrophy</italic>. The fibers of the PDL of the tooth may undergo a disuse atrophy due to the hypofunction of the tooth, since the implant undertakes the majority of the occlusal forces.<sup><xref ref-type="bibr" rid="bibr155-1758736012462025">155</xref><xref ref-type="bibr" rid="bibr156-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr157-1758736012462025">157</xref></sup></p></list-item><list-item><p><italic>Differential energy dissipation</italic>. There is an osteoclastic activity in the PDL due to very high stress transmitted to the tooth.<sup><xref ref-type="bibr" rid="bibr158-1758736012462025">158</xref>,<xref ref-type="bibr" rid="bibr159-1758736012462025">159</xref></sup> The result of this osteoclastic activity is the intrusion of the abutment tooth.</p></list-item><list-item><p><italic>Impaired rebound memory</italic>. This theory suggests that due to the constant pressure, the PDL loses its elastic memory and remodels in a new position. This position is more apical than the original one. The PDL’s remodeling continues until the tooth is completely out of occlusion and stabilizes in that new position.<sup><xref ref-type="bibr" rid="bibr160-1758736012462025">160</xref><xref ref-type="bibr" rid="bibr161-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr162-1758736012462025">162</xref></sup></p></list-item><list-item><p><italic>Rachet effect</italic>. The abutment tooth moves apically due to the occlusal overload and stays in that new position, maybe because of the binding in the socket or in the semiprecision attachments that are very often used.<sup><xref ref-type="bibr" rid="bibr161-1758736012462025">161</xref>,<xref ref-type="bibr" rid="bibr162-1758736012462025">162</xref></sup></p></list-item><list-item><p><italic>Debris impaction</italic>. Food particles can entrap under telescopic copings or in the connecting attachments and prevent the tooth to return to the original position.</p></list-item><list-item><p><italic>Fixed prosthesis flexure</italic>. All beams flex under a stress. Similarly, a fixed prosthesis can flex and force the abutment tooth out of its original position into the socket.</p></list-item><list-item><p><italic>Mandibular flexure</italic>. This theory applies only to mandibular tooth-to-implant prostheses.<sup><xref ref-type="bibr" rid="bibr163-1758736012462025">163</xref><xref ref-type="bibr" rid="bibr164-1758736012462025"></xref><xref ref-type="bibr" rid="bibr165-1758736012462025"></xref><xref ref-type="bibr" rid="bibr166-1758736012462025"></xref><xref ref-type="bibr" rid="bibr167-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr168-1758736012462025">168</xref></sup> Mandibular flexure is a phenomenon that occurs when the mouth opens, due to the contraction of the muscles of mastication and the flexibility of the mandibular bone. This flexure is clinically observed as a posterior narrowing. Therefore, teeth and implants change their relative positions and an internal stress is generated. This repeated stress can result in the abutment tooth movement.</p></list-item></list><p>Fabrication of a rigid connection between implants and teeth was questioned in the past because of the differences in their supporting tissues. Skalak’s<sup><xref ref-type="bibr" rid="bibr169-1758736012462025">169</xref></sup> theory involved lateral stiffness, torsion and bending of the prosthesis, and bending or rotation of the fastening screws and other prosthetic components. According to this hypothesis, distribution of the forces largely depends on the stiffness of the interacting units. The stiffnesses involved are lateral, torsional, and bending.</p><p>When a lateral force <italic>F</italic> is applied to a unit (regardless whether it is an implant or a natural tooth), it causes a deflection <italic>δ</italic><sub>1</sub>. The elastic constant of this unit in lateral stiffness is <inline-formula id="inline-formula11-1758736012462025"><mml:math id="math51-1758736012462025"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>. Stiffness of an implant is bigger than that of a natural tooth, mainly because of the different physical characteristics between the bone and the PDL. Use of a compliant material was suggested in order to level the behavioral differences of the two dissimilar units and obtain an equal load distribution and a deflection <inline-formula id="inline-formula12-1758736012462025"><mml:math id="math52-1758736012462025"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>δ</mml:mi><mml:mo>′</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> that is less steep than <italic>δ</italic><sub>1</sub>.</p><p>Torsional and bending stiffnesses are associated with torque and bending moments in the fixed prosthesis and the way these are transferred to the supporting units (teeth or implants). This requires a very complex analysis due to the fact that the torsional stiffnesses of the implants and teeth as well as the geometric curvature of the prosthesis should also be taken into account. However, Skalak<sup><xref ref-type="bibr" rid="bibr169-1758736012462025">169</xref></sup> believed that incorporation of a compliant material into the prosthesis could provide beneficiary long-term results as the stiffnesses, the implants, and the teeth would be in equilibrium.</p><p>Use of polyoxymethylene as a compliant material, to absorb a big part of the forces applied to the implant and resemble in this way the action of the PDL, has been suggested.<sup><xref ref-type="bibr" rid="bibr170-1758736012462025">170</xref><xref ref-type="bibr" rid="bibr171-1758736012462025"></xref><xref ref-type="bibr" rid="bibr172-1758736012462025"></xref><xref ref-type="bibr" rid="bibr173-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr174-1758736012462025">174</xref></sup> However, clinical data indicated that this material did not serve successfully its purpose and required frequent replacement.<sup><xref ref-type="bibr" rid="bibr175-1758736012462025">175</xref>,<xref ref-type="bibr" rid="bibr176-1758736012462025">176</xref></sup></p></sec><sec id="section12-1758736012462025"><title>Type of connection (rigid or nonrigid)</title><p>Connection of teeth with implants may take place either with an attachment system or with a telescopic crown. This type of connection is dictated by the need of implant prostheses retrievability for reservicing, replacement, or salvaging of the restorations and the implants.</p><p>Attachment systems connecting two parts of a fixed prosthesis can be either rigid or nonrigid. In rigid connectors, there is usually a fastening screw that fixes the patrix and the matrix parts rigidly. In nonrigid attachments, there is a key and a keyway part that slide one into the other, but there is no screw to fix these two parts. Likewise if, instead of an attachment system, a telescopic crown is used, then this can be either fixed rigidly to the suprastructure with a screw or with a definitive cement.</p><p>Clinical studies have shown that a rigid type of connection should be preferred, since with the use of a nonrigid connection, an intrusion may occur in 3%–4% of the cases.<sup><xref ref-type="bibr" rid="bibr8-1758736012462025">8</xref>,<xref ref-type="bibr" rid="bibr12-1758736012462025">12</xref></sup> Nevertheless, a finite element analysis and photoelastic study have demonstrated that with a rigid type of connection, more stress is concentrated around the implant.<sup><xref ref-type="bibr" rid="bibr177-1758736012462025">177</xref></sup> Likewise, a radiographical study has indicated that more bone loss is observed in the area around implants when the tooth–implant connection is rigid.<sup><xref ref-type="bibr" rid="bibr9-1758736012462025">9</xref></sup></p></sec><sec id="section13-1758736012462025"><title>Type of the bone</title><p>An area that has not been investigated at all is related to the type of the bone and its contributory effects to the tooth–implant connection. As stated by Lekholm and Zarb,<sup><xref ref-type="bibr" rid="bibr97-1758736012462025">97</xref></sup> there are four different types of bone, with variations in the cortical and trabecular bone contents. The elastic moduli of cortical and trabecular bones are quite different. The first one has a Young’s modulus of 15,000 MPa, while the cancellous bone has a modulus of elasticity of 1500 MPa.<sup><xref ref-type="bibr" rid="bibr178-1758736012462025">178</xref><xref ref-type="bibr" rid="bibr179-1758736012462025"></xref><xref ref-type="bibr" rid="bibr180-1758736012462025"></xref>–<xref ref-type="bibr" rid="bibr181-1758736012462025">181</xref></sup> The PDL has an elastic modulus of 1.18–2 MPa.<sup><xref ref-type="bibr" rid="bibr52-1758736012462025">52</xref>,<xref ref-type="bibr" rid="bibr182-1758736012462025">182</xref></sup> It should be mentioned, however, that the bone values mentioned do not take into account differences that may occur due to variations of primary and secondary cortical bones, as well as differences due to the density and the organization of the trabeculae in cancellous bone. Research has demonstrated that the mechanical properties and the elastic modulus of the bone are influenced by these factors.<sup><xref ref-type="bibr" rid="bibr94-1758736012462025">94</xref>,<xref ref-type="bibr" rid="bibr96-1758736012462025">96</xref></sup> Consequently, differences in the bone type and its architecture—at the microscopic level—may influence the outcome of tooth–implant connection. Research is required in order to confirm or reject this hypothesis.</p></sec></sec><sec sec-type="conclusions" id="section14-1758736012462025"><title>Conclusion</title><p>Clinicians should be aware of the limitations and disadvantages of the teeth connected to implant prostheses, so that they can plan prosthetic treatments accordingly. The risks related to teeth connected to implants prostheses result from differences in the biomechanics of the involved anatomical structures (i.e. the PDL and the bone) and of the biomechanics of teeth-implants-supported fixed prostheses.</p></sec></body><back><fn-group><fn fn-type="financial-disclosure"><p><bold>Funding:</bold> This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.</p></fn></fn-group><ref-list><title>References</title><ref id="bibr1-1758736012462025"><label>1.</label><mixed-citation publication-type="book"><person-group person-group-type="author"><name><surname>Brånemark</surname><given-names>P-I</given-names></name><name><surname>Zarb</surname><given-names>GA</given-names></name><name><surname>Albrekstsson</surname><given-names>T</given-names></name><etal></etal></person-group><source>Osseointegration in clinical dentistry</source>. <publisher-loc>Chicago, IL</publisher-loc>: <publisher-name>Quintessence</publisher-name>, <year>1985</year>, pp. <fpage>241</fpage>–<lpage>282</lpage></mixed-citation></ref><ref id="bibr2-1758736012462025"><label>2.</label><mixed-citation publication-type="confproc"><person-group person-group-type="author"><name><surname>Adell</surname><given-names>R</given-names></name></person-group><article-title>Clinical results of osseointegrated implants supporting fixed prostheses in edentulous jaws</article-title>. 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