<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.1 20151215//EN"  "JATS-archivearticle1.dtd"><article article-type="research-article" dtd-version="1.1" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><front><journal-meta><journal-id journal-id-type="nlm-ta">elife</journal-id><journal-id journal-id-type="publisher-id">eLife</journal-id><journal-title-group><journal-title>eLife</journal-title></journal-title-group><issn pub-type="epub" publication-format="electronic">2050-084X</issn><publisher><publisher-name>eLife Sciences Publications, Ltd</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">60683</article-id><article-id pub-id-type="doi">10.7554/eLife.60683</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research Article</subject></subj-group><subj-group subj-group-type="heading"><subject>Computational and Systems Biology</subject></subj-group><subj-group subj-group-type="heading"><subject>Developmental Biology</subject></subj-group></article-categories><title-group><article-title>Vascular dimorphism ensured by regulated proteoglycan dynamics favors rapid umbilical artery closure at birth</article-title></title-group><contrib-group><contrib contrib-type="author" id="author-197031"><name><surname>Nandadasa</surname><given-names>Sumeda</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">https://orcid.org/0000-0002-4954-6376</contrib-id><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="other" rid="fund6"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197032"><name><surname>Szafron</surname><given-names>Jason M</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">https://orcid.org/0000-0001-9476-5175</contrib-id><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197033"><name><surname>Pathak</surname><given-names>Vai</given-names></name><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="fn" rid="con3"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197034"><name><surname>Murtada</surname><given-names>Sae-Il</given-names></name><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="fn" rid="con4"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197035"><name><surname>Kraft</surname><given-names>Caroline M</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="fn" rid="con5"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197036"><name><surname>O'Donnell</surname><given-names>Anna</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="fn" rid="con6"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197037"><name><surname>Norvik</surname><given-names>Christian</given-names></name><xref ref-type="aff" rid="aff4">4</xref><xref ref-type="fn" rid="con7"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197038"><name><surname>Hughes</surname><given-names>Clare</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">http://orcid.org/0000-0003-4726-5877</contrib-id><xref ref-type="aff" rid="aff5">5</xref><xref ref-type="fn" rid="con8"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197039"><name><surname>Caterson</surname><given-names>Bruce</given-names></name><xref ref-type="aff" rid="aff5">5</xref><xref ref-type="fn" rid="con9"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197051"><name><surname>Domowicz</surname><given-names>Miriam S</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">http://orcid.org/0000-0001-7860-4427</contrib-id><xref ref-type="aff" rid="aff6">6</xref><xref ref-type="fn" rid="con10"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-99178"><name><surname>Schwartz</surname><given-names>Nancy B</given-names></name><xref ref-type="aff" rid="aff6">6</xref><xref ref-type="fn" rid="con11"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197041"><name><surname>Tran-Lundmark</surname><given-names>Karin</given-names></name><xref ref-type="aff" rid="aff4">4</xref><xref ref-type="other" rid="fund8"/><xref ref-type="fn" rid="con12"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197042"><name><surname>Veigl</surname><given-names>Martina</given-names></name><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="aff" rid="aff7">7</xref><xref ref-type="other" rid="fund5"/><xref ref-type="other" rid="fund9"/><xref ref-type="other" rid="fund7"/><xref ref-type="fn" rid="con13"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197043"><name><surname>Sedwick</surname><given-names>David</given-names></name><xref ref-type="aff" rid="aff7">7</xref><xref ref-type="other" rid="fund5"/><xref ref-type="other" rid="fund9"/><xref ref-type="fn" rid="con14"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-197044"><name><surname>Philipson</surname><given-names>Elliot H</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff8">8</xref><xref ref-type="other" rid="fund4"/><xref ref-type="fn" rid="con15"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-18850"><name><surname>Humphrey</surname><given-names>Jay D</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">https://orcid.org/0000-0003-1011-2025</contrib-id><email>jay.humphrey@yale.edu</email><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="fn" rid="con16"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-66542"><name><surname>Apte</surname><given-names>Suneel S</given-names></name><contrib-id authenticated="true" contrib-id-type="orcid">https://orcid.org/0000-0001-8441-1226</contrib-id><email>aptes@ccf.org</email><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="other" rid="fund1"/><xref ref-type="other" rid="fund2"/><xref ref-type="other" rid="fund3"/><xref ref-type="fn" rid="con17"/><xref ref-type="fn" rid="conf1"/></contrib><aff id="aff1"><label>1</label><institution>Department of Biomedical Engineering, Cleveland Clinic Lerner Research Institute</institution><addr-line><named-content content-type="city">Cleveland</named-content></addr-line><country>United States</country></aff><aff id="aff2"><label>2</label><institution>Department of Biomedical Engineering, Yale University</institution><addr-line><named-content content-type="city">New Haven</named-content></addr-line><country>United States</country></aff><aff id="aff3"><label>3</label><institution>Case Comprehensive Cancer Center, Case Western Reserve University</institution><addr-line><named-content content-type="city">Cleveland</named-content></addr-line><country>United States</country></aff><aff id="aff4"><label>4</label><institution>Department of Experimental Medical Science and Wallenberg Center for Molecular Medicine, Lund University</institution><addr-line><named-content content-type="city">Lund</named-content></addr-line><country>Sweden</country></aff><aff id="aff5"><label>5</label><institution>The Sir Martin Evans Building, School of Biosciences, Cardiff University</institution><addr-line><named-content content-type="city">Cardiff</named-content></addr-line><country>United Kingdom</country></aff><aff id="aff6"><label>6</label><institution>Department of Pediatrics, University of Chicago</institution><addr-line><named-content content-type="city">Chicago</named-content></addr-line><country>United States</country></aff><aff id="aff7"><label>7</label><institution>Department of Medicine, Case Western Reserve University</institution><addr-line><named-content content-type="city">Cleveland</named-content></addr-line><country>United States</country></aff><aff id="aff8"><label>8</label><institution>The Women's Health Institute, Department of Obstetrics and Gynecology, Cleveland Clinic</institution><addr-line><named-content content-type="city">Cleveland</named-content></addr-line><country>United States</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Downs</surname><given-names>Karen</given-names></name><role>Reviewing Editor</role><aff><institution>University of Wisconsin-Madison School of Medicine and Public Health</institution><country>United States</country></aff></contrib><contrib contrib-type="senior_editor"><name><surname>Stainier</surname><given-names>Didier YR</given-names></name><role>Senior Editor</role><aff><institution>Max Planck Institute for Heart and Lung Research</institution><country>Germany</country></aff></contrib></contrib-group><pub-date date-type="publication" publication-format="electronic"><day>10</day><month>09</month><year>2020</year></pub-date><pub-date pub-type="collection"><year>2020</year></pub-date><volume>9</volume><elocation-id>e60683</elocation-id><history><date date-type="received" iso-8601-date="2020-07-02"><day>02</day><month>07</month><year>2020</year></date><date date-type="accepted" iso-8601-date="2020-09-09"><day>09</day><month>09</month><year>2020</year></date></history><permissions><copyright-statement>© 2020, Nandadasa et al</copyright-statement><copyright-year>2020</copyright-year><copyright-holder>Nandadasa et al</copyright-holder><ali:free_to_read/><license xlink:href="http://creativecommons.org/licenses/by/4.0/"><ali:license_ref>http://creativecommons.org/licenses/by/4.0/</ali:license_ref><license-p>This article is distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use and redistribution provided that the original author and source are credited.</license-p></license></permissions><self-uri content-type="pdf" xlink:href="elife-60683-v2.pdf"/><related-article ext-link-type="doi" id="ra1" related-article-type="commentary" xlink:href="10.7554/eLife.63128"/><abstract><p>The umbilical artery lumen closes rapidly at birth, preventing neonatal blood loss, whereas the umbilical vein remains patent longer. Here, analysis of umbilical cords from humans and other mammals identified differential arterial-venous proteoglycan dynamics as a determinant of these contrasting vascular responses. The umbilical artery, but not the vein, has an inner layer enriched in the hydrated proteoglycan aggrecan, external to which lie contraction-primed smooth muscle cells (SMC). At birth, SMC contraction drives inner layer buckling and centripetal displacement to occlude the arterial lumen, a mechanism revealed by biomechanical observations and confirmed by computational analyses. This vascular dimorphism arises from spatially regulated proteoglycan expression and breakdown. Mice lacking aggrecan or the metalloprotease ADAMTS1, which degrades proteoglycans, demonstrate their opposing roles in umbilical vascular dimorphism, including effects on SMC differentiation. Umbilical vessel dimorphism is conserved in mammals, suggesting that differential proteoglycan dynamics and inner layer buckling were positively selected during evolution.</p></abstract><kwd-group kwd-group-type="author-keywords"><kwd>umbilical cord</kwd><kwd>proteoglycans</kwd><kwd>extracellular matrix</kwd><kwd>vascular smooth muscle</kwd><kwd>birth</kwd><kwd>vascular engineering</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>Human</kwd><kwd>Mouse</kwd></kwd-group><funding-group><award-group id="fund1"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000002</institution-id><institution>National Institutes of Health</institution></institution-wrap></funding-source><award-id>HL107147</award-id><principal-award-recipient><name><surname>Apte</surname><given-names>Suneel S</given-names></name></principal-award-recipient></award-group><award-group id="fund2"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000002</institution-id><institution>National Institutes of Health</institution></institution-wrap></funding-source><award-id>HL141130</award-id><principal-award-recipient><name><surname>Apte</surname><given-names>Suneel S</given-names></name></principal-award-recipient></award-group><award-group id="fund3"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000968</institution-id><institution>American Heart Association</institution></institution-wrap></funding-source><award-id>17DIA33820024</award-id><principal-award-recipient><name><surname>Apte</surname><given-names>Suneel S</given-names></name></principal-award-recipient></award-group><award-group id="fund4"><funding-source><institution-wrap><institution>Sabrina's Foundation</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Philipson</surname><given-names>Elliot H</given-names></name></principal-award-recipient></award-group><award-group id="fund5"><funding-source><institution-wrap><institution>National Children's Study</institution></institution-wrap></funding-source><award-id>Formative Research Project L01-3-RT-01-E</award-id><principal-award-recipient><name><surname>Veigl</surname><given-names>Martina</given-names></name><name><surname>Sedwick</surname><given-names>David</given-names></name></principal-award-recipient></award-group><award-group id="fund6"><funding-source><institution-wrap><institution>Mark Lauer Pediatric Research Grant</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Nandadasa</surname><given-names>Sumeda</given-names></name></principal-award-recipient></award-group><award-group id="fund7"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/100000002</institution-id><institution>National Institutes of Health</institution></institution-wrap></funding-source><award-id>CA43703</award-id><principal-award-recipient><name><surname>Veigl</surname><given-names>Martina</given-names></name></principal-award-recipient></award-group><award-group id="fund8"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100003793</institution-id><institution>Swedish Heart-Lung Foundation</institution></institution-wrap></funding-source><principal-award-recipient><name><surname>Tran-Lundmark</surname><given-names>Karin</given-names></name></principal-award-recipient></award-group><award-group id="fund9"><funding-source><institution-wrap><institution>National Children's Study</institution></institution-wrap></funding-source><award-id>Contract # HHSN272500800009C</award-id><principal-award-recipient><name><surname>Veigl</surname><given-names>Martina</given-names></name><name><surname>Sedwick</surname><given-names>David</given-names></name></principal-award-recipient></award-group><funding-statement>The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.</funding-statement></funding-group><custom-meta-group><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>Morphologic, molecular, biomechanical and computational analyses show that the specialized extracellular matrix architecture of the umbilical artery contributes to its rapid closure at birth and regulates smooth muscle cell differentiation.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="s1" sec-type="intro"><title>Introduction</title><p>The umbilical cord, typically containing two arteries and one vein in humans, is a crucial fetal structure in placental mammals. Umbilical arteries carry fetal blood to the placental vascular bed, whereas the umbilical vein returns oxygenated blood to the fetus. Neonatal respiration at birth renders the maternal oxygen supply redundant. Umbilical arteries commence closure rapidly after delivery of the newborn whereas the veins remain open longer. The cord is routinely clamped following delivery and divided between the clamps in modern obstetric practice. Timing of cord clamping after birth, whether early or late, is extensively debated (<xref ref-type="bibr" rid="bib27">Niermeyer, 2015</xref>; <xref ref-type="bibr" rid="bib38">Tarnow-Mordi et al., 2017</xref>). A recent recommendation suggested clamping no earlier than 30–60 s after birth to facilitate the placental transfusion (<xref ref-type="bibr" rid="bib37">TACoOaG, 2017</xref>). Although the necessity of clamping is rarely questioned, it appears to be a modern practice (<xref ref-type="bibr" rid="bib7">Downey and Bewley, 2012</xref>). Cord clamping is rarely practiced in domesticated animals and certainly not in wild animals, yet all current mammalian species have survived evolutionarily. We hypothesized that intrinsic design characteristics of mammalian umbilical arteries prevent blood loss at birth without clamping.</p><p>Prior histological work revealed that umbilical arteries have a bilaminar structure (<xref ref-type="bibr" rid="bib23">Meyer et al., 1978</xref>) but lack elastic lamellae, which endow large arteries with resilience during cyclic loading (<xref ref-type="bibr" rid="bib39">Wagenseil and Mecham, 2009</xref>). However, the molecular mechanism underlying the bilayered structure and its relationship to arterial occlusion remains obscure. Here, we used a multi-disciplinary approach integrating a variety of morphologic approaches with mechanical testing, computational analysis and mouse mutants to demonstrate the molecular and biomechanical basis for rapid umbilical artery closure. The findings emphasize the dual importance of extracellular matrix proteoglycans in regulation of cell differentiation and conferment of desirable tissue mechanical characteristics.</p></sec><sec id="s2" sec-type="results"><title>Results</title><sec id="s2-1"><title>The umbilical artery has a bilaminar wall</title><p>Three-dimensional imaging of term human umbilical cords, using synchrotron-based phase contrast micro-CT with effective pixel size 1.63 × 1.63 μm<sup>2</sup> (<xref ref-type="bibr" rid="bib28">Norvik et al., 2020</xref>) and histology, identified a much thicker tunica media (TM) in the umbilical artery than in the vein, with a visibly different structure (<xref ref-type="fig" rid="fig1">Figure 1a–c</xref>, <xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1a–c</xref>, <xref ref-type="video" rid="fig1video1">Figure 1—videos 1</xref>, <xref ref-type="video" rid="fig1video2">2</xref>). Most umbilical arteries were occluded at birth independent of delivery method or cord region analyzed, whereas umbilical veins remained patent (<xref ref-type="fig" rid="fig1">Figure 1b</xref>, <xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1b</xref>). Smooth muscle cell (SMC) markers showed similar staining intensities within inner and outer TM of the umbilical arteries and TM of the vein with alternating layers of longitudinal and circumferentially oriented SMCs (<xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1c,d</xref>). The veins showed fewer layers of SMCs compared to the arteries (<xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1c,d</xref>).</p><fig-group><fig id="fig1" position="float"><label>Figure 1.</label><caption><title>Dimorphism of the human umbilical artery and vein.</title><p>(<bold>a</bold>) Synchrotron imaging of umbilical vessels at birth illustrates a bilayered arterial wall comprising an inner buckled tunica media (TM) (red) and outer TM (purple) but no distinct inner layer or buckling in the vein. X-Y and Y-Z image planes are indicated by red dashed lines (n = 3 umbilical cords). (<bold>b</bold>) Quantitation of luminal cross-sectional area at birth shows that the umbilical arteries are occluded whereas the veins remain patent (top) and have significantly thicker walls (bottom) (n = 20 cords, error bars indicate mean ± S.E.M., whiskers indicate minimum and maximum values. ***, p&lt;0.001). (<bold>c</bold>) Alcian blue, eosin (pink) and nuclear fast red staining of umbilical vessel cross-sections shows a proteoglycan-rich (blue) inner TM in the umbilical artery but not the vein. Quantified staining intensity is shown on the right (n = 6 umbilical cords, whiskers indicate minimum and maximum values, ***, p&lt;0.001). (<bold>d</bold>) Chondroitin sulfate (CS), heparan sulfate (HS), aggrecan and versican immunofluorescence (n = 4 cords for each antibody) showing that CS staining corresponds with aggrecan and versican staining and alcian blue in (<bold>c</bold>). (<bold>e</bold>) Volcano plots illustrating differential gene expression between human umbilical artery (red) and vein (green) (top, n = 4 umbilical arteries and veins) and differential gene expression between human umbilical artery inner TM (red) and the outer TM (green) (bottom, n = 2). (<bold>f</bold>) RNA in situ hybridization shows robust <italic>ACAN</italic> and <italic>VCAN</italic> expression (red signal) in the inner artery TM and weak expression in the vein (n = 3 umbilical cords for each in situ probe). * marks the vessel lumen. Brackets in c,d,f mark the TM. Wj, Wharton’s jelly. Scale bars = 100 μm in c,d,f.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig1-v2.tif"/></fig><fig id="fig1s1" position="float" specific-use="child-fig"><label>Figure 1—figure supplement 1.</label><caption><title>Morphological and cellular characteristics of the human umbilical arteries and veins at birth.</title><p>(<bold>a</bold>) Synchrotron image from <xref ref-type="fig" rid="fig1">Figure 1a</xref> without red and purple shading illustrates contrast between inner and outer arterial tunica media (TM) as well as inner TM buckling, compared with a uniform appearance of venous TM and lack of buckling (n = 3 umbilical cords). (<bold>b</bold>) Hematoxylin and eosin stained human umbilical artery cross-sections collected from the placental and fetal ends show occlusion of the umbilical artery with buckling of its interior, while the vein remains patent and lacks buckling (n = 25 umbilical cords). (<bold>c</bold>) Hematoxylin and eosin stained cross-sections show multiple cell layers composed of alternating circumferentially (C) or longitudinally (L) oriented smooth muscle cells (SMC) in the umbilical artery. Curved white arrows indicate internal protrusion of the inner TM of the umbilical artery arising from buckling. (<bold>d</bold>) α-SMA (α-smooth muscle actin, red) and DAPI (blue) staining of the umbilical artery and vein shows distinct orientation of the SMC layers as in (<bold>c</bold>) (n = 3 umbilical cords). (<bold>e</bold>) (<italic>Left</italic>) Alcian blue staining shows intense staining of the inner arterial TM, with radially-oriented rounded cells contrasting with outer TM cells having elongated morphology. Eosin (pink) and nuclear fast red counterstaining. (<italic>Right</italic>) Sox9 immunostaining shows nuclear staining of SMCs of the inner umbilical artery TM (n = 6 umbilical cords, * marks the vessel lumen in panels (<bold>c,d,e</bold>). White brackets in (<bold>c,d</bold> and <bold>e</bold>) mark the TM. Tm, tunica media). Scale bars = 200 μm in (<bold>b</bold>) and <bold>c</bold>, 100 μm in <bold>d</bold> and 50 μm in <bold>e</bold>.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig1-figsupp1-v2.tif"/></fig><fig id="fig1s2" position="float" specific-use="child-fig"><label>Figure 1—figure supplement 2.</label><caption><title>Transcriptome comparison and pathway analysis of differences in the human umbilical artery and vein.</title><p>(<bold>a</bold>) Heat map of a subset of differentially expressed genes (1.9 fold-change) from umbilical artery vs vein (n = 4 umbilical cords). (<bold>b</bold>) Ingenuity pathway analysis (IPA) summary of the most significantly different pathways (n = 4 umbilical cords).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig1-figsupp2-v2.tif"/></fig><fig id="fig1s3" position="float" specific-use="child-fig"><label>Figure 1—figure supplement 3.</label><caption><title>Transcriptome comparison and pathway analysis of differences in the human inner umbilical artery tunica media (TM) vs the outer tunica media.</title><p>(<bold>a</bold>) Heat map of a subset of differentially expressed genes (1.9 fold-change) (n = 2 umbilical arteries, two inner TM samples matched to two outer TM samples). (<bold>b</bold>) Ingenuity pathway analysis (IPA) showing significantly different pathways in the umbilical artery inner and outer tunica media (n = 2 umbilical arteries, two inner TM samples matched to two outer TM samples).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig1-figsupp3-v2.tif"/></fig><media id="fig1video1" mime-subtype="mp4" mimetype="video" xlink:href="elife-60683-fig1-video1.mp4"><label>Figure 1—video 1.</label><caption><title>Synchrotron image stack of an umbilical artery.</title><p>A human umbilical artery was imaged cross-sectionally and the images reconstructed along the luminal axis using Amira.</p></caption></media><media id="fig1video2" mime-subtype="mp4" mimetype="video" xlink:href="elife-60683-fig1-video2.mp4"><label>Figure 1—video 2.</label><caption><title>Synchrotron image stack of an umbilical vein.</title><p>A human umbilical vein was imaged cross-sectionally and the images reconstructed along the luminal axis using Amira.</p></caption></media></fig-group><p>Alcian blue, which binds sulfated glycosaminoglycans (GAGs), intensely stained the inner layer of the bilayered arterial TM but only the innermost three to four cell layers of the venous TM (<xref ref-type="fig" rid="fig1">Figure 1c</xref>). SMCs in this GAG-rich region of the arteries were radially oriented and round, with nuclear-localized Sox9, a chondrogenic factor (<xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1e</xref>; <xref ref-type="bibr" rid="bib26">Ng et al., 1997</xref>). The distribution of chondroitin sulfate (CS) coincided with Alcian blue staining (<xref ref-type="fig" rid="fig1">Figure 1c–d</xref>, <xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1e</xref>), whereas heparan sulfate was more abundant in the outer arterial TM (<xref ref-type="fig" rid="fig1">Figure 1d</xref>), suggesting that the inner TM was enriched in CS-proteoglycans (CSPGs). RNA microarray data from matched human umbilical arteries and veins showed, among many differentially expressed genes (<xref ref-type="fig" rid="fig1">Figure 1e</xref>, <xref ref-type="fig" rid="fig1s2">Figure 1—figure supplement 2</xref>, Sup. array data-1), arterial prevalence of mRNAs for <italic>ACAN</italic> and <italic>VCAN</italic> encoding CSPGs bearing the most CS-chains, aggrecan and versican, respectively (<xref ref-type="fig" rid="fig1">Figure 1e</xref>). Microarray analysis of the inner versus outer arterial TM identified stronger <italic>ACAN</italic> and <italic>VCAN</italic> expression in the inner TM, among other differences (<xref ref-type="fig" rid="fig1">Figure 1e</xref>, <xref ref-type="fig" rid="fig1s3">Figure 1—figure supplement 3</xref>, Sup. array data-2). RNA in situ hybridization (RNA-ISH) localized strong <italic>ACAN</italic> and <italic>VCAN</italic> expression in inner arterial TM SMC, and immunostaining showed versican and aggrecan core proteins in a similar distribution as alcian blue and anti-CS staining (<xref ref-type="fig" rid="fig1">Figure 1c,d,f</xref>). Versican is a well-characterized vascular component (<xref ref-type="bibr" rid="bib40">Wight and Merrilees, 2004</xref>), and aggrecan, which is known as a cartilage and neural proteoglycan (<xref ref-type="bibr" rid="bib18">Lauing et al., 2014</xref>; <xref ref-type="bibr" rid="bib34">Schwartz and Domowicz, 2014</xref>), is emerging as a significant CSPG in vascular disease (reviewed in <xref ref-type="bibr" rid="bib15">Koch et al., 2020</xref>).</p></sec><sec id="s2-2"><title>ADAMTS proteoglycanases are differentially expressed in the umbilical artery and vein</title><p>Aggrecan and versican are proteolytically cleaved by ADAMTS1, 4, 5, and 9 (<xref ref-type="bibr" rid="bib3">Dancevic et al., 2016</xref>). <italic>ADAMTS1</italic> and <italic>ADAMTS4</italic> mRNAs had higher levels in the venous wall in microarrays (<xref ref-type="fig" rid="fig1">Figure 1e</xref>, Sup. Array data-1), and RNA-ISH demonstrated stronger expression of <italic>ADAMTS1, ADAMTS4, ADAMTS5,</italic> and <italic>ADAMTS9</italic> in the veins (<xref ref-type="fig" rid="fig2">Figure 2a</xref>). <italic>ADAMTS1</italic> was the most strongly expressed, localizing to venous endothelium and TM, with stronger umbilical artery expression seen in the outer than inner TM (<xref ref-type="fig" rid="fig2">Figure 2a</xref>). <italic>ADAMTS9</italic> was similarly expressed as <italic>ADAMTS1</italic>, whereas <italic>ADAMTS4</italic> and <italic>ADAMTS5</italic> mRNAs were restricted to umbilical vein endothelium and some venous SMC (<xref ref-type="fig" rid="fig2">Figure 2a</xref>). Neo-epitope antibodies detecting ADAMTS-cleaved aggrecan and versican (anti-NITEGE and anti-DPEAAE, respectively) (<xref ref-type="bibr" rid="bib17">Lark et al., 1995</xref>; <xref ref-type="bibr" rid="bib31">Sandy et al., 1992</xref>; <xref ref-type="bibr" rid="bib32">Sandy et al., 2001</xref>) showed strong staining throughout the venous TM and in the outer arterial TM, but not the inner arterial TM (<xref ref-type="fig" rid="fig2">Figure 2b</xref>). Thus, proteoglycan accumulation in the inner TM of the umbilical artery may result from higher <italic>ACAN</italic> and <italic>VCAN</italic> expression and less proteolysis. In contrast, lower <italic>ACAN</italic> and <italic>VCAN</italic> expression and greater ADAMTS levels within the umbilical vein may preclude proteoglycan accumulation.</p><fig id="fig2" position="float"><label>Figure 2.</label><caption><title>ADAMTS proteoglycanases are highly expressed and active in the human umbilical vein.</title><p>(<bold>a</bold>) RNA in situ hybridization shows robust <italic>ADAMTS1</italic> and <italic>ADAMTS9</italic> expression in umbilical vein endothelium and tunica media (TM) and in outer arterial TM. Robust <italic>ADAMTS4</italic> and <italic>ADAMTS5</italic> expression was confined to the venous endothelium, with moderate ADAMTS4 expression and minimal ADAMTS5 expression in SMC (n = 3 umbilical cords for each probe). (<bold>b</bold>) ADAMTS-cleaved aggrecan (anti-NITEGE, red) and versican (anti-DPEAAE, red) both showed strong ADAMTS proteolytic activity throughout the venous wall and the outer artery TM. Unlike aggrecan, extensive versican proteolysis is seen in the arterial intima and sub-intima (n = 4 umbilical cords for each antibody). Wj, Wharton’s jelly. The brackets mark TM boundaries. Scale bars in <bold>a-b</bold> = 100 μm.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig2-v2.tif"/></fig></sec><sec id="s2-3"><title>ADAMTS-mediated differential proteoglycan abundance in the umbilical artery and vein is evolutionarily conserved</title><p>We postulated that abundant hydrated proteoglycans in the inner arterial TM provided compressive stiffness that could not only prevent kinking and premature occlusion but could potentially facilitate rapid umbilical artery closure at birth. If so, similar adaptations should be present in other mammals. Analysis of umbilical cords from nine large primate and non-primate mammals disclosed similar dimorphism, namely, umbilical arteries were occluded and had thicker walls with similar infolding of the inner arterial TM (<xref ref-type="fig" rid="fig3">Figure 3a</xref>) and strong Alcian blue and CS-staining, contrasting with veins (<xref ref-type="fig" rid="fig3">Figure 3a,b</xref>). Anti-aggrecan and anti-NITEGE stained several animal species, confirming aggrecan abundance in the inner arterial TM and robust aggrecan cleavage resulting from ADAMTS protease activity in the outer TM of the artery and the TM of the vein (<xref ref-type="fig" rid="fig3">Figure 3c–d</xref>, <xref ref-type="fig" rid="fig3s1">Figure 3—figure supplement 1</xref>).</p><fig-group><fig id="fig3" position="float"><label>Figure 3.</label><caption><title>Aggrecan enrichment in the inner umbilical artery tunica media (TM) and its proteolysis in the umbilical vein is a characteristic of large mammals.</title><p>(<bold>a</bold>) Alcian blue-eosin staining of umbilical cord sections shows proteoglycan enrichment (blue) in the inner arterial tunica media (TM). The elephant umbilical vein was unavailable. (<bold>b</bold>) Anti-CS immunofluorescence (7D4, green) shows enrichment in the inner arterial TM. Bonobo cords lacked 7D4 reactivity. (<bold>c,d</bold>) Aggrecan and anti-NITEGE immunostaining from reactive species showed aggrecan enrichment in the inner arterial TM and aggrecan proteolysis in the vein and outer artery TM. n = 3 for Gazelle and n = 1 for other mammals. Triplicate sections were stained from each animal cord. Scale bars = 100 μm and 200 μm. * indicates the vessel lumen.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig3-v2.tif"/></fig><fig id="fig3s1" position="float" specific-use="child-fig"><label>Figure 3—figure supplement 1.</label><caption><title>Aggrecan cleavage site conservation in mammals.</title><p>The cleavage site location is shown over the sequence alignment by scissors, and the critical Glu(E) residue essential for cleavage is shown by red highlighting. Bold text identifies the cleavage-revealed neo-epitope NITEGE. Residue numbers are indicated for human and mouse aggrecan only.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig3-figsupp1-v2.tif"/></fig></fig-group></sec><sec id="s2-4"><title>Aggrecan and ADAMTS1 are necessary for normal umbilical cord morphogenesis</title><p>Mouse umbilical cords also demonstrated vascular dimorphism (<xref ref-type="fig" rid="fig4">Figure 4a</xref>), suggesting that genetically modified mice would provide mechanistic insights into proteoglycan dynamics and its impact. Aggrecan and versican immunofluorescence showed strong staining in the mouse umbilical artery inner TM and adventitia, with weaker staining in the veins (<xref ref-type="fig" rid="fig4">Figure 4b</xref>). <italic>Acan, Vcan</italic> and <italic>Adamts1,4,5,9</italic> RNA-ISH at early (E12.5) and late (E18.5) gestational stages showed that <italic>Acan</italic> and <italic>Vcan</italic> were strongly expressed in the umbilical arteries (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1a</xref>). <italic>Adamts1</italic> was the most highly expressed proteoglycanase in the mouse umbilical vein just prior to parturition (E18.5) (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1a</xref>), evidenced by strong β-gal staining in venous TM, adventitia and endothelium; the inner umbilical artery TM and endothelium of <italic>Adamts1</italic><sup>lacZ/+</sup> embryos lacked β-gal staining (<xref ref-type="fig" rid="fig4">Figure 4c</xref>). Although <italic>Adamts9</italic> mutant embryos were previously observed to have short umbilical cords, abnormal umbilical artery development, and to die by 14.5 days of gestation (<xref ref-type="bibr" rid="bib25">Nandadasa et al., 2015</xref>), umbilical cord development was not previously investigated in mutants of the two genes implicated here as potentially critical for umbilical cord vascular dimorphism, <italic>Acan</italic> and <italic>Adamts1</italic>.</p><fig-group><fig id="fig4" position="float"><label>Figure 4.</label><caption><title>Defective morphogenesis in <italic>Acan</italic> and <italic>Adamts1</italic> mutant mouse umbilical cords.</title><p>(<bold>a</bold>) H and E staining of E18.5 wild-type cords showing thicker umbilical arterial (A) and thinner venous (V) wall (n = 6 umbilical cords). (<bold>b</bold>) Aggrecan and versican localization (red, DAPI counterstain blue) in E18.5 wild-type cords showing staining in the arterial inner tunica media (TM) and adventitia but not the vein (n = 3 umbilical cords). (<bold>c</bold>) β-gal (blue) and eosin (red) staining of E18.5 <italic>Adamts1<sup>LacZ</sup></italic><sup>/+</sup> (<italic>Adamts1<sup>+/-</sup></italic>) cord showing strong <italic>Adamts1</italic> expression in venous endothelium and TM and outer artery TM (n = 3 umbilical cords). (<bold>d</bold>) Short umbilical cords in E18.5 <italic>Adamts1<sup>-/-</sup></italic> and <italic>Acan<sup>-/-</sup></italic> embryos compared to wild type. Red arrowhead indicates an omphalocele in <italic>Adamts1<sup>-/-</sup></italic> embryos. (<bold>e</bold>) H &amp; E staining of E18.5 wild type, <italic>Acan<sup>-/-</sup></italic> and <italic>Adamts1<sup>-/-</sup></italic> cord cross-sections showing thinner walls in <italic>Acan<sup>-/-</sup></italic> umbilical vessels and thicker walls in <italic>Adamts1<sup>-/-</sup></italic> umbilical vessels. (<bold>f–g</bold>) Cord length, TM thickness and vessel luminal area quantifications for <italic>Adamts1<sup>-/-</sup></italic> (<bold>f</bold>) and <italic>Acan<sup>-/-</sup></italic> mice (<bold>g</bold>) at E18.5 compared to wild-type littermates. <italic>Acan<sup>-/-</sup></italic> umbilical cords show larger lumens and <italic>Adamts1<sup>-/-</sup></italic> vessels show smaller lumens in (n = 7–11 umbilical cords each, whiskers indicate minimum and maximum values, *, p&lt;0.05; **, p&lt;0.01; ***, p&lt;0.001; ****, p&lt;0.0001). (<bold>h</bold>) Phospho-histone H3 (pHH3) staining shows significantly fewer proliferating cells (white arrowheads) in <italic>Acan<sup>-/-</sup></italic> umbilical vessels. Dotted white lines mark the boundaries of vessel lumens (n = 4 cords each, whiskers indicate minimum and maximum values, **, p&lt;0.001; *, p&lt;0.05). Scale bars = 100 μm in (<bold>a</bold>), 25 μm in (<bold>c</bold>), 100 μm and 50 μm in (<bold>e</bold>).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig4-v2.tif"/></fig><fig id="fig4s1" position="float" specific-use="child-fig"><label>Figure 4—figure supplement 1.</label><caption><title>ADAMTS, <italic>Vcan</italic> and <italic>Acan</italic> expression and impact of aggrecan loss on early mouse umbilical cord and vessel development.</title><p>(<bold>a</bold>) RNA in situ hybridization analysis of E12.5 and E18.5 mouse umbilical cord sections <italic>Acan, Vcan</italic> and relevant ADAMTS genes (n = 3 umbilical cords for each probe). (<bold>b</bold>) E12.5 and E14.5 wildtype and <italic>Acan<sup>-/-</sup></italic> embryos have a comparable umbilical cord length (n = 2 <italic>Acan<sup>-/-</sup></italic> at E12.5 and n = 3 at E14.5). (<bold>c</bold>) Observed and expected genotype ratios at different embryonic stages from <italic>Acan<sup>+/-</sup></italic> and <italic>Adamts1<sup>+/-</sup></italic> intercrosses. (<bold>d</bold>) Hematoxylin and eosin staining of E14.5 wild type and <italic>Acan<sup>-/-</sup></italic> umbilical cords shows completion of longitudinal to circumferential reorientation of smooth muscle cells in the mutant arterial tunica media (TM). Upper panels show tangential longitudinal sections whereas the lower panels are taken through the approximate center of each vessel (n = 3 umbilical cords of each genotype). Adv, adventitia; Tm, tunica media. Scale bars in <bold>a</bold> = 100 μm, 3 mm and 7 mm in (<bold>b</bold>) and 50 μm in (<bold>d</bold>).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig4-figsupp1-v2.tif"/></fig></fig-group><p>The <italic>Adamts1</italic><sup>-/-</sup> mutant is an insertion of an IRES lacZ-bearing cassette into intron 1 of the gene (<xref ref-type="bibr" rid="bib30">Oller et al., 2017</xref>). This insertion reveals <italic>Adamts1</italic> expression via staining for ß-galactosidase activity, and eliminated expression from the targeted allele but in its hemizygous state, the insertion led to reduction in both mRNA and protein (<xref ref-type="bibr" rid="bib30">Oller et al., 2017</xref>). The <italic>Acan</italic><sup>cmd-Bc</sup> allele is a spontaneous mutation found in a BALB/C colony (<xref ref-type="bibr" rid="bib2">Bell et al., 1986</xref>) and resulted from deletion of exon 2 through exon 18 (<xref ref-type="bibr" rid="bib16">Krueger et al., 1999</xref>). <italic>Acan<sup>-/-</sup></italic> embryos do not survive past birth (<xref ref-type="bibr" rid="bib16">Krueger et al., 1999</xref>; <xref ref-type="bibr" rid="bib18">Lauing et al., 2014</xref>) and few surviving <italic>Adamts1<sup>-/-</sup></italic> mice were identified at the time of weaning (<xref ref-type="bibr" rid="bib30">Oller et al., 2017</xref>). <italic>Acan</italic> mutants are thought to succumb to respiratory failure resulting from soft tracheal cartilages and ribs, whereas the cause of <italic>Adamts1<sup>-/-</sup></italic> lethality is unknown. At E18.5, <italic>Acan<sup>-/-</sup></italic> and <italic>Adamts1<sup>-/-</sup></italic> mutants each had significantly short umbilical cords (<xref ref-type="fig" rid="fig4">Figure 4d</xref>) demonstrating their requirement for proper umbilical cord development. Umbilical cord histology showed thinner vascular walls in <italic>Acan<sup>-/-</sup></italic> umbilical vessels, and conversely, thicker vascular walls in <italic>Adamts1<sup>-/-</sup></italic> umbilical vessels (<xref ref-type="fig" rid="fig4">Figure 4e–g</xref>). At earlier developmental stages (E12.5 to E14.5), lack of aggrecan did not affect either umbilical cord length or circumferential SMC reorientation (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1b–d</xref>), which occurs around E13.5 and is defective in <italic>Adamts9</italic> mutants (<xref ref-type="bibr" rid="bib25">Nandadasa et al., 2015</xref>). Furthermore, lack of aggrecan did not impair the survival of mouse embryos until parturition, since <italic>Acan<sup>-/-</sup></italic> embryos were observed at the expected Mendelian ratio at E18.5 (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1c</xref>). Thus, <italic>Acan</italic> and <italic>Adamts1</italic> appear to be involved in umbilical vessel development from early gestation, but their functions manifest near parturition.</p></sec><sec id="s2-5"><title>Contrasting SMC phenotypes in <italic>Acan</italic> and <italic>Adamts1</italic>-deficient umbilical cords</title><p>The arterial and venous lumina were smaller in <italic>Adamts1<sup>-/-</sup></italic> mice relative to wild-type, indicating that their thicker vascular walls compromised luminal diameter, and larger in <italic>Acan<sup>-/-</sup></italic> mice (<xref ref-type="fig" rid="fig4">Figure 4e–g</xref>). Phospho-histone H3 staining revealed fewer proliferating cells in <italic>Acan<sup>-/-</sup></italic> umbilical cords at E18.5 (<xref ref-type="fig" rid="fig4">Figure 4h</xref>). Immunostaining for SMC markers smooth muscle α-actin (SMA), smooth muscle myosin heavy chain (SMMHC), and phosphorylated myosin light chain (pMLC) showed weaker intensity in <italic>Acan<sup>-/-</sup></italic> umbilical arteries compared to wild-type (<xref ref-type="fig" rid="fig5">Figure 5a,b</xref>). In contrast, <italic>Adamts1<sup>-/-</sup></italic> umbilical vessels showed stronger SMA, SMMHC and pMLC staining than wild-type littermates and apparent overgrowth of the arterial and venous walls (<xref ref-type="fig" rid="fig5">Figure 5c</xref>). Intriguingly, endomucin, a venous endothelium-specific marker (<xref ref-type="bibr" rid="bib4">dela Paz and D'Amore, 2009</xref>), also stained <italic>Adamts1<sup>-/-</sup></italic> umbilical arterial endothelium (<xref ref-type="fig" rid="fig5">Figure 5c</xref>) suggesting that ADAMTS1 may have a role in specifying artery/vein identity. Immunostaining of E17.5 <italic>Adamts1<sup>-/-</sup></italic> umbilical cords indicated a crucial role for ADAMTS1 in regulating proteoglycan dynamics in the mouse umbilical cord. Specifically, we observed robust aggrecan and versican accumulation in the <italic>Adamts1<sup>-/-</sup></italic> umbilical vein and in the outer TM of the <italic>Adamts1<sup>-/-</sup></italic> umbilical artery (<xref ref-type="fig" rid="fig6">Figure 6a–d</xref>) with severe reduction of aggrecan and versican neo-epitope staining (<xref ref-type="fig" rid="fig6">Figure 6a–d</xref>).</p><fig id="fig5" position="float"><label>Figure 5.</label><caption><title>Contrasting smooth muscle cell (SMC) phenotype modulation in <italic>Acan</italic> and <italic>Adamts1</italic>-deficient umbilical vessels.</title><p>(<bold>a</bold>) Aggrecan (green) and α-SMA staining (red) in E18.5 umbilical cords show loss of aggrecan and weak α-SMA staining in <italic>Acan<sup>-/-</sup></italic> vessels (n = 3 umbilical cords each genotype). (<bold>b</bold>) Smooth muscle myosin heavy chain (SMMHC, red) and phosphorylated myosin light chain (pMLC, green) staining in E18.5 umbilical cords showing dramatic signal attenuation in the <italic>Acan<sup>-/-</sup></italic> vessels (n = 3 umbilical cords each genotype) (<bold>c</bold>) pMLC (green), endomucin (red), α-SMA (red, center panels) and SMMHC (red, right-hand panels) staining shows blunted dimorphism of <italic>Adamts1</italic><sup>-/-</sup> umbilical artery and vein with stronger expression of differentiated SMC markers in <italic>Adamts1</italic><sup>-/-</sup> umbilical vessels and acquisition of endomucin, a venous endothelium marker, by arterial endothelium (n = 3 umbilical cords each genotype) Scale bars = 100 μm in (<bold>a–c</bold>).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig5-v2.tif"/></fig><fig id="fig6" position="float"><label>Figure 6.</label><caption><title>Reduced aggrecan and versican proteolysis in <italic>Adamts1</italic><sup>-/-</sup>umbilical vessels.</title><p>(<bold>a,b</bold>) E17.5 <italic>Adamts1</italic><sup>-/-</sup> umbilical vessels show increased aggrecan staining and reduced anti-NITEGE staining in (<bold>a</bold>), quantified in (<bold>b</bold>) (n = 3 cords each genotype, error bars indicate mean ±S.D.*, p&lt;0.05; **, p&lt;0.01; ***, p&lt;0.001). (<bold>c,d</bold>) <italic>Adamts1</italic><sup>-/-</sup> umbilical vessels show increased versican (<bold>c</bold>) and reduced anti-DPEAAE staining quantified in (<bold>d</bold>) (n = 3 cords each genotype, error bars indicate mean ±S.D. *, p&lt;0.05; **, p&lt;0.01). Scale bars = 50 μm in (<bold>a–c</bold>).</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig6-v2.tif"/></fig></sec><sec id="s2-6"><title>Differential SMC contraction in the bilayered umbilical arteries</title><p>Despite uniform staining with SMC markers in human umbilical vascular SMC, co-staining with serine<sup>20</sup>-phosphorylated myosin light chain (pMLC) marking contractile SMCs (<xref ref-type="bibr" rid="bib6">Dougherty et al., 2014</xref>) revealed that human umbilical arteries had more contractile SMCs than the vein, predominantly in the outer TM (<xref ref-type="fig" rid="fig7">Figure 7a,b</xref>). This suggests that outer umbilical artery SMCs are principally responsive to vasoconstriction stimuli at birth, whereas inner SMCs are relatively non-contractile. We hypothesized that an outer ring of contracting SMCs could drive the CSPG-rich inner arterial TM centripetally, occluding the lumen, and addressed this possibility initially using ex vivo biomechanical testing of late-gestation mouse umbilical vessels (<xref ref-type="fig" rid="fig7">Figure 7c</xref>, <xref ref-type="fig" rid="fig7s1">Figure 7—figure supplement 1</xref>). Mouse umbilical arteries had a smaller lumen, as expected at E18.5, and deformed less when loaded mechanically, namely, they exhibited lower (circumferential) distensibility and especially (axial) extensibility under passive conditions compared to the umbilical veins (<xref ref-type="fig" rid="fig7s1">Figure 7—figure supplement 1a–d</xref>). Umbilical arteries constricted significantly (30–50% reduction in measured outer diameter at 25 mm Hg fixed pressure), causing complete luminal occlusion verified by optical coherence (OCT) imaging, which was not observed in the umbilical veins (<xref ref-type="fig" rid="fig7">Figure 7c</xref>). Cross-sectional area measurements at fixed lengths revealed wall volume reductions during vasoconstriction (<xref ref-type="fig" rid="fig7s1">Figure 7—figure supplement 1e</xref>), less in the umbilical vein (~35%) than the umbilical artery (~50%), suggesting fluid exudation from the wall under forceful SMC contraction.</p><fig-group><fig id="fig7" position="float"><label>Figure 7.</label><caption><title>Contraction-induced buckling ensures effective closure of the umbilical artery at birth.</title><p>(<bold>a</bold>) Smooth muscle myosin heavy chain (SMMHC-red) and serine-20 phosphorylated myosin light-chain (pMLC-green) show more contraction-primed SMCs in the outer arterial tunica media (TM, white brackets) than the umbilical vein. Scale bars are 100 µm. (<bold>b</bold>) Quantitation of pMLC<sup>+</sup> SMCs in the artery (red) and vein (blue) (top, n = 5 arteries, four veins, whiskers indicate minimum and maximum values, *, p&lt;0.05) and inner and outer TM of both reveal similar distributions but more pMLC<sup>+</sup> SMC in the outer artery TM (bottom, n = 3 arteries, four veins, error bars indicate mean ±S.D. *, p&lt;0.05; **, p&lt;0.01). (<bold>c</bold>) Differential contraction of murine umbilical artery and vein stimulated by 100 mM potassium chloride (KCl) under biaxial loading confirms greater contractility in the artery, with OCT images prior to and following contraction-induced arterial closure (n = 4 arteries and n = 4 veins). (<bold>d</bold>) Computational simulations of a bilayered artery with contractile SMCs in the outer layer and swollen inner layer: critical contractile stress values leading to buckling for (<italic>left</italic>) different numbers of folds for a normalized inner layer volume of 0.5 and (<italic>right)</italic> decreasing values of normalized volume of the inner layer for seven folds. (<bold>e</bold>) Normalized inner radius as a function of contractile stress for inner layer volume change of 0.5 and 7 folds. The states for the inflection (square) and critical active stress (star) are illustrated by the schematics; complete closure achieved with contraction-induced buckling. All simulations were run for 25 mmHg pressure. Due to the linear stability analysis, the amplitude of the folds in the buckled schematics is illustrative. (<bold>f</bold>) Number of buckles observed in human (top, indicated as umbilical cord (UC)1–25) and other large mammalian (bottom) umbilical arteries. Both arteries per cord were included. Open vessel lumens are indicated where observed.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig7-v2.tif"/></fig><fig id="fig7s1" position="float" specific-use="child-fig"><label>Figure 7—figure supplement 1.</label><caption><title>Differential biomechanical properties of arteries and veins of the mouse umbilical cord.</title><p>Biaxial structural (<bold>a,b</bold>) and material (<bold>c,d</bold>) behaviors exhibited by umbilical arteries and veins harvested from wild type E18.5 umbilical cords. The curves show model predicted results for the mean behaviors based on best-fit parameters determined from seven different testing protocols performed individually on n = 4 arteries and n = 4 veins. Though based on data from all seven protocols, for illustrative purposes results are shown for pressure-diameter behaviors during quasi-static pressurization at a fixed vessel-specific in vivo length and quasi-static axial extension at fixed pressures (25 mmHg for the arteries, and 5 mmHg for the veins). Grey-shaded regions show S.D. (<bold>e</bold>) Change in normalized cross-sectional area for the umbilical vein at 5 mm Hg and for the umbilical artery at 25 mm Hg during isobaric contraction. Error bars indicate S.E.M, **, p&lt;0.01.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig7-figsupp1-v2.tif"/></fig><fig id="fig7s2" position="float" specific-use="child-fig"><label>Figure 7—figure supplement 2.</label><caption><title>Computational results for a model umbilical artery.</title><p>(<bold>a</bold>) Normalized inner radius as a function of normalized volume of the inner layer alone for different fixed values of the active stress parameter T<sub>act</sub> and (<bold>b</bold>) inner radius as a function of the active stress parameter T<sub>act</sub> for different fixed values of normalized volume in the inner layer. Loaded inner radius a was normalized by the unloaded inner radius A; the current volume of the vessel v was normalized by the original volume of the vessel V. (<bold>c</bold>) Critical value of the active stress parameter T<sub>act</sub> as a function of the normalized volume in the inner layer for different numbers of buckling-induced luminal folds n. (<bold>d</bold>) Mean circumferential stress t<sub>θθ</sub> across the umbilical artery wall for varying normalized volumes: (<bold>i</bold>) and (ii) show circumferential stress for the case of shrinkage of the inner layer alone with v/V = 0.5 while (iii) and (iv) show the case of no swelling with v/V = 1.0. All simulations use the loading conditions, luminal pressure p=25 mmHg and fixed axial stretch λ<sub>z</sub>=1.28.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig7-figsupp2-v2.tif"/></fig></fig-group></sec><sec id="s2-7"><title>Computational modeling of arterial occlusion</title><p>These biomechanical tests of mouse umbilical cords, together with histological and immunostaining findings from human cords, motivated and informed a novel computational model of the umbilical artery incorporating its complex bilayered, multi-constituent structure (GAG-rich inner layer and contractile SMC-rich outer layer; <xref ref-type="fig" rid="fig7">Figure 7d</xref>) and multiaxial mechanical loading: axial extension, luminal pressurization, active contraction by SMCs, and intramural swelling of the inner layer that regulates tissue volume locally based on GAG content. Nonlinear regression of biaxial mechanical data from passive tests of the murine vessels identified best-fit values of the material parameters in the baseline constitutive model, while data from active contraction studies guided the selection of the associated active constitutive parameters (<xref ref-type="table" rid="table1">Table 1</xref>). Model-based parametric studies examined combinations of different levels of GAG-driven swelling and SMC-generated active stress to identify their roles in umbilical artery closure at different levels of fixed luminal pressure. Increasing inner layer swelling in the absence of active outer layer stress narrowed the lumen at a fixed pressure, as expected given the constraining effect of the outer stiff passive matrix (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2a</xref>). This trend reversed in the presence of active stresses, with increasing inner layer GAGs able to oppose vasoconstriction if overall wall volume remained constant (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2b</xref>). Thus, increased inner layer swelling attenuates the ability of SMC contraction to prematurely reduce luminal radius, as revealed by varying the active stress parametrically for different fixed values of inner layer swelling. Importantly, the model predicted a sharp transition from a widely patent to a narrowed lumen due to small changes in active stress at lower values of swelling whereas radius changes were more gradual at higher values of swelling (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2b</xref>). This transition, at which a decrease in volume of the inner layer associates with larger or smaller luminal radii for values of active stress below or above <inline-formula><mml:math id="inf1"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>≅</mml:mo><mml:mn>50</mml:mn></mml:math></inline-formula> kPa (a key parameter of active stress generation) appears to be close to the in vivo value. Hence, consistent with ex vivo findings (<xref ref-type="fig" rid="fig7s1">Figure 7—figure supplement 1</xref>), it appears that inner layer volume loss during increased SMC contraction aids vessel narrowing. Regardless, the inner radius reached nearly constant values for increasing levels of active stress (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2b</xref>). Thus, contraction alone is insufficient to occlude the vessel.</p><table-wrap id="table1" position="float"><label>Table 1.</label><caption><title>Model parameters fixed for all simulations of the umbilical artery, determined primarily from the biaxial biomechanical data and histological findings.</title></caption><table frame="hsides" rules="groups"><thead><tr><th valign="top">Parameters</th><th valign="top">Description</th><th valign="top">Values</th></tr></thead><tbody><tr><td><inline-formula><mml:math id="inf2"><mml:mi>A</mml:mi><mml:mo>,</mml:mo> <mml:mi/><mml:mi>B</mml:mi><mml:mo>,</mml:mo> <mml:mi/><mml:mi>C</mml:mi></mml:math></inline-formula></td><td valign="top">Unloaded inner, interface, outer radius</td><td valign="top">161.77, 206.86, 236.92 µm</td></tr><tr><td><inline-formula><mml:math id="inf3"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula></td><td valign="top">Loaded axial stretch</td><td valign="top">1.28</td></tr><tr><td><inline-formula><mml:math id="inf4"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo> <mml:mi/><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula></td><td valign="top">GAG/matrix shear modulus inner, outer layer</td><td valign="top">3.0 kPa, 0.1 kPa</td></tr><tr><td><inline-formula><mml:math id="inf5"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula></td><td valign="top">Axial fiber family material parameters</td><td valign="top">0.013 kPa, 11.65</td></tr><tr><td><inline-formula><mml:math id="inf6"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula></td><td valign="top">Circumferential fiber family material parameters</td><td valign="top">2.66 kPa, 1.20</td></tr><tr><td><inline-formula><mml:math id="inf7"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3,4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>3,4</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula></td><td valign="top">Diagonal fiber families’ material parameters</td><td valign="top">3.04 kPa, 4.23</td></tr><tr><td><inline-formula><mml:math id="inf8"><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula></td><td valign="top">Diagonal fiber families’ alignment parameter</td><td valign="top">41.92°, −41.92°</td></tr><tr><td><inline-formula><mml:math id="inf9"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula></td><td valign="top">Maximum, minimum contractile stretch</td><td valign="top">2.5, 0.2</td></tr></tbody></table></table-wrap></sec><sec id="s2-8"><title>Folding of the arterial inner layer is necessary for vascular occlusion</title><p>Given the consistent histological finding of inner arterial TM infolding following birth, we modeled the biomechanics of superimposed inner layer buckling in the bilayered arterial model. Buckling can release energy stored in the inner layer during vasoconstriction-induced compression, thereby reducing the structural stiffness and resistance to SMC contraction. This analysis parametrically considered possible perturbations to the cylindrical geometry achieved at various levels of fixed luminal pressure and different values of swelling and actively generated wall stress. Examining the influence of the number of inner layer folds for different values of swelling disclosed higher inward buckling probability with more folds (<xref ref-type="fig" rid="fig7">Figure 7d</xref>). Since <inline-formula><mml:math id="inf10"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> needed to cause buckling tended to plateau at 7 folds, we used 7 folds subsequently for illustrative purposes. <inline-formula><mml:math id="inf11"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> needed to cause buckling decreased for a more swollen inner layer <xref ref-type="fig" rid="fig7">Figure (7d</xref><xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2c</xref>) and increased exponentially with inner layer volume loss. This finding was likely due to the less negative values of circumferential wall stress in the inner layer occurring with shrinkage (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2d</xref>). We found that an arterial wall consisting solely of SMCs and uniform matrix maintained a mean positive circumferential stress in the inner layer during contraction, that prevented buckling. Thus, a delicate biomechanical balance exists – reduced inner layer volume allows a smaller radius to be achieved via SMC contraction prior to buckling (<xref ref-type="fig" rid="fig7s2">Figure 7—figure supplement 2b</xref>), thus aiding closure, yet excess volume reduction of the inner layer increases the active stress requirement for buckling and achieving complete vessel closure (<xref ref-type="fig" rid="fig7">Figure 7d</xref>). The umbilical artery can reduce its cylindrical radius dramatically at <inline-formula><mml:math id="inf12"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> near a basal value of ~50 kPa, progressing to buckling and closure via a subsequent near-maximal contraction (<xref ref-type="fig" rid="fig7">Figure 7e</xref>). In agreement with the computational modeling, 20/25 of human umbilical arteries had 4 or more buckles, whereas those with no buckles in the area analyzed by histology were patent (<xref ref-type="fig" rid="fig7">Figure 7f</xref>). Other large mammalian species analyzed showed a similar phenomenon (<xref ref-type="fig" rid="fig7">Figure 7f</xref>). Thus, buckling of the proteoglycan-rich inner tunica media may be a crucial and evolutionarily conserved mechanism employed by all mammals for rapid arterial occlusion at birth.</p></sec></sec><sec id="s3" sec-type="discussion"><title>Discussion</title><p>We report two distinct umbilical cord blood vessel specializations that may facilitate rapid umbilical artery occlusion at birth: a distinct proteoglycan-rich inner arterial TM, generating a bilayered arterial wall, and selective contraction of SMCs in the outer layer (<xref ref-type="fig" rid="fig8">Figure 8</xref>). The rounded SMCs of the inner TM may be specialized for CSPG production rather than contraction, consistent with nuclear Sox9 staining, a function they exert prior to delivery. During delivery, lack of pMLC staining suggests that despite differentiated SMC marker expression, the inner cells play a passive role. Biomechanical testing and computational analysis confirm that selective proteoglycan enrichment in the inner arterial TM ensures that contracting SMCs in the outer TM can effectively occlude the arterial lumen at birth (<xref ref-type="fig" rid="fig8">Figure 8</xref>). Histologic and computational analysis showed buckling of the inner TM and fluid redirection into the resulting TM protrusions as critical mechanisms resulting from specialization of the inner and outer arterial TM. By in silico simulations of umbilical arteries with modulation of the contractile outer layer and proteoglycan-rich inner core, we demonstrate that complete occlusion can be achieved.</p><fig id="fig8" position="float"><label>Figure 8.</label><caption><title>The unique bilayered design of the umbilical artery underlies its rapid occlusion at birth.</title><p>(<bold>a</bold>) Differential expression of ADAMTS proteases and large CS-proteoglycans during development results in a bilayered artery with a hydrated proteoglycan-rich inner layer and most contractile SMCs located in the outer layer, contrasting with the umbilical vein (see key at top of figure identifying the illustrated major elements). (<bold>b</bold>) At birth, SMC contraction in the outer layer and fluid movement-induced inner layer buckling redirects the inner layer into the lumen. The single-layered vein does not undergo buckling. (<bold>c</bold>) Umbilical artery occlusion at birth prevents neonatal exsanguination, whereas the patent vein allows a final transfusion from the placenta.</p></caption><graphic mime-subtype="tiff" mimetype="image" xlink:href="elife-60683-fig8-v2.tif"/></fig><p>Our work suggests that a principal mechanism governing umbilical cord vascular dimorphism resides in extracellular matrix, namely, differentially regulated dynamics of aggrecan and versican, which may then modulate SMC development and differentiation. Other mammalian umbilical vessels examined, from animals as large as the walrus and elephant to as small as the mouse, showed similar CSPG and aggrecan modulation as in humans. Although <italic>Vcan</italic> mutant mice die before umbilical cord development is completed (<xref ref-type="bibr" rid="bib41">Yamamura et al., 1997</xref>) and could not be studied, <italic>Acan</italic> and <italic>Adamts1</italic> mutants demonstrated their mechanistic contributions to the observed dimorphism. Our emphasis on aggrecan in the inner layer, with its abundant GAGs and their high fixed charge density, was relevant to the computational findings of the importance of inner layer swelling and buckling in response to SMC contraction. The umbilical vein contains fewer contractile SMCs and has scant aggrecan and versican. Hence, SMC activation in the umbilical vein does not occlude the lumen to the same degree as in the arteries, a contention supported by computational analysis. Given the evolutionary pressure to achieve hemostasis urgently in the artery rather than the vein, these findings suggest a highly evolved mechanism for preventing exsanguination of the newborn that is potentially relevant to other embryonic shunts that occlude rapidly at birth.</p><p>The computational model was built on a long history of studying murine arteries and veins (<xref ref-type="bibr" rid="bib8">Ferruzzi et al., 2013</xref>), but was specialized to the GAG-rich inner layer and SMC-rich outer layer of the umbilical artery. Modeling the dynamics of associated volume changes would have required a mixture or poroelastic model and significantly more experimental data, including measurement of layer-specific permeabilities and fixed-charge densities. Instead, we modeled the quasi-equilibrated states using a well-accepted approach wherein the degree of swelling can be adjusted for each simulation (<xref ref-type="bibr" rid="bib5">Demirkoparan and Pence, 2007</xref>; <xref ref-type="bibr" rid="bib36">Szafron et al., 2017</xref>). Interestingly, prior results by others showed that swelling of an initially unloaded, unilayered cylindrical tube consisting of a neo-Hookean material (which we used to model GAG-rich tissue) increases luminal diameter if the tube is unconstrained (<xref ref-type="bibr" rid="bib5">Demirkoparan and Pence, 2007</xref>). The tube must be constrained, such as by a stiff outer layer surrounding the swollen layer if swelling is to decrease luminal diameter (<xref ref-type="bibr" rid="bib36">Szafron et al., 2017</xref>). The abundance of contractile SMCs in the stiffer outer layer and their basal tone may in fact enhance outer layer stiffness contributed by extracellular matrix and hence buckling appears to be essential to augment contraction-induced closure of the umbilical artery (cf [<xref ref-type="bibr" rid="bib24">Moulton and Goriely, 2011</xref>]).</p><p>This observed architecture of the umbilical cord is likely to have supported survival of mammalian species, humans included, well before formal obstetric involvement in labor. Cord clamping is the default practice today and has the sanction of convention, offering the option of immediate neonatal resuscitation if needed. In regard to the umbilical artery, it would replicate the effect of a natural and apparently conserved physiologic process that interrupts its blood flow during transition from fetal to neonatal life. The latest recommendation to clamp the cord later rather than immediately after birth appears to align better with the delayed closure of the umbilical vein. The present studies in humans and other mammals show that evolution has devised an umbilical cord-intrinsic mechanism that facilitates rapid arterial occlusion at birth, leaving the vein patent and permitting a placental infusion. This unfailing sequence ensures continuation of mammalian species without other formal intervention, since evolutionary success is not about minimizing poor outcomes, but ensuring survival of a significant majority.</p></sec><sec id="s4" sec-type="materials|methods"><title>Materials and methods</title><table-wrap id="keyresource" position="anchor"><label>Key resources table</label><table frame="hsides" rules="groups"><thead><tr><th>Reagent type <break/>(species) or resource</th><th>Designation</th><th>Source or reference</th><th>Identifiers</th><th>Additional <break/>information</th></tr></thead><tbody><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>ACAN</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:319">HGNC:319</ext-link></td><td>Chondroitin sulphate proteoglycan 1</td></tr><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>VCAN</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:2464">HGNC:2464</ext-link></td><td>Chondroitin sulphate proteoglycan 2</td></tr><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>ADAMTS1</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:217">HGNC:217</ext-link></td><td>ADAM metallopeptidase with thrombospondin type 1 motif 1</td></tr><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>ADAMTS4</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:220">HGNC:220</ext-link></td><td>ADAM metallopeptidase with thrombospondin type 1 motif 4</td></tr><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>ADAMTS5</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:221">HGNC:221</ext-link></td><td>ADAM metallopeptidase with thrombospondin type 1 motif 5</td></tr><tr><td>Gene <break/>(<italic>Homo sapiens</italic>)</td><td><italic>ADAMTS9</italic></td><td>GenBank</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/HGNC:13202">HGNC:13202</ext-link></td><td>ADAM metallopeptidase with thrombospondin type 1 motif 9</td></tr><tr><td>Genetic reagent <break/>(<italic>Mus musculus</italic>)</td><td><italic>Acan</italic><sup>cmd-Bc</sup> <break/>(C57BL/6J background)</td><td><xref ref-type="bibr" rid="bib16">Krueger et al., 1999</xref></td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/MGI:1855999">MGI:1855999</ext-link></td><td><italic>Acan</italic> null allele</td></tr><tr><td>Genetic reagent (<italic>Mus musculus</italic>)</td><td><italic>Adamts1</italic><sup>tm1Dgen</sup> <break/>(C57BL/6J background)</td><td><xref ref-type="bibr" rid="bib30">Oller et al., 2017</xref></td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/MGI:5427602">MGI:5427602</ext-link></td><td><italic>Adamts1</italic> null and <italic>LacZ</italic> reporter allele</td></tr><tr><td>Antibody</td><td>Mouse monoclonal smooth muscle α-actin (α-SMA) Cy3 conjugated</td><td>Millipore Sigma C6198</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_476856">AB_476856</ext-link></td><td>IF (1:400)</td></tr><tr><td>Antibody</td><td>Rat monoclonal smooth muscle myosin heavy chain (SMMHC)</td><td>Kamiya Biomedical <break/>MC352</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_1241986">AB_1241986</ext-link></td><td>IF (1:400)</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal Serine-20 phosphorylated myosin light chain (pMLC)</td><td>Abcam Ab2480</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_303094">AB_303094</ext-link></td><td>IF (1:200)</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-serine-10 phosphorylated histone H3 (pHH3)</td><td>Millipore Sigma <break/>06–570</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_310177">AB_310177</ext-link></td><td>IF (1:200)</td></tr><tr><td>Antibody</td><td>Mouse monoclonal anti-chondroitin sulfate (7D4) antibody</td><td>Bruce Caterson/Clare Hughes laboratory <break/>(<xref ref-type="bibr" rid="bib35">Sorrell et al., 1990</xref>)</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_2864328">AB_2864328</ext-link></td><td>IF (1:200)</td></tr><tr><td>Antibody</td><td>Mouse monoclonal FITC-conjugated anti-heparan sulfate (10E4) antibody</td><td>US Biological <break/>H-1890</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_10013601">AB_10013601</ext-link></td><td>IF (1:200)</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal <break/>Anti-versican (pVC)</td><td>Apte laboratory <break/>(<xref ref-type="bibr" rid="bib10">Foulcer et al., 2014</xref>)</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_2864327">AB_2864327</ext-link></td><td>IF (1:400) human tissue</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-versican GAG-beta</td><td>Millipore Sigma AB1033</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_90462">AB_90462</ext-link></td><td>IF (1:400) mouse tissue</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-versican V0/V1 neo epitope DPEAAE</td><td>Invitrogen <break/>PA1-1748A</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_2304324">AB_2304324</ext-link></td><td>IF (1:200) human/mouse</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-aggrecan</td><td>Millipore Sigma <break/>AB1031</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_90460">AB_90460</ext-link></td><td>IF (1:400) all species</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-aggrecan neo epitope NITEGE</td><td>Invitrogen <break/>PA1-1746</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_2242021">AB_2242021</ext-link></td><td>IF (1:200) all species</td></tr><tr><td>Antibody</td><td>Rat monoclonal anti-endomucin antibody (clone eBioV.7C7)</td><td>Invitrogen <break/>14-5851-85</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_891531">AB_891531</ext-link></td><td>IF (1:400)</td></tr><tr><td>Antibody</td><td>Rabbit polyclonal anti-SOX9 antibody</td><td>Millipore Sigma AB5535</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/AB_2239761">AB_2239761</ext-link></td><td>IF (1:200)</td></tr><tr><td>Commercial assay or kit</td><td><italic>ACAN</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>506841</td><td>Human probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>Acan</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>439101</td><td>Mouse probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>VCAN</italic>-E8 RNAscope <break/>In situ probe</td><td>ACD bio</td><td>452241</td><td>Human probe detects exon 8</td></tr><tr><td>Commercial assay or kit</td><td><italic>Vcan</italic>-E8 RNAscope <break/>In situ probe</td><td>ACD bio</td><td>428321</td><td>Mouse probe detects exon 7</td></tr><tr><td>Commercial assay or kit</td><td><italic>ADAMTS1</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>524501</td><td>Human probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>Adamts1</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>463361</td><td>Mouse probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>ADAMTS4</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>537341</td><td>Human probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>Adamts4</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>497161</td><td>Mouse probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>ADAMTS5</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>427611</td><td>Human probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>Adamts5</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>427621</td><td>Mouse probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>ADAMTS9</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>445321</td><td>Human probe</td></tr><tr><td>Commercial assay or kit</td><td><italic>Adamts9</italic> RNAscope <break/>In situ probe</td><td>ACD bio</td><td>400441</td><td>Mouse probe</td></tr><tr><td>Commercial assay or kit</td><td>RNAscope 2.5 HD Red In situ detection kit</td><td>ACD bio</td><td>322350</td><td>Used for detecting all probes in this study</td></tr><tr><td>Software, algorithm</td><td>Affymetrix Transcriptome Analysis Console, RMA-SST sketch algorithm</td><td>Affymetrix <break/>TAC 4.0</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/SCR_018718">SCR_018718</ext-link></td><td>Used for gene expression analysis for all microarray experiments in the study</td></tr><tr><td>Software, algorithm</td><td>R</td><td>Bell Laboratories/R Foundation for Statistical Computing <break/>Ver. 3.5.2.</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/SCR_001905">SCR_001905</ext-link></td><td>Used for statistical computing of microarray data</td></tr><tr><td>Software, algorithm</td><td>GraphPad Prism</td><td>GraphPad</td><td>RRID:<ext-link ext-link-type="uri" xlink:href="https://scicrunch.org/resolver/SCR_002798">SCR_002798</ext-link></td><td>Used for statistical computing of other experimental data</td></tr></tbody></table><table-wrap-foot><fn><p>Abbreviations, IF, Immunofluorescence.</p></fn></table-wrap-foot></table-wrap><sec id="s4-1"><title>Human and large mammal cords</title><p>Twenty-five human umbilical cords were obtained from uncomplicated term pregnancies either after vaginal birth (n = 13) or Cesarean section for obstetric indications (n = 12), that is, malpresentation or repeat Cesarean section. The samples were collected under an IRB exemption from Cleveland Clinic (EX-0118) for use of discarded tissue without patient identifiers. These cords were used for histological/immunohistologic analysis, synchrotron imaging, RNA in situ hybridization, and transcriptomics of inner versus outer umbilical artery TM. Animal cord sections were provided by Disease Investigations, Institute for Conservation Research, San Diego Zoo Global from the Benirschke archive.</p></sec><sec id="s4-2"><title>Mutant mice</title><p>The <italic>Adamts1</italic> transgenic allele (<italic>Adamts1</italic><sup>tm1Dgen</sup>), referred to herein as <italic>Adamts1</italic><sup>-/-</sup>, was produced by insertion of an IRES-lacZ cassette into intron 1 of <italic>Adamts1</italic> using homologous recombination in mouse embryonic stem cells (<xref ref-type="bibr" rid="bib30">Oller et al., 2017</xref>). The <italic>Acan</italic><sup>cmd-Bc</sup> allele was previously described (<xref ref-type="bibr" rid="bib16">Krueger et al., 1999</xref>) and is referred to herein as <italic>Acan</italic><sup>-/-</sup>. Mice were handled under standard conditions under approved IACUC protocols at the Cleveland Clinic (IACUC protocol nos. 18–1996 and 18–2045) and University of Chicago (IACUC protocol no. 43751). Mutant mouse embryos were collected by timed matings of heterozygous mice by the detection of copulation plugs (taken as day 0.5 of gestation). Embryos were dissected out immediately following CO<sub>2</sub> mediated euthanasia and cervical dislocation of pregnant mice. Dissected whole embryos, with umbilical cords and placentas attached, were fixed in 4% paraformaldehyde at 4°C overnight. Umbilical cords were dissected out the following day and washed thrice in PBS and embedded in paraffin or in 4% agarose for vibratome sectioning as previously described (<xref ref-type="bibr" rid="bib25">Nandadasa et al., 2015</xref>).</p></sec><sec id="s4-3"><title>Biomechanical and computational analysis</title><p>The umbilical artery and vein were obtained at E18.5 from mouse embryos (n = 4) following approval by the Yale University IACUC (protocol no 2018–11508), then mounted within a custom computer-controlled biaxial device designed specifically for biomechanical testing of murine vessels (<xref ref-type="bibr" rid="bib11">Gleason et al., 2004</xref>). Vessel maintenance, pre-conditioning, biaxial loading protocols and data collection are described in Appendix 1. The umbilical artery was modeled computationally as a thick-walled, bilayered cylindrical tube subjected to swelling of the GAG-rich inner layer and active contraction of the smooth muscle-rich outer layer; the model also included a passive contribution of extracellular matrix as revealed by biomechanical tests.</p></sec><sec id="s4-4"><title>Statistical analysis</title><p>Statistical analyses were carried out using GraphPad Prism analytical software (versions 6–8, GraphPad, San Diego, CA) in determining statistical significance using two-tailed Student’s <italic>t</italic>-test. Statistical details including <italic>N</italic> and p values are provided in each corresponding figure legend. Statistical analyses for microarray gene expression were performed using Affymetrix’s Transcriptome Analysis Console (TAC 4.0) through the RMA-SST sketch algorithm and R version 3.5.2. Fold changes were calculated by an empirical Bayes ANOVA method through the TAC 4.0 software. Details of these and additional methods are provided in Appendix 1.</p></sec></sec></body><back><ack id="ack"><title>Acknowledgements</title><p>We acknowledge the Paul Scherrer Institut, Villigen, Switzerland for provision of synchrotron radiation beamtime at the TOMCAT beamline X02DA of the SLS and thank Goran Lovric for assistance. We are grateful to the Disease Investigations team at San Diego Zoo Global for use of sections from the Benirschke archive, and to the late Dr. Kurt Benirschke for collecting the animal umbilical cords used in this study.</p></ack><sec id="s5" sec-type="additional-information"><title>Additional information</title><fn-group content-type="competing-interest"><title>Competing interests</title><fn fn-type="COI-statement" id="conf1"><p>No competing interests declared</p></fn></fn-group><fn-group content-type="author-contribution"><title>Author contributions</title><fn fn-type="con" id="con1"><p>Conceptualization, Data curation, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing</p></fn><fn fn-type="con" id="con2"><p>Conceptualization, Formal analysis, Investigation, Visualization, Methodology, Writing - review and editing</p></fn><fn fn-type="con" id="con3"><p>Data curation, Formal analysis, Visualization, Writing - review and editing</p></fn><fn fn-type="con" id="con4"><p>Formal analysis, Visualization, Methodology, Writing - review and editing</p></fn><fn fn-type="con" id="con5"><p>Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con6"><p>Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con7"><p>Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con8"><p>Resources, Writing - review and editing</p></fn><fn fn-type="con" id="con9"><p>Resources</p></fn><fn fn-type="con" id="con10"><p>Resources, Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con11"><p>Resources, Writing - review and editing</p></fn><fn fn-type="con" id="con12"><p>Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con13"><p>Resources, Methodology, Writing - review and editing</p></fn><fn fn-type="con" id="con14"><p>Resources, Investigation, Writing - review and editing</p></fn><fn fn-type="con" id="con15"><p>Conceptualization, Resources, Supervision, Funding acquisition, Investigation, Writing - original draft, Writing - review and editing</p></fn><fn fn-type="con" id="con16"><p>Conceptualization, Data curation, Formal analysis, Supervision, Funding acquisition, Visualization, Writing - original draft, Writing - review and editing</p></fn><fn fn-type="con" id="con17"><p>Conceptualization, Data curation, Supervision, Funding acquisition, Investigation, Methodology, Writing - original draft, Project administration, Writing - review and editing</p></fn></fn-group><fn-group content-type="ethics-information"><title>Ethics</title><fn fn-type="other"><p>Human subjects: Human umbilical cord samples were collected under an IRB exemption (EX-0118) from Cleveland Clinic for use of discarded tissue without patient identifiers. These cords were used for histological/immunohistologic analysis, in situ hybridization, and transcriptomics of inner vs outer umbilical artery TM. For microarray analysis of umbilical cord artery versus vein, human umbilical cords were collected separately through the National Children's Study under University Hospitals-Case Medical Center approved IRB protocol 01-11-28.</p></fn><fn fn-type="other"><p>Animal experimentation: This study was performed in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. All of the animals were handled according to approved institutional animal care and use committee (IACUC) protocols: 18-1996 and 18-2045 (Cleveland Clinic IACUC), 2018-11508 (Yale University IACUC) and 43751 (University of Chicago IACUC).</p></fn></fn-group></sec><sec id="s6" sec-type="supplementary-material"><title>Additional files</title><supplementary-material id="supp1"><label>Supplementary file 1.</label><caption><title>Microarray comparison of transcriptome of the human umbilical artery and vein.</title></caption><media mime-subtype="xlsx" mimetype="application" xlink:href="elife-60683-supp1-v2.xlsx"/></supplementary-material><supplementary-material id="supp2"><label>Supplementary file 2.</label><caption><title>Microarray comparison of the transcriptome of the inner umbilical artery tunica media with the outer umbilical artery tunica media.</title></caption><media mime-subtype="xlsx" mimetype="application" xlink:href="elife-60683-supp2-v2.xlsx"/></supplementary-material><supplementary-material id="transrepform"><label>Transparent reporting form</label><media mime-subtype="docx" mimetype="application" xlink:href="elife-60683-transrepform-v2.docx"/></supplementary-material></sec><sec id="s7" sec-type="data-availability"><title>Data availability</title><p>All data generated or analysed during this study are included in the manuscript and supporting files.</p><p>The following datasets were generated:</p><p><element-citation id="dataset1" publication-type="data" specific-use="isSupplementedBy"><person-group person-group-type="author"><name><surname>Nandadasa</surname><given-names>S</given-names></name><name><surname>Szafron</surname><given-names>JM</given-names></name><name><surname>Pathak</surname><given-names>V</given-names></name><name><surname>Murtada</surname><given-names>S-I</given-names></name><name><surname>Kraft</surname><given-names>CM</given-names></name><name><surname>O'Donnell</surname><given-names>A</given-names></name><name><surname>Norvik</surname><given-names>C</given-names></name><name><surname>Hughes</surname><given-names>C</given-names></name><name><surname>Caterson</surname><given-names>B</given-names></name><name><surname>Domowicz</surname><given-names>MS</given-names></name><name><surname>Schwartz</surname><given-names>NB</given-names></name><name><surname>Tran-Lundmark</surname><given-names>K</given-names></name><name><surname>Veigl</surname><given-names>M</given-names></name><name><surname>Sedwick</surname><given-names>D</given-names></name><name><surname>Philipson</surname><given-names>EH</given-names></name><name><surname>Humphrey</surname><given-names>JD</given-names></name><name><surname>Apte</surname><given-names>SS</given-names></name></person-group><year iso-8601-date="2020">2020</year><data-title>Human umbilical cord artery inner tunica media vs outer tunica media.</data-title><source>Dryad Digital Repository</source><pub-id assigning-authority="Dryad" pub-id-type="doi">10.5061/dryad.4j0zpc88k</pub-id></element-citation></p><p><element-citation id="dataset2" publication-type="data" specific-use="isSupplementedBy"><person-group person-group-type="author"><name><surname>Nandadasa</surname><given-names>S</given-names></name><name><surname>Szafron</surname><given-names>JM</given-names></name><name><surname>Pathak</surname><given-names>V</given-names></name><name><surname>Murtada</surname><given-names>S-I</given-names></name><name><surname>Kraft</surname><given-names>CM</given-names></name><name><surname>O'Donnell</surname><given-names>A</given-names></name><name><surname>Norvik</surname><given-names>C</given-names></name><name><surname>Hughes</surname><given-names>C</given-names></name><name><surname>Caterson</surname><given-names>B</given-names></name><name><surname>Domowicz</surname><given-names>MS</given-names></name><name><surname>Schwartz</surname><given-names>NB</given-names></name><name><surname>Tran-Lundmark</surname><given-names>K</given-names></name><name><surname>Veigl</surname><given-names>M</given-names></name><name><surname>Sedwick</surname><given-names>D</given-names></name><name><surname>Philipson</surname><given-names>EH</given-names></name><name><surname>Humphrey</surname><given-names>JD</given-names></name><name><surname>Apte</surname><given-names>SS</given-names></name></person-group><year 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The time from delivery to cord clamping varied between deliveries and in general ranged from 5 to 30 s. Two additional clamps were placed on the cord after a fetal blood sample was obtained for Rhesus typing, and scissors were used to cut between these two clamps to obtain a segment of umbilical cord for analysis. Umbilical cord samples were fixed in 4% paraformaldehyde for 48 hr at 4°C and embedded in paraffin. All human umbilical cord sections analyzed contained all three vessels and both arteries were imaged in addition to the vein. For microarray analysis of umbilical artery versus vein, human umbilical cords were separately collected as a part of the National Children’s Study (NCS) (UHCMC IRB# 01-11-28). Immediately upon delivery, cords were sectioned into 1-inch segments (for a total of 3 segments per cord at 0 hr) and flash frozen in liquid nitrogen. <italic>RNA Later-ICE</italic> (Ambion) was used to maintain nucleic acid viability during freeze-thaw of tissue. The umbilical vein and both umbilical arteries were dissected from each segment for homogenization. RNA from homogenized vein and artery tissue was extracted using Qiagen’s RNeasy nucleic acid isolation kit. For microarray analysis of the inner versus outer arterial tunica media, human umbilical arteries were dissected immediately upon neonatal delivery and sectioned into 1-inch segments (for a total of three segments per artery) and their inner and outer tunica media were carefully dissected under a dissecting microscope and flash frozen in liquid nitrogen in Trizol reagent (Ambion). RNA from homogenized arteries was extracted using the chloroform-isopropanol precipitation method.</p></sec><sec id="s8-2"><title>Histological and immuno-staining</title><p>Seven-micron-thick paraffin sections were collected using a Leica RM 2255 microtome and stained with hematoxylin and eosin, Alcian blue, or Masson’s trichrome after deparaffinization. For immunostaining, sections were deparaffinized, and boiled in citrate buffer (10 mM citric acid, 0.05% Tween 20, pH 6.0) for 90 s for antigen retrieval, washed with PBST and blocked in 10% normal goat serum before incubation with the following primary antibodies overnight at 4°C: Cy3-conjugated smooth muscle α-actin (α-SMA) (1:400, Sigma C6198), anti-smooth muscle myosin heavy chain (SMMHC) (1:400 Kamiya Biomedical, MC352), anti-serine<sup>20</sup>-phosphorylated myosin light chain (pMLC) (Abcam, ab2480 1:200), anti-phospho histone H3 (ser10)(1:200, Millipore, 06–570), anti-chondroitin sulfate (7D4) antibody (<xref ref-type="bibr" rid="bib35">Sorrell et al., 1990</xref>) (1:200), FITC-conjugated anti-heparan sulfate (10E4) antibody (1:200, US biological H-1890), polyclonal anti-versican (<xref ref-type="bibr" rid="bib10">Foulcer et al., 2014</xref>) (anti-VC; 1:400), polyclonal anti-versican GAG-β (1:400, Milipore Sigma, AB1033), anti- DPEAAE (versican V0/V1 neo epitope antibody, 1:200, Invitrogen, PA1-1748A), polyclonal anti-aggrecan (1:400, Milipore Sigma, AB1031), anti- NITEGE (aggrecan neo epitope antibody, 1:200, Invitrogen, PA1-1746), rat monoclonal anti-endomucin (1:400, Invitrogen, 14-5851-85), anti-Sox9 (1:200, Milipore Sigma Ab5535). Alexa-488 or Alexa-568 conjugated secondary antibodies against rabbit and mouse IgG, respectively, were used at 1:400 dilution. Vectashield mounting medium (H-1200) contained DAPI for staining nuclei. All images were taken using an Olympus BX51 microscope connected to a Leica DFC 7000T camera using bright field or fluorescence modes. Multi-channel fluorescent images were merged using the NIH Image J software.</p></sec><sec id="s8-3"><title>Synchrotron-based phase contrast micro-CT</title><p>Imaging of arteries and veins from three umbilical cords was performed at the X02DA TOMCAT beamline of the Swiss Light Source at the Paul Scherrer Institute (Villigen, Switzerland). A 4x magnifying objective was used, resulting in a field-of-view of 4.2 × 3.5 mm<sup>2</sup> and an effective pixel size of 1.63 × 1.63 μm<sup>2</sup>. For each scanned vessel, a stack of 1080 tomographic images was acquired. Data analysis was performed using NIH Image J and Amira. Amira allowed for visualization of the vessels from any angle and images were created by combining two different tomographic imaging planes at a 90-degree angle. NIH Image J was used for creating <xref ref-type="video" rid="fig1video1">Figure 1—videos 1</xref> and <xref ref-type="video" rid="fig1video2">2</xref>.</p></sec><sec id="s8-4"><title>Microarray analysis of human umbilical cords</title><p>NCS samples to evaluate changes in expression in artery versus vein were run on the Affymetrix Hu-Gene U219 microarray Peg-plate; 150 ng of input RNA from each sample was labeled using an Affymetrix 3’ IVT labeling protocol. Samples hybridized to the PEG arrays were washed, stained, and scanned by the Affymetrix Gene Titan. RNA from two of the four cords was also evaluated on the Affymetrix Hu-Gene 1.1 ST Peg-plate microarray, using the whole transcriptome (WT) labeling protocol also starting with 150 ng of input total RNA. The hybridized PEG Arrays were washed, stained, and scanned by the Affymetrix Gene Titan according to standard protocols. RNA (150 ng) from inner and outer TM samples was labeled using Affymetrix’s WT PLUS protocol. Labeled samples were hybridized overnight to Affymetrix Hu-Gene 2.0 ST microarray cartridges. The sample was removed, then the microarray cartridges were washed and stained on the Affymetrix GeneChip Fluidics Station450 and scanned by the Affymetrix GeneChip Scanner 3000. Total RNA was labeled using Affymetrix’s FLASH Tag Protocol. Labeled samples were hybridized overnight to Affymetrix GeneChip miRNA 2.0 Array, which interrogates all the mature human miRNA sequences in miRBase Release 20microRNAs. The sample was then removed and the microarray cartridges were washed and stained on the Affymetrix GeneChip Fluidics Station450 and scanned by the Affymetrix GeneChip Scanner 3000. Gene expression analysis for each microarray study was performed using Affymetrix’s Transcriptome Analysis Console (TAC 4.0) through the RMA-SST sketch algorithm and R version 3.5.2. Fold changes were calculated by an empirical Bayes ANOVA method through the TAC 4.0 software. Parameters for gene expression changes include a p-value≤0.05 and a fold change value ≥1.5 and≤−1.5. Data collected from samples labeled by different labeling protocols, obtained using different scanners or hybridized to different arrays were analyzed as independent data sets.</p></sec><sec id="s8-5"><title>Additional details of transgenic mice</title><p>The <italic>Adamts1</italic> transgenic allele used here (referred to as <italic>Adamts1</italic><sup>-</sup>) was generated by Deltagen Inc (San Carlos, CA; Deltagen identifier T1288; MGI:5427602 (B6;129P2-Adamts1 &lt; tm1Dgen&gt;/H;)) by inserting an IRES-lacZ-neomycin resistance gene cassette into intron 1. RT-PCR using PCR primers bridging exon 1 and exon two showed that the insertion eliminated gene expression (Deltagen, unpublished data). The allele was deposited in the MRC Harwell, Frozen Embryo and Sperm Archive (Harwell, UK). Frozen <italic>Adamts1</italic><sup>+/-</sup> embryos harvested at the two-cell stage of development from matings of wild-type C57BL/6 females with hemizygous males at MRC Harwell were obtained under an academic use license signed by the Cleveland Clinic with Deltagen. Frozen embryos were implanted into pseudo-pregnant female recipient mice at the Case Transgenic and Targeting Facility (Cleveland, OH). Subsequently, the <italic>Adamts1<sup>-</sup></italic> allele was crossed into the C57BL/6 strain for at least 10 generations and is maintained in this strain. The <italic>Acan</italic><sup>cmd-Bc</sup> allele (referred to here as <italic>Acan</italic><sup>+/-</sup>) was back-crossed to C57BL/6 for over 20 generations at the University of Chicago (<xref ref-type="bibr" rid="bib18">Lauing et al., 2014</xref>) and subsequently transferred to the Cleveland Clinic Lerner Research Institute. β-Galactosidase staining of <italic>Adamts1</italic><sup>+/-</sup> umbilical cords was done as previously described (<xref ref-type="bibr" rid="bib21">McCulloch et al., 2009</xref>). Both β-galactosidase staining and genotyping were used to identify <italic>Adamts1<sup>+/-</sup></italic> mice obtained from hemizygous with wild-type matings, whereas PCR genotyping was used to distinguish the three genotypes possibly arising from crosses of <italic>Adamts1</italic> or <italic>Acan</italic> hemizygous animals which were used to generate homozygous embryos.</p><p>For genotyping, genomic DNA was isolated from clipped toes between 7–10 days after birth or from embryo tails using 100 μl or 50 μl Direct PCR Tail Lysis reagent, respectively (Viagen, catalog number 102 T) containing 1 μl of proteinase K (Milipore Sigma, catalog number 3115879001) followed by digestion overnight at 55°C. Alternatively, tail DNA was extracted using QuickExtract DNA (Lucigen) at 65°C for 15 min and reaction was stopped by incubation at 100°C for 5 min.</p><p>The targeted <italic>Adamts1</italic> allele was detected by PCR using forward primer 5' <named-content content-type="sequence">GGGCCAGCTCATTCCTCCCACTCAT</named-content> 3' and reverse primer 5'<named-content content-type="sequence">GCCATCGGGGTCAGCTTTTCAAATG</named-content> 3' (generating a 356 bp product) and the wild-type <italic>Adamts1</italic> allele was detected using forward primer 5' <named-content content-type="sequence">GGTTGTAGTTTCGCGCTGAGTTTTG3</named-content>' and reverse primer 5'<named-content content-type="sequence">GCCATCGGGGTCAGCTTTTCAAATG</named-content> 3' (generating a 189 bp product). <italic>Acan</italic> genotyping was performed using the following primer pairs: <italic>Acan<sup>cmd</sup></italic> forward primer 5' <named-content content-type="sequence">ATCAAGACCCTCAGCTTTTATTAATCTTTA</named-content> 3' and reverse primer 5' <named-content content-type="sequence">CATAAGATGAGAGGAGATGGTTTAGAGTAT</named-content> 3' (expected product 811 bp); <italic>Acan</italic> wild type allele forward primer 5' <named-content content-type="sequence">TCCTATTTACACAAAGTCTGAAATTAATGC</named-content> 3' and reverse primer 5' <named-content content-type="sequence">GAGAATTGGCTATAGCTGTTTATGACTC</named-content> 3' (expected product 332 bp).</p></sec><sec id="s8-6"><title>RNA in situ hybridization</title><p>Six-micron-thick paraffin sections were probed with RNAscope probes according to manufacturer’s guidelines. The following probes were used: <italic>ACAN</italic> (Advanced Cell Diagnostics (ACD) Cat. No. 506841), <italic>VCAN-exon 8</italic> (ACD Cat. No. 452241), <italic>ADAMTS1</italic> (ACD Cat. No. 524501), <italic>ADAMTS4</italic> (ACD Cat. No. 537341), <italic>ADAMTS5</italic> (ACD Cat. No. 427611), <italic>ADAMTS9</italic> (ACD Cat. No. 445321), <italic>Acan</italic> (ACD Cat. No. 439101), <italic>Vcan-exon 8</italic> (ACD Cat. No. 428321), <italic>Adamts1</italic> (ACD Cat. No. 463361), <italic>Adamts4</italic> (ACD Cat. No. 497161), <italic>Adamts5</italic> (ACD Cat. No. 427621) and <italic>Adamts9</italic> (ACD Cat. No. 400441). The probes were detected using RNAscope 2.5 HD Red reagent kits (ACD Cat. No. 322350) essentially as recently described (<xref ref-type="bibr" rid="bib22">Mead and Apte, 2020</xref>).</p></sec><sec id="s8-7"><title>Biomechanical characterization of mouse umbilical arteries and veins</title><p>Umbilical cords were excised and separated from the placenta from wild-type C57BL/6J mice at embryonic day E18.5 (n = 4). The umbilical arteries and veins were identified by opening the abdomen of the embryo and locating an artery by its connection to the iliac artery and a vein by its connection to the inferior vena cava through the ductus venosus. After separating and cleaning the umbilical arteries and veins from excessive adipose tissue, the intact vessels were cannulated on custom-drawn glass micropipettes, secured with silk sutures at each end, and mounted within a custom computer-controlled biaxial test device designed specifically for testing murine vessels. These cylindrical specimens were then immersed in a Krebs-Ringer bicarbonate solution (Krebs) and oxygenated with 95% O<sub>2</sub> and 5% CO<sub>2</sub> while maintained at 37°C. The lumen of the umbilical artery closed immediately after separating the umbilical cord from the placenta but was relaxed by acclimation within the testing chamber for 5–15 min at 10 mmHg and wash-out with Krebs solution up to three times. The vessels were then subjected to two isobaric (luminal pressure of 5 mmHg, then 10 mmHg) - axially isometric (fixed specimen-specific in vivo axial stretch) contractions by adding 100 mM KCl to the bath to ensure viability of the SMC. The transmural organization of the vessel wall was monitored during different stages of contraction using an optical coherence tomography (OCT) system having an axial (depth) resolution &lt;7 microns and lateral resolution of 8 microns (Callisto Model, Thorlabs, Newton, NJ).</p><p>For the subsequent passive tests, the normal Krebs solution was replaced with a Ca<sup>2+</sup>-free Krebs solution. The vessels were preconditioned via four cycles of pressurization (artery: 0–40 mmHg and vein: 0–25 mmHg) while held fixed at their individual in vivo axial stretch. Subsequently, the vessels were subjected to a series of seven biaxial testing protocols: cyclic pressurization over ranges noted above at three different fixed values of axial stretch and cyclic axial stretching at four different fixed values of luminal pressure (artery at 10, 20, 30 or 40 mmHg and vein at 1, 2, 10 or 20 mmHg). Data collected online included outer diameter, luminal pressure, axial length, and axial force, which facilitated robust parameter estimation of the biaxial biomechanical behavior (<xref ref-type="bibr" rid="bib9">Ferruzzi et al., 2015</xref>). Toward this end, we used a validated ‘four-fiber family’ type constitutive relation (see below) that has proved useful in characterizing murine arteries (systemic and pulmonary) and veins.</p><p>Note that these biaxial data were used to build a baseline, bilayered cylindrical model for the purposes of parametrically exploring possible effects of different levels of smooth muscle contractility in the outer tunica media, GAG-induced swelling of the inner tunica media, and possible buckling of this innermost layer in silico. Hence, rather than focus on specimen-to-specimen differences, we sought ‘mean’ properties of the normal murine umbilical artery. Data from all seven passive biaxial testing protocols from all four mice were grouped as a single data set and best-fit values of the associated eight material parameters were determined simultaneously by minimizing the sum-of-the-squares differences between predicted and measured pressures and axial forces, as described previously (<xref ref-type="bibr" rid="bib9">Ferruzzi et al., 2015</xref>). To facilitate a global, rather than local, minimization, we used multiple randomly generated initial guesses for the parameter estimation, accomplished using the MATLAB routine (lsqnonlin function). It is important to note that best-fit parameters in exponential constitutive relations need not be unique, hence what is most important is that together they yield correct values of biaxial stress (verified by comparison to measured values) and predict appropriate levels of material stiffness and stored energy. Finally, it is noted that initial parameter estimation was based on data from a representative specimen rather than all data combined; the subsequent model simulations were similar in both cases, suggesting that the baseline model captured salient features as desired.</p></sec></sec><sec id="s9" sec-type="appendix"><title>Computational modeling of the umbilical artery</title><sec id="s9-1"><title>Swelling and contraction</title><p>The traction-free, non-swollen configuration <inline-formula><mml:math id="inf13"><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>,</mml:mo><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> is treated as the reference with inner radius, interfacial radius, and outer radius denoted <inline-formula><mml:math id="inf14"><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo> <mml:mi/><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>, respectively. Swelling is accounted for via previously outlined methods (<xref ref-type="bibr" rid="bib36">Szafron et al., 2017</xref>; <xref ref-type="bibr" rid="bib5">Demirkoparan and Pence, 2007</xref>), which leads to a residually stressed, traction-free configuration with corresponding material points mapped to <inline-formula><mml:math id="inf15"><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo> <mml:mi/><mml:msup><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math></inline-formula> in <inline-formula><mml:math id="inf16"><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math></inline-formula>. The final loaded configuration, pressurized and axially stretched, is then characterized by <inline-formula><mml:math id="inf17"><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo> <mml:mi/><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula> in <inline-formula><mml:math id="inf18"><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">θ</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>.</p><p>Consider a deformation from <inline-formula><mml:math id="inf19"><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>,</mml:mo><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> to <inline-formula><mml:math id="inf20"><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math></inline-formula> via the deformation gradient tensor <inline-formula><mml:math id="inf21"><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">g</mml:mi><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:math></inline-formula>, where volume change is imposed by <inline-formula><mml:math id="inf22"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>v</mml:mi><mml:mo>/</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula> with <inline-formula><mml:math id="inf23"><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> the ratio of volume <inline-formula><mml:math id="inf24"><mml:mi>v</mml:mi></mml:math></inline-formula> in <inline-formula><mml:math id="inf25"><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math></inline-formula> to volume <inline-formula><mml:math id="inf26"><mml:mi>V</mml:mi></mml:math></inline-formula> in <inline-formula><mml:math id="inf27"><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">Θ</mml:mi><mml:mo>,</mml:mo><mml:mi>Z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>. With <inline-formula><mml:math id="inf28"><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, there is no swelling and the vessel is considered to have no residual stresses; in contrast, <inline-formula><mml:math id="inf29"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula> (expansion) and <inline-formula><mml:math id="inf30"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula> (shrinkage) yields self-equilibrating wall stresses in the absence of external loading. Note that <inline-formula><mml:math id="inf31"><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msup><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msubsup><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>. Further deformation to any loaded configuration <inline-formula><mml:math id="inf32"><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">θ</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> from the swollen configuration <inline-formula><mml:math id="inf33"><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:math></inline-formula> is then described by <inline-formula><mml:math id="inf34"><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">g</mml:mi><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>,</mml:mo> <mml:mi mathvariant="normal"/><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:math></inline-formula>, where the assumption of incompressibility during transient external loading requires <inline-formula><mml:math id="inf35"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>, thus <inline-formula><mml:math id="inf36"><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>. A multiplicative decomposition of the deformations yields <inline-formula><mml:math id="inf37"><mml:mi mathvariant="bold">F</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold">*</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">g</mml:mi><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mi>r</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>/</mml:mo><mml:mi>R</mml:mi></mml:math></inline-formula>,<inline-formula><mml:math id="inf38"> <mml:mi mathvariant="normal"/><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>) with <inline-formula><mml:math id="inf39"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">z</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi mathvariant="normal">Λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> for convenience. The matrix component <inline-formula><mml:math id="inf40"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> can also be expressed in terms of the original derivatives, such that <inline-formula><mml:math id="inf41"><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>)</mml:mo><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>, which allows us to apply the chain rule and integrate <inline-formula><mml:math id="inf42"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>r</mml:mi><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>∂</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula> to find any radial point <inline-formula><mml:math id="inf43"><mml:mi>r</mml:mi></mml:math></inline-formula> within the vessel wall.</p><p>The vessel is assumed to be quasi-equilibrated in any state, such that linear momentum balance requires <inline-formula><mml:math id="inf44"><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">v</mml:mi> <mml:mi mathvariant="normal"/><mml:mi mathvariant="bold">t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, where <inline-formula><mml:math id="inf45"><mml:mi mathvariant="bold">t</mml:mi></mml:math></inline-formula> is the Cauchy stress tensor. Circumferential and axial equilibrium is satisfied identically at each <inline-formula><mml:math id="inf46"><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="normal">θ</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>. Radial equilibrium requires <inline-formula><mml:math id="inf47"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi mathvariant="normal">∂</mml:mi><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant="normal">∂</mml:mi><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula>. The Cauchy stress is specialized as <inline-formula><mml:math id="inf48"><mml:mi mathvariant="bold">t</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">t</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mi mathvariant="bold">I</mml:mi></mml:math></inline-formula> with <inline-formula><mml:math id="inf49"><mml:mi>p</mml:mi></mml:math></inline-formula> the Lagrange multiplier enforcing incompressibility during transient loading, <inline-formula><mml:math id="inf50"><mml:mi mathvariant="bold">I</mml:mi></mml:math></inline-formula> the identity tensor, and <inline-formula><mml:math id="inf51"><mml:msup><mml:mrow><mml:mi mathvariant="bold">t</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> the 'extra' part of the stress due to deformation and the constitutive response. Integration yields<disp-formula id="equ1"><mml:math id="m1"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi>r</mml:mi><mml:mo>+</mml:mo> <mml:mi/><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="inf52"><mml:mi>P</mml:mi></mml:math></inline-formula> is the transmural pressure across the vessel wall, with <inline-formula><mml:math id="inf53"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>P</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mstyle></mml:math></inline-formula> indicating internal pressurization. We also calculate the overall axial load required for overall equilibrium,<disp-formula id="equ2"><mml:math id="m2"><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced><mml:mi>r</mml:mi><mml:mi>d</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></disp-formula></p><p>See <xref ref-type="fig" rid="fig7">Figure 7d</xref>. The equilibrium problem is solved iteratively for the loaded inner radius <inline-formula><mml:math id="inf54"><mml:mi>a</mml:mi></mml:math></inline-formula> for each luminal pressure <inline-formula><mml:math id="inf55"><mml:mi>P</mml:mi></mml:math></inline-formula> and axial extension <inline-formula><mml:math id="inf56"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>.</p><p>Constitutively, the extra part of the Cauchy stress can be computed from a stored energy density function <inline-formula><mml:math id="inf57"><mml:mi>W</mml:mi></mml:math></inline-formula> for the vessel, with <inline-formula><mml:math id="inf58"><mml:mi mathvariant="bold">t</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="bold">F</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>W</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi mathvariant="bold">C</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow></mml:mrow><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mi mathvariant="bold">I</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="inf59"><mml:mi mathvariant="bold">C</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant="bold">F</mml:mi></mml:math></inline-formula> the right Cauchy-Green tensor. Due to the microstructure of the umbilical vessels, the GAG-rich inner layer is modeled as a neo-Hookean matrix that can swell (<xref ref-type="bibr" rid="bib5">Demirkoparan and Pence, 2007</xref>), with<disp-formula id="equ3"><mml:math id="m3"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">r</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant="bold">C</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:msup><mml:mi/><mml:mrow><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mi mathvariant="normal">∀</mml:mi><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula>with <inline-formula><mml:math id="inf60"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> a shear modulus for this inner layer. As there is no evidence of collagen with a preferred orientation or smooth muscle cells capable of contraction within the inner layer, it is considered isotropic and passive. Fewer GAGs are present in the outer layer, but we include the possibility of a swellable matrix for illustrative purposes and to provide radial stiffness. The outer layer is then modeled using a modified four-fiber family model for a passive nonlinear stress-stretch behavior and a Rachev-type model for SMC contractility (i.e. active behavior), with a potential function<disp-formula id="equ4"><mml:math id="m4"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mtable columnalign="left left" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:mi>W</mml:mi></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">r</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant="bold">C</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>ν</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:msup><mml:mi/><mml:mrow><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mn>4</mml:mn><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">x</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>λ</mml:mi><mml:mrow><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi/><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mfrac><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mtext> </mml:mtext><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>λ</mml:mi><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mi mathvariant="normal">∀</mml:mi><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>b</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mstyle></mml:math></disp-formula>where <inline-formula><mml:math id="inf61"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> is shear modulus for the outer layer, <inline-formula><mml:math id="inf62"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> and <inline-formula><mml:math id="inf63"><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> are material parameters for each fiber family <inline-formula><mml:math id="inf64"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1,2</mml:mn><mml:mo>,</mml:mo><mml:mn>3,4</mml:mn></mml:math></inline-formula>, <inline-formula><mml:math id="inf65"><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant="normal">s</mml:mi><mml:mi mathvariant="normal">q</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mi mathvariant="normal">t</mml:mi><mml:mo>(</mml:mo><mml:mi mathvariant="bold">C</mml:mi><mml:mo>∶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">M</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo>⨂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">M</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>), <inline-formula><mml:math id="inf66"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the magnitude of the active stress, <inline-formula><mml:math id="inf67"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is the stretch at which contraction is maximum, and <inline-formula><mml:math id="inf68"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> is the stretch at which contraction ceases (<xref ref-type="bibr" rid="bib1">Baek et al., 2007</xref>). Each fiber family has an orientation vector <inline-formula><mml:math id="inf69"><mml:msup><mml:mrow><mml:mi mathvariant="bold">M</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="normal">sin</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi mathvariant="normal">c</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">s</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mi mathvariant="bold">e</mml:mi></mml:mrow><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> with <inline-formula><mml:math id="inf70"><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> the fiber angle relative to the axial direction. The fiber families are assumed to lie in the circumferential direction (<inline-formula><mml:math id="inf71"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo> <mml:mi/><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>90</mml:mn></mml:mrow><mml:mrow><mml:mo>°</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>), the axial direction (<inline-formula><mml:math id="inf72"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo> <mml:mi/><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>°</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>), and symmetric diagonal directions about the <inline-formula><mml:math id="inf73"><mml:mi>z</mml:mi></mml:math></inline-formula>-axis with <inline-formula><mml:math id="inf74"><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> fit from the experimental data (<inline-formula><mml:math id="inf75"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo> <mml:mi/><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="inf76"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo> <mml:mi/><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>). Parameter values are given in <xref ref-type="table" rid="table1">Table 1</xref> based on the aforementioned nonlinear least squares approach (<xref ref-type="bibr" rid="bib8">Ferruzzi et al., 2013</xref>).</p></sec><sec id="s9-2"><title>Bifurcation analysis – basic approach</title><p>To examine the potential for unstable equilibria (i.e. bifurcations in the solutions) leading to buckling of the wall, consider an incremental deformation added to the finite deformation (<xref ref-type="bibr" rid="bib29">Ogden, 1984</xref>) with a notation similar to that previously used to describe bifurcation behaviors in growing elastic solids (<xref ref-type="bibr" rid="bib19">Li et al., 2011</xref>; <xref ref-type="bibr" rid="bib24">Moulton and Goriely, 2011</xref>), which is mathematically similar to swelling. Quantities related to the intermediate finite deformation are given as <inline-formula><mml:math id="inf77"><mml:msup><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mo>⋅</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> while those related to the incremental motions are denoted as <inline-formula><mml:math id="inf78"><mml:msup><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mo>⋅</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>. The current position <inline-formula><mml:math id="inf79"><mml:mi mathvariant="bold-italic">x</mml:mi></mml:math></inline-formula> from position <inline-formula><mml:math id="inf80"><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">x</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> is specified as <inline-formula><mml:math id="inf81"><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">x</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">u</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> with <inline-formula><mml:math id="inf82"><mml:mi>ϵ</mml:mi><mml:mo>≪</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> scaling the displacement <inline-formula><mml:math id="inf83"><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">u</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, which allows the deformation gradient to be written as <inline-formula><mml:math id="inf84"><mml:mi mathvariant="bold">F</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> with <inline-formula><mml:math id="inf85"><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi>P</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold">*</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> the deformation gradient from the reference to the finitely deformed configuration and <inline-formula><mml:math id="inf86"><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> the incremental displacement gradient with respect to that configuration. The nominal stress <inline-formula><mml:math id="inf87"><mml:mi mathvariant="bold">P</mml:mi></mml:math></inline-formula>, defined through <inline-formula><mml:math id="inf88"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="bold">t</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mo form="prefix" movablelimits="true">det</mml:mo><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula>, follows as <inline-formula><mml:math id="inf89"><mml:mi mathvariant="bold">P</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>. As noted previously (<xref ref-type="bibr" rid="bib12">Haughton and Ogden, 1978</xref>; <xref ref-type="bibr" rid="bib33">Sanft et al., 2019</xref>), it is convenient to update the reference configuration to the current configuration yielding <inline-formula><mml:math id="inf90"><mml:msub><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>, which, along with considering the Lagrange multiplier as <inline-formula><mml:math id="inf91"><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, allows us to write <inline-formula><mml:math id="inf92"><mml:msubsup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi mathvariant="bold-script">B</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mi mathvariant="bold">I</mml:mi></mml:math></inline-formula> with <inline-formula><mml:math id="inf93"><mml:mi mathvariant="bold-script">B</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>W</mml:mi><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mo>∂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mo>)</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">F</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup></mml:math></inline-formula> the fourth-order stiffness tensor calculated in the finitely deformed configuration and <inline-formula><mml:math id="inf94"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> the increment in the Lagrange multiplier. Linear momentum balance then requires <inline-formula><mml:math id="inf95"><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">v</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, which is satisfied by the finite deformation, leaving <inline-formula><mml:math id="inf96"><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">v</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> to be resolved. We proceed by assuming a form for the incremental displacement describing buckling as <inline-formula><mml:math id="inf97"><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">u</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>[</mml:mo><mml:mi mathvariant="normal">u</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi mathvariant="normal">v</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>]</mml:mo></mml:math></inline-formula>, with no incremental motion in the axial direction. Displacements in the <inline-formula><mml:math id="inf98"><mml:mi>r</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="inf99"><mml:mi>θ</mml:mi></mml:math></inline-formula> directions are not a function of <inline-formula><mml:math id="inf100"><mml:mi>z</mml:mi></mml:math></inline-formula>. The incremental displacement gradient then becomes<disp-formula id="equ5"><mml:math id="m5"><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi mathvariant="normal">v</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi mathvariant="normal">v</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">u</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">v</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math></disp-formula>where <inline-formula><mml:math id="inf101"><mml:msub><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mo>∙</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo>∂</mml:mo><mml:mfenced separators="|"><mml:mrow><mml:mo>∙</mml:mo></mml:mrow></mml:mfenced><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="inf102"><mml:msub><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mo>∙</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo>∂</mml:mo><mml:mfenced separators="|"><mml:mrow><mml:mo>∙</mml:mo></mml:mrow></mml:mfenced><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>θ</mml:mi></mml:math></inline-formula>. The incremental deformation is assumed to be isochoric, requiring <inline-formula><mml:math id="inf103"><mml:mi mathvariant="normal">t</mml:mi><mml:mi mathvariant="normal">r</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold">H</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. As there is no axial dependence in the incremental deformation, the equilibrium equations reduce to a system of two differential equations in <inline-formula><mml:math id="inf104"><mml:mi mathvariant="normal">u</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="inf105"><mml:mi mathvariant="normal">v</mml:mi></mml:math></inline-formula>, and <inline-formula><mml:math id="inf106"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>. These functions are assumed to have sinusoidal forms in the buckled configuration, where <inline-formula><mml:math id="inf107"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">u</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula>, <inline-formula><mml:math id="inf108"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">v</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math></inline-formula>, and <inline-formula><mml:math id="inf109"><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mrow><mml:mi mathvariant="normal">cos</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:math></inline-formula> with <inline-formula><mml:math id="inf110"><mml:mi>n</mml:mi></mml:math></inline-formula> the buckling mode (i.e. the number of folds in the inner portion of the vessel wall). Using the incompressibility condition, it is possible to rewrite <inline-formula><mml:math id="inf111"><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> in terms of <inline-formula><mml:math id="inf112"><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>, and the two equilibrium equations can be combined to eliminate <inline-formula><mml:math id="inf113"><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>, yielding a single fourth order, ordinary differential equation in <inline-formula><mml:math id="inf114"><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula> (<xref ref-type="bibr" rid="bib12">Haughton and Ogden, 1978</xref>; <xref ref-type="bibr" rid="bib33">Sanft et al., 2019</xref>) of the form<disp-formula id="equ6"><mml:math id="m6"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>where coefficients <inline-formula><mml:math id="inf115"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>-<inline-formula><mml:math id="inf116"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> are given below and <inline-formula><mml:math id="inf117"><mml:mo>(</mml:mo><mml:mo>∙</mml:mo><mml:msup><mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mo>(</mml:mo><mml:mo>∙</mml:mo><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>d</mml:mi><mml:mi>r</mml:mi></mml:math></inline-formula>. We assume that the incremental tractions on the inner surface are zero with <inline-formula><mml:math id="inf118"><mml:mi mathvariant="bold">n</mml:mi><mml:mo>∙</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="bold">t</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>, which yields two equations<disp-formula id="equ7"><mml:math id="m7"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:msup><mml:mi/><mml:mo>‴</mml:mo></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mrow><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">n</mml:mi><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula>with coefficients <inline-formula><mml:math id="inf119"><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> given below. As there is little evidence of buckling in the outer layer, and the Rachev-type contractility model generally yields tensile stresses that would inhibit buckling, we specify that the incremental deformation vanishes at the interface of the two layers with <inline-formula><mml:math id="inf120"><mml:mi>f</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and <inline-formula><mml:math id="inf121"><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mfenced separators="|"><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula> and that the incremental shear traction is zero (<xref ref-type="bibr" rid="bib42">Yang et al., 2007</xref>), which gives<disp-formula id="equ8"><mml:math id="m8"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mrow><mml:mi mathvariant="normal">a</mml:mi><mml:mi mathvariant="normal">n</mml:mi><mml:mi mathvariant="normal">d</mml:mi></mml:mrow><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mtext> </mml:mtext><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula>with coefficients <inline-formula><mml:math id="inf122"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> given below.</p><p>We used the compound matrix method (<xref ref-type="bibr" rid="bib14">Haughton and Orr, 1997</xref>; <xref ref-type="bibr" rid="bib20">Lindsay and Rooney, 1992</xref>), a modification of the determinantal method commonly used for linear bifurcation analysis (<xref ref-type="bibr" rid="bib13">Haughton and Ogden, 1979</xref>), to solve the differential equation numerically. The fourth order equation above was rewritten as a system of four, first order differential equations, <inline-formula><mml:math id="inf123"><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mi mathvariant="bold-italic">'</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="bold-italic">A</mml:mi><mml:mi mathvariant="bold-italic">y</mml:mi></mml:math></inline-formula>, where <inline-formula><mml:math id="inf124"><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="inf125"><mml:mi mathvariant="bold-italic">A</mml:mi></mml:math></inline-formula> is the corresponding coefficient matrix. Boundary conditions at the inner surface and interface were similarly re-written as <inline-formula><mml:math id="inf126"><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi mathvariant="bold-italic">B</mml:mi><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and <inline-formula><mml:math id="inf127"><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi mathvariant="bold-italic">C</mml:mi><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, respectively. We define two linearly independent initial conditions at <inline-formula><mml:math id="inf128"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi></mml:math></inline-formula>, which can be integrated to <inline-formula><mml:math id="inf129"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:math></inline-formula> to create linearly independent solutions <inline-formula><mml:math id="inf130"><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">y</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> and <inline-formula><mml:math id="inf131"><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">y</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> such that <inline-formula><mml:math id="inf132"><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">y</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">y</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant="bold-italic">M</mml:mi><mml:mi mathvariant="bold-italic">k</mml:mi></mml:math></inline-formula>. For the determinantal method, one iterates on <inline-formula><mml:math id="inf133"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> until <inline-formula><mml:math id="inf134"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mi mathvariant="bold-italic">C</mml:mi><mml:mi mathvariant="bold-italic">M</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:math></inline-formula> However, this method can fail for stiff systems, thus we use Laplace expansions to write a new bifurcation condition equivalent to the original with <inline-formula><mml:math id="inf135"><mml:mrow><mml:mrow><mml:mi mathvariant="normal">det</mml:mi></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mi mathvariant="bold-italic">C</mml:mi><mml:mi mathvariant="bold-italic">M</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> where<disp-formula id="equ9"><mml:math id="m9"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced close="|" open="|" separators="|"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced separators="|"><mml:mrow><mml:mn>1,2</mml:mn></mml:mrow></mml:mfenced><mml:mo>,</mml:mo> <mml:mi/> <mml:mi/><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced close="|" open="|" separators="|"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced separators="|"><mml:mrow><mml:mn>1,3</mml:mn></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:math></disp-formula>and similarly, <inline-formula><mml:math id="inf136"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>1,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf137"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>2,3</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf138"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>2,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf139"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>3,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>. We evaluate <inline-formula><mml:math id="inf140"><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo></mml:math></inline-formula> with<disp-formula id="equ10"><mml:math id="m10"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula>and similarly, <inline-formula><mml:math id="inf141"><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>1,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf142"><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>2,3</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf143"><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>2,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, <inline-formula><mml:math id="inf144"><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn>3,4</mml:mn><mml:mo>)</mml:mo></mml:math></inline-formula>, where<disp-formula id="equ11"><mml:math id="m11"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula></p><p>To evaluate the new bifurcation condition, we create a system of equations <inline-formula><mml:math id="inf145"><mml:mi mathvariant="bold-italic">ϕ</mml:mi><mml:mi mathvariant="bold-italic">'</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="bold-script">A</mml:mi><mml:mi mathvariant="bold-italic">ϕ</mml:mi></mml:math></inline-formula> and identify the components of <inline-formula><mml:math id="inf146"><mml:mi mathvariant="bold-italic">ϕ</mml:mi><mml:mi mathvariant="bold-italic">'</mml:mi></mml:math></inline-formula> as, for example,<disp-formula id="equ12"><mml:math id="m12"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msubsup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msup><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd><mml:msup><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msup><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mtd><mml:mtd><mml:msup><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mtable columnalign="center center" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable><mml:mo>|</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mstyle></mml:math></disp-formula></p><p>The components of <inline-formula><mml:math id="inf147"><mml:mi mathvariant="bold-script">A</mml:mi></mml:math></inline-formula> can thus be conveniently defined in terms of the original components of <inline-formula><mml:math id="inf148"><mml:mi mathvariant="bold-italic">A</mml:mi></mml:math></inline-formula> (<xref ref-type="bibr" rid="bib14">Haughton and Orr, 1997</xref>), as listed below. This new system is then integrated from <inline-formula><mml:math id="inf149"><mml:mi>a</mml:mi></mml:math></inline-formula> to <inline-formula><mml:math id="inf150"><mml:mi>b</mml:mi></mml:math></inline-formula> using a fourth-order Runge-Kutta method, and <inline-formula><mml:math id="inf151"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> is varied iteratively until the boundary condition at <inline-formula><mml:math id="inf152"><mml:mi>b</mml:mi></mml:math></inline-formula> is satisfied, namely <inline-formula><mml:math id="inf153"><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="script">𝒞</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>. Loading conditions, including the volume change <inline-formula><mml:math id="inf154"><mml:msup><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, luminal pressure <inline-formula><mml:math id="inf155"><mml:mi>P</mml:mi></mml:math></inline-formula>, and axial stretch <inline-formula><mml:math id="inf156"><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, can be varied parametrically to understand their effects on the critical value of <inline-formula><mml:math id="inf157"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> needed to induce buckling. Note, one may also fix the value of <inline-formula><mml:math id="inf158"><mml:msub><mml:mrow><mml:mi mathvariant="normal">T</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> and identify the critical value of a different loading variable of interest.</p></sec></sec><sec id="s10" sec-type="appendix"><title>Bifurcation analysis – specific functions</title><p>For the fourth order governing differential equation for the incremental displacement:<disp-formula id="equ13"><mml:math id="m13"><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>we have<disp-formula id="equ14"><mml:math id="m14"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mtable columnalign="left left" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>∗</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mo>′</mml:mo></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mo>′</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:msup><mml:mi/><mml:mo>″</mml:mo></mml:msup></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>7</mml:mn><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mstyle></mml:math></disp-formula></p><p>Note: These coefficients include only the non-zero components of <inline-formula><mml:math id="inf159"><mml:mi mathvariant="bold-script">B</mml:mi></mml:math></inline-formula> for the considered stored energy density function.</p><p>For the fourth order differential equation<disp-formula id="equ15"><mml:math id="m15"><mml:mi mathvariant="bold-italic">y</mml:mi><mml:mi mathvariant="bold-italic">'</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="bold-italic">A</mml:mi><mml:mi mathvariant="bold-italic">y</mml:mi></mml:math></disp-formula>note that<disp-formula id="equ16"><mml:math id="m16"><mml:mi mathvariant="bold-italic">A</mml:mi><mml:mo>=</mml:mo><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>1</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>1</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>1</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>-</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></disp-formula></p><p>For boundary conditions on the governing equation at the inner surface:<disp-formula id="equ17"><mml:math id="m17"><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>1,3</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>1,2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>1,1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>1,0</mml:mn></mml:mrow></mml:msub><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow> <mml:mi/><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula>we have,</p><p><inline-formula><mml:math id="inf160"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mtable columnalign="left left" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:msubsup><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mi>r</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mstyle></mml:math></inline-formula></p><p>For boundary conditions on the governing equation at the interface:<disp-formula id="equ18"><mml:math id="m18"><mml:msub><mml:mrow><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1,2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1,1</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>'</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mrow> <mml:mi/><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula>we have</p><p><inline-formula><mml:math id="inf161"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mtable columnalign="left left" columnspacing="1em" rowspacing="4pt"><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mi>r</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">B</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mi>r</mml:mi><mml:mi>θ</mml:mi><mml:mtext> </mml:mtext></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mstyle></mml:math></inline-formula></p><p>Finally, for the compound matrix method component matrix<disp-formula id="equ19"><mml:math id="m19"><mml:mi mathvariant="bold-script">A</mml:mi><mml:mo>=</mml:mo><mml:mfenced close="]" open="[" separators="|"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>32</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>42</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>43</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>44</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>31</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>34</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>24</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>41</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>43</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>22</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>44</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>23</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>41</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>31</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>42</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>32</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>33</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>44</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></disp-formula>with the components of <inline-formula><mml:math id="inf162"><mml:mi mathvariant="bold-italic">A</mml:mi></mml:math></inline-formula> given above.</p></sec></boxed-text></app></app-group></back><sub-article article-type="decision-letter" id="sa1"><front-stub><article-id pub-id-type="doi">10.7554/eLife.60683.sa1</article-id><title-group><article-title>Decision letter</article-title></title-group><contrib-group><contrib contrib-type="editor"><name><surname>Downs</surname><given-names>Karen</given-names></name><role>Reviewing Editor</role><aff><institution>University of Wisconsin-Madison School of Medicine and Public Health</institution><country>United States</country></aff></contrib></contrib-group><contrib-group><contrib contrib-type="reviewer"><name><surname>Wagenseil</surname><given-names>Jessica</given-names> </name><role>Reviewer</role><aff><institution/></aff></contrib></contrib-group></front-stub><body><boxed-text><p>In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.</p></boxed-text><p><bold>Acceptance summary:</bold></p><p>We agree that study of the mature umbilical cord has been neglected - especially lacking is understanding how it prevents fetal blood loss during passage from the birth canal. The ground-breaking analyses described in this manuscript, leading to your unprecedented model of umbilical arterial closure at birth, will provide a wealth of new insight into the biology and biomechanics of one of Placentalia's most important vascular structures.</p><p><bold>Decision letter after peer review:</bold></p><p>Thank you for submitting your article &quot;Vascular dimorphism ensured by regulated proteoglycan dynamics favors rapid umbilical artery closure at birth&quot; for consideration by eLife. Your article has been reviewed by Didier Stainier as the Senior Editor, a Reviewing Editor, and two reviewers. The following individuals involved in review of your submission have agreed to reveal their identity: Jessica Wagenseil (Reviewer #2).</p><p>The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.</p><p>We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.</p><p>Summary:</p><p>A comprehensive and elegant combination of gene/protein expression, cross-species analyses, genetic mutants, and biomechanical testing/computer analysis was used to study a universal but generally neglected biological event common to all Placentalia: rapid umbilical closure at birth. The phenomenon is critical to fetal survival because, as the purveyor of fetal blood to the chorionic bed of the placenta, the artery must rapidly close to prevent ex-sanguination of the fetus as it exits the birth canal. By contrast, the vein carries blood from the chorionic bed to the fetus and thus, at least in this regard, is less significant.</p><p>This is a highly significant and universally relevant study to all Placentalia, providing new insight into the cell/developmental biology and biomechanics of one of this group's most important vascular structures. The study has been beautifully executed and generally well presented. Every finding is novel. And now, for the first time, closure of the umbilical cord at birth has an experimental precedent. The authors conclude that evolution has made sure that a healthy placenta permits a healthy transition from fetal to neo-natal life.</p><p>Essential revisions:</p><p>1) Computational Model</p><p>Variability in the experimental data, challenges in testing and fitting material parameters for these vessels, and how that might affect the conclusions of the computational model should be briefly discussed. The authors already have all the data and are using it for the computational model to draw conclusions, but they left out a lot of details for the modeling and did not address how some of the assumptions and fitted parameters used in the model would affect their conclusions.</p><p>2) Logic of Presentation</p><p>Because the biomechanical testing in Figure 4 assumed similarities between mouse and human umbilical arteries and veins in advance of the confirmation that was ultimately presented in Figure 5 (mouse studies), could the authors reverse the order of presentation, both in the text and in the figures which follow, i.e., place the whole of their (descriptive) mouse data immediately after the large animal analysis (current Figure 3) and before the functional biomechanical testing and computer modeling (current Figure 4)?</p><p>3) All Figures:</p><p>For all of the Figures involving synchrotron, histological, immunofluorescence, and RNA-ISH, please state in the corresponding legend how many samples were imaged.</p><p>4) Figure 1.</p><p>a) Given that the human umbilical cord has two arteries and a vein, did the authors ever separate out the arteries for analysis and, if not, why not? Please address this somewhere in the manuscript.</p><p>b) Similarly, in the Supplemental methods, &quot;Veins and arteries were dissected....&quot; - please clarify here whether both arteries were dissected.</p><p>5) Figure 1—figure supplement 1C,D.</p><p>These panels appear to show that the smooth muscle cells (SMC) of both vessels are similarly organized with alternating circumferential and longitudinal layers, but that the major difference between them is fewer layers of SMC in the vein. It is not clear from the text or legend whether the authors conclude this, too - could they clarify their interpretation in the text?</p><p>6) Figure 4F.</p><p>What do the authors make of the differences in the number of buckles within and between human umbilical cords - statistically significant? Can they say anything about those differences in the other mammals?</p><p>7) Figure 5. The mutant analyses:</p><p>a) In the text, legends, and figures, the notation is inconsistent for the knockout mouse models: Adamts1-/-, Adamts1 KO, Acan-/-, Acancmd/cmd, Acan mutants, and Acan KO are all used. Please choose a consistent notation for each model.</p><p>b) Please provide information on genotyping mutant litters in the Methods/Supplemental Methods, and their ratios at each embryonic day examined, especially as the authors claim that mutants did not show intrauterine growth retardation or death (see Figure 5—figure supplement 1). If possible, please indicate the background resorption level in these strains, which would be evident by having included the genotype of the resorptions.</p><p>c) Could the authors group presentation of aggregan and adamts1 mutants rather than intermingle the data within the text? - it was difficult to follow which result correlated with which mutant.</p><p>d) Figure 5G. The PHH3 staining examples are not very good. How are they being normalized to calculate a percentage?</p><p>8) Figure 4—figure supplement 1.</p><p>a) Need SD or SEM.</p><p>b) The number of samples used for mechanical testing needs to be included.</p><p>c) Please include circumferential and axial stress-strain curves which are directly related to the fitted material parameters used for the modeling.</p><p>d) Figure 4—figure supplement 1B, subsection “Differential SMC contraction in the bilayered umbilical arteries and vein”of the text state that the artery has a smaller lumen, citing this panel, but this panel shows the outer diameter, and not the inner one.</p><p>e) Figure 4—figure supplement 1C and subsection “Differential SMC contraction in the bilayered umbilical arteries and vein”, text: Please define &quot;distensibility&quot; and &quot;extensibility&quot; as used in the text. At what pressures or axial stretches are you comparing the distensibility and extensibility? How do the axial stretch values in Figure 4—figure supplement 1C compare to the in vivo axial stretches?</p><p>9) Figure 5—figure supplement 1.</p><p>Please support the important conclusion, subsection “Aggrecan and Adamts1 are necessary for normal umbilical cord morphogenesis” based on this figure that &quot;Neither mutant showed intrauterine growth retardation, and intrauterine death was infrequent, suggesting adequate cord circulation.&quot;, with actual measurements, especially as the acan KO mutant in Figure 5—figure supplement 1B appears smaller at E14.5 than its wildtype counterpart - assuming the same magnifications, which should be indicated.</p><p>10) Figure 8.</p><p>- What are the blue cells? Something other than SMCs?</p><p>11) Table 1.</p><p>a) As c1 for the diagonal fibers is 2-3 orders of magnitude below the circ and axial fibers, are the diagonal fibers really necessary in the constitutive model?</p><p>b) Also, the c2 values are 1-2 orders of magnitude higher than values that the group has published previously for mouse elastic arteries. Can the authors comment on the suitability of the constitutive model for fitting such nonlinear data and differences between the mechanical behavior of umbilical artery/vein and elastic arteries in the mouse?</p><p>c) Are the values in Table 1 averages of individual values from multiple arteries, fit from a combination of data from multiple arteries, or representative values from a single artery?</p><p>[Editors' note: further revisions were suggested prior to acceptance, as described below.]</p><p>Thank you for re-submitting your article &quot;Vascular dimorphism ensured by regulated proteoglycan dynamics favors rapid umbilical artery closure at birth&quot; for re-consideration by eLife. Your revised manuscript has been re-reviewed Didier Stainier as the Senior Editor, a Reviewing Editor, and two reviewers. The following individuals involved in review of your submission have agreed to reveal their identity: Jessica Wagenseil (Reviewer #2).</p><p>The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.</p><p>We would like to draw your attention to changes in our revision policy that we have made in response to COVID-19 (https://elifesciences.org/articles/57162). Specifically, we are asking editors to accept without delay manuscripts, like yours, that they judge can stand as eLife papers without additional data, even if they feel that they would make the manuscript stronger. Thus the revisions requested below only address clarity and presentation.</p><p>Summary:</p><p>In their revised manuscript concerning the mechanism of umbilical arterial closure at birth in Placentalia, Nandadasa et al., have satisfactorily addressed the majority of the reviewers' concerns. However, there remain two major concerns: (1) the mouse mutant analyses, and (2) the number of specimens used per experiment; a small number of minor revisions; and requested changes to the Abstract, in accord with eLife's policies.</p><p>Essential revisions:</p><p>1) Genetic mutants. The reviewers had requested the following (copied from the previous letter to the authors): (b) Please provide information on genotyping mutant litters in the Materials and methods section/Supplemental methods, and their ratios at each embryonic day examined, especially as the authors claim that mutants did not show intrauterine growth retardation or death (see Figure 5—figure supplement 1). If possible, please indicate the background resorption level in these strains, which would be evident by having included the genotype of the resorptions.</p><p>Mouse mutant Adamts-/-. As the authors did not provide their method for genotyping this mutant strain, as requested, more investigation was needed on the part of the reviewers to understand what exactly this strain is to consider why the authors ignored the request.</p><p>Background (Oller et al., 2017): The Adamts-/- mutant used in this study was described by Oller et al., 2017 as an insertion of a lacZ-bearing cassette into intron 1 of the gene. According to this previous paper, not only does this insertion reveal where Adamts1 is expressed via staining for ß-galactosidase activity, but in its hemizygous state, the insertion also causes a reduction in both mRNA and protein. In its homozygous state, ß-galactosidase activity is still detectable, but the Adamts1 protein is not (Figure 1a of Oller et al., 2017). Oller et al did not provide details on how the animals were maintained and mated (which they really should have been asked to do, alas), but they did explain how their litters were genotyped, the specific sequence used in those PCR genotyping experiments, and the genotypic ratios of their animals at weaning (Supplemental Figure 1b of Oller et al., 2017).</p><p>In the current study, the authors distinguished hemizygotes and homozygotes in the figure panels (e.g., Figure 4C versus the other panels), and wildtype and homozygotes in the others that relate to the mutant analysis, implying that they genotyped this material. However, they ignored the reviewers' procedural request, which is repeated and expanded as follows:</p><p>Please provide information on</p><p>i) genotyping method used, including the exact DNA sequence for PCR analysis;</p><p>ii) genotypic ratios according to gestational day;</p><p>iii) the genetic background on which the Adamts animals were maintained (according to Oller et al., it seems to be a B/6 background, but please confirm);</p><p>iv) the parental genotypes used to produce the specific genotypes (perhaps hemizygous by wildtype matings produced hemizygous embryos detectable by X-gal staining, whilst homozygous embryos were obtained by crossing hemizygous animals and using PCR genotyping to distinguish the three genotypes? - please confirm/clarify);</p><p>v) how gestational age was determined. Were timed matings used? if not timed matings, then how was gestational age determined?;</p><p>vi) the protocol used for X-gal staining the hemizygotes.</p><p>While Oller et al., 2017 used this transgenic mouse line, they did not create it. Please clarify the specific origin of this mutant mouse strain - that information was impossible to locate on the EMMA site.</p><p>It would be most helpful for the reader if the authors would introduce the Adamts mutant in the Results section by summarizing Oller et al.'s results concerning the levels of mRNA versus protein in hemizygotes and homozygotes, as under &quot;Background&quot;, above.</p><p>Finally, if the authors can, would they comment on whether hemizygotes exhibited foreshortened umbilical cords, too, and did their lengths fall between those of the wildtype and homozygous mutants?</p><p>Mouse mutant Acan-/-.</p><p>Although the authors provided the genotypic ratios of the Acan-/- mutants as requested by the reviewers (new Figure 4—figure supplement 1 Panel 4C), they did not indicate the genotyping procedure. Please add it to Detailed Methods, to include how the DNA was obtained, and the DNA sequence used to PCR the littermates' DNA.</p><p>The mutant allele was originally described by Krueger et al., 1999, but from where did the authors procure this mouse strain?</p><p>Litters were obtained at E12.5, E14.5, and E18.5; please indicate how matings were carried out to ascertain the timing of gestation, including parental genotypes that produced the Acan litters.</p><p>2) The reviewers had requested the number of specimens (n) for every experiment. The following are still missing:</p><p>Figure 1D. n, the number of immunostained specimens?</p><p>Figure 2B, line 545. &quot;n=3 umbilical cords&quot; - for each probe?, or for both?</p><p>Figure 4—figure supplement 1.</p><p>S4b. n = ?</p><p>S4d. n = ?</p></body></sub-article><sub-article article-type="reply" id="sa2"><front-stub><article-id pub-id-type="doi">10.7554/eLife.60683.sa2</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><disp-quote content-type="editor-comment"><p>Summary:</p><p>A comprehensive and elegant combination of gene/protein expression, cross-species analyses, genetic mutants, and biomechanical testing/computer analysis was used to study a universal but generally neglected biological event common to all Placentalia: rapid umbilical closure at birth. The phenomenon is critical to fetal survival because, as the purveyor of fetal blood to the chorionic bed of the placenta, the artery must rapidly close to prevent ex-sanguination of the fetus as it exits the birth canal. By contrast, the vein carries blood from the chorionic bed to the fetus and thus, at least in this regard, is less significant.</p><p>This is a highly significant and universally relevant study to all Placentalia, providing new insight into the cell/developmental biology and biomechanics of one of this group's most important vascular structures. The study has been beautifully executed and generally well presented. Every finding is novel. And now, for the first time, closure of the umbilical cord at birth has an experimental precedent. The authors conclude that evolution has made sure that a healthy placenta permits a healthy transition from fetal to neo-natal life.</p><p>Essential revisions:</p><p>1) Computational Model</p><p>Variability in the experimental data, challenges in testing and fitting material parameters for these vessels, and how that might affect the conclusions of the computational model should be briefly discussed. The authors already have all the data and are using it for the computational model to draw conclusions, but they left out a lot of details for the modeling and did not address how some of the assumptions and fitted parameters used in the model would affect their conclusions.</p></disp-quote><p>Thank you for this important comment. Our primary goal was to understand how the umbilical artery closes at birth, which necessitated computational modeling of swelling, contraction, and buckling. We thus focused on the model, though indeed we needed new data to inform the model. We now provide much more detail on these data and how they were collected. We nevertheless emphasize that we sought a general mechanism, not subject-to-subject differences, hence we used “mean data” to inform the modeling, which was then explored via extensive numerical parametric studies in silico.</p><p>We used a custom computer-controlled biaxial testing system designed specifically for testing murine arteries, which has proven highly reliable. We used a nonlinear, anisotropic constitutive relation that we have found to be robust in describing diverse murine arteries (systemic and pulmonary) and veins, a relation that has been validated independently by multiple groups. Best-fit material parameters for the 4-fiber family constitutive model (Table 1) were determined using a standard nonlinear regression approach in MATLAB (lsqnonlin function), where differences in predicted vs. measured pressure-diameter data and force-length data were minimized. Multiple random initial guesses were used to ensure convergence of the nonlinear regression to the same minimum. The primary parameters were necessarily obtained from passive, tensile data, yet a key parameter in this study of buckling was the compressive stiffness. We thus studied parametrically the effects of the associated neoHookean parameter in the inner layer, which is greater in compression due to the increased presence of GAGs identified in the immunoassays. Table 1 reports values from our data analysis and many preliminary simulations.</p><p>Computational results were initially presented for a representative umbilical artery sample, but, motivated by the reviewer’s excellent suggestion, we went back and fit simultaneously all of the data from all 4 umbilical artery samples, hence yielding best-fit parameters for the truly mean behavior (which is different from using mean values of the individually determined parameters, which we never do). There were some changes in the individual parameter values (based on the mean data) from those used previously (for a single representative sample), particularly for the one fiber-family parameters (which resolved a concern of the reviewer noted below), but these changes in parameter values did not change any of the conclusions from resulting simulation outputs, suggesting further that the buckling phenomenon occurs across different parameter values and is required for closure.</p><disp-quote content-type="editor-comment"><p>2) Logic of Presentation</p><p>Because the biomechanical testing in Figure 4 assumed similarities between mouse and human umbilical arteries and veins in advance of the confirmation that was ultimately presented in Figure 5 (mouse studies), could the authors reverse the order of presentation, both in the text and in the figures which follow, i.e., place the whole of their (descriptive) mouse data immediately after the large animal analysis (current Figure 3) and before the functional biomechanical testing and computer modeling (current Figure 4)?</p></disp-quote><p>We agree with the reviewer’s suggestion and have made the requested changes to both the figures and the text. The mouse umbilical cord data is now presented immediately after the large mammal data followed by computational modeling.</p><disp-quote content-type="editor-comment"><p>3) All Figures:</p><p>For all of the Figures involving synchrotron, histological, immunofluorescence, and RNA-ISH, please state in the corresponding legend how many samples were imaged.</p></disp-quote><p>We have added the missing <italic>n</italic> information to the legend of each figure panel.</p><disp-quote content-type="editor-comment"><p>4) Figure 1.</p><p>a) Given that the human umbilical cord has two arteries and a vein, did the authors ever separate out the arteries for analysis and, if not, why not? Please address this somewhere in the manuscript.</p></disp-quote><p>In all our experiments the two arteries of human cords appeared identical and indistinguishable in histology and immunostaining. Since it is not possible to identify a distinction between the arteries (assigning for example in each cord, an artery A or B designation), such a comparison would have little basis. Therefore, we did not dissect the two arteries and study them individually. All human umbilical cord sections analyzed contained three vessels and both arteries were imaged in addition to the vein. We have added this information to the methods section of the manuscript. The only differences observed between the two arteries from an individual cord are the number of folds in the areas analyzed. We have illustrated this information in Figure 7F (cord #12,15,16 and 19).</p><disp-quote content-type="editor-comment"><p>b) Similarly, in the Supplemental Methods, &quot;Veins and arteries were dissected....&quot; - please clarify here whether both arteries were dissected.</p></disp-quote><p>We have updated the supplemental methods section to reflect both arteries were used.</p><disp-quote content-type="editor-comment"><p>5) Figure 1—figure supplement 1C,D.</p><p>These panels appear to show that the smooth muscle cells (SMC) of both vessels are similarly organized with alternating circumferential and longitudinal layers, but that the major difference between them is fewer layers of SMC in the vein. It is not clear from the text or legend whether the authors conclude this, too - could they clarify their interpretation in the text?</p></disp-quote><p>We agree with the reviewer’s interpretation. Histologically, the major difference observed between the arteries and the vein was the thickness of the tunica media, with the vein having fewer SMC layers overall, as clarified in the revised text.</p><disp-quote content-type="editor-comment"><p>6) Figure 4F.</p><p>What do the authors make of the differences in the number of buckles within and between human umbilical cords - statistically significant? Can they say anything about those differences in the other mammals?</p></disp-quote><p>Our conclusions were limited to the very short length of each vessel analyzed (sectioned) in each umbilical cord. The correlation of the number of buckles and the patency of the vessel should also therefore be limited to the small area we analyzed. i.e. an open artery with fewer folds in the area analyzed for a specific specimen may have an area with more buckles and an occluded lumen in a different region of the cord which we have not analyzed. A future, more focused study analyzing the formation of buckles along the length of the cord using a 3D imaging technique may be necessary for completely understanding the number of buckles needed for driving vessel occlusion. However, from the analysis of the cohort of cords in this study, at least 4 buckles observed in a small segment are sufficient to enable arterial occlusion. The computational modeling predicts 3-7 buckles. Since a large number of cords would be required for statistical conclusions in large mammals, and we were only able to obtain a small number, that too with difficulty, we have not done a statistical analysis on these mammalian cords.</p><disp-quote content-type="editor-comment"><p>7) Figure 5. The mutant analyses:</p><p>a) In the text, legends, and figures, the notation is inconsistent for the knockout mouse models: Adamts1-/-, Adamts1 KO, Acan-/-, Acancmd/cmd, Acan mutants, and Acan KO are all used. Please choose a consistent notation for each model.</p></disp-quote><p>We have updated both the text and the figures to consistently use the <italic>Adamts1<sup>-/-</sup></italic> and <italic>Acan<sup>-/-</sup> </italic>notations throughout the manuscript.</p><disp-quote content-type="editor-comment"><p>b) Please provide information on genotyping mutant litters in the Methods/Supplemental Methods, and their ratios at each embryonic day examined, especially as the authors claim that mutants did not show intrauterine growth retardation or death (see Figure 5—figure supplement 1). If possible, please indicate the background resorption level in these strains, which would be evident by having included the genotype of the resorptions.</p></disp-quote><p>We have added the observed genotype information as a new figure panel (Figure 4—figure supplement 1C) and rewritten the section of the manuscript better describing the <italic>Acan<sup>-/-</sup> embryos</italic>, limiting our conclusions to their umbilical cords. Defective cartilage and impaired skeletal development the <italic>Acan<sup>-/-</sup></italic> embryos was previously extensively characterized. Deficiencies in these processes alter embryo dimensions and we therefore have removed any conclusions or suggestions related to overall growth, and limited the focus to the umbilical cord phenotype. <italic>Acan<sup>-/-</sup></italic> embryos are observed at the expected Mendelian ratio at E18.5 and hence do not die in utero prior to E18.5. We have not observed a higher rate of resorptions of embryos in our crossings in agreement with observations made by others.</p><p>In the initial analysis of <italic>Acan<sup>-/-</sup></italic> mice, from a total of 733 offspring, 180 were homozygous (24.6%) as reported in Rittenhouse et al., 1978. The background strain of these mice (C57BL/6J) is reported to have an average litter size of 6.2 pups (Verley et al., 1967) and average resorption sites from embryonic day 11 to term of 1.54 + 0.15 in 3-7 month old mothers and 2.94 + 0.28 in 11-12 month old mothers (cf. Holinka, Tseng and Finch, 1979).</p><disp-quote content-type="editor-comment"><p>c) Could the authors group presentation of aggregan and adamts1 mutants rather than intermingle the data within the text? - it was difficult to follow which result correlated with which mutant.</p></disp-quote><p>We have rearranged this figure and grouped the <italic>Acan</italic> and <italic>Adamts1</italic> data panels (Figure 4F for <italic>Adamts1</italic> and Figure 4G for <italic>Acan</italic>) separately. The manuscript text reflects this change.</p><disp-quote content-type="editor-comment"><p>d) Figure 5G. The PHH3 staining examples are not very good. How are they being normalized to calculate a percentage?</p></disp-quote><p>We enlarged this figure panel and clearly marked pHH3 positive cells with white arrowheads and indicated the vessel lumen using a white dotted line (Figure 4H). The samples were counterstained with DAPI to identify all nuclei. The percentage of pHH3 positive nuclei was determined. At least two sections were stained and quantified from each umbilical cord (a total of 4 umbilical cords for each genotype (16 sections)).</p><disp-quote content-type="editor-comment"><p>8) Figure 4—figure supplement 1.</p></disp-quote><p>As noted above, our primary goal was to determine salient characteristics that drive umbilical artery closure at birth using a (new) computational model via parametric studies. Yet, we needed baseline passive biomechanical properties. Although we tested N=4 umbilical arteries (and N=4 veins), we previously used best-fit material parameters for a single “representative” sample. In the revised manuscript, however, we re-performed all data analysis, now based on a rigorous mean behavior (all pressure-diameter, axial force-length data for all 4 samples were combined into a single large data set and best-fit values were determined). As can be seen from the new Fig 7—figure supplement 2, the model-predicted behavior describes very well the mean responses, noting that the grey regions show standard deviations (not standard errors of the mean) to reveal the full extent of the specimen-to-specimen differences.</p><disp-quote content-type="editor-comment"><p>a) Need SD or SEM.</p></disp-quote><p>Since we informed the model with mean properties, we show the SD as a grey region to enable easy visual comparison of the computed mean against this backdrop.</p><disp-quote content-type="editor-comment"><p>b) The number of samples used for mechanical testing needs to be included.</p></disp-quote><p>Now noted, N=4 umbilical veins and N=4 umbilical arteries were analyzed, now noted.</p><disp-quote content-type="editor-comment"><p>c) Please include circumferential and axial stress-strain curves which are directly related to the fitted material parameters used for the modeling.</p></disp-quote><p>These figures have been added to the supplemental figure.</p><disp-quote content-type="editor-comment"><p>d) Figure 4—figure supplement 1B, subsection “Differential SMC contraction in the bilayered umbilical arteries and vein”of the text state that the artery has a smaller lumen, citing this panel, but this panel shows the outer diameter, and not the inner one.</p></disp-quote><p>Thank you. This has now been corrected to show the inner diameter in Figure 5—figure supplement 1.</p><disp-quote content-type="editor-comment"><p>e) Figure 4—figure supplement 1C and subsection “Differential SMC contraction in the bilayered umbilical arteries and vein”, text: Please define &quot;distensibility&quot; and &quot;extensibility&quot; as used in the text. At what pressures or axial stretches are you comparing the distensibility and extensibility? How do the axial stretch values in Figure 4—figure supplement 1C compare to the in vivo axial stretches?</p></disp-quote><p>Indeed, we needed to be clearer. Unfortunately, there is a clinical definition of “distensibility” that is actually a measure of structural compliance (normalized changes in diameter divided by pulse pressure). Herein, distensibility (circumferential deformation, referring to an enlargement) and extensibility (axial deformation, lengthening) are kinematic measures. Values were computed at the vessel-specific in vivo axial stretch and near physiological pressures (UA: 20 mmHg, UV: 5mmHg).</p><disp-quote content-type="editor-comment"><p>9) Figure 5—figure supplement 1.</p><p>Please support the important conclusion, subsection “Aggrecan and Adamts1 are necessary for normal umbilical cord morphogenesis” based on this figure that &quot;Neither mutant showed intrauterine growth retardation, and intrauterine death was infrequent, suggesting adequate cord circulation.&quot;, with actual measurements, especially as the acan KO mutant in Figure 5—figure supplement 1B appears smaller at E14.5 than its wildtype counterpart - assuming the same magnifications, which should be indicated.</p></disp-quote><p>Please see response to point 7b above. Both embryos were imaged at the same magnification and we have now added scale bars to this image. We have also limited our conclusions to umbilical cord development and removed any references and conclusions related to growth retardation of these mutants in our study. <italic>Acan</italic> mutants have a very abnormal skeletal system with short limbs and craniofacial defects as previously characterized, constituting an overall embryo dysmorphology and growth retardation. We observe them at the expected Mendelian ratio at E18.5 just prior to birth and our conclusions from the E14.5 embryos are limited to the reorientation of the umbilical cord SMCs which takes place from E12.5-E14.5.</p><disp-quote content-type="editor-comment"><p>10) Figure 8.</p><p>- What are the blue cells? Something other than SMCs?</p></disp-quote><p>The blue cells represent proteoglycan-rich non-contractile SMCs that get redirected centripetally to occlude the lumen. We have modified the cartoon legend to clarify this.</p><disp-quote content-type="editor-comment"><p>11) Table 1.</p><p>a) As c1 for the diagonal fibers is 2-3 orders of magnitude below the circ and axial fibers, are the diagonal fibers really necessary in the constitutive model?</p></disp-quote><p>It is well known that best-fit values of material parameters in exponential relations are not unique (with high c1 balancing low c2 and vice versa), hence it is most important to be consistent in the estimation (using random initial guesses, using constrained optimization to ensure non-negative values, using biaxial data, etc.) as we were. It is also important not to ascribe much physiological meaning to individual parameters in phenomenological models, but rather to focus on their collective contributions to calculated stresses, stiffnesses, energy, etc. That said, when re-fitting the data (mean data, not single representative), we found increased c1 values for the diagonal fiber families, whereas the axial family’s value of c1 decreased. The inclusion of all 4 families is, we think, prudent to ensure good fits to account for possible variability in the experimental data and is justified based on prior good fits to diverse murine data.</p><disp-quote content-type="editor-comment"><p>b) Also, the c2 values are 1-2 orders of magnitude higher than values that the group has published previously for mouse elastic arteries. Can the authors comment on the suitability of the constitutive model for fitting such nonlinear data and differences between the mechanical behavior of umbilical artery/vein and elastic arteries in the mouse?</p></disp-quote><p>The magnitude of the c2 values are consistent with those of some past works, cf. Supplemental Table 3 from Bersi et al., 2016 which includes some c2 values even higher than those in this work. While these nonlinear constitutive equations were initially developed for elastic arteries, they have since been used to describe the behavior of pulmonary arteries, veins, and tissue engineered vascular constructs, suggesting that they are a reasonable first approach for understanding the behavior of a new vessel. Furthermore, the key simulations in this work relate to the buckling phenomenon observed experimentally. We sought to determine whether active stress present in an external layer could drive closure and if eventual buckling of a GAG-rich inner layer was necessary for complete occlusion, which were studied parametrically. Using this constitutive approach allowed us to determine the feasibility of such a hypothesis and to understand how changes in volume related to swelling could impact the degree of contractility necessary to cause buckling.</p><disp-quote content-type="editor-comment"><p>c) Are the values in Table 1 averages of individual values from multiple arteries, fit from a combination of data from multiple arteries, or representative values from a single artery?</p></disp-quote><p>The initially presented values were those for a representative sample. We have since re-parameterized for the mean behavior by re-running all estimations. The value of axial stretch was also updated to 1.28</p><p>[Editors' note: further revisions were suggested prior to acceptance, as described below.]</p><disp-quote content-type="editor-comment"><p>In their revised manuscript concerning the mechanism of umbilical arterial closure at birth in Placentalia, Nandadasa et al., have satisfactorily addressed the majority of the reviewers' concerns. However, there remain two major concerns: (1) the mouse mutant analyses, and (2) the number of specimens used per experiment; a small number of minor revisions; and requested changes to the Abstract, in accord with eLife's policies.</p><p>Essential revisions:</p><p>1) Genetic mutants. The reviewers had requested the following (copied from the previous letter to the authors): (b) Please provide information on genotyping mutant litters in the Materials and methods section/Supplemental methods, and their ratios at each embryonic day examined, especially as the authors claim that mutants did not show intrauterine growth retardation or death (see Figure 5—figure supplement 1). If possible, please indicate the background resorption level in these strains, which would be evident by having included the genotype of the resorptions.</p><p>Mouse mutant Adamts-/-. As the authors did not provide their method for genotyping this mutant strain, as requested, more investigation was needed on the part of the reviewers to understand what exactly this strain is to consider why the authors ignored the request.</p><p>Background (Oller et al., 2017): The Adamts-/- mutant used in this study was described by Oller et al., 2017 as an insertion of a lacZ-bearing cassette into intron 1 of the gene. According to this previous paper, not only does this insertion reveal where Adamts1 is expressed via staining for ß-galactosidase activity, but in its hemizygous state, the insertion also causes a reduction in both mRNA and protein. In its homozygous state, ß-galactosidase activity is still detectable, but the Adamts1 protein is not (Figure 1a of Oller et al., 2017). Oller et al did not provide details on how the animals were maintained and mated (which they really should have been asked to do, alas), but they did explain how their litters were genotyped, the specific sequence used in those PCR genotyping experiments, and the genotypic ratios of their animals at weaning (Supplemental Figure 1b of Oller et al., 2017).</p><p>In the current study, the authors distinguished hemizygotes and homozygotes in the figure panels (e.g., Figure 4C versus the other panels), and wildtype and homozygotes in the others that relate to the mutant analysis, implying that they genotyped this material. However, they ignored the reviewers' procedural request, which is repeated and expanded as follows:</p><p>Please provide information on</p><p>i) genotyping method used, including the exact DNA sequence for PCR analysis;</p></disp-quote><p>This (DNA isolation method, and primer sequences, expected PCR products) is now provided in detailed methods in Appendix 1</p><disp-quote content-type="editor-comment"><p>ii) genotypic ratios according to gestational day;</p></disp-quote><p>This data for both Acan and Adamt1s mutants is in revised Figure 4—figure supplement 1 panel C.</p><disp-quote content-type="editor-comment"><p>iii) the genetic background on which the Adamts animals were maintained (according to Oller et al., it seems to be a B/6 background, but please confirm);</p></disp-quote><p>We maintained them in C57BL/6, as now written in detailed Materials and methods section.</p><disp-quote content-type="editor-comment"><p>iv) the parental genotypes used to produce the specific genotypes (perhaps hemizygous by wildtype matings produced hemizygous embryos detectable by X-gal staining, whilst homozygous embryos were obtained by crossing hemizygous animals and using PCR genotyping to distinguish the three genotypes? - please confirm/clarify);</p></disp-quote><p>Yes, this is correct and added to Appendix 1. Either lacZ staining and genotyping were used to identify hemizygotes obtained from hemizygous X wild-type matings, whereas PCR genotyping was used to distinguish the three genotypes arising from crosses of hemizygous animals, which were used to generate homozygous embryos.</p><disp-quote content-type="editor-comment"><p>v) how gestational age was determined. Were timed matings used? if not timed matings, then how was gestational age determined?;</p></disp-quote><p>Yes, timed matings were used, as specified in the revised Materials and methods section.</p><disp-quote content-type="editor-comment"><p>vi) the protocol used for X-gal staining the hemizygotes.</p></disp-quote><p>We cite one of our previous manuscripts t where the detailed protocol was described.</p><disp-quote content-type="editor-comment"><p>- While Oller et al., 2017 used this transgenic mouse line, they did not create it. Please clarify the specific origin of this mutant mouse strain - that information was impossible to locate on the EMMA site.</p></disp-quote><p>These details are provided in the expanded Appendix 1 subsection “Additional details of transgenic mice”. The details of the mice are no longer listed in EMMA, unfortunately.</p><disp-quote content-type="editor-comment"><p>- It would be most helpful for the reader if the authors would introduce the Adamts mutant in the Results section by summarizing Oller et al.'s results concerning the levels of mRNA versus protein in hemizygotes and homozygotes, as under &quot;Background&quot;, above.</p></disp-quote><p>This is now included in the revised results section.</p><disp-quote content-type="editor-comment"><p>- Finally, if the authors can, would they comment on whether hemizygotes exhibited foreshortened umbilical cords, too, and did their lengths fall between those of the wildtype and homozygous mutants?</p></disp-quote><p>We did not measure the lengths of hemizygous cords, but they were visually undistinguishable from wild-type cords.</p><disp-quote content-type="editor-comment"><p>Mouse mutant Acan-/-.</p><p>- Although the authors provided the genotypic ratios of the Acan-/- mutants as requested by the reviewers (new Figure 4—figure supplement 1 Panel 4C), they did not indicate the genotyping procedure. Please add it to Detailed Methods, to include how the DNA was obtained, and the DNA sequence used to PCR the littermates' DNA.</p></disp-quote><p>The details of the Acan genotyping are now provided in Appendix 1.</p><disp-quote content-type="editor-comment"><p>- The mutant allele was originally described by Krueger et al., 1999, but from where did the authors procure this mouse strain?</p></disp-quote><p>The colony has long been established in the laboratory of Nancy Schwartz and Miriam Domowicz at the University of Chicago, and mice from that colony were transferred to the Cleveland Clinic Lerner Research Institute. Please see subsection “Additional details of transgenic mice” for details of this allele.</p><disp-quote content-type="editor-comment"><p>- Litters were obtained at E12.5, E14.5, and E18.5; please indicate how matings were carried out to ascertain the timing of gestation, including parental genotypes that produced the Acan litters.</p></disp-quote><p>The homozygous mutants were produced by inter-crossing hemizygous parents. We have specified this in the expanded section in Appendix 1 to explain this as part of this sentence: PCR genotyping was used to distinguish the three genotypes possibly arising from crosses of Adamts1 or Acan hemizygous animals which were used to generate homozygous embryos.</p><disp-quote content-type="editor-comment"><p>2) The reviewers had requested the number of specimens (n) for every experiment. The following are still missing:</p><p>- Figure 1D. n, the number of immunostained specimens?</p></disp-quote><p>n=4 cords for each antibody, added to figure legend.</p><disp-quote content-type="editor-comment"><p>- Figure 2B, line 545. &quot;n=3 umbilical cords&quot; - for each probe?, or for both?</p></disp-quote><p>(a) n=3 umbilical cords for each in situ probe</p><p>(b) n=4 umbilical cords for each antibody staining</p><p>added to figure legend</p><disp-quote content-type="editor-comment"><p>- Figure 4—figure supplement 1.</p><p>S4b. n = ?</p></disp-quote><p>n=2 Acan KO at E E12.5 and n=3 for E14.5, added to figure legend.</p><disp-quote content-type="editor-comment"><p>S4d. n = ?</p></disp-quote><p>n=3 UC each genotype, added to figure legend.</p></body></sub-article></article>