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<journal-id journal-id-type="publisher-id">Front. Bioeng. Biotechnol.</journal-id>
<journal-title>Frontiers in Bioengineering and Biotechnology</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Bioeng. Biotechnol.</abbrev-journal-title>
<issn pub-type="epub">2296-4185</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
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<article-id pub-id-type="publisher-id">1476473</article-id>
<article-id pub-id-type="doi">10.3389/fbioe.2024.1476473</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Bioengineering and Biotechnology</subject>
<subj-group>
<subject>Original Research</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>On the influence of structural and chemical properties on the elastic modulus of woven bone under healing</article-title>
<alt-title alt-title-type="left-running-head">Bl&#xe1;zquez-Carmona et al.</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/fbioe.2024.1476473">10.3389/fbioe.2024.1476473</ext-link>
</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Bl&#xe1;zquez-Carmona</surname>
<given-names>Pablo</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
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</contrib>
<contrib contrib-type="author">
<name>
<surname>Mora-Mac&#xed;as</surname>
<given-names>Juan</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
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<contrib contrib-type="author">
<name>
<surname>Pajares</surname>
<given-names>Antonia</given-names>
</name>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
<uri xlink:href="https://loop.frontiersin.org/people/2841758/overview"/>
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<contrib contrib-type="author">
<name>
<surname>M&#xe1;rmol</surname>
<given-names>&#xc1;lvaro</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
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<contrib contrib-type="author">
<name>
<surname>Reina-Romo</surname>
<given-names>Esther</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
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<aff id="aff1">
<sup>1</sup>
<institution>Escuela T&#xe9;cnica Superior de Ingenier&#xed;a</institution>, <institution>Universidad de Sevilla</institution>, <addr-line>Sevilla</addr-line>, <country>Spain</country>
</aff>
<aff id="aff2">
<sup>2</sup>
<institution>Escuela T&#xe9;cnica Superior de Ingenier&#xed;a</institution>, <institution>Universidad de Huelva</institution>, <addr-line>Huelva</addr-line>, <country>Spain</country>
</aff>
<aff id="aff3">
<sup>3</sup>
<institution>Departamento de Ingenier&#xed;a Mec&#xe1;nica</institution>, <institution>Energ&#xe9;tica y de los Materiales</institution>, <institution>Universidad de Extremadura</institution>, <addr-line>Badajoz</addr-line>, <country>Spain</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1586776/overview">Jingwei Zhang</ext-link>, Shanghai Jiao Tong University, China</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2107346/overview">Sacha Cavelier</ext-link>, Queensland University of Technology, Australia</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/116494/overview">Christian Hellmich</ext-link>, Vienna University of Technology, Austria</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Pablo Bl&#xe1;zquez-Carmona, <email>pbcarmona@us.es</email>
</corresp>
</author-notes>
<pub-date pub-type="epub">
<day>01</day>
<month>10</month>
<year>2024</year>
</pub-date>
<pub-date pub-type="collection">
<year>2024</year>
</pub-date>
<volume>12</volume>
<elocation-id>1476473</elocation-id>
<history>
<date date-type="received">
<day>05</day>
<month>08</month>
<year>2024</year>
</date>
<date date-type="accepted">
<day>19</day>
<month>09</month>
<year>2024</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2024 Bl&#xe1;zquez-Carmona, Mora-Mac&#xed;as, Pajares, M&#xe1;rmol and Reina-Romo.</copyright-statement>
<copyright-year>2024</copyright-year>
<copyright-holder>Bl&#xe1;zquez-Carmona, Mora-Mac&#xed;as, Pajares, M&#xe1;rmol and Reina-Romo</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p>
</license>
</permissions>
<abstract>
<sec>
<title>Introduction</title>
<p>Woven bone, a heterogeneous and temporary tissue in bone regeneration, is remodeled by osteoblastic and osteoclastic activity and shaped by mechanical stress to restore healthy tissue properties. Characterizing this tissue at different length scales is crucial for developing micromechanical models that optimize mechanical parameters, thereby controlling regeneration and preventing non-unions.</p>
</sec>
<sec>
<title>Methods</title>
<p>This study examines the temporal evolution of the mechanical properties of bone distraction callus using nanoindentation, ash analysis, micro-CT for trabecular microarchitecture, and Raman spectroscopy for mineral quality. It also establishes single- and two-parameter power laws based on experimental data to predict tissue-level and bulk mechanical properties.</p>
</sec>
<sec>
<title>Results</title>
<p>At the macro-scale, the tissue exhibited a considerable increase in bone fraction, controlled by the widening of trabeculae. The Raman mineral-to-matrix ratios increased to cortical levels during regeneration, but the local elastic modulus remained lower. During healing, the tissue underwent changes in ash fraction and in the percentages of Calcium and Phosphorus. Six statistically significant power laws were identified based on the ash fraction, bone fraction, and chemical and Raman parameters.</p>
</sec>
<sec>
<title>Discussion</title>
<p>The microarchitecture of woven bone plays a more significant role than its chemical composition in determining the apparent elastic modulus of the tissue. Raman parameters were demonstrated to provide more significant power laws correlations with the micro-scale elastic modulus than mineral content from ash analysis.</p>
</sec>
</abstract>
<kwd-group>
<kwd>woven bone</kwd>
<kwd>nanoindentation</kwd>
<kwd>micro-CT</kwd>
<kwd>Raman sprectroscopy</kwd>
<kwd>composition</kwd>
<kwd>distraction osteogenesis</kwd>
<kwd>elastic modulus</kwd>
<kwd>power law</kwd>
</kwd-group>
<contract-sponsor id="cn001">Ministerio de Ciencia e Innovaci&#xf3;n<named-content content-type="fundref-id">10.13039/501100004837</named-content>
</contract-sponsor>
<custom-meta-wrap>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Biomechanics</meta-value>
</custom-meta>
</custom-meta-wrap>
</article-meta>
</front>
<body>
<sec id="s1">
<title>1 Introduction</title>
<p>During skeletal tissue differentiation processes such as distraction osteogenesis, fracture healing, adaptation to skeletal implants, bone development and regeneration, the structure and composition of the mineralized bone tissue, referred to as woven bone or immature bone, continually change due to growth and responses to the tissue environment, including physical, chemical, and mechanical factors (<xref ref-type="bibr" rid="B93">Wehrle et al., 2014</xref>; <xref ref-type="bibr" rid="B88">Tourolle N&#xe9; Betts et al., 2020</xref>; <xref ref-type="bibr" rid="B56">Mora-Mac&#xed;as et al., 2021</xref>; <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al., 2021a</xref>; <xref ref-type="bibr" rid="B69">Paul et al., 2022</xref>). The organic and inorganic matrices of woven bone are highly dynamic and undergo alterations in their structure and composition (<xref ref-type="bibr" rid="B39">Isaksson et al., 2010</xref>; <xref ref-type="bibr" rid="B90">Turunen et al., 2011</xref>; <xref ref-type="bibr" rid="B53">Mart&#xed;nez-Reina et al., 2018</xref>; <xref ref-type="bibr" rid="B72">Reznikov et al., 2018</xref>). The inorganic phase is primarily composed of nonstoichiometric hydroxyapatite crystals. This mineral content is deposited in two phases: an initial phase in which 70% of the final mineral content is incorporated within a few weeks, followed by a second phase in which the mineral content increases over months and years (<xref ref-type="bibr" rid="B78">Roschger et al., 2020</xref>). The organic phase is predominantly composed of type I collagen, along with various biomacromolecules, including proteoglycans and non-collagenous proteins (<xref ref-type="bibr" rid="B14">Boskey and Robey, 2013</xref>). During the initial stages of healing, na&#xef;ve collagen fibers are randomly oriented due to their rapid formation. As the maturation progresses, these fibers increase in density, reorient, and structurally mature (<xref ref-type="bibr" rid="B12">Bl&#xe1;zquez-Carmona et al., 2022</xref>). This maturation is understood as the process in which the fibers are crosslinked and packaged to accommodate mineralization. These microstructure arrangements result in a time-dependent mechanical behavior of the tissue, with initial stiffness much lower than that of the matured cortical and trabecular bones.</p>
<p>The microstructure and composition of bone play crucial roles in its macroscopic structural and mechanical behavior. Their relationship has been extensively studied in cortical and trabecular bones (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B73">Rho et al., 1995</xref>; <xref ref-type="bibr" rid="B81">Sevostianov and Kachanov, 2000</xref>; <xref ref-type="bibr" rid="B17">Budyn et al., 2012</xref>). It has been shown that the elastic modulus of the mature bones strongly depends on mineral content, while toughness correlates with the quality of the collagen matrix (<xref ref-type="bibr" rid="B96">Zioupos and Currey, 1998</xref>). Correlations between the elastic modulus and the bone mineral density have been established to estimate mechanical properties from the image-based measurements of the bone mineral density at both the organ and tissue scales (<xref ref-type="bibr" rid="B63">Nobakhti and Shefelbine, 2018</xref>). At the organ level, the elastic modulus (<italic>E</italic>) is related to the apparent mineral density <inline-formula id="inf1">
<mml:math id="m1">
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, mineralized bone mass/bulk volume, through a power law relationship <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x221d;</mml:mo>
<mml:mspace width="0.3333em"/>
<mml:msup>
<mml:mrow>
<mml:mi>&#x3c1;</mml:mi>
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</mml:msup>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. For cortical bones, <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> ranges between 4 and 7.4 (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B21">Currey, 1988</xref>) and for trabecular bones, it is normally lower, <inline-formula id="inf4">
<mml:math id="m4">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 1.27-2.57 (<xref ref-type="bibr" rid="B76">Rice et al., 1988</xref>; <xref ref-type="bibr" rid="B43">Keller, 1994</xref>; <xref ref-type="bibr" rid="B15">Bouxsein and Radloff, 1997</xref>), indicating lower apparent stiffness. One limitation of these models is that they do not distinguish the influence of bone volume fraction from ash fraction. Therefore, at the macro- level, these correlations are improved when porosity (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B21">Currey, 1988</xref>) or fabric orientation are included (<xref ref-type="bibr" rid="B37">Hodgskinson and Currey, 1993</xref>; <xref ref-type="bibr" rid="B52">Martin and Boardman, 1993</xref>; <xref ref-type="bibr" rid="B62">Nicholson et al., 1997</xref>). However, this relationship is weak at the tissue level, likely due to the effect of microstructural features at small length scales (<xref ref-type="bibr" rid="B63">Nobakhti and Shefelbine, 2018</xref>). In cortical bone, hierarchical micromechanical models have been also developed to explain how mechanical interactions between components at different observation scales regulate the tissue&#x2019;s effective elastoplasticity (<xref ref-type="bibr" rid="B27">Fritsch and Hellmich, 2007</xref>; <xref ref-type="bibr" rid="B32">Hamed et al., 2010</xref>; <xref ref-type="bibr" rid="B46">Kumbolder et al., 2024</xref>). However, they traditionally assume invariant structural and mechanical properties of these elementary components, including the collagen and hydroxyapatite (<xref ref-type="bibr" rid="B33">Hellmich et al., 2022</xref>). This assumption is not valid for an evolving tissue like woven bone. In the woven bone, 6th-degree polynomial equations were proven to best fits the elastic modulus versus the ash fraction, predicted by a multiscale computational homogeneization model (<xref ref-type="bibr" rid="B29">Garc&#xed;a-Rodr&#xed;guez and Mart&#xed;nez-Reina, 2017</xref>). In the <italic>in silico</italic> model, Garc&#xed;a-Rondr&#xed;guez and Mart&#xed;nez-Reina (<xref ref-type="bibr" rid="B29">Garc&#xed;a-Rodr&#xed;guez and Mart&#xed;nez-Reina, 2017</xref>) assumed isotropic orientation of collagen fibrils and used the ash fraction reported in the literature (<xref ref-type="bibr" rid="B53">Mart&#xed;nez-Reina et al., 2018</xref>). A simple exponential regression equation based on the ash fraction <inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf6">
<mml:math id="m6">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>12.88</mml:mn>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
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<mml:mn>2.75</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, was also provided with a significant correlation coefficient. To date, no previous work has proposed power laws based on experimental data to predict the apparent mechanical properties of woven bone at micro- or macro-scales considering the evolutionary structure of this tissue.</p>
<p>Various techniques have been developed to assess the bone inorganic matrix at macroscopic scale to measure bone mineral density or mineral content: computed tomography (<xref ref-type="bibr" rid="B83">Snyder and Schneider, 1991</xref>; <xref ref-type="bibr" rid="B73">Rho et al., 1995</xref>), micro-computed tomography, dual-energy X-ray absorptiometry (<xref ref-type="bibr" rid="B37">Hodgskinson and Currey, 1993</xref>; <xref ref-type="bibr" rid="B15">Bouxsein and Radloff, 1997</xref>), ashing the sample (<xref ref-type="bibr" rid="B21">Currey, 1988</xref>; <xref ref-type="bibr" rid="B53">Mart&#xed;nez-Reina et al., 2018</xref>) or weighing it (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B52">Martin and Boardman, 1993</xref>; <xref ref-type="bibr" rid="B43">Keller, 1994</xref>; <xref ref-type="bibr" rid="B62">Nicholson et al., 1997</xref>). To account for local variations in mineral content, scanning small angle X-ray scattering and wide angle X-ray scattering have been used to measure mineral crystal length and thickness (<xref ref-type="bibr" rid="B78">Roschger et al., 2020</xref>). Quantitative back-scattered electron microscopy is a validated method for determining spatially resolved calcium content (<xref ref-type="bibr" rid="B78">Roschger et al., 2020</xref>), and Fourier Transform infrared microspectroscopy and Raman microspectroscopy have been widely used to study the inorganic chemical compositional changes of the bone (<xref ref-type="bibr" rid="B85">Tarnowski et al., 2002</xref>; <xref ref-type="bibr" rid="B3">Akkus et al., 2004</xref>; <xref ref-type="bibr" rid="B54">McCreadie et al., 2006</xref>; <xref ref-type="bibr" rid="B95">Yerramshetty et al., 2006</xref>; <xref ref-type="bibr" rid="B60">Morris and Mandair, 2011</xref>), including mineralization, crystallinity, or carbonate substitution amongst others. For instance, investigating the inorganic chemical compositional changes during bone ageing using infrared and Raman microspectroscopy, it was concluded that Raman microspectroscopy is more sensitive for the inorganic matrix (<xref ref-type="bibr" rid="B90">Turunen et al., 2011</xref>). Bone healing processes have also been monitored with Raman spectroscopy (<xref ref-type="bibr" rid="B92">Uthgenant et al., 2007</xref>; <xref ref-type="bibr" rid="B28">Gamulin et al., 2013</xref>; <xref ref-type="bibr" rid="B2">Ahmed et al., 2018</xref>). <xref ref-type="bibr" rid="B2">Ahmed et al. (2018)</xref> studied early healing in calvarial defects, which heal spontaneously, with Raman Spectroscopy. They showed an increase in mineral/matrix and crystallinity ratios and a reduction in the carbonate/phosphate ratio after 14 days of healing. However, the long-term temporal evolution of the mechanical, chemical and morphological properties of woven bone during healing in critical-sized bone defects remains unknown. Beyond ash analysis, using Raman spectra or back-scattered electron signals as reliable input data for building power laws to predict the mechanical properties of bone tissue has not been explored to date.</p>
<p>Methods for evaluating the mechanical properties of bone tissue are more standardized. Macroscopic mechanical properties of cortical and trabecular bones have been traditionally evaluated through different mechanical testing including tension, compression, torsion, three-point bending, or buckling (<xref ref-type="bibr" rid="B89">Townsend et al., 1975</xref>; <xref ref-type="bibr" rid="B79">Ryan and Williams, 1989</xref>; <xref ref-type="bibr" rid="B94">Woo et al., 1991</xref>; <xref ref-type="bibr" rid="B6">Bayraktar et al., 2004</xref>; <xref ref-type="bibr" rid="B7">Beaupied et al., 2007</xref>). Local mechanics of mature tissue has been also assessed with ultrasound microscopy (<xref ref-type="bibr" rid="B24">Eriksen et al., 1994</xref>; <xref ref-type="bibr" rid="B34">Hengsberger et al., 2002</xref>; <xref ref-type="bibr" rid="B5">Bala et al., 2013</xref>), nanoindentation (<xref ref-type="bibr" rid="B73">Rho et al., 1995</xref>; <xref ref-type="bibr" rid="B75">1999</xref>; <xref ref-type="bibr" rid="B74">Rho and Pharr, 1999</xref>; <xref ref-type="bibr" rid="B97">Zysset et al., 1999</xref>; <xref ref-type="bibr" rid="B87">Thurner, 2009</xref>) or atomic force microscopy (<xref ref-type="bibr" rid="B4">Asgari et al., 2019</xref>). However, for woven bone and its evolutionary properties, studies have used both <italic>in vivo</italic> (<xref ref-type="bibr" rid="B23">Dwyer et al., 1996</xref>; <xref ref-type="bibr" rid="B1">Aarnes et al., 2005</xref>; <xref ref-type="bibr" rid="B58">Mora-Mac&#xed;as et al., 2015a</xref>; <xref ref-type="bibr" rid="B59">b</xref>; <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al., 2021a</xref>) and <italic>ex vivo</italic> approaches (<xref ref-type="bibr" rid="B65">Ohyama et al., 1994</xref>; <xref ref-type="bibr" rid="B25">Floerkemeier et al., 2010</xref>; <xref ref-type="bibr" rid="B12">Bl&#xe1;zquez-Carmona et al., 2022</xref>) to provide values of local and apparent stiffness at different time-points during healing. Typically, <italic>in vivo</italic> mechanical properties are indirectly measured through instrumentation of surgically implanted fixations (<xref ref-type="bibr" rid="B23">Dwyer et al., 1996</xref>; <xref ref-type="bibr" rid="B1">Aarnes et al., 2005</xref>; <xref ref-type="bibr" rid="B58">Mora-Mac&#xed;as et al., 2015a</xref>; <xref ref-type="bibr" rid="B59">b</xref>; <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al., 2021a</xref>). For <italic>ex vivo</italic> techniques, nanoindentation tests have proven valid to account for nanoscale heterogeneity and examine bone quality at a lower scale (<xref ref-type="bibr" rid="B48">Leong and Morgan, 2008</xref>; <xref ref-type="bibr" rid="B51">Manjubala et al., 2009</xref>; <xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al., 2017</xref>; <xref ref-type="bibr" rid="B56">Mora-Mac&#xed;as et al., 2021</xref>). For instance, <xref ref-type="bibr" rid="B51">Manjubala et al. (2009)</xref> and <xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al. (2017)</xref> measured the nanoindentation modulus for woven bone during fracture healing and bone transport processes, respectively.</p>
<p>
<italic>In silico</italic> models are valuable for the understanding the course of healing from a mechanobiological perspective. Many researchers have sought to establish relationships between the mechanical properties of undifferentiated tissue and ultimate tissue phenotype formed in a wide variety of bone regeneration processes, such as fracture healing (<xref ref-type="bibr" rid="B13">Borgiani et al., 2017</xref>; <xref ref-type="bibr" rid="B30">Ghiasi et al., 2017</xref>), distraction osteogenesis (<xref ref-type="bibr" rid="B38">Isaksson et al., 2007</xref>; <xref ref-type="bibr" rid="B71">Reina-Romo et al., 2011</xref>; <xref ref-type="bibr" rid="B70">2012</xref>) or tissue engineering (<xref ref-type="bibr" rid="B19">Byrne et al., 2007</xref>; <xref ref-type="bibr" rid="B84">Stops et al., 2010</xref>), amongst others. As a limitation, all these works develop phenomenological rules based on empirical observations and input parameters that are not sufficiently validated with experimental data, leaving room for improvement in accuracy. In particular, most of these models assumed constant values for the elastic modulus and porosity of the woven bone, regardless of its mineralization state or degree of matrix organization, which is not explicitly modeled. A constant porosity value of 80% and an elastic modulus of 1,000&#xa0;MPa are typically assigned to the woven bone in most of these mechanobiological evolutionary models (<xref ref-type="bibr" rid="B47">Lacroix and Prendergast, 2002</xref>; <xref ref-type="bibr" rid="B38">Isaksson et al., 2007</xref>; <xref ref-type="bibr" rid="B18">Burke and Kelly, 2012</xref>).</p>
<p>The main aim of this study is to identify time-related changes at the macro- and micro-scales at the chemical, mechanical, and morphological properties of the woven bone during healing of a critical-size bone defect. There is a need for a better understanding of the woven bone micromechanics and the development of a reliable mechanical model of this tissue to improve the prediction of its stiffness. For this purpose, at the microscopic scale, micromechanical properties will be measured with nanoindentation; mineral and organic characteristics will be quantified through Raman spectroscopy; and structural features through micro-CT. At the macroscopic scale, ash and element analyses will be assessed together with a numerical model to reproduce the woven bone bulk mechanical behavior based on physical measurements. All these experimental data will be used to build power laws relating the elastic modulus with the structural and chemical properties of the woven bone for the first time.</p>
</sec>
<sec sec-type="materials|methods" id="s2">
<title>2 Materials and methods</title>
<sec id="s2-1">
<title>2.1 Origin and preparation of the woven bone samples</title>
<p>The woven bone samples analyzed in the current study originate from previous <italic>in vivo</italic> distraction osteogenesis experiments conducted on the right-back metatarsi of six skeletally-mature female Merino sheep (<xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al., 2021a</xref>; <xref ref-type="bibr" rid="B10">b</xref>). The animals followed a surgical intervention and a bone lengthening protocol approved by the Animal Ethics of the University of C&#xf3;rdoba (Reference 2021PI/21), in compliance with the European (2010/63/UE) and national (RD 1201/2005) regulations. During the surgery, an Ilizarov-type external fixator was implanted in the metatarsus, and a cross-sectional osteotomy was performed in an intermediate section of the diaphysis. Consequently, each metatarsus was divided into two initially unconnected bone fragments, with their alignment ensured by the fixator device. After a 15-day latency period during which the bone healing begins by way of a preliminary soft callus formation, the fragments were distracted at rate of 1&#xa0;mm/day during 15 days (<xref ref-type="fig" rid="F1">Figure 1A</xref>). The resulting 15-mm bone callus was then allowed to mineralize over time. The specimens were sacrificed at different time-points of the distraction and consolidation phases to analyze <italic>ex vivo</italic> different states of callus ossification: days 18, 29, 47, 64, 112 and 161 after surgery. More details on the design of the external fixator and the distraction protocol were provided in previous works (<xref ref-type="bibr" rid="B11">Bl&#xe1;zquez-Carmona et al., 2020</xref>; <xref ref-type="bibr" rid="B10">2021b</xref>). After sacrifice, operated limbs were immediately stored at &#x2212;80&#xb0;C.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>
<bold>(A)</bold> Scheme of the osteotomized metatarsus and the distracion protocol, <bold>(B)</bold> scheme of the frontal plane sliced and analyzed from each operated limb, <bold>(C)</bold> image of a bone callus sample being imaged in the tomography, <bold>(D)</bold> 3D reconstruction of one of the bone calluses and volume of interest structurally analyzed, <bold>(E)</bold> visualization of microscope image of a bone callus sample. An example of cortical and woven bone areas to be analyzed by Raman Spectroscopy and nanoindentation are indicated, <bold>(F)</bold> scheme of the embedded bone callus samples and the areas analyzed, <bold>(G)</bold> example of a raw Raman spectra of the bone and the embedding resin, <bold>(H)</bold> comparison of corrected Raman spectra between cortical and woven bones, <bold>(I)</bold> load function applied by the indenter during a nanoindentation measure, <bold>(J)</bold> comparison of the indentation curve between cortical and woven bones, <bold>(K)</bold> nanoindentation of a cortical bone area, <bold>(L)</bold> nanoindentation of a woven bone area.</p>
</caption>
<graphic xlink:href="fbioe-12-1476473-g001.tif"/>
</fig>
<p>Two slices of each bone callus, each approximately 3&#xa0;mm thick, were cut in the frontal plane to address potential differences in the lateral-medial direction (<xref ref-type="fig" rid="F1">Figure 1B</xref>). These cuts were performed using a Femi FM-785XL band saw (Femi, Castel Guelfo, Bologna, Italy) while the sample was freshly taken out of the freezer to maintain the integrity of both the surrounding soft tissues and the callus itself. The first slice was used for micro-CT, Raman Spectroscopy and nanoindentation analysis. For micro-CT, the sample was analyzed in the same condition as cut. However, the other techniques required embedding and subsequent surface polishing to achieve flat and parallel surfaces. For embedding, a 40-mm diameter cylindrical polypropylene mold (FixiForm, Struers, California, United States) and a slow-curing epoxy resin (Epofix, Struers, California, United States) were employed. This resin hardens at room temperature in 24&#xa0;h, thus avoiding alterations in the mechanical or chemical properties of the sample due to high curing temperatures. Following a bone polishing guidelines provided in the literature (<xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al., 2017</xref>), the surfaces of the embedded samples were polished by means of carbide papers (P600 to P4000) and diamond slurry (from 3 to 0.25&#xa0;&#xb5;m), cleaning them ultrasonically with distilled water after each polishing step. The second slice was used to analyze the chemical composition of each bone callus, for which the cortical fragments were excised. No additional preparation was needed for this second slice. After the slicing, samples were stored in PBS-soaked gauze and plastic wrap at &#x2212;80&#xb0;C until the testing day. They were thawed by placement in PBS at room temperature for 1&#xa0;h prior to every test. Throughout the brief intervals between micro-CT, nanoindentation, and Raman spectroscopy tests, the samples were consistently kept hydrated in PBS-soaked gauze and plastic wrap at 4&#xb0;C. As a reference for the interpretation and discussion of the micro-CT results, six trabecular and six cortical samples from the operated ovine metatarsus were also cut and preserved as detailed above. Similarly, as a control for the chemical analysis, cortical fragments of each operated metatarsus were prepared.</p>
</sec>
<sec id="s2-2">
<title>2.2 Micro-computed tomography of the bone calluses</title>
<p>Micro-CT images were taken using a Cougar tomography<sup>&#xae;</sup> (Y.Cougar SMT, Yxlon, Hudson, Ohio, United States) with the following settings: &#xd7;4 optical magnification; exposure time of 9 s; voxel size of 3.72&#xa0;&#xb5;m); source setting of 50&#xa0;kV and 90.02&#xa0;&#xb5;m); and a physical size of approximately 25 &#xd7; 25 &#xd7; 25&#xa0;mm. An image of one of the samples in the micro-CT during imaging is provided for reference (<xref ref-type="fig" rid="F1">Figure 1C</xref>). The tomographies were processed using the image analysis platform Thermo Scientific Avizo Software (Thermo Fisher Scientific, Waltham, Massachusetts, United States). A median filter (3D interpretation, 6 neighborhood pixels, 3 iterations) was initially applied to reduce the contrast, soften the edges of the woven bone, and facilitate segmentation. A mixed automatic-manually interactive thresholding was performed, followed by a small noise spot removal with a maximum size of 100 px. The 3D structures were cropped into an inner cube covering the entire bone callus volume. An example of a thresholded sample and the cropped bone callus volume of interest is provided (<xref ref-type="fig" rid="F1">Figure 1D</xref>). From each 3D reconstructed callus, several parameters were determined: bone volume over total volume (BV/TV, module <italic>Volume Fraction</italic>), average trabecular thickness (Tb.Th, module <italic>Average Object Thickness</italic>), average trabecular separation (Tb.Sp, module <italic>Average Space Thickness</italic>), trabecular number (Tb.Nm, module <italic>Average Object Number per slice</italic>), degree of anisotropy (DA, module <italic>Degree of Anisotropy</italic>), and Structural Model Index (SMI, module <italic>Structure Model Index</italic>), which defines the relative prevalence of rods and plates in the structure.</p>
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<p>Control trabecular and cortical bone samples (n &#x3d; 6 per tissue type) were also measured and analyzed using micro-CT. However, BV/TV was the only parameter calculated in cortical samples due to the absence of trabecular structures in this tissue.</p>
</sec>
<sec id="s2-3">
<title>2.3 Material analysis using Raman Spectroscopy</title>
<p>For the spectroscopy and nanoindentation tests, six rectangular regions of interest (800 &#xd7; 800&#xa0;&#xb5;m) were selected in both woven and cortical tissues from each of the embedded and polished bone callus samples. An example locations of a composition of images taken with the microscope to select these areas of interest is provided for reference (<xref ref-type="fig" rid="F1">Figure 1E</xref>), highlighting examples of cortical and woven tissue. One area was always located in the proximal cortical fragment as a control reference (<xref ref-type="fig" rid="F1">Figure 1F</xref>). The other five zones were positioned to cover the proximal-distal (A-B-C) and medial-distal (B-D-E) directions.</p>
<p>Raman Spectroscopy analysis was performed using a LabRAM Horiba Jobin Yvon (HORIBA, Kyoto, Japan) confocal Raman microscope. The embedded samples were periodically sprayed with PBS on their surface to maintain hydration. Spectra were collected with a 785&#xa0;nm red laser at &#xd7;100 magnification, with a 5-s integration time and 20 accumulations. The spectral range was from 200 to 1800 <inline-formula id="inf14">
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<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> band is also interesting due to its inversely proportionality to mineral crystalline length and, therefore, is an indirect measure of mineral crystallinity. The most widely used bands to measure the matrix content are Proline at <inline-formula id="inf27">
<mml:math id="m29">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>853 <inline-formula id="inf28">
<mml:math id="m30">
<mml:mrow>
<mml:msup>
<mml:mrow>
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<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
<mml:mrow>
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<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, Hydroxyproline at <inline-formula id="inf29">
<mml:math id="m31">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>872 <inline-formula id="inf30">
<mml:math id="m32">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">c</mml:mi>
<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, Amide III at <inline-formula id="inf31">
<mml:math id="m33">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>1250 <inline-formula id="inf32">
<mml:math id="m34">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">c</mml:mi>
<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, and Tyrosine at <inline-formula id="inf33">
<mml:math id="m35">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>1607 <inline-formula id="inf34">
<mml:math id="m36">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="normal">c</mml:mi>
<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B50">Mandair and Morris, 2015</xref>; <xref ref-type="bibr" rid="B26">Fraulob et al., 2020</xref>). The spectra were processed in MATLAB (R2022a, MathWorks, Natick, Massachusetts, United States) to calculate intensity peaks (I), areas under the peaks (A), and broadness as full width at half maximum (FWHM). The parameters and ratios calculated are detailed in <xref ref-type="table" rid="T1">Table 1</xref>, indicating if they are related to the mineral or oganic phase.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Raman parameters in cortical and woven bone tissues calculated for each Raman spectra. I: maximum peak intensity, A: area under the peaks, FWHM: full width at half maximum. the subscript indicates the Raman shift position (<inline-formula id="inf35">
<mml:math id="m37">
<mml:mrow>
<mml:msup>
<mml:mrow>
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<mml:mi mathvariant="normal">m</mml:mi>
</mml:mrow>
<mml:mrow>
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<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>).</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="center">Raman paramenter</th>
<th align="center">Tissue phase</th>
<th align="center">Ratio</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">Crystallinity</td>
<td align="center">Mineral</td>
<td align="center">
<inline-formula id="inf36">
<mml:math id="m38">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mi>W</mml:mi>
<mml:mi>H</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>M</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>959</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">Carbonate-to-phosphate</td>
<td align="center">Mineral</td>
<td align="center">
<inline-formula id="inf37">
<mml:math id="m39">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1070</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>959</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">
<inline-formula id="inf38">
<mml:math id="m40">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>PO<inline-formula id="inf39">
<mml:math id="m41">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/Proline mineral-to-matrix</td>
<td align="center">Mineral/Organic</td>
<td align="center">
<inline-formula id="inf40">
<mml:math id="m42">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
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<mml:mrow>
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</mml:mrow>
<mml:mrow>
<mml:mn>959</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>853</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">
<inline-formula id="inf41">
<mml:math id="m43">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>PO<inline-formula id="inf42">
<mml:math id="m44">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/Tyrosine mineral-to-matrix</td>
<td align="center">Mineral/Organic</td>
<td align="center">
<inline-formula id="inf43">
<mml:math id="m45">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>959</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1607</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">
<inline-formula id="inf44">
<mml:math id="m46">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>PO<inline-formula id="inf45">
<mml:math id="m47">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/Amide III mineral-to-matrix</td>
<td align="center">Mineral/Organic</td>
<td align="center">
<inline-formula id="inf46">
<mml:math id="m48">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>422</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1250</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">Carbonate-to-matrix</td>
<td align="center">Mineral/Organic</td>
<td align="center">
<inline-formula id="inf47">
<mml:math id="m49">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1070</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">1620&#x2212;1700</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">Hydroxyproline-to-proline</td>
<td align="center">Organic</td>
<td align="center">
<inline-formula id="inf48">
<mml:math id="m50">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>872</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>853</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
<tr>
<td align="center">Amide I-to-Amide III</td>
<td align="center">Organic</td>
<td align="center">
<inline-formula id="inf49">
<mml:math id="m51">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">1620&#x2212;1700</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1250</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec id="s2-4">
<title>2.4 Mechanical properties using nanoindentation</title>
<p>The elastic modulus of the tissue was determined by nanoindentation tests on the same regions of interest described in <xref ref-type="sec" rid="s2-3">section 2.3</xref>, including the cortical zones. Nanoindentation was performed using a Nanotest indenter (Nanotest, Micro Materials Ltd. Wrexham, United Kingdom) equipped with a Berkovich diamond indenter. In each area, 16 &#xd7; 16 indentations were made, spaced 50&#xa0;&#xb5;m apart in both directions, corresponding to the minimum trabecula thickness of na&#xef;ve woven bone tissue (<xref ref-type="bibr" rid="B49">L&#xf3;pez-Pliego et al., 2016</xref>). Thus, a total of 256 indentations were performed per area, covering the entire region of interest. Samples were sprayed with PBS from time to time to preserve their hydration. The load applied to each indentation point was increased at a rate of 0.5&#xa0;mN/s to a maximum of 5&#xa0;mN (<xref ref-type="fig" rid="F1">Figure 1I</xref>). Once the maximum load was reached in approximately 10 s, it was held constant for 40&#xa0;s. The unloading was performed at the same rate. The indentation depth, or the indenter&#x2019;s displacement, is a variable parameter that depends on the hardness of the tissue, being lower in cortical bone (<xref ref-type="fig" rid="F1">Figure 1J</xref>). The Oliver and Pharr method was used to calculate the elastic modulus <inline-formula id="inf50">
<mml:math id="m52">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>E</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> from the load-depth data (<xref ref-type="bibr" rid="B66">Oliver and Pharr, 1992</xref>). Examples of nanoindentation maps measured on cortical and woven bone regions are provided (<xref ref-type="fig" rid="F1">Figures 1K, L</xref>, zones of interest marked in <xref ref-type="fig" rid="F1">Figure 1E</xref>). The average elastic modulus per zone and its standard deviation were calculated by eliminating from the calculation those areas of porosity identified by microscope images or with elastic modulus less than 2&#xa0;GPa.</p>
</sec>
<sec id="s2-5">
<title>2.5 Chemical analysis of the tissue</title>
<p>The second slice was employed for ash analysis and elemental analysis. The samples were initially ground with a sterilized pestle and mortar. They were then immediately dried in a BINDER VD 23 vacuum drying chamber (BINDER GmbH, Tuttlingen, Germany) using heating cycles at 105 <inline-formula id="inf51">
<mml:math id="m53">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 2&#xb0;C for 1&#xa0;h each, measuring the samples&#x2019; weight at the end of each cycle until it stabilized. The resulting weight combines that corresponding to the mineral and organic phases <inline-formula id="inf52">
<mml:math id="m54">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mrow>
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</mml:mrow>
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</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> of the woven bone tissue after removing water. To obtain the ash fraction, the samples were ashed in a Nabertherm Muffle Furnace (Nabertherm GmbH, Lilienthal Germany) following a protocol published in the literature (<xref ref-type="bibr" rid="B53">Mart&#xed;nez-Reina et al., 2018</xref>): (1) the temperature was initially increased from room temperature to 250&#xb0;C over 30&#xa0;min, and held at this temperature for 1&#xa0;h; (2) the temperature was gradually increased to 650&#xb0;C over 30&#xa0;min, and held for 2&#xa0;h; (3) the sample was then taken out of furnace and weighed; (4) the sample was reintroduced into the furnace at 650&#xb0;C and maintained at this temperature for 30&#xa0;min. Steps 3 and 4 were repeated until a constant weight was achieved. This process allowed removing the organic material. Therefore, the weight obtained at the end of the protocol corresponds to the mineral phase exclusively <inline-formula id="inf53">
<mml:math id="m55">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
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<mml:mrow>
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</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. The ash fraction <inline-formula id="inf54">
<mml:math id="m56">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> was finally calculated using <xref ref-type="disp-formula" rid="e3">Equation 3</xref>.<disp-formula id="e3">
<mml:math id="m57">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
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<mml:mi>m</mml:mi>
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<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>m</mml:mi>
</mml:mrow>
<mml:mrow>
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</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>
</p>
<p>An elemental analysis was also performed on the ashed samples to measure the calcium (Ca) mass percentage and trace element substitutions, mainly stable isotopes of carbon with oxygen in the carbonate (<inline-formula id="inf55">
<mml:math id="m58">
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">C</mml:mi>
<mml:mi mathvariant="normal">O</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula>) and hydrogen phosphate (<inline-formula id="inf56">
<mml:math id="m59">
<mml:mrow>
<mml:msubsup>
<mml:mrow>
<mml:mi mathvariant="normal">H</mml:mi>
<mml:mi mathvariant="normal">P</mml:mi>
<mml:mi mathvariant="normal">O</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula>) (<xref ref-type="bibr" rid="B53">Mart&#xed;nez-Reina et al., 2018</xref>; <xref ref-type="bibr" rid="B42">Katzenberg, 2020</xref>). Hence, the mass percentages of carbon (C) and phosphorus (P) were evaluated as an indirect measure of these impurities. Other possible calcium substitutes measured in the current study were sodium (Na), potassium (K) and other alkaline earth elements similar to calcium, including magnesium (Mg) or strontium (Sr) (<xref ref-type="bibr" rid="B8">Bergstrom and Wallace, 1954</xref>; <xref ref-type="bibr" rid="B42">Katzenberg, 2020</xref>).</p>
<p>The elemental analysis of the carbon content was conducted using a TruSpec Micro analyzer (LECO Corporation, Michigan, United States). The calcium and phosphorus contents were measured with an inductively coupled plasma optical emission spectrometer SpectroBLUE (Spectro Analytical Instruments, Kleve, Germany) after the samples were dissolved in hydrochloric acid. As stated in <xref ref-type="sec" rid="s2-1">section 2.1</xref>, one cortical slice from each metatarsal diaphysis was also prepared and analyzed using the methodology detailed above (six cortical fragments in total).</p>
</sec>
<sec id="s2-6">
<title>2.6 Definition of power laws for woven bone</title>
<p>The <italic>ex vivo</italic> experiments described in the previous sections will elucidate the evolution of the mechanical and chemical properties of woven bone during healing at the tissue scale, as well as the structural changes at the apparent level towards a cortical microarchitecture. In the literature, ash fraction or calcium content have been shown to be effective predictors of cancellous and cortical bone&#x2019;s mechanical properties (<xref ref-type="bibr" rid="B21">Currey, 1988</xref>; <xref ref-type="bibr" rid="B35">Hernandez et al., 2001</xref>; <xref ref-type="bibr" rid="B29">Garc&#xed;a-Rodr&#xed;guez and Mart&#xed;nez-Reina, 2017</xref>). Therefore, these parameters were used to define single-parameter powers laws that predict the woven bone elastic modulus over the regeneration time, following <xref ref-type="disp-formula" rid="e4">Equation 4</xref>:<disp-formula id="e4">
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<label>(4)</label>
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</inline-formula> are empirical constants derived from experimental data, and the variable <inline-formula id="inf60">
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</inline-formula> represents the predictor parameter. Specifically, empirical constants were fitted using the mean elastic modulus per sample measured by nanoindentation, evaluating the following experimental parameters as a predictors: ash fraction <inline-formula id="inf61">
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</inline-formula>/Amide III mineral-to-matrix <inline-formula id="inf66">
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</inline-formula>/Amide III ratio has been extensively shown to be proportional to calcium content, as measured by quantitative backscattered electron microscopy (<xref ref-type="bibr" rid="B77">Roschger et al., 2014</xref>; <xref ref-type="bibr" rid="B50">Mandair and Morris, 2015</xref>). The phosphate <inline-formula id="inf72">
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</inline-formula>, as a ratio indicative of organic matrix content, is also widely accepted as a measure of mineral content.</p>
<p>Given that the experimental data used to fit the empirical coefficients were measured at the micro-scale, the previous models predict the elastic modulus at the tissue scale. Consequently, a two-parameter power law function was also proposed in <xref ref-type="disp-formula" rid="e5">Equation 5</xref> to differentiate the influence of bone volume (BV/TV) and ash fraction <inline-formula id="inf74">
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</inline-formula> on an apparent macro-scale elastic modulus, as has been previously applied in cortical bone studies (<xref ref-type="bibr" rid="B21">Currey, 1988</xref>; <xref ref-type="bibr" rid="B35">Hernandez et al., 2001</xref>):<disp-formula id="e5">
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<p>To adjust the coefficients of this second model, the volume fraction (BV/TV) measured by micro-CT were employed. Moreover, the apparent elastic modulus was taken by the model provided by <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al. (2021a)</xref> and shown in <xref ref-type="disp-formula" rid="e6">Equation 6</xref>, which is based on experimental gait force data measured <italic>in vivo</italic> through instrumented external fixators and a load platform during gait tests on the same animals whose bone calluses were used in this work (R-square &#x3d; 0.933, <italic>p</italic>-value <inline-formula id="inf75">
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</mml:mrow>
</mml:math>
</inline-formula> 0.001):<disp-formula id="e6">
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<mml:mn>3.76</mml:mn>
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<mml:msup>
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<mml:mrow>
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</mml:mrow>
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<label>(6)</label>
</disp-formula>where t is the day after surgery, and <inline-formula id="inf76">
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</inline-formula> is the apparent elastic modulus in MPa. Evaluating the time-points analyzed in this study, the apparent elastic modulus ranges between 0.01&#x2013;8.53&#xa0;GPa. The determination coefficients, R-square and <italic>p</italic>-values, were also calculated for all power laws defined above.</p>
</sec>
</sec>
<sec sec-type="results" id="s3">
<title>3 Results</title>
<sec id="s3-1">
<title>3.1 Evolution of the woven bone microarchitecture</title>
<p>The evolution of the structural parameters in the 3D reconstructed woven bone calluses were measured from the micro-tomographic images (<xref ref-type="fig" rid="F2">Figure 2</xref>). The BV/TV raised during the mineralization process (<xref ref-type="fig" rid="F2">Figure 2A</xref>). The calluses were partially mineralized shortly after the surgery, comprising 44.59<inline-formula id="inf77">
<mml:math id="m83">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> of the total callus volume after 18 days. Bone volume increased slightly over time, reaching 68.89<inline-formula id="inf78">
<mml:math id="m84">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> after 161 days. Throughout the analyzed period, the calluses had a volumetric fraction above that of the spongy bone in the proximal epiphysis (35.89 <inline-formula id="inf79">
<mml:math id="m85">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 4.42%, red data) but below the surrounding cortical tissue (96.80 <inline-formula id="inf80">
<mml:math id="m86">
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</inline-formula> 1.52%, green data). The size of the trabeculae increased substantially during the regeneration period from approximately 0.11 mm, similar to that of trabecular bone, to 0.29&#xa0;mm (<xref ref-type="fig" rid="F2">Figure 2B</xref>). In contrast, trabecular separation did not show a specific evolution, consistently remaining below the separation in the proximal cancellous bone (<xref ref-type="fig" rid="F2">Figure 2C</xref>). Both the trabecular number Tb. Nm (<xref ref-type="fig" rid="F2">Figure 2D</xref>) and the connectivity Conn. D (<xref ref-type="fig" rid="F2">Figure 2E</xref>) exhibited a similar pattern. A high number of trabecular structural structures with high connectivity were observed shortly after surgery. However, as trabeculae increased in size, these parameters decreased rapidly and stabilized at similar values to trabecular bone values. The degree of anisotropy (DA) remained constant between 0.45-0.51 (<xref ref-type="fig" rid="F2">Figure 2F</xref>). Thus, similar to trabecular bone, although the structures showed a certain preferred orientation during trabecular formation, they were neither perfectly isotropic (DA &#x3d; 0) nor completely anisotropic (DA &#x3d; 1). Finally, the structural model index (SMI) negatively increase, indicating an increase in the concavity of the trabecular surfaces (<xref ref-type="fig" rid="F2">Figure 2G</xref>).</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Evolution of the woven bone&#x2019;s microarchiture measured from micro-tomographic images: <bold>(A)</bold> volume fraction, <bold>(B)</bold> trabecular thickness, <bold>(C)</bold> trabecular separation, <bold>(D)</bold> trabecular number, <bold>(E)</bold> connectivity density, <bold>(F)</bold> degree of anisotropy, <bold>(G)</bold> structural model index. The average value and standard deviation of the cortical and trabecular data is provided as a reference in green and red, respectively.</p>
</caption>
<graphic xlink:href="fbioe-12-1476473-g002.tif"/>
</fig>
</sec>
<sec id="s3-2">
<title>3.2 Elastic modulus of the bone callus</title>
<p>The average and standard deviation of the elastic modulus were calculated in each analyzed area, including the cortical region (<xref ref-type="fig" rid="F3">Figure 3A</xref>). Additionally, nanoindentation maps of the analyzed zones, along with their corresponding microscopic reference images, are provided in the <xref ref-type="sec" rid="s11">Supplementary Material</xref> (<xref ref-type="sec" rid="s11">Supplementary Figure S1</xref>). Overall, no remarkable differences were found between the areas in any analyzed direction of the callus, either the proximal-distal direction (A-B-C) or the medial-lateral direction (B-D-E). In all samples, the elastic modulus of cortical tissue is double or triple that of woven bone. The temporal evolution of the elastic modulus of the samples was also analyzed as the global average of all regions of interest (<xref ref-type="fig" rid="F3">Figure 3B</xref>). Note that the control cortical data provided in <xref ref-type="fig" rid="F3">Figure 3B</xref> encompasses the mean and standard deviation from all cortical nanoindentations across all samples/animals (256 cortical indentations per sample, 1,536 in total). Initially, the elastic modulus ranged from 3.07 to 6.32&#xa0;GPa in the first weeks after surgery, and it increased slightly to 9.25&#xa0;GPa after 6&#xa0;months. During the analyzed period, the mechanical properties of woven bone remained notably lower than those of cortical bone, whose elastic modulus was 16.44 <inline-formula id="inf81">
<mml:math id="m87">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 2.79&#xa0;GPa.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Evolution of the micro-scale woven bone&#x2019;s elastic modulus and chemical composition measured by nanoindentation and Raman Spectroscopy, respectively. Mechanical properties: <bold>(A)</bold> elastic modulus by callus zones in each of the samples analyzed, <bold>(B)</bold> average elastic modulus of all the indentations made in all analyzed areas of the sample. Chemical structure: <bold>(C)</bold> Cristallinity, <bold>(D)</bold> Carbonate-to-phosphate, <bold>(E)</bold> Proline mineral-to-matrix, <bold>(F)</bold> Tyrosine mineral-to-matrix, <bold>(G)</bold> Amide III mineral-to-matrix, <bold>(H)</bold> Carbonate-to-matrix, <bold>(I)</bold> Hydroxyproline-to-proline, <bold>(J)</bold> Amide I-to-Amide III. Data from cortical bone (mean and standard deviation) is provided in green.</p>
</caption>
<graphic xlink:href="fbioe-12-1476473-g003.tif"/>
</fig>
</sec>
<sec id="s3-3">
<title>3.3 Chemical composition of the woven bone</title>
<p>In this study, the chemical structures of woven and cortical bones were analyzed on both micro- and macro-scales. On the micro-scale, the evolution of the Raman parameters were calculated for each woven bone sample (<xref ref-type="fig" rid="F3">Figure 3</xref>). The data is provided as the average ratios of all zones and all spectra taken from each. The control cortical data provides the average ratios of all spectra measured in all samples (6 spectra per sample, 36 in total). The results per areas of each sample is also provided in the <xref ref-type="sec" rid="s11">Supplementary Material</xref> (<xref ref-type="sec" rid="s11">Supplementary Figure S2</xref>). In both figures, the ratios calculated for the reference cortical tissue are shown in green. The na&#xef;ve woven bone tissue exhibited a degree of crystallinity comparable to that of cortical tissue from few days after surgery, without appreciable evolution over time (<xref ref-type="fig" rid="F3">Figure 3C</xref>). During regeneration, the carbonate-to-phosphate ratio was also similar to that of cortical bone, without noteworthy differences in their standard deviation (<xref ref-type="fig" rid="F3">Figure 3D</xref>). The mineral amount of Phosphate v1 and v2 was normalized by the organic components of the matrix: Proline (<xref ref-type="fig" rid="F3">Figure 3E</xref>), Tyrosine (<xref ref-type="fig" rid="F3">Figure 3F</xref>), and Amide III (<xref ref-type="fig" rid="F3">Figure 3G</xref>), respectively. Although no trend was observed when normalizing by Proline compared to the cortical tissue, the ratios against the other organic phases began notably lower in woven bone compared to mature bone tissue, tending to approach cortical values during the regeneration phase. The carbonate content of the newly formed bone tissue also increased relative to the organic phase until reaching cortical values (<xref ref-type="fig" rid="F3">Figure 3H</xref>). No trend was found in the Hydroxyproline-to-Proline (<xref ref-type="fig" rid="F3">Figure 3I</xref>) and Amide I-to-Amide III (<xref ref-type="fig" rid="F3">Figure 3J</xref>) ratios, but their standard deviations were lower than those calculated for the same ratios in cortical bone.</p>
<p>Regarding the macro-scale chemical composition, the evolution of the ash fraction over the regeneration time was investigated (<xref ref-type="fig" rid="F4">Figure 4A</xref>). The results obtained from the six control cortical fragments are also provided as a mean and standard deviation (green data). The mass fraction of the mineral component versus the organic component of the tissue increased as the collagen template mineralized over time. Consequently, the ash fraction increased progressively from 0.15 to 0.29 in the less mature calluses to 0.57 in the most ossified ones. This more consolidated callus reached an ash fraction similar to that of cortical tissue, 0.66 <inline-formula id="inf82">
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<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Evolution of the macro-scale woven bone&#x2019;s chemical composition measured with micro-analysis: <bold>(A)</bold> ash fraction, <bold>(B)</bold> carbon percentage, <bold>(C)</bold> calcium percentage, <bold>(D)</bold> potassium percentage, <bold>(E)</bold> magnesium percentage, <bold>(F)</bold> sodium percentage, <bold>(G)</bold> phosphorus percentage, <bold>(H)</bold> strontium percentage. Cortical data (mean and standard deviation) is provided as a reference in green lines.</p>
</caption>
<graphic xlink:href="fbioe-12-1476473-g004.tif"/>
</fig>
<p>The micro-elemental determinations of the chemical components specified in <xref ref-type="sec" rid="s2-5">section 2.5</xref> are also provided (<xref ref-type="fig" rid="F4">Figures 4B&#x2013;H</xref>). The carbon content (C<inline-formula id="inf84">
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</mml:math>
</inline-formula>) slightly increased from 0.20 to 0.65<inline-formula id="inf85">
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</inline-formula>. However, the percentage of this element in all samples, including their cortical areas, remained relatively low (<inline-formula id="inf86">
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</mml:mrow>
</mml:math>
</inline-formula>) was the predominant element in all samples, with percentages ranging from 35.71 to 47.96<inline-formula id="inf88">
<mml:math id="m94">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="fig" rid="F4">Figure 4B</xref>). Note that the calcium content in a cortical bone sample was estimated at 47.25 <inline-formula id="inf89">
<mml:math id="m95">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 1.38<inline-formula id="inf90">
<mml:math id="m96">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Although a slight increase is seen throughout the regeneration phase, some of the least mature samples already showed calcium levels similar to cortical levels. Potassium (K<inline-formula id="inf91">
<mml:math id="m97">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <xref ref-type="fig" rid="F4">Figure 4D</xref>), magnesium (Mg<inline-formula id="inf92">
<mml:math id="m98">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <xref ref-type="fig" rid="F4">Figure 4E</xref>), and sodium (Na<inline-formula id="inf93">
<mml:math id="m99">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <xref ref-type="fig" rid="F4">Figure 4F</xref>) followed a similar trend. All of these impurities began with percentage contents (0.48-0.71<inline-formula id="inf94">
<mml:math id="m100">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for K; 0.57-0.89<inline-formula id="inf95">
<mml:math id="m101">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for Mg; 1.80-2.71<inline-formula id="inf96">
<mml:math id="m102">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for Na) above the cortical values of 0.08 <inline-formula id="inf97">
<mml:math id="m103">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 0.01<inline-formula id="inf98">
<mml:math id="m104">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, 0.45 <inline-formula id="inf99">
<mml:math id="m105">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 0.04<inline-formula id="inf100">
<mml:math id="m106">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, and 1.04 <inline-formula id="inf101">
<mml:math id="m107">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 0.01<inline-formula id="inf102">
<mml:math id="m108">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, respectively. However, all of them normalized over the period analyzed. Phosphorus (P<inline-formula id="inf103">
<mml:math id="m109">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>) was the second most abundant element (<xref ref-type="fig" rid="F4">Figure 4G</xref>), maintaining a relatively constant percentage between 20.03 and 24.16<inline-formula id="inf104">
<mml:math id="m110">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, similar to that of cortical bone (23.74 <inline-formula id="inf105">
<mml:math id="m111">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 0.55<inline-formula id="inf106">
<mml:math id="m112">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>). Finally, the percentage of strontium (Sr<inline-formula id="inf107">
<mml:math id="m113">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>) was negligible in all samples, remaining around 0.02<inline-formula id="inf108">
<mml:math id="m114">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="fig" rid="F4">Figure 4H</xref>).</p>
</sec>
<sec id="s3-4">
<title>3.4 Power laws: influence of structural and chemical properties of the elastic modulus</title>
<p>
<xref ref-type="table" rid="T2">Table 2</xref> presents the different power laws, the value of the empirical constants after adjustment with the experimental data, as well as the goodness-of-fit parameters: the R-square and the <italic>p</italic>-value. Additionally, these best-fit power law models of the tissue and apparent elastic modulus was also graphically represented as functions of the predictor parameters, plotted against the experimental data (<xref ref-type="fig" rid="F5">Figure 5</xref>). All single-parameter adjustments demonstrated high predictive strength of the tissue elastic modulus, with all R-square values exceeding 0.5 and most of <italic>p</italic>-values below 0.05. The highest degrees of significance were observed in correlation based on Raman parameters. The largest resulting predictor exponent values were found in the Calcium content measured by chemical analysis (b<inline-formula id="inf109">
<mml:math id="m115">
<mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.86</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>) and by Raman spectroscopy (Amide III mineral-to-matrix, b<inline-formula id="inf110">
<mml:math id="m116">
<mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.64</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>).</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Definition of the power laws of the tissue and apparent elastic modulus (<inline-formula id="inf111">
<mml:math id="m117">
<mml:mrow>
<mml:mi>E</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in GPa) based of different structural and chemical measured parameters, fitting constants, and goodness-of-fit parameters, R-square and <italic>p</italic>-values. Note that Calcium (Ca) and Phosphorus (P) in the power laws are expressed on a per unit basis. &#x2a;Apparent elastic modulus measured <italic>in vivo</italic> by Bl&#xe1;zquez-Carmona et al. (<xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al., 2021a</xref>).</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Parameter</th>
<th rowspan="2" align="center">Power law</th>
<th colspan="3" align="center">Constants</th>
<th rowspan="2" align="center">R-square</th>
<th rowspan="2" align="center">
<italic>p</italic>-value</th>
</tr>
<tr>
<th align="center">a</th>
<th align="center">b</th>
<th align="center">c</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">Ash fraction</td>
<td align="center">
<inline-formula id="inf112">
<mml:math id="m118">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">20.84</td>
<td align="center">1.26</td>
<td align="center">&#x2014;</td>
<td align="center">0.55</td>
<td align="center">0.091</td>
</tr>
<tr>
<td align="center">Calcium</td>
<td align="center">
<inline-formula id="inf113">
<mml:math id="m119">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:mi>C</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">71.56</td>
<td align="center">2.86</td>
<td align="center">&#x2014;</td>
<td align="center">0.72</td>
<td align="center">0.031</td>
</tr>
<tr>
<td align="center">Phosphorus</td>
<td align="center">
<inline-formula id="inf114">
<mml:math id="m120">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>P</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">349.85</td>
<td align="center">2.62</td>
<td align="center">&#x2014;</td>
<td align="center">0.56</td>
<td align="center">0.104</td>
</tr>
<tr>
<td align="center">Ash and volume fraction</td>
<td align="center">
<inline-formula id="inf115">
<mml:math id="m121">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>E</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2a;</mml:mo>
</mml:mrow>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>&#x22c5;</mml:mo>
<mml:mi>B</mml:mi>
<mml:mi>V</mml:mi>
<mml:mo>/</mml:mo>
<mml:mi>T</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mi>V</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">2.74e3</td>
<td align="center">3.88</td>
<td align="center">9.98</td>
<td align="center">0.88</td>
<td align="center">0.005</td>
</tr>
<tr>
<td align="center">Amide III mineral-to-matrix</td>
<td align="center">
<inline-formula id="inf116">
<mml:math id="m122">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>422</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>A</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1250</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">2.79</td>
<td align="center">2.64</td>
<td align="center">&#x2014;</td>
<td align="center">0.83</td>
<td align="center">0.011</td>
</tr>
<tr>
<td align="center">Tyrosine mineral-to-matrix</td>
<td align="center">
<inline-formula id="inf117">
<mml:math id="m123">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x22c5;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1607</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>/</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>I</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>853</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">0.12</td>
<td align="center">1.89</td>
<td align="center">&#x2014;</td>
<td align="center">0.90</td>
<td align="center">0.003</td>
</tr>
</tbody>
</table>
</table-wrap>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Correlations of the power laws defined in <xref ref-type="table" rid="T2">Table 2</xref> to predict the tissue or apparent elastic modulus from the following structural and chemical parameters measured <italic>ex vivo</italic>: <bold>(A)</bold> ash fraction from chemical analysis, <bold>(B)</bold> calcium content from chemical analysis, <bold>(C)</bold> phosphorus content from chemical analysis, <bold>(D)</bold> ash fraction from chemical analysis and volume fraction from micro-CT, <bold>(E)</bold> <inline-formula id="inf118">
<mml:math id="m124">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>PO<inline-formula id="inf119">
<mml:math id="m125">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/Amide III mineral-to-matrix from Raman Spectroscopy, undertood as the calcium content, <bold>(F)</bold> <inline-formula id="inf120">
<mml:math id="m126">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>PO<inline-formula id="inf121">
<mml:math id="m127">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>&#x2212;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/Tyrosine mineral-to-matrix from Raman Spectroscopy.</p>
</caption>
<graphic xlink:href="fbioe-12-1476473-g005.tif"/>
</fig>
<p>The two-parameter power law model based on the ash fraction and bone volume fraction measured in the current study, alongside the macro-scale apparent elastic modulus provided by <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al. (2021a)</xref> (<xref ref-type="fig" rid="F5">Figure 5D</xref>), was also found to be highly significant (R-square &#x3d; 0.88 and <italic>p</italic>-value &#x3d; 0.005). Furthermore, the best-fit volume fraction exponent value was nearly 3-fold the ash fraction exponent value, at 9.98 versus 3.88.</p>
</sec>
</sec>
<sec sec-type="discussion" id="s4">
<title>4 Discussion</title>
<p>Quantifying the structural, chemical, and mechanical properties of the woven bone is crucial for understanding the mechanobiology behind bone regeneration and assessing the effectiveness of ortophaedic approaches for healing critical-sized bone defects. Equally important is the parallel development of numerical models that optimize key engineering parameters to control this biological process and prevent non-unions. There is a notable lack of micromechanical models of woven bone in the literature based on <italic>ex vivo</italic> experimental data that consider changes in tissue properties at different regeneration phases. The multiscale research presented here analyzed the evolution of the trabeculae organization in this immature tissue through micro-CT, its elastic modulus using nanoindentation, its mineral quality via Raman spectroscopy, and its chemical composition with ash analysis. From these experiments, power laws based on physical measurements were defined for the first time, relating the elastic modulus of the woven bone with its porosity and mineralization state. In addition, the data and power laws provided in this study are highly extrapolated to human bone tissues due to the ovine nature of the samples. High similarities in bone composition were reported between mature sheep and humans in the literature (<xref ref-type="bibr" rid="B61">Newman et al., 1995</xref>). Similarly, comparable metabolic and bone remodeling rates were found using dual energy X-ray absorptiometry (<xref ref-type="bibr" rid="B22">den Boer et al., 1999</xref>). The results obtained in this study also try to address some open questions in the field: Why is the stiffness of woven bone lower than that of cortical bone? Is it solely due to the mineral content, or is it influenced by the structural factors as well? Do structural and mineral changes equally contribute to the bulk mechanical properties of woven bone? Are Raman spectra valid inputs for defining power laws?</p>
<p>The mechanical properties of the woven bone tissue exhibited a slight increase over the analyzed regeneration period, as indicated by the elastic modulus measured by nanoindentation, which ranged from 3.07 to 9.25&#xa0;GPa (<xref ref-type="fig" rid="F3">Figures 3A, B</xref>; <xref ref-type="sec" rid="s11">Supplementary Figure S1</xref>). However, the stiffness levels characteristic of cortical tissue, approximately 16.44 <inline-formula id="inf122">
<mml:math id="m128">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 2.79 GPa, were not reached even after 161 days post-surgery. Previous studies also used nanoindentation to measure the elastic modulus of bone calluses formed during other regeneration processes (<xref ref-type="bibr" rid="B48">Leong and Morgan, 2008</xref>; <xref ref-type="bibr" rid="B51">Manjubala et al., 2009</xref>; <xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al., 2017</xref>). <xref ref-type="bibr" rid="B51">Manjubala et al. (2009)</xref> reported values in the range of 2&#x2013;13&#xa0;GPa over the first 9&#xa0;weeks of a fracture healing experiment in an ovine tibia. Thus, the woven tissue associated to this regeneration process reached higher stiffness at a greater speed, likely due to the smaller size of the bone gap in fracture healing, approximately 3&#xa0;mm in this study. Similarly, Leong and Morgan (<xref ref-type="bibr" rid="B48">Leong and Morgan, 2008</xref>) conducted indentation tests on rodent fracture healing calluses, reporting an elastic modulus ranging from 0.03 to 1.01&#xa0;GPa at day 35 after fracture. These results are considerably lower than those observed in the current study, possibly due to the different animal model used and the earlier healing stage analyzed. In another study, <xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al. (2017)</xref> also measured elastic modulus values ranging from 7 to 14&#xa0;GPa in bone transport calluses of the ovine metatarsus with the same critical size of 15&#xa0;mm. Specifically, they reported an elastic modulus of 11&#xa0;GPa after 161 days in bone transport (<xref ref-type="bibr" rid="B57">Mora-Mac&#xed;as et al., 2017</xref>), closely matching the 9.25 <inline-formula id="inf123">
<mml:math id="m129">
<mml:mrow>
<mml:mo>&#xb1;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 2.93&#xa0;GPa measured in this study during bone lengthening at the same time-point. The possible lower degree of mechanical maturation in the bone lengthening callus could be explained by a lower mechanical stimulation associated to the less mobility and bearing capacity of the specimens after the indirect elongation of the surrounding soft tissues (<xref ref-type="bibr" rid="B10">Bl&#xe1;zquez-Carmona et al., 2021b</xref>). These findings reflect a slower recovery of the mechanical properties of the woven bone at the tissue scale throughout its remodeling phase. It is important to highlight that nanoindentation tests specifically excluded the influence of inter-trabecular macroporosities on the average data provided. Consequently, the observed lower stiffness can only be attributed to the presence of microporosities or other microstructural features at smaller length scales, or to differences in the chemical composition and organization of the deposited mineral phase, as discussed later.</p>
<p>The mineral content in woven bone has been previously examinated in the literature. <xref ref-type="bibr" rid="B51">Manjubala et al. (2009)</xref> measured calcium content directly from back-scattered electron microscopic images (2D mineral content), showing inhomogeneity in the callus mineralization with a maximum Ca content of 20% of Ca that saturates at week 9 after surgery. In cortical bone, <xref ref-type="bibr" rid="B90">Turunen et al. (2011)</xref> investigated the temporal evolution of mineral content during aging with Infrared and Raman spectroscopy. They found that while mineralization increases over time, hydroxyapatite crystals mature more slowly. <xref ref-type="bibr" rid="B2">Ahmed et al. (2018)</xref> reported an increase in the mineral-to-matrix and crystallinity ratios during the first 14 days of healing in calvarial defects through Raman Spectroscopy. In our study, bone mineralization during healing was quantified at the microscopic scale using several Raman parameters, including mineral-to-matrix and carbonate-to-matrix ratios. <xref ref-type="fig" rid="F3">Figure 3</xref> demonstrates the positive correlation between the Tyrosine and Amide III mineral-to-matrix and healing time, reaching values comparable to those of cortical bone at day 161. These measurements provide a quantitative assessment of the extent of bone mineralization, directly related to ash weight (<xref ref-type="bibr" rid="B31">Gourion-Arsiquaud et al., 2009</xref>). Crystallinity, which is a measure of the crystal size development (<xref ref-type="bibr" rid="B90">Turunen et al., 2011</xref>), remained nearly constant during the consolidation phase, with values similar to those of the cortical bone (<xref ref-type="fig" rid="F3">Figure 3C</xref>). Given that osteoid is deposited by osteoblasts at different locations and timing within the callus, this finding suggests that the maximum crystal size is obtained in an early stage of regeneration. Despite the increase in the carbonate-to-matrix ratio (<xref ref-type="fig" rid="F3">Figure 3H</xref>), the carbonate-to-phosphate ratio (<xref ref-type="fig" rid="F3">Figure 3D</xref>) showed no trend during the consolidation phase, maintaining values akin to those of cortical bone. Similar results were reported in calvarian healing (<xref ref-type="bibr" rid="B2">Ahmed et al., 2018</xref>). This ratio represents type-B carbonate substitution into hydroxyapatite, a process not fully understood but associated with increased solubility and decreased mechanical performance of the tissue (<xref ref-type="bibr" rid="B67">Pan and Darvell, 2010</xref>). Literature suggests this substitution is more related to aging, as carbonate peak areas and intensities increase with age (<xref ref-type="bibr" rid="B14">Boskey and Robey, 2013</xref>; <xref ref-type="bibr" rid="B91">Turunen et al., 2013</xref>). Regarding the hydroxyproline-to-proline ratio, the literature proved that it negatively correlates with the maturation of the tissue (<xref ref-type="bibr" rid="B82">Shah et al., 2019</xref>). However, it keeps constant in woven bone and within the standard deviation range of cortical bone (<xref ref-type="fig" rid="F3">Figure 3G</xref>). A similar trend was observed for the Amide I-to-Amide III ratio, which remained approximately constant above 1 (<xref ref-type="fig" rid="F3">Figure 3J</xref>). It follows a very similar trend to crystallinity, as both processes are closely correlated with the bone formation. Initially, na&#xef;ve collagen fibers are created followed by an enzymatically crosslinking (<xref ref-type="bibr" rid="B44">Knott and Bailey, 1998</xref>). These fibers then mineralized and served as scaffolds for the nucleation and growth of the additional mineral crystals (<xref ref-type="bibr" rid="B31">Gourion-Arsiquaud et al., 2009</xref>). A high Amide I-to-Amide III ratio indicates a predominance of functional and mature collagen crosslinks, whereas a predominance of Amide III suggests high activity of nonfunctional precursor collagen with reducible crosslinks (<xref ref-type="bibr" rid="B20">Coelho et al., 2014</xref>; <xref ref-type="bibr" rid="B50">Mandair and Morris, 2015</xref>; <xref ref-type="bibr" rid="B26">Fraulob et al., 2020</xref>). Thus, woven bone exhibits an increase in procollagen molecules and immature crosslinks during the analyzed regeneration phase.</p>
<p>
<xref ref-type="fig" rid="F4">Figure 4</xref> also reported the chemical analysis performed by ash analysis and elemental study. The ash fraction, phosphorous and calcium content increased nonlinearly with healing time with a plateau nearly the cortical reference values. Phosphorous and calcium contents are also in line with the infrared and X-ray analysis on natural hydroxyapatite (<xref ref-type="bibr" rid="B40">Janicki et al., 2012</xref>). Numerous substitutors were reported in hydroxyapatite crystals in the literature, including carbon, potassium, magnesium, sodium, or strontium (<xref ref-type="bibr" rid="B8">Bergstrom and Wallace, 1954</xref>; <xref ref-type="bibr" rid="B16">Buddhachat et al., 2016</xref>; <xref ref-type="bibr" rid="B42">Katzenberg, 2020</xref>). For potassium and sodium, the decrease with time was quantified (<xref ref-type="fig" rid="F4">Figures 4D, F</xref>). However, none of the above substitutes seems to play a key role during the bone regeneration process.</p>
<p>Regarding the structural features of the woven bone at the trabecular scale, as measured by micro-CT, the BV/TV increased up to 0.69. This increase is primarily due to an increased in the trabecuale thickness (Tb.Th) as the different mineralization fronts advance (<xref ref-type="fig" rid="F2">Figure 2</xref>). Previous studies have already examined bone volume fraction using different techniques and length scales. For instance, <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al. (2021a)</xref> and <xref ref-type="bibr" rid="B56">Mora-Mac&#xed;as et al. (2021)</xref> measured the bone volume fraction from CT macroscopic images in bone lengthening and bone transport, respectively. In these studies, the volume fraction ranged from 88.18% to 93.97% in bone lengthening and 87.55%&#x2013;100% in bone transport at days 121&#x2013;161 after surgery. The lower values in our study are more consistent with estimates reported in bone transport calluses (<xref ref-type="bibr" rid="B56">Mora-Mac&#xed;as et al., 2021</xref>), which used direct segmentation of 2D micrographs based on x-ray grey-scale images. Thus, the porosity at the trabecular scale appears to significantly impact the estimation of the bulk volume fraction. This structural feature likely contributes to the differences found in the same period of time between the micro-scale elastic modulus measured in this study (3.07&#x2013;9.25&#xa0;GPa), and the lower apparent elastic modulus measured <italic>in vivo</italic> by <xref ref-type="bibr" rid="B9">Bl&#xe1;zquez-Carmona et al. (2021a)</xref> (0.01&#x2013;8.53&#xa0;GPa), input in a later discussed power law. Compared to the trabecular tissue, woven bone only exhibits BV/TV levels similar to those observed in this study or those reported in femoral bone at the early stages of regeneration (<xref ref-type="bibr" rid="B55">Mittra et al., 2005</xref>; <xref ref-type="bibr" rid="B86">Teo et al., 2007</xref>). The other three-dimensional microarchitectural parameters of woven bone (<xref ref-type="fig" rid="F2">Figure 2</xref>) could not be directly compared to other regenerating tissues due to the lack of reported data in the literature, to our knowledge. Therefore, comparisons are limited to cancellous bone tissue data from this study or literature sources (<xref ref-type="bibr" rid="B55">Mittra et al., 2005</xref>; <xref ref-type="bibr" rid="B86">Teo et al., 2007</xref>; <xref ref-type="bibr" rid="B91">Turunen et al., 2013</xref>). The aforementioned increase in Tb.Th above cancellous values occurs alongside a progressive reduction of the number (Tb.Nm) and interconnectivity (Conn.D) of woven trabeculae, more similar to that of healthy metatarsal trabecular tissue, 3.26 <inline-formula id="inf124">
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<mml:math id="m131">
<mml:mrow>
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<mml:mrow>
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</inline-formula> 3.96, respectively. These reference trabecular values are slightly higher than those of healthy ovine or human femoral trabecular tissue reported in the literature (<xref ref-type="bibr" rid="B55">Mittra et al., 2005</xref>; <xref ref-type="bibr" rid="B86">Teo et al., 2007</xref>; <xref ref-type="bibr" rid="B91">Turunen et al., 2013</xref>), probably due to physiological differences between bones. The stabilization of the woven trabecular separation around 0.13&#x2013;0.17&#xa0;mm (<xref ref-type="fig" rid="F2">Figure 2C</xref>) contrasts with the constant increase in Tb.Th, which can only be explained by the remodeling of already mineralized tissue in the medullary cavity or the outer interzones of the callus farthest from the original cortical bone. The degree of anisotropy (DA) remained similar to that of trabecular tissue in the same bone (<xref ref-type="fig" rid="F2">Figure 2F</xref>) or to values reported in human femoral bone by <xref ref-type="bibr" rid="B91">Turunen et al. (2013)</xref>, Finally, SMI evolved negatively from around 0, typical for trabecular bone (<xref ref-type="bibr" rid="B55">Mittra et al., 2005</xref>; <xref ref-type="bibr" rid="B91">Turunen et al., 2013</xref>), indicating a transition from a rod-and-plate structure typical of trabecular bone to a denser structure characterized by void spaces (<xref ref-type="bibr" rid="B36">Hildebrand and R&#xfc;egsegger, 1997</xref>). These results highlight the dynamic evolution of woven bone microstructure, from numerous disorganized thin trabeculae at the beginning of the consolidation phase to a structure with fewer but thicker and better-interconnected trabeculae.</p>
<p>The previous experimental data facilitated the definition of the power laws presented in <xref ref-type="table" rid="T2">Table 2</xref>, which serve as valuable tools for predicting the elastic modulus of woven bone in future <italic>in silico</italic> models at both macro- and micro-scales. The varied laws allow for the assignment of woven bone tissue mechanical properties based on different experimental results, not necessarily mechanical testing. Except for the current study, woven bone correlations have never been established using mechanical, structural and chemical parameters measured directly from <italic>ex vivo</italic> samples. The exponents of these expressions (<inline-formula id="inf127">
<mml:math id="m133">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
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</inline-formula> and <inline-formula id="inf128">
<mml:math id="m134">
<mml:mrow>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> fitting constants in <xref ref-type="table" rid="T2">Table 2</xref>) provide insights into the degree of influence of chemical composition (measured at the macro- scale by ash and elemental analyses and at the micro-scale by Raman spectroscopy) and microstructural morphology on the mechanical properties of the woven bone at both macro- and micro-scale. For instance, the ash fraction appears to have a lower relative influence on the elastic modulus measured by nanoindentation <inline-formula id="inf129">
<mml:math id="m135">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>b</mml:mi>
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</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
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</inline-formula> over a broad range of ash fraction values compared to the calcium or phosphorous contents <inline-formula id="inf130">
<mml:math id="m136">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>b</mml:mi>
<mml:mo>&#x3d;</mml:mo>
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</mml:mrow>
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</inline-formula>. This exponent is also slightly lower than those in equivalent power laws reported in the literature (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B21">Currey, 1988</xref>; <xref ref-type="bibr" rid="B35">Hernandez et al., 2001</xref>). The differences can be attributed to the use of compact and cancellous bone tissues from different mammalian species, the range of ash fractions analyzed, and the different loading conditions in the mechanical characterization. The degree of significance of all correlation (R-square in range of 0.55&#x2013;0.90 and most of <italic>p</italic>-values below 0.05) is consistent with previous works (<xref ref-type="bibr" rid="B80">Schaffler and Burr, 1988</xref>; <xref ref-type="bibr" rid="B21">Currey, 1988</xref>; <xref ref-type="bibr" rid="B35">Hernandez et al., 2001</xref>). However, Raman parameters, Amide III mineral-to-matrix and Tyrosine mineral-to-matrix, were found to correlate more significantly with elastic modulus than ash, calcium or phosphorus fractions (<xref ref-type="fig" rid="F3">Figures 3A&#x2013;C, E, J</xref>; <xref ref-type="table" rid="T2">Table 2</xref>). We hypothesize it could also be due to significant chemical heterogeneity in the tissue. Note that the Raman spectra were taken in the same areas of woven bone as those measured by nanoindentation. We also define a two-parameter function to explain more of the variance in the bulk mechanical properties of the woven bone than single-parameter functions based solely on chemical factors. Thus, the combination of ash fraction with the evolution of the BV/TV provides a significant correlation with the apparent elastic modulus (<xref ref-type="fig" rid="F3">Figure 3D</xref>; <xref ref-type="table" rid="T2">Table 2</xref>), representing mechanical behavior at a macro-scale. In this case, the substantial difference between the ash fraction <inline-formula id="inf131">
<mml:math id="m137">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>b</mml:mi>
<mml:mo>&#x3d;</mml:mo>
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</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
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</inline-formula> and the BV/TV exponent values <inline-formula id="inf132">
<mml:math id="m138">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>c</mml:mi>
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<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
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</inline-formula> indicates a greater influence of the bone volume evolution over the total callus volume, encompassing both mineralized and non-mineralized phases, on the bone callus&#x2019; apparent mechanical properties during healing, compared to the influence of the ash fraction. This finding contrasts with previous reports on cortical and trabecular tissue in the literature, where ash fraction seems to have a significant greater influence on bone strength and modulus (<xref ref-type="bibr" rid="B35">Hernandez et al., 2001</xref>), possibly due to the less structural remodeling and time depending nature of healthy bone tissues.</p>
<p>There are several limitations associated with the methodology employed in this study. Firstly, it is expected that the mechanical, structural and chemical parameters analyzed <italic>ex vivo</italic> are not significantly influenced by differences in bone maturity of sheep during the regeneration period <italic>in vivo</italic> since the longest experiment (161 days) represents only around 4% of the animal life cycle. Secondly, the methods used for the indentation test (<xref ref-type="bibr" rid="B66">Oliver and Pharr, 1992</xref>) assume linear elastic behavior in the tissue, despite the possibility that the woven bone could exhibit viscoelastic behavior in some callus interzones, particularly during early stage of mineralization. However, the mechanical characterization performed in this study is focused on the mineralized phase, where the influence of the viscoelastic behavior should not be significant compared to a soft tissue (<xref ref-type="bibr" rid="B10">Bl&#xe1;zquez-Carmona et al., 2021b</xref>). Using the Oliver-Pharr method in the evaluation of the nanoindentations also presents several limitations. The elastic modulus of surrounding tissue sub-domains, along with the presence of microcracks or defects, may influence the homogenized and averaged elastic modulus measured by the indenter (<xref ref-type="bibr" rid="B41">Kariem et al., 2015</xref>; <xref ref-type="bibr" rid="B68">Pastrama et al., 2018</xref>; <xref ref-type="bibr" rid="B45">K&#xf6;nigsberger et al., 2022</xref>). Additionally, the use of different specimens for each time-point introduces an inevitable source of variation in the measurements. Despite this, the results appear to follow logical trends, and the power laws derived are significant. Finally, a higher number of animals would have enhanced the statistical significance of the conclusions of this study. As previously mentioned, the authors prioritized the use of an ovine animal model, whose results are crucial for construction of more realistic micromechanical models and whose conclusions can be readily extrapolated to human clinical cases.</p>
<p>In conclusion, this work contributes to the development of novel micromechanical models applicable to numerical simulations of bone regeneration processes. It also enhances the understanding of the relationship between chemical composition, structural morphology, and the mechanical properties of woven bone at different length scales by combining multiple <italic>ex vivo</italic> techniques: micro-CT, nanoindentation, Raman spectroscopy, and chemical and elemental analysis. Addressing the questions posed at the beginning of the discussion, woven bone exhibited a significant increase in bone volume, primarily controlled by the thickening and unification of trabeculae, alongside a simultaneous increase in apparent ash fraction, calcium, and phosphate content. The power law established for the bulk elastic modulus highlights a more significant influence of the evolving trabecular microarchitecture on the apparent mechanical behavior of the bone callus. At the tissue scale, the Raman mineral-to-matrix ratio increased over the analyzed regeneration period, reaching values and crystallinity levels comparable to those of cortical bone. However, the local elastic modulus did not increase sufficiently to reach cortical values, potentially due to the role of microporosities within the trabeculae. Power laws predicting the evolution of the woven bone elastic modulus at this scale demonstrated that Raman parameters correlate better with the elastic modulus than apparent mineral content measured through ash analysis and elemental studies. Therefore, we advocate for the combined use of Raman spectroscopy and nanoindentation in constructing future power laws for regenerating tissue, potentially strengthened by including a second parameter to account for structural features at the microscale.</p>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s5">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/<xref ref-type="sec" rid="s11">Supplementary Material</xref>, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec id="s6">
<title>Ethics statement</title>
<p>The animal study was approved by Animal Ethics of the University of C&#xf3;rdoba (Reference 2021PI/21). The study was conducted in accordance with the local legislation and institutional requirements.</p>
</sec>
<sec id="s7">
<title>Author contributions</title>
<p>PB-C: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Visualization, Writing&#x2013;original draft, Writing&#x2013;review and editing. JM-M: Conceptualization, Funding acquisition, Methodology, Project administration, Supervision, Writing&#x2013;review and editing. AP: Investigation, Methodology, Writing&#x2013;review and editing. &#xc1;M: Data curation, Writing&#x2013;review and editing. ER-R: Conceptualization, Formal Analysis, Funding acquisition, Project administration, Supervision, Writing&#x2013;original draft, Writing&#x2013;review and editing.</p>
</sec>
<sec sec-type="funding-information" id="s8">
<title>Funding</title>
<p>The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. Grant PID 2020-113790RB-I00 funded by MICIU/AEI/10.13039/501100011033.</p>
</sec>
<ack>
<p>Micro-CT imaging and chemical analysis of the bone samples were performed in the University of Seville Research, Technology and Innovation Centre (CITIUS).</p>
</ack>
<sec sec-type="COI-statement" id="s9">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s10">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<sec id="s11">
<title>Supplementary material</title>
<p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="https://www.frontiersin.org/articles/10.3389/fbioe.2024.1476473/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/fbioe.2024.1476473/full&#x23;supplementary-material</ext-link>
</p>
<supplementary-material xlink:href="DataSheet1.PDF" id="SM1" mimetype="application/PDF" xmlns:xlink="http://www.w3.org/1999/xlink"/>
</sec>
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