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<front>
<journal-meta>
<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>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">846665</article-id>
<article-id pub-id-type="doi">10.3389/fbioe.2022.846665</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>The Degradation of Synthetic Polymeric Scaffolds With Strut-like Architecture Influences the Mechanics-dependent Repair Process of an Osteochondral Defect <italic>in Silico</italic>
</article-title>
<alt-title alt-title-type="left-running-head">Tortorici et&#x20;al.</alt-title>
<alt-title alt-title-type="right-running-head">Scaffold Degradation Influences Osteochondral Repair</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Tortorici</surname>
<given-names>Martina</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/1129317/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Petersen</surname>
<given-names>Ansgar</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/432540/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Duda</surname>
<given-names>Georg N.</given-names>
</name>
<uri xlink:href="https://loop.frontiersin.org/people/436461/overview"/>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Checa</surname>
<given-names>Sara</given-names>
</name>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
<uri xlink:href="https://loop.frontiersin.org/people/400817/overview"/>
</contrib>
</contrib-group>
<aff>
<institution>Julius Wolff Institute</institution>, <institution>Berlin Institute of Health at Charit&#xe9;&#x2013;Universit&#xe4;tsmedizin Berlin</institution>, <addr-line>Berlin</addr-line>, <country>Germany</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/452642/overview">Jin Nam</ext-link>, University of California, Riverside, United&#x20;States</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/109400/overview">Magali Cucchiarini</ext-link>, Center of Experimental Orthopaedics and Osteoarthritis Research, Saarland University Medical Center, Germany</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/883418/overview">Syam P Nukavarapu</ext-link>, University of Connecticut, United&#x20;States</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Sara Checa, <email>sara.checa@bih-charite.de</email>
</corresp>
<fn fn-type="other">
<p>This article was submitted to Biomechanics, a section of the journal Frontiers in Bioengineering and Biotechnology</p>
</fn>
</author-notes>
<pub-date pub-type="epub">
<day>10</day>
<month>03</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="collection">
<year>2022</year>
</pub-date>
<volume>10</volume>
<elocation-id>846665</elocation-id>
<history>
<date date-type="received">
<day>31</day>
<month>12</month>
<year>2021</year>
</date>
<date date-type="accepted">
<day>21</day>
<month>02</month>
<year>2022</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2022 Tortorici, Petersen, Duda and Checa.</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Tortorici, Petersen, Duda and Checa</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&#x20;terms.</p>
</license>
</permissions>
<abstract>
<p>Current clinical treatments of osteochondral defects in articulating joints are frequently not successful in restoring articular surfaces. Novel scaffold-based tissue engineering strategies may help to improve current treatment options and foster a true regeneration of articulating structures. A frequently desired property of scaffolds is their ability to degrade over time and allow a full restoration of tissue and function. However, it remains largely unknown how scaffold degradation influences the mechanical stability of the tissue in a defect region and, in turn, the regenerative process. Such differing goals&#x2013;supporting regeneration by degrading its own structure&#x2013;can hardly be analyzed for tissue engineered constructs in clinical trials and <italic>in vivo</italic> preclinical experiments. Using an <italic>in silico</italic> analysis, we investigated the degradation-induced modifications in material and architectural properties of a scaffold with strut-like architecture over the healing course and their influence on the mechanics-dependent tissue formation in osteochondral defects. The repair outcome greatly varied depending on the degradation modality, i.e. surface erosion or bulk degradation with and without autocatalysis, and of the degradation speed, i.e. faster, equal or slower than the expected repair time. Bulk degradation with autocatalysis, independently of degradation speed, caused the mechanical failure of the scaffold prior to osteochondral defect repair and was thereby deemed inappropriate for further application. On the other hand, scaffolds with strut-like architecture degrading by both surface erosion and bulk degradation with slow degradation speed resulted in comparably good repair outcomes, thereby indicating such degradation modalities as favorable for the application in osteochondral defects.</p>
</abstract>
<kwd-group>
<kwd>osteochondral defect</kwd>
<kwd>tissue engineering</kwd>
<kwd>scaffold degradation</kwd>
<kwd>computer model</kwd>
<kwd>mechanobiolgy</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="s1">
<title>Introduction</title>
<p>Osteochondral defects affect the articular cartilage and the subchondral bone and are frequently a result of traumatic events or degenerative processes (<xref ref-type="bibr" rid="B12">Davis et&#x20;al., 2021</xref>). As cartilage presents limited natural regenerative potential, the pre-injury tissue configuration is rarely restored by spontaneous repair processes (<xref ref-type="bibr" rid="B21">Hunziker et&#x20;al., 2015</xref>). Osteochondral defects are associated with relevant pain and limit the joint function and mobility of patients. Additionally, they might initiate degenerative processes in surrounding tissues, eventually leading to further degeneration of the complete joint (<xref ref-type="bibr" rid="B21">Hunziker et&#x20;al., 2015</xref>). To stop the degenerative progression of osteochondral defects, a timely and effective treatment is of great importance. However, current clinical treatments are unable to restore healthy articulation or are associated with severe limitations (<xref ref-type="bibr" rid="B12">Davis et&#x20;al., 2021</xref>), such as the need for multiple interventions or a triggering of degeneration in previously unaffected cartilage areas (<xref ref-type="bibr" rid="B33">Nukavarapu &#x26; Dorcemus, 2013</xref>; <xref ref-type="bibr" rid="B21">Hunziker et&#x20;al., 2015</xref>). The development of scaffold-based tissue engineering (TE) strategies has the potential of supporting osteochondral defect healing, thereby complementing current clinical treatment options with a truly regenerative strategy.</p>
<p>The regeneration of osteochondral defects might be achieved by TE scaffolds imparting specific mechanical cues to guide the differentiation of mesenchymal stromal cells (MSCs) (<xref ref-type="bibr" rid="B12">Davis et&#x20;al., 2021</xref>). In fact, numerous <italic>in&#x20;vitro</italic> evaluations have shown that MSCs can sense and respond to mechanical stimulation (<xref ref-type="bibr" rid="B13">Delaine-Smith &#x26; Reilly, 2012</xref>; <xref ref-type="bibr" rid="B41">Schreivogel et&#x20;al., 2019</xref>). Moreover, computational models employing mechano-biological rules to determine tissue formation have highlighted the importance of mechanics on the healing process of osteochondral defects. For example, the repair of osteochondral defects in minipigs was reproduced by describing tissue formation under specific ranges of minimum principal strain (<xref ref-type="bibr" rid="B14">Duda et&#x20;al., 2005</xref>). Moreover, typical features of the <italic>in vivo</italic> repair pattern were obtained simulating tissue formation based on a mechanical stimulus, which was calculated from octahedral shear strain and fluid velocity (<xref ref-type="bibr" rid="B24">Kelly &#x26; Prendergast, 2005</xref>; <xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). To support the mechanics-dependent healing of osteochondral defects, scaffolds with different mechanical properties have been compared <italic>in vivo</italic>. For example, poly (lactide-co-glycolide) (PLGA) scaffolds with high and low stiffness were implanted in osteochondral defects in sheep, resulting in better subchondral bone formation with the stiffer scaffolds, although without improvements of cartilage formation (<xref ref-type="bibr" rid="B40">Schlichting et&#x20;al., 2008</xref>). In another case, three PLGA scaffolds with different elastic moduli were tested in osteochondral defects in rabbit, observing that the two scaffolds with lower elastic moduli better supported the repair process compared to the stiffer one (<xref ref-type="bibr" rid="B22">Ikeda et&#x20;al., 2009</xref>). In both cited examples, however, the different mechanical properties of the scaffolds were achieved by different scaffold porosities. Therefore, a clear distinction of mechanical stimuli from morphological cues (e.g., pore size) was not possible. To separately evaluate the influence of scaffold stiffness and morphology on osteochondral defect repair, an <italic>in silico</italic> model was recently developed (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). By investigating <italic>in silico</italic> the mechanical behavior of simple scaffold designs upon implantation into an osteochondral defect, it was possible to suggest scaffold properties that are supportive for the treatment of osteochondral defects. In fact, the computational results indicated that a scaffold with material elastic modulus in the low GPa range and an architecture reducing both compressive and radial displacements has the potential to improve osteochondral defect repair compared to an untreated defect (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). However, only non-degradable scaffolds were investigated in this&#x20;model.</p>
<p>The ability to degrade in the environment of the body (see <xref ref-type="table" rid="T1">Table&#x20;1</xref> for definitions) is often listed amongst the properties of an ideal scaffold or material for TE (<xref ref-type="bibr" rid="B6">Bose et&#x20;al., 2012</xref>; <xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>; <xref ref-type="bibr" rid="B48">Turnbull et&#x20;al., 2018</xref>). In fact, a biodegradable material has the advantage of not requiring a second surgical intervention for its removal (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>), while avoiding concerns on its long term influence at the site of implantation, e.g., the elicitation of a foreign body response (<xref ref-type="bibr" rid="B17">Griffith, 2002</xref>) or the stress shielding of the adjacent tissues (<xref ref-type="bibr" rid="B31">Muschler et&#x20;al., 2004</xref>). Material degradation can be even employed for the timely delivery of specific chemical cues or drugs (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). However, the process of degradation may result in modifications of the chemical, morphological, biological, and mechanical properties of a material (<xref ref-type="bibr" rid="B32">Nair &#x26; Laurencin, 2007</xref>), and thereby needs to be carefully tuned for the individual application. Synthetic biocompatible and biodegradable polymers (see <xref ref-type="table" rid="T1">Table&#x20;1</xref> for definitions) typically degrade by hydrolysis (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>), i.e.,&#x20;by a shortening of the polymeric chains caused by a reaction with water molecules (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). Depending on degradation kinetic and small molecule diffusion, two types of hydrolytic degradation are possible: surface erosion or bulk degradation (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). In surface erosion, the hydrolytic reaction is faster than water diffusion into the material, resulting in a thinning of the scaffold features without a reduction in molecular weight (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>; <xref ref-type="bibr" rid="B53">Woodruff &#x26; Hutmacher, 2010</xref>). Surface erosion is typically observed in polyanhydrides and polyorthoesters (<xref ref-type="bibr" rid="B32">Nair &#x26; Laurencin, 2007</xref>). In bulk degradation, the hydrolytic reaction is slower than water diffusion into the material, resulting in a uniform reduction of molecular weight in the scaffold without weight loss nor volume modifications (<xref ref-type="bibr" rid="B53">Woodruff &#x26; Hutmacher, 2010</xref>; <xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). Polyesters are known to degrade by bulk degradation with acidic degradation by-products (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). If the production of the acidic degradation by-products is faster than their diffusion away from the material, these acidic by-products speed up the reduction in polymeric molecular weight in the bulk of the device (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>), a phenomenon known as autocatalysis. The establishment of autocatalysis depends not only on polymer chemistry, but also on architectural features of the device, e.g., its thickness (<xref ref-type="bibr" rid="B18">Grizzi et&#x20;al., 1995</xref>). Bulk degradation with autocatalysis has been observed <italic>in vivo</italic> in devices made from poly (lactide-co-glycolide) (PLGA) and poly (lactic acid) (PLA) (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>), but not from poly (&#x3b5;-caprolactone) (PCL) (<xref ref-type="bibr" rid="B53">Woodruff &#x26; Hutmacher, 2010</xref>). The experimentally observed degradation behaviors of some of the polymers that are commonly investigated for TE applications are summarized in <xref ref-type="table" rid="T2">Table&#x20;2</xref>.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Definitions of degradation-related&#x20;terms.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Property</th>
<th align="center">Definition</th>
<th align="center">References</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Degradation</td>
<td align="left">&#x201c;Chain scission process during which polymer chains are cleaved to form oligomers and finally to form monomers&#x201d;</td>
<td align="left">
<xref ref-type="bibr" rid="B15">G&#xf6;pferich, (1996)</xref>
</td>
</tr>
<tr>
<td align="left">Erosion</td>
<td align="left">&#x201c;Loss of material owing to monomers and oligomers leaving the polymer&#x201d;</td>
<td align="left">
<xref ref-type="bibr" rid="B15">G&#xf6;pferich, (1996)</xref>
</td>
</tr>
<tr>
<td align="left">Biodegradable</td>
<td align="left">&#x201c;Solid polymeric devices which break down to macromolecule degradation with dispersion in an animal body but no proof for elimination from the body&#x201d;</td>
<td align="left">
<xref ref-type="bibr" rid="B50">Vert et&#x20;al. (1992)</xref>
</td>
</tr>
<tr>
<td align="left">Bioresorbable</td>
<td align="left">&#x201c;Solid materials which can degrade and further resorb <italic>in vivo</italic>, i.e. which are eliminated through natural pathways either because of simple filtration of degradation by-products or after their metabolization&#x201d;</td>
<td align="left">
<xref ref-type="bibr" rid="B50">Vert et&#x20;al. (1992)</xref>
</td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Degradation behaviors of some of the polymers that are commonly investigated in tissue engineering.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Polymer</th>
<th align="center">Abbreviation</th>
<th align="center">Degradation modality</th>
<th align="center">Degradation time</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Poly (carboxyphenoxypropane-sebacic acid)</td>
<td align="left">P(CPP-SA)</td>
<td align="left">Surface degradation (<xref ref-type="bibr" rid="B44">Tamada &#x26; Langer, 1993</xref>)</td>
<td align="left">6&#x2013;8&#xa0;weeks (<xref ref-type="bibr" rid="B23">Katti et&#x20;al., 2002</xref>)</td>
</tr>
<tr>
<td align="left">Poly (&#x3b5;-caprolactone)</td>
<td align="left">PCL</td>
<td align="left">Bulk degradation without autocatalysis (<xref ref-type="bibr" rid="B25">Lam et&#x20;al., 2008</xref>)</td>
<td align="left">2&#x2013;4&#xa0;years, depending on molecular weight (<xref ref-type="bibr" rid="B53">Woodruff &#x26; Hutmacher, 2010</xref>)</td>
</tr>
<tr>
<td align="left">Poly (D,L-lactic acid)</td>
<td align="left">PDLLA</td>
<td align="left">Bulk degradation with autocatalysis (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>)</td>
<td align="left">12&#x2013;16&#xa0;months (<xref ref-type="bibr" rid="B32">Nair &#x26; Laurencin, 2007</xref>)</td>
</tr>
<tr>
<td align="left">Poly (lactide-co-glycolide)</td>
<td align="left">PLGA</td>
<td align="left">Bulk degradation (<xref ref-type="bibr" rid="B44">Tamada &#x26; Langer, 1993</xref>) with autocatalysis (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>)</td>
<td align="left">1&#x2013;6&#xa0;months, depending on co-polymer ratio (<xref ref-type="bibr" rid="B32">Nair &#x26; Laurencin, 2007</xref>)</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>As architecture is, next to the material type, the most important factor determining the mechanical properties of a scaffold, e.g., its compressive modulus (<xref ref-type="bibr" rid="B28">Marchiori et&#x20;al., 2019</xref>), modifications in mechanical behavior can be expected as a consequence of a surface erosion process. Moreover, the molecular weight of polymers is closely related to their mechanical properties (<xref ref-type="bibr" rid="B34">Nunes et&#x20;al., 1982</xref>). Consequently, a reduction in molecular weight resulting from a bulk degradation process (with or without autocatalysis) may modify the mechanical properties of a scaffold, as previously observed experimentally, e.g., in PLGA (<xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>) and PCL (<xref ref-type="bibr" rid="B25">Lam et&#x20;al., 2008</xref>) scaffolds. Given the link between scaffold-dependent mechanical cues and tissue formation discussed above, it is important to evaluate how modifications in scaffold mechanical properties due to degradation would influence the healing process. However, an experimental investigation of this phenomenon presents significant technical challenges, such as the need for long term <italic>in&#x20;vitro</italic> and <italic>in vivo</italic> experiments (<xref ref-type="bibr" rid="B35">Pan, 2015</xref>). Computational models can offer first evaluations in a simplified environment, providing indications for future, more complex experimental tests. A number of computational studies have already explored tissue growth in presence of a degradable scaffold in the context of bone formation (<xref ref-type="bibr" rid="B1">Adachi et&#x20;al., 2006</xref>; <xref ref-type="bibr" rid="B11">Chen et&#x20;al., 2011</xref>; <xref ref-type="bibr" rid="B16">Gorriz et&#x20;al., 2015</xref>). To the best of our knowledge, such an evaluation is so far lacking for osteochondral defects.</p>
<p>Here, we employed a previously developed computational model of scaffold-supported osteochondral defect repair (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>) to study the consequences of scaffold degradation on the mechanics-dependent repair process. To do so, an &#x201c;artificial&#x201d; scaffold with strut-like architecture was modelled, whose material and architectural properties prior to degradation were those of the simplest design fostering an improved osteochondral defect repair compared to the untreated case, as previously established (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). The scaffold was defined &#x201c;artificial&#x201d; because its axisymmetric architecture and its material properties do not model a specific scaffold nor material currently investigated for tissue engineering. Three types of polymeric scaffold degradation, i.e. surface erosion, bulk degradation, and bulk degradation with autocatalysis, as well as different degradation rates were studied. Although ceramic (<xref ref-type="bibr" rid="B29">Marques et&#x20;al., 2021</xref>) and metallic (<xref ref-type="bibr" rid="B27">Lv et&#x20;al., 2021</xref>) degradable scaffolds have also been developed, their degradation modalities are different from the ones of polymeric materials and were not investigated&#x20;here.</p>
</sec>
<sec sec-type="materials|methods" id="s2">
<title>Materials and Methods</title>
<p>An iterative computational model simulated the mechanics-dependent tissue formation within an osteochondral defect implanted with a scaffold in a knee femoral condyle. The model represented a focal osteochondral defect resulting from surgical intervention to treat damaged chondral or bone tissue, e.g., in consequence of trauma, early osteoarthritis or disease-related bone lesions. Therefore, the tissues outside the defect were simulated as healthy. A detailed description of the model has been provided elsewhere (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>) and is here briefly summarized. Amongst the scaffolds evaluated in the cited model, the scaffold with the simplest architecture that resulted in improved repair compared to the untreated osteochondral defect was selected for the present investigation. As an additional feature compared to the previously published model, scaffold degradation was implemented here, as described in the following sections.</p>
<sec id="s2-1">
<title>Model of Osteochondral Defect Repair With Scaffold</title>
<p>A simplified axisymmetric geometry of a knee femoral condyle laying on a meniscus and a tibial plateau was built in a finite element (FE) solver (Abaqus, Dassault Syst&#xe8;me) (<xref ref-type="fig" rid="F1">Figure&#x20;1A</xref>). The femoral condyle was composed of healthy cartilage, subchondral bone, cancellous bone, and an osteochondral defect with a radius and a depth of 5&#xa0;mm. Moreover, a scaffold composed of three vertical struts (thickness of 0.5&#xa0;mm) was modelled in the defect. The scaffold material had the following properties: porosity of 50%; elastic modulus of 1,000&#xa0;MPa; permeability of 3.63 &#xd7; 10<sup>&#x2013;8</sup>&#xa0;mm/s; void ratio of 4; bulk modulus of grain of 0&#xa0;MPa. Moreover, cells could diffuse through the scaffold material with a diffusion coefficient of 0.01&#xa0;mm<sup>2</sup>/day. At the beginning of the simulation, the areas of the osteochondral defect that were not occupied by the scaffold were filled with granulation tissue. All biological tissues were modelled as isotropic and poroelastic, except healthy cartilage, which was isotropic and hyperelastic. However, newly formed cartilage within the osteochondral defect was isotropic and poroelastic, as later described. In addition, the meniscus was modelled as transversally isotropic and poroelastic. The values assigned to the material properties are listed in <xref ref-type="table" rid="T3">Table&#x20;3</xref>. The tibial plateau was modelled as a rigid&#x20;wire.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Model of osteochondral defect repair. <bold>(A)</bold> Geometry (left) and mesh (right) of the axisymmetric model of knee femoral condyle. Black arrows and the black triangle show the applied pressure and the encastre boundary condition in the finite element (FE) analysis, respectively; <bold>(B)</bold> workflow of the model. <italic>S</italic>&#x20;&#x3d; mechanics-dependent stimulus of tissue formation.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g001.tif"/>
</fig>
<table-wrap id="T3" position="float">
<label>TABLE 3</label>
<caption>
<p>Properties of biological tissues (<xref ref-type="bibr" rid="B24">Kelly &#x26; Prendergast, 2005</xref>; <xref ref-type="bibr" rid="B11">Chen et&#x20;al., 2011</xref>; <xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). All poroelastic tissues had a specific weight of the wetting liquid of 9.74 &#xd7; 10<sup>&#x2013;6</sup>&#xa0;N/mm<sup>3</sup> and a bulk modulus of fluid of 2,300&#xa0;MPa. The axial, radial, and circumferential directions are indicated by 1, 2, and 3, respectively. <italic>E</italic>: elastic modulus; <italic>&#x3bd;</italic>: Poisson&#x2019;s ratio; <italic>G</italic>: shear modulus.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th colspan="8" align="left">Isotropic and poroelastic tissues</th>
</tr>
<tr>
<th align="left">Tissue</th>
<th align="center">Diffusion coefficient (mm<sup>2</sup>/day)</th>
<th align="center">Elastic modulus (MPa)</th>
<th align="center">Poisson&#x2019;s ratio</th>
<th align="center">Permeability (mm/s)</th>
<th align="center">Void ratio</th>
<th colspan="2" align="center">Bulk modulus of grain (MPa)</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Subchondral bone</td>
<td align="center">0.01</td>
<td align="center">17,000</td>
<td align="center">0.3</td>
<td align="center">9.74 &#xd7; 10<sup>&#x2013;11</sup>
</td>
<td align="center">0.042</td>
<td colspan="2" align="center">13,920</td>
</tr>
<tr>
<td align="left">Cancellous bone</td>
<td align="center">0.01</td>
<td align="center">6,000</td>
<td align="center">0.3</td>
<td align="center">3.63 &#xd7; 10<sup>&#x2013;8</sup>
</td>
<td align="center">4</td>
<td colspan="2" align="center">13,920</td>
</tr>
<tr>
<td align="left">Poroelastic cartilage</td>
<td align="center">0.05</td>
<td align="center">10</td>
<td align="center">0.167</td>
<td align="center">4.87 &#xd7; 10<sup>&#x2013;8</sup>
</td>
<td align="center">4</td>
<td colspan="2" align="center">3,700</td>
</tr>
<tr>
<td align="left">Fibrous tissue</td>
<td align="center">0.10</td>
<td align="center">2</td>
<td align="center">0.167</td>
<td align="center">9.74 &#xd7; 10<sup>&#x2013;8</sup>
</td>
<td align="center">4</td>
<td colspan="2" align="center">2,300</td>
</tr>
<tr>
<td align="left">Granulation tissue</td>
<td align="center">0.80</td>
<td align="center">0.2</td>
<td align="center">0.167</td>
<td align="center">9.74 &#xd7; 10<sup>&#x2013;8</sup>
</td>
<td align="center">4</td>
<td colspan="2" align="center">2,300</td>
</tr>
<tr>
<td colspan="8" align="left">
<bold>Transversally Isotropic and poroelastic tissue</bold>
</td>
</tr>
<tr>
<td align="left">&#x2003;<bold>Tissue</bold>
</td>
<td colspan="2" align="center">
<bold>Elastic modulus (MPa)</bold>
</td>
<td align="center">
<bold>Poisson&#x2019;s ratio</bold>
</td>
<td align="center">
<bold>Shear modulus (MPa)</bold>
</td>
<td align="center">
<bold>Permeability (mm/s)</bold>
</td>
<td align="center">
<bold>Void ratio</bold>
</td>
<td align="center">
<bold>Bulk modulus of grain (MPa)</bold>
</td>
</tr>
<tr>
<td rowspan="3" align="left">&#x2003;Meniscus</td>
<td colspan="2" align="center">&#x2022; <italic>E<sub>1</sub>
</italic> &#x3d; 0.5</td>
<td align="center">&#x2022; <italic>&#x3bd;<sub>12</sub>
</italic> &#x3d; 0.5</td>
<td align="center">&#x2022; <italic>G<sub>12</sub>
</italic> &#x3d; 0.167</td>
<td rowspan="3" align="center">4.87 &#xd7; 10<sup>&#x2013;8</sup>
</td>
<td rowspan="3" align="center">4</td>
<td rowspan="3" align="center">3,700</td>
</tr>
<tr>
<td colspan="2" align="center">&#x2022; <italic>E<sub>2</sub>
</italic> &#x3d; 0.5</td>
<td align="center">&#x2022; <italic>&#x3bd;<sub>23</sub>
</italic> &#x3d; <italic>&#x3bd;<sub>31</sub>
</italic> &#x3d; 0.0015</td>
<td align="center">&#x2022; <italic>G<sub>23</sub>
</italic> &#x3d; <italic>G<sub>31</sub>
</italic> &#x3d; 0.05</td>
</tr>
<tr>
<td colspan="2" align="center">&#x2022; <italic>E<sub>3</sub>
</italic> &#x3d; 100</td>
<td align="left"/>
<td align="left"/>
</tr>
<tr>
<td colspan="8" align="left">
<bold>Isotropic and hyperelastic tissue</bold>
</td>
</tr>
<tr>
<td align="left">&#x2003;<bold>Tissue</bold>
</td>
<td colspan="2" align="center">
<bold>Strain energy potential</bold>
</td>
<td align="center">
<bold>C10 (MPa)</bold>
</td>
<td align="center">
<bold>D1 (MPa)</bold>
</td>
<td colspan="3" align="center">
<bold>Permeability (mm/s)</bold>
</td>
</tr>
<tr>
<td align="left">&#x2003;Hyperelastic cartilage</td>
<td colspan="2" align="center">Neo-Hookean</td>
<td align="center">2.14</td>
<td align="center">0.399</td>
<td colspan="3" align="center">4.87 &#xd7; 10&#x2013;8</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>The femoral condyle-tibial plateau, femoral condyle-meniscus, and meniscus-tibial plateau contacts were modelled as &#x201c;hard contact&#x201d; in the normal direction and as frictionless in the tangential one. The model was meshed with elements type CAX8RP. Specifically, the osteochondral defect was meshed with 1,600 elements having a seed size of 0.125&#xa0;mm, while a coarser mesh with seed size up to 0.8&#xa0;mm was used in the rest of the model (<xref ref-type="fig" rid="F1">Figure&#x20;1A</xref> right).</p>
<p>The model underwent a 1&#xa0;s compression step, followed by a 0.5&#x20;s consolidation step. The load was applied on the upper surface of the cancellous bone in the form of a 0.637&#xa0;MPa pressure (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). All displacement and rotation degrees of freedom were restrained in the tibial plateau at the axis of symmetry (<xref ref-type="fig" rid="F1">Figure&#x20;1A</xref>). The following initial conditions were implemented in the whole model: pore pressure of 0&#xa0;MPa; and saturation of 1&#xa0;mm<sup>3</sup>/mm<sup>3</sup>. During the consolidation step, a pore pressure of 0&#xa0;MPa was set at the free cartilage&#x20;edges.</p>
<p>The octahedral shear strain (&#x3b3;) and the fluid velocity (v) in the defect were used to compute a mechanical stimulus (<italic>S</italic>) as indicated by <xref ref-type="disp-formula" rid="e1">Equation 1</xref>:<disp-formula id="e1">
<mml:math id="m1">
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>a</mml:mi>
</mml:mfrac>
<mml:mo>&#x2b;</mml:mo>
<mml:mfrac>
<mml:mi>v</mml:mi>
<mml:mi>b</mml:mi>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(1)</label>
</disp-formula>Where <italic>a</italic>&#x20;&#x3d; 3.75% and <italic>b</italic>&#x20;&#x3d; 3&#x20;&#xd7; 10<sup>&#x2013;3</sup>&#xa0;mm/s were empirical constants (<xref ref-type="bibr" rid="B24">Kelly &#x26; Prendergast, 2005</xref>). Tissue formation was described by thresholds of <italic>S</italic>: bone resorption if 0&#x20;&#x2264; <italic>S</italic>&#x20;&#x3c; 0.01; bone formation if 0.01 &#x2264; <italic>S</italic>&#x20;&#x3c; 1; cartilage formation if 1&#x20;&#x2264; <italic>S</italic>&#x20;&#x3c; 3; fibrous tissue formation if <italic>S</italic>&#x20;&#x2265;&#x20;3.</p>
<p>The mechanics-dependent cellular activities, specifically mitosis, apoptosis, and MSCs differentiation, were simulated based on <italic>S</italic> using a Matlab (MathWorks) script. The defect was represented by a 40 x 40 elements matrix, with elements corresponding to the ones in the FE mesh. Each element could be occupied by a maximum of 100 cells. Cells could also populate elements belonging to the struts of the scaffold due to the porosity of the scaffold material. In addition to MSCs, three cellular phenotypes were simulated: osteoblasts, chondrocytes, and fibroblasts, corresponding to bone, cartilage, and fibrous tissue, respectively. If the value of <italic>S</italic> in an element was within the thresholds fostering the formation of a tissue, 5% of the MSCs differentiated into the cell phenotype of that specific tissue. Moreover, the already existing cells of the tissue performed mitosis by increasing of 5%, while all other cell phenotypes underwent apoptosis by decreasing of 15% in number. MSCs had a mechanics-independent mitosis rate of 15%. The mechanics-dependent bone resorption was simulated by a 10% reduction in the number of osteoblasts.</p>
<p>At the initial time, the elements neighboring cancellous bone were completely filled with MSCs, while the elements in the rest of the defect were empty of cells. MSCs migration, simulated as a diffusion process, progressively populated the whole defect with cells. The diffusion process was modelled by solving a mass diffusion problem in a second FE model, where only the defect region was implemented and meshed with 1,600 elements type DC2D4.</p>
<p>It was assumed that each cell phenotype would produce the corresponding tissue proportionally to its number. Therefore, material properties in the defect changed as a consequence of cellular activities and needed to be updated at every iteration of the model (<xref ref-type="fig" rid="F1">Figure&#x20;1B</xref>). For each element, the elastic modulus, the Poisson&#x2019;s ratio, the permeability, the bulk modulus of grain and the diffusion coefficient were calculated as the weighted average of the materials occupying the element itself, as indicated by <xref ref-type="disp-formula" rid="e2">Equation 2</xref>:<disp-formula id="e2">
<mml:math id="m2">
<mml:mrow>
<mml:mi>X</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mi>A</mml:mi>
<mml:mi>X</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mrow>
<mml:mo>[</mml:mo>
<mml:mrow>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mi>A</mml:mi>
<mml:mi>X</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mstyle>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>&#x22c5;</mml:mo>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mstyle displaystyle="true">
<mml:msubsup>
<mml:mo>&#x2211;</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msubsup>
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#x22c5;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mstyle>
</mml:mrow>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
<label>(2)</label>
</disp-formula>Where <italic>X</italic> is one of the parameters listed above, <italic>N</italic>
<sub>
<italic>MAX</italic>
</sub> is the maximum number of cells allowed in each element, <italic>n</italic>
<sub>
<italic>t</italic>
</sub> indicates the species (granulation tissue, bone, cartilage, fibrous tissue, and scaffold), <italic>N</italic> is the volume fraction occupied by the species, and <italic>X</italic>
<sub>
<italic>Gran</italic>
</sub> is the value that the parameter assumes for granulation tissue. Moreover, the properties of the scaffold (specifically, elastic modulus and volume fraction) varied in consequence of its degradation, as described in detail in the following section.</p>
<p>After the update of the material properties in the defect region, a new FE analysis of the femoral condyle begun, marking the beginning of a new iteration (<xref ref-type="fig" rid="F1">Figure&#x20;1B</xref>). One iteration roughly corresponded to 1&#xa0;day of the <italic>in vivo</italic> repair process. For untreated osteochondral defects and osteochondral defects implanted with non-degradable scaffolds, repair was calculated to be achieved at day 50 (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). To enable all models implementing degradable scaffolds to reach the equilibrium state, i.e. the absence of variations in tissue distributions, the simulations in this work were run until day 125, i.e. until 25&#xa0;days after the expected time of complete degradation of the scaffold with the lowest degradation speed (see following section).</p>
</sec>
<sec id="s2-2">
<title>Model of Polymeric Scaffold Degradation</title>
<p>The degradation of the scaffold material was modelled by varying the properties of the scaffold as a function of time, i.e. as a function of the model iteration. Three types of polymeric scaffold degradation by hydrolysis were modelled, i.e. surface erosion, bulk degradation, and bulk degradation with autocatalysis. Experimentally, the degradation of a scaffold by one of these three modalities is determined by both the chemistry of the material and the architecture of the device (<xref ref-type="bibr" rid="B7">von Burkersroda et&#x20;al., 2002</xref>). Thus, scaffolds degrading by different modalities would most likely have different properties in their non-degraded state. To evaluate whether one of these degradation modalities was particularly advantageous or disadvantageous in supporting the repair of osteochondral defects, an &#x201c;artificial&#x201d; scaffold with strut-like architecture was investigated (<xref ref-type="fig" rid="F2">Figures 2A,C</xref>), having the same material properties and architectural features prior to the onset of degradation, but the possibility to degrade by each one of the three modalities in turn. When comparing the three different degradation modalities, a linear degradation behavior was employed. Moreover, the influence of the degradation rate was evaluated by defining three speeds: fast, matched, and slow. When the matched degradation speed was implemented, the complete degradation of the scaffold happened in the same time span of osteochondral defect repair, i.e. 50&#xa0;days (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). The fast and slow speeds were chosen to result in complete scaffold degradation in half (25&#xa0;days) and double (100&#xa0;days) the time, respectively, compared to the matched speed. The degradation rates of the &#x201c;artificial&#x201d; scaffold were calculated to fulfil the imposed degradation&#x20;times.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Model of scaffold degradation in osteochondral defect. <bold>(A)</bold> and <bold>(C)</bold> Elastic modulus and volume fraction, respectively, of the scaffold material prior to degradation; <bold>(B)</bold> and <bold>(D)</bold> Elastic modulus and volume fraction, respectively, of the &#x201c;artificial&#x201d; scaffold with matched degradation speed at day 25 (complete scaffold degradation set to day 50). The left, middle, and right columns show the scaffold degrading by surface erosion, bulk degradation, and bulk degradation with autocatalysis, respectively. At the time point shown in the plots, the surface and bulk elements of the scaffold degrading by surface erosion and bulk degradation with autocatalysis, respectively, were completely degraded. The dash-dot and solid red lines mark the axis of symmetry and the articular interface, respectively. The legends to interpret the plots are on the right side of the corresponding rows. The volume fraction is expressed as percentage of the total volume in one element.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g002.tif"/>
</fig>
<p>Experimentally, the bulk degradation of several polymers, such as PCL, PDLLA, and PLGA, has been shown to follow a non-linear behavior (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B38">Pitt G et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>). Therefore, a simulation of scaffold bulk degradation by hydrolysis that was closer to the <italic>in vivo</italic> or <italic>in&#x20;vitro</italic> cases was subsequently implemented based on the experimental data reported in literature.</p>
<sec id="s2-2-1">
<title>Hydrolytic Degradation by Surface Erosion</title>
<p>Experimentally, a hydrolytic degradation process by surface erosion results in a thinning of the scaffold features without alterations of the bulk properties (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>). Here, surface erosion was modelled by reducing the amount of scaffold material without changes in the scaffold material properties. Each element could be occupied by a mixture of tissues and scaffold material, where the parameter quantifying the amount of each species was the corresponding volume fraction. Thus, surface erosion was simulated by a linear decrease in the volume fraction of the scaffold material (expressed as percentage of the total volume of an individual element), as indicated by <xref ref-type="disp-formula" rid="e3">Equation 3</xref>:<disp-formula id="e3">
<mml:math id="m3">
<mml:mrow>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>Where <italic>N</italic>
<sub>
<italic>t</italic>
</sub> and <italic>N</italic>
<sub>
<italic>0</italic>
</sub> (&#x3d; 50% due to the porosity of the scaffold material) are the volume fraction of the scaffold at a time <italic>t</italic> and at the beginning, respectively, and <italic>k</italic>
<sub>
<italic>N</italic>
</sub> is the degradation rate, whose values are reported in <xref ref-type="table" rid="T4">Table&#x20;4</xref>. Moreover, the reduction in volume fraction was applied only to the elements situated on the scaffold surface (<xref ref-type="fig" rid="F2">Figure&#x20;2D</xref> left). When the volume fraction of the scaffold in an element was zero (completely degraded), the elastic modulus of the scaffold material in that element was set to zero (<xref ref-type="fig" rid="F2">Figure&#x20;2B</xref> left).</p>
<table-wrap id="T4" position="float">
<label>TABLE 4</label>
<caption>
<p>Degradation rates of volume fraction (<italic>k<sub>N</sub>
</italic>) and elastic modulus (<italic>k<sub>E</sub>
</italic>) of scaffold material for the three degradation modalities and the three degradation speeds investigated with the &#x201c;artificial&#x201d; scaffold. The time of complete scaffold degradation for each degradation speed is indicated in brackets. The volume fraction is expressed as percentage of the volume of one element.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Degradation modality</th>
<th align="center">Fast (25&#xa0;days)</th>
<th align="center">Matched (50&#xa0;days)</th>
<th align="center">Slow (100&#xa0;days)</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Surface erosion</td>
<td align="left">Surface elements: <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 4%/day</td>
<td align="left">Surface elements: <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 2%/day</td>
<td align="left">Surface elements: <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 1%/day</td>
</tr>
<tr>
<td rowspan="3" align="left">Bulk degradation</td>
<td align="left">All elements</td>
<td align="left">All elements</td>
<td align="left">All elements</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 40&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 20&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 10&#xa0;MPa/day</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 20%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 10%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 5%/day</td>
</tr>
<tr>
<td rowspan="6" align="left">Bulk degradation with autocatalysis</td>
<td align="left">Surface elements</td>
<td align="left">Surface elements</td>
<td align="left">Surface elements</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 40&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 20&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 10&#xa0;MPa/day</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 20%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 10%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 5%/day</td>
</tr>
<tr>
<td align="left">Bulk elements</td>
<td align="left">Bulk elements</td>
<td align="left">Bulk elements</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>2k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 80&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>2k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 40&#xa0;MPa/day</td>
<td align="left">&#x2022; <italic>2k</italic>
<sub>
<italic>E</italic>
</sub> &#x3d; 20&#xa0;MPa/day</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 40%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 20%/day</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 10%/day</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec id="s2-2-2">
<title>Hydrolytic Degradation by Bulk Degradation</title>
<p>In a scaffold undergoing bulk degradation, the molecular weight is homogeneously reduced without alterations in volume (<xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). Moreover, the mechanical properties of a polymer are closely related to its molecular weight (<xref ref-type="bibr" rid="B34">Nunes et&#x20;al., 1982</xref>). Here, direct proportionality was assumed between molecular weight and the compressive elastic modulus of the scaffold material, as indicated by <xref ref-type="disp-formula" rid="e4">Equation 4</xref>:<disp-formula id="e4">
<mml:math id="m4">
<mml:mrow>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x221d;</mml:mo>
<mml:mi>M</mml:mi>
<mml:msub>
<mml:mi>w</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
<label>(4)</label>
</disp-formula>Where <italic>E</italic>
<sub>
<italic>t</italic>
</sub> and <italic>Mw</italic>
<sub>
<italic>t</italic>
</sub> are the elastic modulus and the molecular weight, respectively, of the scaffold material at a time <italic>t</italic>. Thus, bulk degradation was simulated by a linear and homogeneous reduction of the elastic modulus of the scaffold material (<xref ref-type="fig" rid="F2">Figure&#x20;2B</xref> middle), following <xref ref-type="disp-formula" rid="e5">Equation 5</xref>:<disp-formula id="e5">
<mml:math id="m5">
<mml:mrow>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>E</mml:mi>
</mml:msub>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math>
<label>(5)</label>
</disp-formula>Where <italic>E</italic>
<sub>
<italic>t</italic>
</sub> and <italic>E</italic>
<sub>
<italic>0</italic>
</sub> (&#x3d; 1,000&#xa0;MPa) are the elastic modulus of the scaffold material at a time <italic>t</italic> and at the initial time, respectively, and <italic>k</italic>
<sub>
<italic>E</italic>
</sub> is the degradation rate (<xref ref-type="table" rid="T4">Table&#x20;4</xref>).</p>
<p>Eventually, also scaffolds undergoing bulk degradation begin losing mass and volume, i.e.,&#x20;begin eroding (see definition in <xref ref-type="table" rid="T1">Table&#x20;1</xref>). For example, the onset of mass loss for PCL has been observed at a molecular weight of 5,000&#xa0;g/mol (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>). This result was reported to be independent of the initial molecular weight. As the degradation behavior of PCL with an initial molecular weight of approximately 50,000&#xa0;g/mol was described in detail (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>), a threshold of 10% of the initial molecular weight was here taken as reference for a simplified model of the onset of erosion. Taking into account the direct proportionality between elastic modulus and molecular weight assumed here (see <xref ref-type="disp-formula" rid="e4">Equation 4</xref>), the loss of scaffold volume fraction in the model of bulk degradation begun when <italic>E</italic>
<sub>
<italic>t</italic>
</sub> was 10% of the initial material elastic modulus. In this case, the erosion process was modelled by applying <xref ref-type="disp-formula" rid="e3">Equation 3</xref> to all elements of the scaffold, independently of their position on the surface or in the bulk of the struts, with a degradation rate <italic>k</italic>
<sub>
<italic>N</italic>
</sub> enabling the complete degradation of the scaffold in the imposed time (<xref ref-type="table" rid="T4">Table&#x20;4</xref>). When the volume fraction of the scaffold in an element was completely degraded, the elastic modulus of the scaffold material in that element was set to&#x20;zero.</p>
</sec>
<sec id="s2-2-3">
<title>Hydrolytic Degradation by Bulk Degradation With Autocatalysis</title>
<p>Autocatalysis is established when the acidic degradation by-products accumulate in the bulk of the polymeric device, fostering a faster reduction of the molecular weight in the inner regions compared to the surface (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>). Taking into account the direct proportionality between molecular weight and elastic modulus assumed here (see <xref ref-type="disp-formula" rid="e4">Equation 4</xref>), the establishment of autocatalysis was modelled by imposing a greater reduction in elastic modulus to the elements in the bulk of the scaffold struts compared to the elements on the surface. It has been reported that after 2&#xa0;weeks of <italic>in vivo</italic> degradation, the molecular weight on the surface of PDLLA devices was almost two times higher than the one in the bulk (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>). Therefore, the reduction in elastic modulus of the elements on the surface of the scaffold struts were calculated with <xref ref-type="disp-formula" rid="e5">Equation 5</xref>, while <xref ref-type="disp-formula" rid="e6">Equation 6</xref> was applied to the elements in the bulk of the struts:<disp-formula id="e6">
<mml:math id="m6">
<mml:mrow>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>2</mml:mn>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>E</mml:mi>
</mml:msub>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:math>
<label>(6)</label>
</disp-formula>
</p>
<p>Also in this case, a reduction in volume fraction of the scaffold material begun when the elastic modulus reached 10% of its initial value (<xref ref-type="fig" rid="F2">Figures 2B,D</xref> right), as described above for bulk degradation without autocatalysis. The degradation rates of volume fraction and elastic modulus of the scaffold material when autocatalysis was implemented are listed in <xref ref-type="table" rid="T4">Table&#x20;4</xref>. Moreover, the elastic modulus of the scaffold material was set to zero in the elements with complete volume fraction degradation.</p>
</sec>
<sec id="s2-2-4">
<title>Hydrolytic Degradation Based on Experimental Observations</title>
<p>After an initial evaluation with simplified degradation behaviors, the model of scaffold degradation was employed to more closely reproduce experimentally observed degradation phenomena. The reduction in polymeric molecular weight during bulk degradation without autocatalysis has been reported to follow an exponential law in several cases: for example, in porous scaffolds produced from PDLLA and PLGA (<xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>) and in film and capsules of PCL (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>) and PDLLA (<xref ref-type="bibr" rid="B38">Pitt G et&#x20;al., 1981</xref>). Based on the assumption of direct proportionality between molecular weight and elastic modulus applied here (see <xref ref-type="disp-formula" rid="e4">Equation 4</xref>), this experimental observation was implemented in the model of scaffold degradation with <xref ref-type="disp-formula" rid="e7">Equation 7</xref>:<disp-formula id="e7">
<mml:math id="m7">
<mml:mrow>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
<mml:mi>t</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
<label>(7)</label>
</disp-formula>Where <italic>k</italic>
<sub>
<italic>e</italic>
</sub> is the exponential degradation rate. <xref ref-type="disp-formula" rid="e7">Equation 7</xref> was applied to the &#x201c;artificial&#x201d; scaffold with strut-like architecture previously investigated with the linear degradation laws described in the sections above. Moreover, <xref ref-type="disp-formula" rid="e7">Equation 7</xref> was used to simulate bulk degradation without autocatalysis, and was thereby applied to all scaffold elements independently of their position on the surface or in the bulk of the struts.</p>
<p>PCL films and capsules were reported to have an exponential degradation rate of 0.003&#x20;days<sup>&#x2212;1</sup> <italic>in vivo</italic> (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>). No autocatalysis was observed in PCL devices <italic>in vivo</italic> (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B26">Lam et&#x20;al., 2009</xref>). Moreover, the degradation of PCL does not depend on the surface area, as shown by comparing the <italic>in&#x20;vitro</italic> degradation of PCL films and micro-particles (<xref ref-type="bibr" rid="B10">Chen et&#x20;al., 2000</xref>). Therefore, the experimentally measured value of the PCL degradation rate was considered appropriate to describe the exponential degradation of the scaffold modelled here. Thus, a value of 0.003&#xa0;MPa/day was assigned to a slow exponential degradation rate (<italic>k</italic>
<sub>
<italic>e,slow</italic>
</sub>).</p>
<p>The degradation rate of PDLLA films and capsules <italic>in vivo</italic> was measured to be 0.00841&#x20;days<sup>&#x2212;1</sup> (<xref ref-type="bibr" rid="B38">Pitt G et&#x20;al., 1981</xref>). However, PDLLA has been observed to undergo autocatalysis (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>). Therefore, the size of a device produced from PDLLA will influence its degradation rate (<xref ref-type="bibr" rid="B18">Grizzi et&#x20;al., 1995</xref>). Porous PDLLA scaffolds (porosity &#x3e;80%) were reported to have an <italic>in&#x20;vitro</italic> exponential degradation rate of 0.023&#x20;weeks<sup>&#x2212;1</sup>, corresponding to 0.003&#x20;days<sup>&#x2212;1</sup>, without the establishment of autocatalysis (<xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>). Therefore, the exponential degradation rate <italic>k</italic>
<sub>
<italic>e,slow</italic>
</sub> with a value of 0.003&#xa0;MPa/day was deemed appropriate to model the degradation of a porous scaffold produced from both PCL and PDLLA.</p>
<p>The <italic>in&#x20;vitro</italic> exponential degradation rate of porous PLGA scaffolds (lactic/glycolic acid molar ratio of 75/25) was measured to be 0.153&#x20;weeks<sup>&#x2212;1</sup>, corresponding to 0.022&#x20;days<sup>&#x2212;1</sup>, without autocatalysis phenomena (<xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>). Therefore, a fast exponential degradation rate (<italic>k</italic>
<sub>
<italic>e,fast</italic>
</sub>) with value of 0.022&#xa0;MPa/day was implemented to simulate the degradation of porous scaffolds produced from&#x20;PLGA.</p>
<p>With the <italic>k</italic>
<sub>
<italic>e,slow</italic>
</sub> exponential degradation rate, the onset of erosion at 10% of the initial elastic modulus value was calculated to take place after more than 2&#xa0;years of degradation (specifically, at day 768). To enable a comparison with the simplified degradation behaviors studied here while keeping the evaluation within reasonable computational times, the model of PCL and PDLLA bulk degradation with exponential degradation law was evaluated until day 125. When the <italic>k</italic>
<sub>
<italic>e,fast</italic>
</sub> exponential degradation rate was implemented, the onset of erosion was calculated to take place at day 105. Given the similar times of complete degradation between the simulated exponential degradation of PLGA and the simplified slow bulk degradation, the same erosion rate (<italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 5%/day) was implemented to model the erosion of the PLGA porous scaffolds. The set up of the models implementing the experimentally-derived exponential bulk degradation is summarized in <xref ref-type="table" rid="T5">Table&#x20;5</xref>.</p>
<table-wrap id="T5" position="float">
<label>TABLE 5</label>
<caption>
<p>Parameters of the models implementing the exponential bulk degradation. PCL: poly (&#x3b5;-caprolactone); PDLLA: poly (D,L-lactic acid); PLGA: poly (lactide-co-glycolide); <italic>k<sub>e</sub>
</italic>: exponential degradation rate; <italic>k<sub>N</sub>
</italic>: erosion rate (as percentage of the total volume fraction of one element/day).</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Modelled material</th>
<th align="center">Degradation rate</th>
<th align="center">Time of complete scaffold degradation</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Porous PCL or PDLLA scaffolds</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>e,slow</italic>
</sub> &#x3d; 0.003&#xa0;MPa/day</td>
<td align="center">&#x3e;2&#xa0;years</td>
</tr>
<tr>
<td rowspan="2" align="left">Porous PLGA scaffold</td>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>e,fast</italic>
</sub> &#x3d; 0.022&#xa0;MPa/day</td>
<td rowspan="2" align="center">Day 115</td>
</tr>
<tr>
<td align="left">&#x2022; <italic>k</italic>
<sub>
<italic>N</italic>
</sub> &#x3d; 5%/day</td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
</sec>
<sec id="s2-3">
<title>Evaluation of Computational Result</title>
<p>In the presented model, tissue formation was determined by cellular number and distribution, while the distribution of <italic>S</italic> (mechanical stimulus) indicated the tissues whose formation or resorption would have been favored by the mechanical environment at a specific day. Due to the applied mechano-biological rules of tissue formation, cellular distributions were not independent from the distribution of <italic>S</italic>, but they might differ from it before reaching the equilibrium state, i.e. the full repair of the defect. Therefore, results were evaluated by comparing the distribution of MSCs, osteoblasts, chondrocytes, and fibroblasts to derive the distribution of granulation tissue, bone, cartilage, and fibrous tissue, respectively. However, it was previously observed that cellular distributions at the equilibrium state matched the prediction of tissue formation based on <italic>S</italic> (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). Thus, the predicted repair outcome was here visualized by the distribution of <italic>S</italic> at the end of the repair process.</p>
<p>Some of the modelled degradation cases never reached the equilibrium state due to non-convergence of the finite element analysis at some day of the iterative repair process. The non-convergence of the model was caused by the excessive deformation of elements belonging to the scaffold during its degradation. This simulation outcome was considered equivalent to a mechanical failure of the scaffold <italic>in vivo</italic> and was not further analyzed.</p>
<p>The stiffness of the scaffold during the degradation process was determined by a separate FE analysis, which modelled the scaffold alone in its three-dimensional form (i.e. three concentric rings, <xref ref-type="sec" rid="s10">Supplementary Figure S1</xref>). The overall elastic modulus of the scaffold material (<italic>E</italic>
<sub>
<italic>Scaffold</italic>
</sub>) was calculated taking into account both the elastic modulus (<italic>E</italic>
<sub>
<italic>t</italic>
</sub>) and volume fraction (<italic>N</italic>
<sub>
<italic>t</italic>
</sub>) at a given time, as indicated by <xref ref-type="disp-formula" rid="e8">Equation 8</xref>:<disp-formula id="e8">
<mml:math id="m8">
<mml:mrow>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mrow>
<mml:mi>S</mml:mi>
<mml:mi>c</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>l</mml:mi>
<mml:mi>d</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x2217;</mml:mo>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
<label>(8)</label>
</disp-formula>
</p>
<p>Therefore, different <italic>E</italic>
<sub>
<italic>Scaffold</italic>
</sub> were assigned to elements on the surface and in the bulk of the struts when modelling non-homogeneous degradation processes, e.g. surface erosion. A 3% compressive displacement (<italic>&#x394;x</italic>) was applied and the resulting reaction force (<italic>RF</italic>) was measured, enabling the calculation of the scaffold stiffness (<italic>K</italic>) with <xref ref-type="disp-formula" rid="e9">Equation 9</xref>:<disp-formula id="e9">
<mml:math id="m9">
<mml:mrow>
<mml:mi>K</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mi>R</mml:mi>
<mml:mi>F</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="normal">&#x394;</mml:mi>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
<label>(9)</label>
</disp-formula>
</p>
</sec>
</sec>
<sec sec-type="results" id="s3">
<title>Results</title>
<p>The influence of polymeric scaffold degradation on the mechanics-dependent repair of osteochondral defects was investigated <italic>in silico</italic> by simulating three modalities of hydrolytic degradation: surface erosion, bulk degradation, and bulk degradation with autocatalysis. For each degradation modality, three speeds were investigated: fast, matched, and slow, causing the complete degradation of the scaffold in 25, 50, and 100&#xa0;days, respectively. The repair outcome was evaluated at day 125. Moreover, bulk degradation with an exponential loss of mechanical competence was implemented, simulating the experimentally-observed degradation rates of porous scaffolds produced from PCL or PDLLA (which were reported to have comparable exponential degradation rates) and from PLGA. The following sections report the results obtained for each degradation modality and each degradation&#x20;speed.</p>
<sec id="s3-1">
<title>Degradation by Surface Erosion</title>
<p>Scaffold degradation by surface erosion was modelled by reducing the volume fraction occupied by the scaffold material in the elements situated on the scaffold surface. The distribution of octahedral shear strain (&#x3b3;) and the prediction of tissue formation over the repair process prior to equilibrium are reported in <xref ref-type="sec" rid="s10">Supplementary Figure&#x20;S2</xref>.</p>
<p>When the fast degradation by surface erosion was implemented, the model predicted the mechanical failure of the scaffold at day 24, i.e. one day before its complete degradation (<xref ref-type="table" rid="T6">Table&#x20;6</xref>). Therefore, the results of this simulation were not further analyzed.</p>
<table-wrap id="T6" position="float">
<label>TABLE 6</label>
<caption>
<p>Models for which the mechanical failure of the scaffold prior to complete defect repair was predicted.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th align="left">Degradation modality</th>
<th align="center">Degradation speed</th>
<th align="center">Day&#x20;of complete scaffold degradation</th>
<th align="center">Model-specific events</th>
<th align="center">Day&#x20;of mechanical failure</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="left">Surface erosion</td>
<td align="left">Fast</td>
<td align="char" char=".">25</td>
<td align="left">&#x2022; Complete degradation of first layer of surface elements: day 13</td>
<td align="char" char=".">24</td>
</tr>
<tr>
<td rowspan="9" align="left">Bulk degradation with autocatalysis</td>
<td rowspan="3" align="left">Fast</td>
<td rowspan="3" align="char" char=".">25</td>
<td align="left">&#x2022; Complete degradation of elements in strut bulk: day 13</td>
<td rowspan="3" align="char" char=".">14</td>
</tr>
<tr>
<td align="left">&#x2022; Onset of erosion: &#x2003;<monospace>o</monospace>Surface: day 23</td>
</tr>
<tr>
<td align="left">&#x2003;&#x2003;<monospace>o</monospace>Bulk: day 12</td>
</tr>
<tr>
<td rowspan="3" align="left">Matched</td>
<td rowspan="3" align="char" char=".">50</td>
<td align="left">&#x2022; Complete degradation of elements in strut bulk: day 25</td>
<td rowspan="3" align="char" char=".">26</td>
</tr>
<tr>
<td align="left">&#x2022; Onset of erosion: &#x2003;<monospace>o</monospace>Surface: day 45</td>
</tr>
<tr>
<td align="left">&#x2003;&#x2003;<monospace>o</monospace>Bulk: day 23</td>
</tr>
<tr>
<td rowspan="3" align="left">Slow</td>
<td rowspan="3" align="char" char=".">100</td>
<td align="left">&#x2022; Complete degradation of elements in strut bulk: day 50</td>
<td rowspan="3" align="char" char=".">67</td>
</tr>
<tr>
<td align="left">&#x2022; Onset of erosion: &#x2003;<monospace>o</monospace>Surface: day 90</td>
</tr>
<tr>
<td align="left">&#x2003;&#x2003;<monospace>o</monospace>Bulk: day 45</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>At the completion of the repair process, the scaffold degrading by surface erosion with matched degradation speed resulted in very low (&#x3c;1%) mechanical strains (&#x3b3;) at the proximal base and side of the defect (<xref ref-type="fig" rid="F3">Figure&#x20;3A</xref> left). Intermediate (&#x2248;5&#x2013;8%) and high (&#x2248;10&#x2013;30%) values of strain were predicted in the middle region and at the articular surface, respectively (<xref ref-type="fig" rid="F3">Figure&#x20;3A</xref> left). The predicted repair outcome resulted in fibrous tissue formation at the articular surface, cartilage formation in the central region, and bone formation at the side and base of the defect (<xref ref-type="fig" rid="F3">Figure&#x20;3B</xref> left). The interfaces between the different formed tissues were not clear-cut, showing ample areas of mixed tissue growth and traces of bone resorption at the base of the defect.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Influence of scaffold degradation by surface erosion on osteochondral defect repair. <bold>(A)</bold> Distribution of octahedral shear strain (<italic>&#x3b3;</italic>); <bold>(B)</bold> prediction of tissue formation based on the mechanical stimulus (<italic>S</italic>). The left and right columns show the repair outcome with matched and slow degradation speed at day 125. The yellow, orange, and green borders indicate the neighboring healthy tissues of cancellous bone, subchondral bone, and cartilage, respectively. The dash-dot and solid black lines highlight the axis of symmetry and the articular surface, respectively; <bold>(C)</bold> Volume fraction (as percentage of the total defect volume) and stiffness of the scaffold during degradation with different speeds. The red cross marks the predicted mechanical failure of the scaffold. Lines and symbols refer to volume fraction and stiffness, respectively.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g003.tif"/>
</fig>
<p>The implementation of the slow degradation speed resulted in very low mechanical strains (&#x3b3; &#x3c; 1%) at the base and side, while the rest of the defect experienced intermediate strains (&#x3b3; &#x2248; 7%) (<xref ref-type="fig" rid="F3">Figure&#x20;3A</xref> right). The regions of low and intermediate strains were separated by an area subjected to high values of &#x3b3;, ranging from 10 to 64% (<xref ref-type="fig" rid="F3">Figure&#x20;3A</xref> right). The corresponding mechanics-dependent prediction of the repair outcome was the formation of cartilage in most of the defect, with bone growing at the proximal base and side and an intermediate region of fibrous tissue formation (<xref ref-type="fig" rid="F3">Figure&#x20;3B</xref> right).</p>
<p>The volume fraction of the scaffold showed a linear decrease for each investigated speed, as imposed by the applied degradation model (<xref ref-type="fig" rid="F3">Figure&#x20;3C</xref>, black lines). Concomitantly, the stiffness of the scaffold decreased from the initial value of 2445&#xa0;N/mm (<xref ref-type="fig" rid="F3">Figure&#x20;3C</xref>, gray symbols). Although at early time points the stiffness of the scaffold was comparable amongst the different degradation speeds (e.g., 1951&#x20;N/mm for fast, 2197&#xa0;N/mm for matched, and 2320&#xa0;N/mm for slow degradation speeds at day 5), it dropped rapidly to 0&#xa0;N/mm when the fast and matched degradation speeds were imposed. With the slow degradation speed, the scaffold stiffness maintained high values (&#x3e;500&#xa0;N/mm) until a late time point of the repair process (day&#x20;75).</p>
</sec>
<sec id="s3-2">
<title>Degradation by Bulk Degradation</title>
<p>The bulk degradation of the scaffold was modelled by a homogenous reduction in the elastic modulus of the scaffold material. When the material elastic modulus reached 10% of its initial value, it determined the onset of scaffold erosion, leading to a reduction in the volume fraction of the scaffold material until its complete disappearance. The distribution of mechanical strain (&#x3b3;) and the prediction of tissue formation based on the mechanical stimulus prior to equilibrium are reported in <xref ref-type="sec" rid="s10">Supplementary Figure&#x20;S3</xref>.</p>
<p>At the completion of the repair process, the scaffold with fast bulk degradation resulted in very low (&#x3c;1%) values of &#x3b3; in most of the defect (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> left). At the articular surface, however, &#x3b3; ranged from 20 to 35% (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> left). The corresponding repair outcome was the formation of fibrous tissue at the articular surface, the growth of bone at the side, and the establishment of a region of mixed bone resorption and bone apposition in the middle and at the proximal base of the defect (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> left). Cartilage formation was predicted only in a small central area (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> left).</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Influence of scaffold degradation by bulk degradation on osteochondral defect repair. <bold>(A)</bold> Distribution of octahedral shear strain (<italic>&#x3b3;</italic>); <bold>(B)</bold> prediction of tissue formation based on the mechanical stimulus (<italic>S</italic>). The left, middle, and right columns show the repair outcome with fast, matched and slow degradation speed, respectively, at day 125. The yellow, orange, and green borders indicate the neighboring healthy tissues of cancellous bone, subchondral bone, and cartilage, respectively. The dash-dot and solid black lines highlight the axis of symmetry and the articular surface, respectively; <bold>(C)</bold> Material elastic modulus and scaffold stiffness during degradation with different degradation speeds. The grey area marks the region of concomitant scaffold erosion. Lines and symbols refer to elastic modulus and stiffness, respectively.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g004.tif"/>
</fig>
<p>When the matched bulk degradation was implemented, &#x3b3; assumed very low (&#x3c;1%) values at the proximal base and side, while most of the defect experienced intermediate strain values of approximately 4&#x2013;8% (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> middle). High values of &#x3b3; (&#x2248;18&#x2013;40%, with local peaks up to 52%) were found in a narrow band crossing the distal half of the defect and at the interface with healthy subchondral bone and cartilage (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> middle). This distribution of &#x3b3; resulted in the prediction of cartilage formation in the defect, with bone apposition at the base and side and fibrous tissue formation in the region of high strain (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> middle).</p>
<p>Similarly, the values of &#x3b3; were very low (&#x3c;1%) at the base and side and intermediate (&#x2248;7%) in most of the defect with slow bulk degradation (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> right). A band of high &#x3b3; (20&#x2013;40%, with local peaks up to 90%) was observed in this case in the proximal half of the defect and at the interface with the surrounding healthy tissues (<xref ref-type="fig" rid="F4">Figure&#x20;4A</xref> right). The corresponding repair prediction was the formation of cartilage in most of the defect, with bone apposition and fibrous tissue formation in the regions of very low and high &#x3b3;, respectively (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> right).</p>
<p>In accordance with the imposed degradation model, the elastic modulus of the scaffold material linearly decreased throughout the repair process (<xref ref-type="fig" rid="F4">Figure&#x20;4C</xref>, black lines). Interestingly, despite the different modality by which the mechanical competence of the scaffold was lost, the stiffness of the scaffold throughout the bulk degradation process (<xref ref-type="fig" rid="F4">Figure&#x20;4C</xref>, grey symbols) matched that observed during the surface erosion process (<xref ref-type="fig" rid="F3">Figure&#x20;3C</xref>, grey symbols).</p>
</sec>
<sec id="s3-3">
<title>Degradation by Bulk Degradation With Autocatalysis</title>
<p>Autocatalysis was modelled by a faster reduction in material elastic modulus in the bulk of the scaffold struts compared to the surface. Also in this case, scaffold erosion begun when the elastic modulus of the scaffold material reached 10% of its initial value. The distribution of &#x3b3; and the prediction of tissue formation based on the mechanical stimulus prior to equilibrium are reported in <xref ref-type="sec" rid="s10">Supplementary Figure&#x20;S4</xref>.</p>
<p>None of the models implementing autocatalysis reached the equilibrium state, i.e. the full repair of the osteochondral defect, independently of the imposed degradation speed (<xref ref-type="table" rid="T6">Table&#x20;6</xref>). Typically, the mechanical failure occurred the day after the complete degradation of the scaffold bulk, when the struts of the scaffold became an &#x201c;empty shell&#x201d;. The model with slow degradation was an exception, with the mechanical failure happening at day 67, i.e. 17&#xa0;days after the complete degradation of the bulk of the struts.</p>
</sec>
<sec id="s3-4">
<title>Degradation Based on Experimental Data</title>
<p>The hydrolytic bulk degradation of porous scaffolds produced from several synthetic polymers, such as PCL, PDLLA, and PLGA, has been reported to follow an exponential law. Therefore, scaffold degradation by bulk degradation was modelled by implementing an exponential reduction in elastic modulus with experimentally-derived degradation rates. The experimentally-derived degradation rates simulated the degradation of porous scaffold produced from PCL or PDLLA, denoted as slow, and from PLGA, denoted as fast. The distribution of &#x3b3; and the prediction of tissue formation based on the mechanical stimulus prior to equilibrium are reported in <xref ref-type="sec" rid="s10">Supplementary Figure&#x20;S5</xref>.</p>
<p>When the slow exponential degradation rate was implemented, the complete degradation of the scaffold material was predicted after more than 2&#xa0;years. The repair process was studied only until day 125 to enable the comparison with the simplified degradation cases. At this time point, &#x3b3; showed very low values (&#x3c;1%) at the base of the defect, intermediate values (&#x2248;4&#x2013;7%) in the central region and at the articular surface, and high values (&#x2248;20&#x2013;30% with a peak of 38%) at the interface with healthy tissues (<xref ref-type="fig" rid="F5">Figure&#x20;5A</xref> right). The prediction of tissue formation based on S was the growth of cartilage in the defect, with a thin layer of subchondral bone at the base and fibrous tissue at the interface with the healthy tissues (<xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> right). At day 125, the scaffold maintained a high material elastic modulus of 687&#xa0;MPa (<xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>, black dashed line) and its volume fraction was 15% of the total defect volume. Moreover, the scaffold maintained a high stiffness of 1680&#xa0;N/mm (<xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>, gray circles). Although the scaffold was not fully degraded at day 125, tissue formation reached an equilibrium already at approximately day 75. In fact, a maximum difference of 5% was found between the amounts of tissues formed at day 75 and at day&#x20;125.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Influence of scaffold degradation with exponential bulk degradation on osteochondral defect repair. <bold>(A)</bold> Distribution of octahedral shear strain (<italic>&#x3b3;</italic>); <bold>(B)</bold> prediction of tissue formation based on the mechanical stimulus (<italic>S</italic>). All images refer to day 125. The right column shows the outcome obtained with the exponential degradation rate of porous PCL and PDLLA scaffolds (slow), while the left column refers to porous PLGA scaffolds (fast). The yellow, orange, and green borders indicate the neighboring healthy tissues of cancellous bone, subchondral bone, and cartilage, respectively. The dash-dot and solid black lines highlight the axis of symmetry and the articular surface, respectively. The black dashed lines mark the scaffold struts; <bold>(C)</bold> Material elastic modulus and scaffold stiffness during degradation with the fast and slow exponential degradation rate. The grey area marks the region of concomitant erosion. Lines and symbols refer to elastic modulus and stiffness, respectively.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g005.tif"/>
</fig>
<p>Implementing the fast exponential degradation rate yielded similar results, although with slightly increased values of &#x3b3;. In fact, &#x3b3; assumed very low (&#x3c;1%), intermediate (&#x2248;7&#x2013;10%), and high (&#x2248;20&#x2013;40%, with local peaks up to 60%) values at the base, middle, and side of the defect, respectively (<xref ref-type="fig" rid="F5">Figure&#x20;5A</xref> left). As a consequence of the higher values of &#x3b3;, a greater amount of fibrous tissue was predicted to form in the defect implanted with the scaffold degrading like PLGA (fast exponential degradation, <xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> left) compared to the one degrading like PCL or PDLLA (slow exponential degradation, <xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> right). Nonetheless, the repair outcome in the two cases was similar, with cartilage forming in the defect and bone growth at the base, although with small regions of bone resorption (<xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> left). The elastic modulus of the scaffold material, in accordance with the imposed degradation model, decreased exponentially until day 114, when the completion of the erosion process caused the scaffold material to completely disappear (<xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>, black solid line). Concomitantly, the stiffness of the scaffold decreased, reaching a value of 271&#xa0;N/mm at day 100 (<xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>, gray squares), shortly prior to the onset of erosion at day&#x20;105.</p>
</sec>
</sec>
<sec sec-type="discussion" id="s4">
<title>Discussion</title>
<p>Scaffold-based tissue engineering strategies aim at supporting the healing of osteochondral defects by overcoming the limitations of current clinical treatments (<xref ref-type="bibr" rid="B12">Davis et&#x20;al., 2021</xref>). However, an identification of the ideal mechanical and architectural features of a scaffold for osteochondral defect healing is still lacking. Promising mechanical and architectural properties of a scaffold for osteochondral defect healing were previously identified <italic>in silico</italic> by employing an axisymmetric model of a knee femoral condyle, in which tissue formation was determined by a mechanical stimulus computed from octahedral shear strain and fluid velocity (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). However, this model did not provide any insights concerning the influence of scaffold degradation, which has the potential to alter both the mechanical and the architectural characteristics of a device. In this study, the cited model was further developed to simulate the degradation of a synthetic polymeric scaffold with strut-like architecture by three modalities (surface erosion, bulk degradation, and bulk degradation with autocatalysis) and with three different degradation speeds (faster, equal or slower than the tissue repair process). The <italic>in silico</italic> evaluations suggested that, in the absence of autocatalysis, the speed of scaffold degradation, rather than its modality, is the factor having a predominant influence on the mechanics-dependent tissue formation. Specifically, times of full scaffold degradation longer than the expected time of completion of the tissue repair process were found to foster the best repair outcomes in the investigated&#x20;model.</p>
<p>Initially, the three different modalities of polymeric scaffold degradation were compared to assess whether one of them presented particular advantages or disadvantages for the repair of osteochondral defects. Specifically, simplified algorithms with linear reductions in either volume fraction or elastic modulus for surface erosion and bulk degradation (with or without autocatalysis), respectively, were investigated. For bulk degradation and bulk degradation with autocatalysis, the scaffold begun the erosion process (leading to the complete disappearance of the device) when its material elastic modulus reached 10% of the initial value. Thus, the simplified algorithms reproduced the experimental observations that surface erosion causes thinning of the scaffold features without alterations of molecular weight (here considered directly proportional to the material elastic modulus, see <xref ref-type="disp-formula" rid="e4">Equation 4</xref>) and that bulk degradation results in a homogeneous reduction in molecular weight, and thereby of elastic modulus according to <xref ref-type="disp-formula" rid="e4">Equation 4</xref>, without alterations in the volume of the device up to an advanced state of degradation (<xref ref-type="bibr" rid="B30">Middleton &#x26; Tipton, 2000</xref>; <xref ref-type="bibr" rid="B9">Casalini, 2017</xref>). When autocatalysis was modelled, the elements in the bulk of the scaffold struts had a faster reduction in molecular weight, i.e. in elastic modulus, than those on the surface, also in this case reproducing experimental observations (<xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>). Therefore, surface erosion and bulk degradation resulted in a transfer of the load-bearing function from the scaffold to the newly formed tissues in the model by two different mechanisms. In surface erosion, the biological tissues could gradually grow in areas previously occupied by the scaffold, whose load-bearing ability was reduced by the physical disappearance of the scaffold material. In bulk degradation, the amount of scaffold material remained unaltered for the majority of the degradation process and the transfer of the load-bearing function to the biological tissues happened by a softening of the scaffold material itself.</p>
<p>Previous analyses performed with the same computational model studied the mechanics-dependent repair process of an untreated osteochondral defect and of an osteochondral defect implanted with a non-degradable scaffold, which had the same mechanical and morphological properties of the scaffold investigated here (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). In the mentioned study, the repair of untreated osteochondral defects was predicted to develop fibrous tissue at the articular surface, with bone resorption mixed with bone formation in the middle and proximal regions, and lateral bone apposition. The non-degradable scaffold fostered an improved repair, as cartilage formed in most of the defect (and specifically at the articular surface) and a thin subchondral bone layer was established throughout the proximal base, although fibrous tissue was predicted to form at the interface with the surrounding healthy tissues. The improved repair outcome fostered by the scaffold was ascribed to its mechanical support, which reduced the strain in the defect, and to its architecture, which enabled load transmission to the defect&#x20;base.</p>
<p>When a fast bulk degradation of the scaffold was implemented, the repair outcome was analogous to the one predicted for the untreated osteochondral defect (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> left), indicating that the mechanical support provided by the scaffold lasted a time too short to significantly steer tissue formation. On the other hand, both matched and slow bulk degradation resulted in a repair outcome similar to the non-degradable scaffold, although with higher amounts of fibrous tissue (<xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> middle and right, respectively). The increased formation of fibrous tissue could be ascribed to higher strains generated by the progressive decrease in stiffness of the scaffold (<xref ref-type="fig" rid="F4">Figure&#x20;4C</xref>). Interestingly, the mechanical failure of the scaffold was never predicted when bulk degradation was implemented, independently of the degradation speed. This result was ascribed to the gradual and homogenous change in mechanical properties of the scaffold, which avoided the excessive straining of individual scaffold elements.</p>
<p>When the scaffold degraded by surface erosion, the fast degradation speed was predicted to induce mechanical failure of the scaffold (<xref ref-type="table" rid="T6">Table&#x20;6</xref>). As the failure occurred 1&#xa0;day prior to complete scaffold degradation, it could be ascribed to the low thickness of the struts and their extremely increased porosity, which derived from the reduction in the volume fraction occupied by the scaffold material (<xref ref-type="fig" rid="F3">Figure&#x20;3C</xref>). Mechanical failures were observed neither with the matched nor with the slow surface erosion. The repair outcome with the matched surface erosion was intermediate between the ones previously obtained in the untreated defect and with the non-degradable scaffold (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). In fact, a continuous subchondral bone layer formed and cartilage was found in the middle of the defect (<xref ref-type="fig" rid="F3">Figure&#x20;3B</xref> left); however, the articular surface was occupied by fibrous tissue. Thus, the scaffold undergoing matched surface erosion did not provide sufficient mechanical support to the defect to avoid major fibrous tissue formation. Interestingly, the repair outcome with slow surface erosion was analogous to the one with slow bulk degradation (compare <xref ref-type="fig" rid="F3">Figure&#x20;3B</xref> right to <xref ref-type="fig" rid="F4">Figure&#x20;4B</xref> right). Although the overall stiffness of surface-eroding and bulk-degrading scaffolds was comparable between all imposed degradation speeds throughout the simulation (compare <xref ref-type="fig" rid="F3">Figure&#x20;3C</xref> and <xref ref-type="fig" rid="F4">Figure&#x20;4C</xref>), a similar repair outcome was achieved only with the implementation of the slow degradation speed. This result seems to indicate that the modality of degradation (either surface erosion or bulk degradation without autocatalysis) for scaffolds with strut-like architecture does not have an influence on the mechanics-dependent repair process, as long as the scaffold provides mechanical support for a sufficiently long time. Notably, such a sufficiently long time was identified in this computational model not as the time of expected defect healing, i.e. 50 days, but as long as 100 days. The similarity of overall scaffold stiffness during bulk degradation and surface erosion observed in this model was ascribed to the low thickness of the strut-like architecture, where the strut width was composed of 50% surface and 50% bulk elements. Thus, the reduction in scaffold material during surface erosion had a comparable influence on the overall mechanical properties to a homogenous reduction in material elastic modulus. This result is not expected to hold true for bulkier devices, for which surface erosion might have a reduced influence on the overall mechanical properties. It is important to notice that when both surface erosion and bulk degradation (without autocatalysis) with slow degradation speed were implemented, a layer of fibrous tissue developed at the interface between soft and hard tissues, probably deriving from increased strain values in this area resulting from the progressive loss of mechanical support from the scaffold. To avoid this effect, the architecture of the scaffold could be modified to one that provides a higher resistance against deformation in the early phases of degradation: for example, a grid-like architecture that reduces displacements both in the direction of the applied load and perpendicular to it has been previously indicated as favorable (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>).</p>
<p>The results of the computational model clearly indicated that the instauration of autocatalysis is critical, as the mechanical failure of the scaffold with strut-like architecture was consistently predicted prior to full repair of the osteochondral defect, independently of the imposed degradation speed (<xref ref-type="table" rid="T6">Table&#x20;6</xref>). Such a markedly negative effect depended also on the architecture of the scaffold, which was composed only of thin (0.5&#xa0;mm thickness) vertical struts (<xref ref-type="fig" rid="F1">Figure&#x20;1A</xref>). As the bulk of the scaffold struts was completely degraded, the two surface layers of each strut effectively became independent pillars with the same height of the original strut (5&#xa0;mm), but a much reduced thickness of 0.125&#xa0;mm, corresponding to the width of one element in the mesh of the finite element model. This alteration of the height-to-thickness ratio, combined with the progressive decrease in material elastic modulus, made the scaffold struts more susceptible to buckling phenomena or to breakages due to excessive deformation. In particular, buckling strongly depends on the aspect ratio of the loaded body and may cause failures even if the yield stress of the material has not been exceeded (<xref ref-type="bibr" rid="B4">Beer et&#x20;al., 2015</xref>). Therefore, scaffold architectures different than the one investigated here might better sustain loads even in presence of autocatalysis, for example if featuring connecting elements between vertical struts and/or having oblique rather than vertical struts. Nonetheless, the transformation of the scaffold into a structure consisting only of a thin outer layer, and the deriving reduction in mechanical competence, could be expected when employing materials that undergo bulk degradation with autocatalysis. The precise thickness of the remaining outer layer depends on several factors, e.g. the diffusion speed of the involved molecules or oligomers and the cleavage rate of the chemical bonds (<xref ref-type="bibr" rid="B8">Cameron &#x26; Kamvari-Moghaddam, 2008</xref>).</p>
<p>Bulk degradation without autocatalysis was also modelled as an exponential decrease in molecular weight (and thereby in elastic modulus, see <xref ref-type="disp-formula" rid="e4">Equation 4</xref>) based on the available literature data. This analysis enabled an evaluation of the influence of degradation that came closer to an experimental case. Specifically, the degradation behavior of porous scaffolds made from PCL or PDLLA and from PLGA was implemented. When the PCL- or PDLLA-like degradation was studied (slow exponential degradation rate), the complete disappearance of the scaffold was calculated to happen after more than 2&#x20;years. Therefore, the scaffold material was still present at the last investigated time point (day 125). However, tissue formation had already reached equilibrium at day 75, meaning that variations in tissue distributions from the results here reported might be expected only after the onset of erosion. As the scaffold material maintained a high elastic modulus (&#x3e;600&#xa0;MPa, <xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>) throughout the investigated time, the mechanics-dependent repair of the osteochondral defect (<xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> right) was analogous to the one obtained with the non-degradable scaffold (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>), which was regarded as a positive outcome. It remains to be ascertained whether the onset of erosion would drastically alter this result. However, in the <italic>in vivo</italic> case, the production of extra-cellular matrix would likely stabilize the tissue configuration that was built during the long time of scaffold degradation. Thus, the repair tissues would gradually become independent of the mechanical support of the scaffold, reducing the risk of large variations in tissue distribution in consequence of the full degradation of the scaffold material.</p>
<p>A similar repair of the osteochondral defect was predicted with the material degrading as a porous PLGA scaffold (fast exponential degradation rate), although more fibrous tissue formation and regions of bone resorption were predicted (<xref ref-type="fig" rid="F5">Figure&#x20;5B</xref> left). In this case, the degradation was faster and the scaffold material had completely disappeared at the last investigated time point (day 125, <xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>). The results obtained with the experimentally-derived bulk degradation rates confirmed the observation performed with the linear degradation rates, i.e. that long times of scaffold degradation are particularly advantageous to the mechanics-dependent tissue formation.</p>
<p>A verification of the computational results by comparison with the <italic>in vivo</italic> tissue formation in osteochondral defects is challenging. Although the <italic>in vivo</italic> degradation behavior of scaffolds or biomaterials has been often studied in subcutaneous (<xref ref-type="bibr" rid="B25">Lam et&#x20;al., 2008</xref>), subdermal (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B38">Pitt G et&#x20;al., 1981</xref>), intramuscular (<xref ref-type="bibr" rid="B25">Lam et&#x20;al., 2008</xref>; <xref ref-type="bibr" rid="B46">Therin et&#x20;al., 1992</xref>) or calvarial (<xref ref-type="bibr" rid="B25">Lam et&#x20;al., 2008</xref>) defects, extensive evaluations of scaffold degradation in osteochondral defects, and thereby of degradation-dependent tissue formation, are, to the best of our knowledge, still lacking. When testing biodegradable or bioresorbable scaffolds for osteochondral defect healing, the presence or absence of scaffold remnants at a specific time point is a frequently reported observation on the <italic>in vivo</italic> degradation behavior of the specimens (<xref ref-type="bibr" rid="B2">Athanasiou et&#x20;al., 1997</xref>; <xref ref-type="bibr" rid="B3">Barron et&#x20;al., 2015</xref>; <xref ref-type="bibr" rid="B19">Holland et&#x20;al., 2005</xref>; <xref ref-type="bibr" rid="B20">Hsieh et&#x20;al., 2018</xref>; <xref ref-type="bibr" rid="B22">Ikeda et&#x20;al., 2009</xref>; <xref ref-type="bibr" rid="B39">Reyes et&#x20;al., 2012</xref>; <xref ref-type="bibr" rid="B40">Schlichting et&#x20;al., 2008</xref>). Although important to assess the time of complete disappearance of the investigated device, this observation is not sufficient to establish a comparison with the results of the here presented model in terms of mechanics-dependent tissue formation. Some studies offer an evaluation of the <italic>in vivo</italic> degradation of scaffolds in osteochondral defects based on previous <italic>in&#x20;vitro</italic> tests. For example, a two-phase PLGA scaffold implanted in goat failed to support tissue growth in the center of the osteochondral defects, an outcome ascribed to the faster degradation of the scaffold core due to the establishment of autocatalysis, as observed <italic>in&#x20;vitro</italic> (<xref ref-type="bibr" rid="B2">Athanasiou et&#x20;al., 1997</xref>). However, some authors reported differences in the <italic>in vivo</italic> degradation of the scaffolds compared to the results expected from previous <italic>in&#x20;vitro</italic> evaluations (<xref ref-type="bibr" rid="B19">Holland et&#x20;al., 2005</xref>), or that expected differences in degradation amongst specimens could not be confirmed <italic>in vivo</italic> by visual inspection only (<xref ref-type="bibr" rid="B51">Wang et&#x20;al., 2014</xref>). In another study, two types of PLGA scaffolds were tested in sheep, with the stiffer scaffold with slower degradation resulting in improved subchondral bone formation (<xref ref-type="bibr" rid="B40">Schlichting et&#x20;al., 2008</xref>). Interestingly, the authors suggested that mechanical failures due to the faster degradation of the softer scaffold may have negatively influenced tissue formation, thereby supporting our <italic>in silico</italic> predictions. However, the differences in the initial stiffness and porosity of the investigated scaffolds prevent a clear comparison with our model in terms of mechanics-dependent tissue growth. Therefore, the results here obtained <italic>in silico</italic> can be considered as an indication, while further experimental assessments are needed to confirm these findings. For example, direct comparisons between <italic>in vivo</italic> and simulated tissue repair have been reported for untreated osteochondral defects (<xref ref-type="bibr" rid="B14">Duda et&#x20;al., 2005</xref>) and scaffold-supported large bone defects (<xref ref-type="bibr" rid="B36">Perier-Metz et&#x20;al., 2020</xref>) and could be in future performed also for osteochondral defects treated with biodegradable architectured scaffolds.</p>
<p>As limitation of the here presented model, the &#x201c;artificial&#x201d; scaffold configuration must be mentioned. The scaffold was defined &#x201c;artificial&#x201d; because it did not model a specific polymer nor scaffold architecture currently investigated in tissue engineering, but its simplified properties were shown to promote regeneration in a previous computational study (<xref ref-type="bibr" rid="B47">Tortorici et&#x20;al., 2021</xref>). The &#x201c;artificial&#x201d; scaffold in each investigated case had the same mechanical and architectural properties in its non-degraded state, but the possibility to degrade by each one of the three investigated modalities in turn. This situation was also &#x201c;artificial&#x201d; because, in reality, the degradation modality of a polymeric device is determined by both its chemical and architectural features (<xref ref-type="bibr" rid="B7">von Burkersroda et&#x20;al., 2002</xref>). Thus, polymeric scaffolds degrading by different modalities can hardly be expected to have equal properties prior to the onset of degradation. For example, changes in architecture might be required to obtain the same overall stiffness in scaffolds produced from different materials. Further limitations of this study include the simplified degradation behaviors that were implemented, as well as the evaluation of only mechanics-related phenomena. A first simplification in the implemented degradation behaviors was the clear-cut distinction between surface erosion and bulk degradation, while in reality polymeric degradation might involve a combination of the two modalities (<xref ref-type="bibr" rid="B15">G&#xf6;pferich, 1996</xref>). A second simplification was the direct proportionality between decrease in polymeric molecular weight and decrease in material elastic modulus assumed here. Although the loss of mechanical competence is a degradation-related phenomenon, the kinetics of molecular weight reduction and of elastic modulus decrease might differ (<xref ref-type="bibr" rid="B15">G&#xf6;pferich, 1996</xref>). For example, in highly porous PLGA scaffolds, the first stages of <italic>in&#x20;vitro</italic> degradation were characterized by a constant or slightly increased elastic modulus, while the molecular weight was already subjected to reductions (<xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>). Furthermore, the linear algorithms and the imposed degradation speeds employed for the comparison of the three degradation modalities were a third simplification, which has been addressed in the second part of this work. In fact, bulk degradation has been reported to follow an exponential behavior for several polymers, such as PCL, PDLLA, and PLGA (<xref ref-type="bibr" rid="B37">Pitt C et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B38">Pitt G et&#x20;al., 1981</xref>; <xref ref-type="bibr" rid="B54">Wu &#x26; Ding, 2004</xref>). Moreover, the degradation coefficients measured experimentally, e.g. for PCL and PDLLA, would cause a much slower scaffold degradation than the one modelled with the fast, matched, and slow degradation speeds, as shown when the experimentally-derived bulk degradation was implemented (<xref ref-type="fig" rid="F5">Figure&#x20;5C</xref>). Concerning the evaluation of the influence of degradation phenomena on osteochondral defect repair, this work focused on mechanics-dependent tissue formation. However, the biological influence of scaffold degradation might derive also from other factors, such as local variations in pH and the release of degradation by-products (<xref ref-type="bibr" rid="B42">Suganuma &#x26; Alexander, 1993</xref>; <xref ref-type="bibr" rid="B45">Taylor et&#x20;al., 1994</xref>; <xref ref-type="bibr" rid="B43">Sung et&#x20;al., 2004</xref>). Therefore, the here presented model does not provide a complete evaluation of the complex events involved in polymeric scaffold degradation and the consequent tissue formation. Nonetheless, the model offers valuable insights on the mechanical influences that can be expected from the degradation of a synthetic polymeric scaffold with strut-like architecture in osteochondral defects. These insights could be used as indications in the selection of biodegradable or bioresorbable materials for scaffolds to support the healing of osteochondral defects. The results of this study are expected to apply also to defects of greater width implanted with a scaffold composed of the same repetitive unit. In fact, a previous <italic>in silico</italic> model showed similar mechanics-dependent repair patterns in untreated osteochondral defects with radii of 5&#xa0;mm, 7&#xa0;mm, and 9&#xa0;mm, although with more fibrous tissue and less bone formation with increasing defect width (<xref ref-type="bibr" rid="B24">Kelly &#x26; Prendergast, 2005</xref>). On the other hand, the repair outcome supported by degradable scaffolds with non-repetitive architectural units and/or different morphologies should be specifically evaluated. Particularly, scaffolds obtained via production techniques that do not enable a precise control on the resulting material distribution might generate diverse local mechanical environments in the defect, e.g., due to walls of varying thickness or pores of different sizes. However, the simulation possibilities of the model are currently limited to scaffolds with axisymmetric geometries, thereby precluding an accurate assessment of devices with non-uniform properties in three dimensions. This issue could be addressed in future by employing a three-dimensional non-axisymmetric&#x20;model.</p>
<p>The here-presented model could be further developed to investigate the role of scaffold-supported tissue formation also for specific subpopulations, e.g., differing in age or sex. In fact, evidence suggests that age and sex might result in differences in tissue growth, e.g., when comparing chondrogenesis and osteogenesis of MSCs seeded in decellularized extra-cellular matrix taken from new-born, juvenile, and adult rabbits (<xref ref-type="bibr" rid="B52">Wang et&#x20;al., 2020</xref>), and in the progression of diseases, e.g., osteoarthritis (<xref ref-type="bibr" rid="B49">Urquhart et&#x20;al., 2008</xref>). The mechanobiological rules employed to simulate tissue formation may be adapted to reproduce the experimental outcomes obtained for various groups, as previously done in the context of bone regeneration when comparing adult and elderly mice (<xref ref-type="bibr" rid="B5">Borgiani et&#x20;al., 2019</xref>). To specifically investigate osteochondral tissue affected by osteoarthritis, additional modifications of the model, such as its geometry and the assigned material properties, might be required. In fact, osteoarthritis has been observed to influence the properties of all the joint tissues and to cause structural changes, e.g., the narrowing of the joint space (<xref ref-type="bibr" rid="B12">Davis et&#x20;al., 2021</xref>).</p>
<p>Taken together, our results suggest that, amongst the three possible degradation modalities of synthetic polymeric scaffolds with strut-like architecture, bulk degradation with autocatalysis and bulk degradation without autocatalysis cause the highest and the lowest mechanical instability of the scaffold, respectively (<xref ref-type="fig" rid="F6">Figure&#x20;6</xref>). As the instauration of autocatalysis consistently resulted in the prediction of mechanical failures, we propose that the choice of material and architecture of a scaffold for osteochondral defect repair should aim at avoiding this phenomenon. Moreover, the results of the simulation indicate that scaffolds with strut-like architectures degrading by surface erosion or by bulk degradation without autocatalysis can provide an equally adequate support to the mechanics-dependent repair of osteochondral defects, but only if the scaffold material degrades in a much longer time than the time of expected completion of the repair process.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Summary of the tissue repair outcomes obtained by simulating the degradation of a synthetic polymeric scaffold with strut-like architecture implanted in an osteochondral defect. The colors red, yellow, and green indicate mechanical failure, poor repair, and good repair, respectively.</p>
</caption>
<graphic xlink:href="fbioe-10-846665-g006.tif"/>
</fig>
</sec>
</body>
<back>
<sec id="s5">
<title>Data Availability Statement</title>
<p>The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.</p>
</sec>
<sec id="s6">
<title>Author Contributions</title>
<p>MT and SC designed the study. MT developed the <italic>in silico</italic> models and collected the data. MT, SC, AP, and GD interpreted the data. MT and SC drafted the manuscript. All authors read and revised the manuscript and approved its content.</p>
</sec>
<sec id="s7">
<title>Funding</title>
<p>This work was funded by the German Federal Ministry of Education and Research (BMBF) through Grant No. 13XP5048D. This work was further supported through the Collaborative Research Center 1444 funded by the German Research Foundation (DFG). The open access pubblication was supported by the Open Access Publication Fund of Charit&#xe9;-Universit&#xe4;tsmedizin Berlin.</p>
</sec>
<sec sec-type="COI-statement" id="s8">
<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="s9">
<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>
<ack>
<p>The authors thank the German Federal Ministry of Education and Research (BMBF) and the German Research Foundation (DFG) for funding this work and the Charit&#xe9;-Universit&#xe4;tsmedizin Berlin for enabling its open access publication.</p>
</ack>
<sec id="s10">
<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.2022.846665/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/fbioe.2022.846665/full&#x23;supplementary-material</ext-link>
</p>
<supplementary-material xlink:href="DataSheet1.docx" id="SM1" mimetype="application/docx" xmlns:xlink="http://www.w3.org/1999/xlink"/>
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