{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,19]],"date-time":"2026-04-19T08:36:19Z","timestamp":1776587779577,"version":"3.51.2"},"reference-count":60,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T00:00:00Z","timestamp":1615507200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian Ministry of Education, University and Research (MIUR)","award":["CUP: E11G18000350001"],"award-info":[{"award-number":["CUP: E11G18000350001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Cell migration in highly constrained environments is fundamental in a wide variety of physiological and pathological phenomena. In particular, it has been experimentally shown that the migratory capacity of most cell lines depends on their ability to transmigrate through narrow constrictions, which in turn relies on their deformation capacity. In this respect, the nucleus, which occupies a large fraction of the cell volume and is substantially stiffer than the surrounding cytoplasm, imposes a major obstacle. This aspect has also been investigated with the use of microfluidic devices formed by dozens of arrays of aligned polymeric pillars that limit the available space for cell movement. Such experimental systems, in particular, in the designs developed by the groups of Denais and of Davidson, were here reproduced with a tailored version of the Cellular Potts model, a grid-based stochastic approach where cell dynamics are established by a Metropolis algorithm for energy minimization. The proposed model allowed quantitatively analyzing selected cell migratory determinants (e.g., the cell and nuclear speed and deformation, and forces acting at the nuclear membrane) in the case of different experimental setups. Most of the numerical results show a remarkable agreement with the corresponding empirical data.<\/jats:p>","DOI":"10.3390\/axioms10010032","type":"journal-article","created":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T11:56:55Z","timestamp":1615550215000},"page":"32","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A Cellular Potts Model for Analyzing Cell Migration across Constraining Pillar Arrays"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3509-9015","authenticated-orcid":false,"given":"Marco","family":"Scianna","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1532-1854","authenticated-orcid":false,"given":"Luigi","family":"Preziosi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1002\/bdrc.20125","article-title":"Cell biology of embryonic migration","volume":"84","author":"Kurosaka","year":"2008","journal-title":"Birth Defects Res. C Embryo Today"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1038\/ni.f.212","article-title":"Interstitial leukocyte migration and immune function","volume":"9","author":"Friedl","year":"2008","journal-title":"Nat. Immunol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1038\/nrc2229","article-title":"Illuminating the metastatic process","volume":"7","author":"Sahai","year":"2007","journal-title":"Nat. Rev. Cancer"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1109\/MEMB.2003.1256271","article-title":"Scaffold-based articular cartilage repair","volume":"22","author":"Capito","year":"2003","journal-title":"IEEE Eng. Med. Biol. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1159\/000075035","article-title":"Optimal degradation rate for collagen chambers used for regeneration of peripheral nerves over long gaps","volume":"176","author":"Harley","year":"2008","journal-title":"Cells Tissues Organs"},{"key":"ref_6","first-page":"843","article-title":"AG-CSF functionalized PLLA scaffold for wound repair: An in vitro preliminary study","volume":"2010","author":"Spadaccio","year":"2010","journal-title":"Conf. Proc. IEEE Eng. Med. Biol. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1073\/pnas.86.3.933","article-title":"Synthesis and characterization of a model extracellular matrix that induces partial regeneration of adult mammalian skin","volume":"86","author":"Yannas","year":"1989","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/S0092-8674(03)00513-0","article-title":"Membrane type I matrix metalloproteinase usurps tumor growth control imposed by the three-dimensional extracellular matrix","volume":"114","author":"Hotary","year":"2003","journal-title":"Cell"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1038\/ncb1616","article-title":"Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion","volume":"9","author":"Wolf","year":"2007","journal-title":"Nat. Cell.Biol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s000180050498","article-title":"The biology of cell locomotion within three-dimensional extracellular matrix","volume":"57","author":"Friedl","year":"2000","journal-title":"Cell. Mol. Life Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1038\/nrc1075","article-title":"Tumour-cell invasion and migration: Diversity and escape mechanisms","volume":"3","author":"Friedl","year":"2003","journal-title":"Nat. Rev. Cancer"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1083\/jcb.200909003","article-title":"Plasticity of cell migration: A multiscale tuning model","volume":"188","author":"Friedl","year":"2009","journal-title":"J. Cell Biol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3357","DOI":"10.1091\/mbc.e08-03-0319","article-title":"The role of myosin ii in glioma invasion of the brain","volume":"19","author":"Beadle","year":"2008","journal-title":"Mol. Biol. Cell"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/j.ceb.2011.03.001","article-title":"Nuclear mechanics during cell migration","volume":"23","author":"Friedl","year":"2011","journal-title":"Curr. Opin. Cell Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1083\/jcb.200209006","article-title":"Compensation mechanism in tumor cell migration mesenchymalamoeboid transition after blocking of pericellular proteolysis","volume":"160","author":"Wolf","year":"2003","journal-title":"J. Cell Biol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1083\/jcb.201210152","article-title":"Physical limits of cell migration: Control by ECM space and nuclear deformation and tuning by proteolysis and traction force","volume":"201","author":"Wolf","year":"2013","journal-title":"J. Cell Biol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.semcdb.2009.08.005","article-title":"Collagen-based cell migration models in vitro and in vivo","volume":"20","author":"Wolf","year":"2009","journal-title":"Semin. Cell Dev. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1534","DOI":"10.1039\/C5IB00200A","article-title":"Design of a microfluidic device to quantify dynamic intra-nuclear deformation during cell migration through confining environments","volume":"7","author":"Davidson","year":"2015","journal-title":"Integr. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1126\/science.aad7297","article-title":"Nuclear envelope rupture and repair during cancer cell migration","volume":"352","author":"Denais","year":"2016","journal-title":"Science"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1783","DOI":"10.1039\/b710524j","article-title":"Polar stimulation and constrained cell migration in microfluidic channels","volume":"7","author":"Irimia","year":"2007","journal-title":"Lab A Chip"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Rolli, C.G., Seufferlein, T., Kemkemer, R., and Spatz, J.P. (2010). Impact of tumor cell cytoskeleton organization on invasiveness and migration: A microchannel-based approach. PLoS ONE, 5.","DOI":"10.1371\/annotation\/9a6b2508-81c8-403f-87bd-071bdcb5b251"},{"key":"ref_22","unstructured":"Anderson, A.R.A., Chaplain, M.A.J., and Rejniak, K.A. (2007). The Glazier-Graner-Hogeweg model: Extensions, future directions, and opportunities for further study. Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkha\u00fcser."},{"key":"ref_23","unstructured":"Anderson, A.R.A., Chaplain, M.A.J., and Rejniak, K.A. (2007). Magnetization to morphogenesis: A brief history of the Glazier-Graner-Hogeweg model. Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkha\u00fcser."},{"key":"ref_24","first-page":"2128","article-title":"Simulation of the differential adhesion driven rearrangement of biological cells","volume":"47","author":"Glazier","year":"1993","journal-title":"Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2013","DOI":"10.1103\/PhysRevLett.69.2013","article-title":"Simulation of biological cell sorting using a two-dimensional extended Potts model","volume":"69","author":"Graner","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","unstructured":"Anderson, A.R.A., Chaplain, M.A.J., and Rejniak, K.A. (2007). The Cellular Potts Model and biophysical properties of cells, tissues and morphogenesis. Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interactions, Birkha\u00fcser."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1137\/100812951","article-title":"Multiscale developments of the Cellular Potts Model","volume":"10","author":"Scianna","year":"2012","journal-title":"Multiscale Model. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.jtbi.2012.11.003","article-title":"Modelling the influence of nucleus elasticity on cell invasion in fiber networks and microchannels","volume":"317","author":"Scianna","year":"2013","journal-title":"J. Theor. Biol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s00466-013-0944-6","article-title":"A cellular Potts model for the MMP-dependent and-independent cancer cell migration in matrix microtracks of different dimensions","volume":"53","author":"Scianna","year":"2014","journal-title":"Comp. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"235","DOI":"10.3934\/mbe.2013.10.235","article-title":"A Cellular Potts Model simulating cell migration on and in matrix environments","volume":"10","author":"Scianna","year":"2013","journal-title":"Math. Biosci. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/BF02980577","article-title":"Beitrag zur theorie des ferromagnetismus","volume":"31","author":"Ising","year":"1925","journal-title":"Z. Physik."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1017\/S0305004100027419","article-title":"Some generalized order-disorder transformations","volume":"48","author":"Potts","year":"1952","journal-title":"Proc. Camb. Phil. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3105","DOI":"10.1529\/biophysj.106.101501","article-title":"A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis","volume":"92","author":"Bauer","year":"2007","journal-title":"Biophys. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5661","DOI":"10.1529\/biophysj.108.140624","article-title":"The role of extracellular matrix in glioma invasion: A Cellular Potts Model approach","volume":"95","author":"Rubenstein","year":"2006","journal-title":"Biophys. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of state calculations by fast computing machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"R721","DOI":"10.1016\/j.cub.2003.08.050","article-title":"Polymer motors: Pushing out the front and pulling up the back","volume":"13","author":"Mogilner","year":"2003","journal-title":"Curr. Biol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/S0092-8674(03)00120-X","article-title":"Cellular motility driven by assembly and disassembly of actin filaments","volume":"112","author":"Pollard","year":"2003","journal-title":"Cell"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1126\/science.1092053","article-title":"Cell migration: Integrating signals from front to back","volume":"302","author":"Ridley","year":"2003","journal-title":"Science"},{"key":"ref_39","first-page":"1253","article-title":"A multiscale hybrid model for pro-angiogenic calcium signals in a vascular endothelial cell","volume":"76","author":"Scianna","year":"2011","journal-title":"Bull. Math. Biol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1126\/science.141.3579.401","article-title":"Reconstruction of tissues by dissociated cells. Some morphogenetic tissue movements and the sorting out of embryonic cells may have a common explanation","volume":"141","author":"Steinberg","year":"1963","journal-title":"Science"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1002\/jez.1401730406","article-title":"Does differential adhesion govern self-assembly processes in histogenesis? Equilibrium configurations and the emergence of a hierarchy among populations of embryonic cells","volume":"171","author":"Steinberg","year":"1970","journal-title":"J. Exp. Zool."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1039\/c0ib00050g","article-title":"Critical review: Cellular mechanobiology and amoeboid migration","volume":"2","author":"Guck","year":"2010","journal-title":"Integr. Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1038\/nature06887","article-title":"Rapid leukocyte migration by integrin-independent flowing and squeezing","volume":"453","author":"Bader","year":"2008","journal-title":"Nature"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1006\/jtbi.1996.0237","article-title":"Modelling morphogenesis: From single cells to crawling slugs","volume":"184","author":"Savill","year":"1997","journal-title":"J. Theor. Biol."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Rens, E.R., and Edelstein-Keshet, L. (2019). From energy to cellular forces in the Cellular Potts Model: An algorithmic approach. PLoS Comput. Biol., 15.","DOI":"10.1371\/journal.pcbi.1007459"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3471","DOI":"10.1152\/jn.00352.2004","article-title":"Diffusion of Epidermal Growth Factor in Rat Brain Extracellular Space Measured by Integrative Optical Imaging","volume":"92","author":"Thorne","year":"2004","journal-title":"J. Neurophysiol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1093\/emboj\/cdg176","article-title":"Modeling the early stages of vascular network assembly","volume":"22","author":"Serini","year":"2003","journal-title":"EMBO J."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1152\/ajpgi.00250.2001","article-title":"Expression and endocytosis of VEGF and its receptors in human colonic vascular endothelial cells","volume":"282","author":"Wang","year":"2002","journal-title":"Am. J. Physiol. Gastrointest. Liver Physiol."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1016\/j.bpj.2016.08.011","article-title":"A Chemomechanical Model for Nuclear Morphology and Stresses during Cell Transendothelial Migration","volume":"111","author":"Cao","year":"2016","journal-title":"Biophys. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1083\/jcb.201308029","article-title":"Nuclear lamin stiffness is a barrier to 3D migration, but softness can limit survival","volume":"204","author":"Harada","year":"2014","journal-title":"J. Cell Biol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"25768","DOI":"10.1074\/jbc.M513511200","article-title":"Lamins A and C but Not Lamin B1 Regulate Nuclear Mechanics","volume":"281","author":"Lammerding","year":"2006","journal-title":"J. Biol. Chem."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"8610","DOI":"10.1074\/jbc.M112.441535","article-title":"Nuclear Envelope Composition Determines the Ability of Neutrophil-Type Cells to Passage Through Micron-Scale Constrictions","volume":"288","author":"Rowat","year":"2013","journal-title":"J. Biol. Chem."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1039\/b908595e","article-title":"Spontaneous migration of cancer cells under conditions of mechanical confinement","volume":"1","author":"Irimia","year":"2009","journal-title":"Integr. Biol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s11538-017-0262-9","article-title":"How nucleus mechanics and ECM microstructure influence the invasion of single cells and multicellular aggregates","volume":"80","author":"Giverso","year":"2018","journal-title":"Bull. Math. Biol."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1038\/nrm2720","article-title":"Collective cell migration in morphogenesis, regeneration and cancer","volume":"10","author":"Friedl","year":"2009","journal-title":"Nat. Rev. Mol. Cell Biol."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"015010","DOI":"10.1088\/1478-3975\/8\/1\/015010","article-title":"Two-photon laser-generated microtracks in 3D collagen lattices: Principles of MMP-dependent and -independent collective cancer cell invasion","volume":"8","author":"Ilina","year":"2011","journal-title":"Phys. Biol."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10266-008-0087-y","article-title":"Toward guided tissue and bone regeneration: Morphology, attachment, proliferation, and migration of cells cultured on collagen barrier membranes. A systematic review","volume":"96","author":"Behring","year":"2008","journal-title":"Odontology"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"4205","DOI":"10.1182\/blood-2006-03-013789","article-title":"2006. Rho GTPase CDC42 regulates directionality and random movement via distinct MAPK pathways in neutrophils","volume":"108","author":"Szczur","year":"2006","journal-title":"Blood"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1002\/jcb.10358","article-title":"HGF\/SF-Met signaling in the control of branching morphogenesis and invasion","volume":"88","author":"Zhang","year":"2003","journal-title":"J. Cell. Biochem."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.pbiomolbio.2011.01.004","article-title":"A multiscale hybrid approach for vasculogenesis and related potential blocking therapies","volume":"106","author":"Scianna","year":"2011","journal-title":"Prog. Biophys. Mol. Biol."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:34:35Z","timestamp":1760160875000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,12]]},"references-count":60,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["axioms10010032"],"URL":"https:\/\/doi.org\/10.3390\/axioms10010032","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,12]]}}}