{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:28:43Z","timestamp":1772252923057,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T00:00:00Z","timestamp":1657843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["642866"],"award-info":[{"award-number":["642866"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["642866"],"award-info":[{"award-number":["642866"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union Horizon 2020 research and innovation programme","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["642866"],"award-info":[{"award-number":["642866"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["1902431"],"award-info":[{"award-number":["1902431"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Health Foundation","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}]},{"name":"Health Foundation","award":["642866"],"award-info":[{"award-number":["642866"]}]},{"name":"Health Foundation","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}]},{"name":"Health Foundation","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}]},{"name":"Health Foundation","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"NIHR","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}]},{"name":"NIHR","award":["642866"],"award-info":[{"award-number":["642866"]}]},{"name":"NIHR","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}]},{"name":"NIHR","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}]},{"name":"NIHR","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"NIHR","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}]},{"name":"NIHR","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}]},{"name":"NIHR","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"NIHR","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Leverhulme Trust Research Project","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}]},{"name":"Leverhulme Trust Research Project","award":["642866"],"award-info":[{"award-number":["642866"]}]},{"name":"Leverhulme Trust Research Project","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}]},{"name":"Leverhulme Trust Research Project","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}]},{"name":"Leverhulme Trust Research Project","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Leverhulme Trust Research Project","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}]},{"name":"Leverhulme Trust Research Project","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}]},{"name":"Leverhulme Trust Research Project","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Leverhulme Trust Research Project","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}]},{"name":"Health Foundation","award":["642866"],"award-info":[{"award-number":["642866"]}]},{"name":"Health Foundation","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}]},{"name":"Health Foundation","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}]},{"name":"Health Foundation","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["EP\/R001588\/1"],"award-info":[{"award-number":["EP\/R001588\/1"]}]},{"name":"Health Foundation","award":["642866"],"award-info":[{"award-number":["642866"]}]},{"name":"Health Foundation","award":["EP\/J016780\/1"],"award-info":[{"award-number":["EP\/J016780\/1"]}]},{"name":"Health Foundation","award":["EP\/T00410X\/1"],"award-info":[{"award-number":["EP\/T00410X\/1"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["NIHR133761"],"award-info":[{"award-number":["NIHR133761"]}]},{"name":"Health Foundation","award":["RPG-2014-149"],"award-info":[{"award-number":["RPG-2014-149"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]},{"name":"Health Foundation","award":["1902431"],"award-info":[{"award-number":["1902431"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.<\/jats:p>","DOI":"10.3390\/jimaging8070199","type":"journal-article","created":{"date-parts":[[2022,7,17]],"date-time":"2022-07-17T22:57:06Z","timestamp":1658098626000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Force Estimation during Cell Migration Using Mathematical Modelling"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0619-0840","authenticated-orcid":false,"given":"Fengwei","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Engineering, University of Warwick, Coventry CV4 7AL, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4039-2056","authenticated-orcid":false,"given":"Chandrasekhar","family":"Venkataraman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK"}]},{"given":"Sai","family":"Gu","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Warwick, Coventry CV4 7AL, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0387-4067","authenticated-orcid":false,"given":"Vanessa","family":"Styles","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9511-8903","authenticated-orcid":false,"given":"Anotida","family":"Madzvamuse","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK"},{"name":"Department of Mathematics, University of British Columbia, 1984 Mathematics Rd., Vancouver, BC V6T 1Z2, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,15]]},"reference":[{"key":"ref_1","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_2","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":"2010","journal-title":"J. Cell Biol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s10585-008-9174-2","article-title":"The actin cytoskeleton in cancer cell motility","volume":"26","author":"Olson","year":"2009","journal-title":"Clin. Exp. Metastasis"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"R485","DOI":"10.1016\/j.cub.2008.04.048","article-title":"Moving towards a better understanding of chemotaxis","volume":"18","author":"Stephens","year":"2008","journal-title":"Curr. Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1083\/jcb.201508047","article-title":"Collective cell migration in development","volume":"212","author":"Scarpa","year":"2016","journal-title":"J. Cell Biol."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"1208","DOI":"10.1126\/science.1175862","article-title":"Actin, A central player in cell shape and movement","volume":"326","author":"Pollard","year":"2009","journal-title":"Science"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1038\/s41467-021-22377-9","article-title":"Two-dimensional TIRF-SIM\u2013traction force microscopy (2D TIRF-SIM-TFM)","volume":"12","author":"Barbieri","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/S0006-3495(98)77808-7","article-title":"Membrane bending modulus and adhesion energy of wild-type and mutant cells of dictyostelium lacing talin or cortexillins","volume":"74","author":"Simson","year":"1998","journal-title":"Biophys. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1016\/j.cub.2013.05.063","article-title":"Membrane tension in rapidly moving cells is determined by cytoskeletal forces","volume":"23","author":"Lieber","year":"2013","journal-title":"Curr. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"054003","DOI":"10.1088\/0960-1317\/21\/5\/054003","article-title":"Microsystems for cellular force measurement: A review","volume":"21","author":"Zheng","year":"2011","journal-title":"J. Micromechanics Microeng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1836","DOI":"10.1529\/biophysj.107.113605","article-title":"Life and times of a cellular bleb","volume":"94","author":"Charras","year":"2008","journal-title":"Biophys. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.1038\/ncb2297","article-title":"MAP4 and CLASP1 operate as a safety mechanism to maintain a stable spindle position in mitosis","volume":"13","author":"Samora","year":"2011","journal-title":"Nat. Cell Biol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1016\/j.bbamcr.2012.11.006","article-title":"Membrane tension homeostasis of epithelial cells through surface area regulation in response to osmotic stress","volume":"1833","author":"Pietuch","year":"2013","journal-title":"Biochim. Biophys. Acta"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1002\/opph.201190249","article-title":"Digital Holographic Microscopy A New Method for Surface Analysis and Marker-Free Dynamic Life Cell Imaging","volume":"2","author":"Kemper","year":"2007","journal-title":"Opt. Photonik"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1364\/OL.23.000817","article-title":"Quantitative optical phase microscopy","volume":"23","author":"Barty","year":"1998","journal-title":"Opt. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1046\/j.0022-2720.2001.00976.x","article-title":"A method for quantifying cell size from differential interference contrast images: Validation and application to osmotically stressed chondrocytes","volume":"205","author":"Alexopoulos","year":"2001","journal-title":"J. Microsc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2503","DOI":"10.1364\/OL.29.002503","article-title":"Fourier phase microscopy for investigation of biological structures and dynamics","volume":"29","author":"Popescu","year":"2004","journal-title":"Opt. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1364\/OL.30.001165","article-title":"Hilbert phase microscopy for investigating fast dynamics in transparent systems","volume":"30","author":"Ikeda","year":"2005","journal-title":"Opt. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2064","DOI":"10.1364\/OL.28.002064","article-title":"High-resolution optical coherence microscopy for high-speed, in vivo cellular imaging","volume":"28","author":"Aguirre","year":"2003","journal-title":"Opt. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1364\/OL.30.000468","article-title":"Digital holographic microscopy: A noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy","volume":"30","author":"Marquet","year":"2005","journal-title":"Opt. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113533","DOI":"10.1016\/j.bios.2021.113533","article-title":"Tracking cell migration by cellular force footprint recorded with a mechano-optical biosensor","volume":"193","author":"Tu","year":"2021","journal-title":"Biosens. Bioelectron."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yu, Z., and Liu, K. (2021). Soft Polymer-Based Technique for Cellular Force Sensing. Polymers, 13.","DOI":"10.3390\/polym13162672"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108104","DOI":"10.1103\/PhysRevLett.105.108104","article-title":"Computational model for cell morphodynamics","volume":"105","author":"Shao","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"048101","DOI":"10.1103\/PhysRevLett.103.048101","article-title":"Effect of Adhesion Geometry and Rigidity on Cellular Force Distributions","volume":"103","author":"Bischofs","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/j.jcp.2015.05.014","article-title":"Whole cell tracking through the optimal control of geometric evolution laws","volume":"297","author":"Blazakis","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"65","DOI":"10.4208\/cicp.240715.080716a","article-title":"A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws","volume":"21","author":"Yang","year":"2017","journal-title":"Commun. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Hau\u00dfer, F., Janssen, S., and Voigt, A. (2012). Control of nanostructures through electric fields and related free boundary problems. Constrained Optimization and Optimal Control for Partial Differential Equations, Springer.","DOI":"10.1007\/978-3-0348-0133-1_29"},{"key":"ref_29","first-page":"3591","article-title":"Electroporation of adherent cells with low sample volumes on a microscope stage","volume":"216","author":"Tsugiyama","year":"2013","journal-title":"J. Exp. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1002\/cm.21100","article-title":"Time-dependent traction force microscopy for cancer cells as a measure of invasiveness","volume":"70","author":"Peschetola","year":"2013","journal-title":"Cytoskeleton"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1083\/jcb.200506152","article-title":"Integrin-dependent actomyosin contraction regulates epithelial cell scattering","volume":"171","author":"Kerstens","year":"2005","journal-title":"J. Cell Biol."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Blowey, J., and Elliott, C. (1993). Curvature dependent phase boundary motion and parabolic double obstacle problems. Degenerate Diffusions, Springer.","DOI":"10.1007\/978-1-4612-0885-3_2"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6213","DOI":"10.1038\/srep06213","article-title":"Quantification of surface tension and internal pressure generated by single mitotic cells","volume":"4","author":"Hyman","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.jcp.2015.01.040","article-title":"Thermalsolute phase field three dimensional simulation of binary alloy solidification","volume":"287","author":"Bollada","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5887","DOI":"10.1103\/PhysRevA.39.5887","article-title":"Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations","volume":"39","author":"Caginalp","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1271","DOI":"10.1016\/j.jcp.2007.01.027","article-title":"A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification","volume":"225","author":"Rosam","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1109\/TSMC.1979.4310076","article-title":"A threshold selection method from gray level histograms","volume":"9","author":"Otsu","year":"1979","journal-title":"IEEE Trans. Systems Man Cybern."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1083\/jcb.200601159","article-title":"Direct measurement of the lamellipodial protrusive force in a migrating cell","volume":"174","author":"Prass","year":"2006","journal-title":"J. Cell Biol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1137\/040606417","article-title":"Retrieving topological information for phase field models","volume":"65","author":"Du","year":"2005","journal-title":"Siam J. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"113002","DOI":"10.1088\/1361-6463\/aa56fe","article-title":"Physical models of collective cell motility: From cell to tissue","volume":"50","author":"Camley","year":"2017","journal-title":"J. Phys. Appl. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Hobson, C.M., and Stephens, A.D. (2020). Modeling of cell nuclear mechanics: Classes, components, and applications. Cells, 9.","DOI":"10.3390\/cells9071623"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"738","DOI":"10.1038\/s41580-019-0172-9","article-title":"Mechanisms of 3D cell migration","volume":"20","author":"Yamada","year":"2019","journal-title":"Nat. Rev. Mol. Cell Biol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"015009","DOI":"10.1088\/1478-3975\/8\/1\/015009","article-title":"The role of the cytoskeleton in cellular force generation in 2D and 3D environments","volume":"8","author":"Carey","year":"2011","journal-title":"Phys. Biol."}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/8\/7\/199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:51:34Z","timestamp":1760140294000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/8\/7\/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,15]]},"references-count":43,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["jimaging8070199"],"URL":"https:\/\/doi.org\/10.3390\/jimaging8070199","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202206.0213.v1","asserted-by":"object"}]},"ISSN":["2313-433X"],"issn-type":[{"value":"2313-433X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,15]]}}}