{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,26]],"date-time":"2025-10-26T15:05:23Z","timestamp":1761491123780,"version":"build-2065373602"},"reference-count":59,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,13]],"date-time":"2020-11-13T00:00:00Z","timestamp":1605225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["GlioMath 05M2016"],"award-info":[{"award-number":["GlioMath 05M2016"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001655","name":"Deutscher Akademischer Austauschdienst","doi-asserted-by":"publisher","award":["57252260"],"award-info":[{"award-number":["57252260"]}],"id":[{"id":"10.13039\/501100001655","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction\u2013diffusion\u2013taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.<\/jats:p>","DOI":"10.3390\/sym12111870","type":"journal-article","created":{"date-parts":[[2020,11,13]],"date-time":"2020-11-13T10:32:47Z","timestamp":1605263567000},"page":"1870","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4600-1028","authenticated-orcid":false,"given":"Pawan","family":"Kumar","sequence":"first","affiliation":[{"name":"Felix-Klein-Zentrum f\u00fcr Mathematik, TU Kaiserslautern, 67663 Kaiserslautern, Germany"}]},{"given":"Christina","family":"Surulescu","sequence":"additional","affiliation":[{"name":"Felix-Klein-Zentrum f\u00fcr Mathematik, TU Kaiserslautern, 67663 Kaiserslautern, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/glia.440150303","article-title":"Histopathology, classification and grading of gliomas","volume":"5","author":"Kleihues","year":"1995","journal-title":"Glia"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Perry, A., and Wesseling, P. (2016). Histologic classification of gliomas. Handbook of Clinical Neurology, Elsevier.","DOI":"10.1016\/B978-0-12-802997-8.00005-0"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1158\/0008-5472.CAN-03-2073","article-title":"Pseudopalisades in glioblastoma are hypoxic, express extracellular matrix proteases, and are formed by an actively migrating cell population","volume":"64","author":"Brat","year":"2004","journal-title":"Cancer Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1038\/labinvest.3700070","article-title":"Vaso-occlusive and prothrombotic mechanisms associated with tumor hypoxia, necrosis, and accelerated growth in glioblastoma","volume":"84","author":"Brat","year":"2004","journal-title":"Lab. Investig."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1097\/00005072-200606000-00001","article-title":"\u2018Pseudopalisading\u2019 necrosis in glioblastoma: A familiar morphologic feature that links vascular pathology, hypoxia, and angiogenesis","volume":"65","author":"Rong","year":"2006","journal-title":"J. Neuropathol. Exp. Neurol."},{"key":"ref_6","first-page":"2037","article-title":"Neuropathology for the neuroradiologist: Palisades and pseudopalisades","volume":"27","author":"Wippold","year":"2006","journal-title":"Am. J. Neuroradiol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.nurt.2009.04.001","article-title":"Biology of angiogenesis and invasion in glioma","volume":"6","author":"Tate","year":"2009","journal-title":"Neurotherapeutics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s10014-010-0007-z","article-title":"Angiogenesis and invasion in glioma","volume":"28","author":"Onishi","year":"2011","journal-title":"Brain Tumor Pathol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1097\/00125480-200201000-00004","article-title":"Genetic and biologic progression in astrocytomas and their relation to angiogenic dysregulation","volume":"9","author":"Brat","year":"2002","journal-title":"Adv. Anat. Pathol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5612","DOI":"10.1158\/1078-0432.CCR-14-0834","article-title":"Antiangiogenic Therapy for Glioblastoma: Current Status and Future Prospects","volume":"20","author":"Batchelor","year":"2014","journal-title":"Clin. Cancer Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.biopha.2018.10.063","article-title":"Chemoresistance caused by the microenvironment of glioblastoma and the corresponding solutions","volume":"109","author":"Zhang","year":"2019","journal-title":"Biomed. Pharmacother."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"879","DOI":"10.3934\/mbe.2015.12.879","article-title":"Mathematically modeling the biological properties of gliomas: A review","volume":"12","author":"Martirosyan","year":"2015","journal-title":"Math. Biosci. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"20170490","DOI":"10.1098\/rsif.2017.0490","article-title":"The biology and mathematical modelling of glioma invasion: A review","volume":"14","author":"Alfonso","year":"2017","journal-title":"J. R. Soc. Interface"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1142\/S0218202505000960","article-title":"Mathematical Modeling of Glioblastoma Tumor Development: A Review","volume":"15","author":"Hatzikirou","year":"2005","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cai, Y., Wu, J., Li, Z., and Long, Q. (2016). Mathematical modelling of a brain tumour initiation and early development: A coupled model of glioblastoma growth, pre-existing vessel co-option, angiogenesis and blood perfusion. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0150296"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.jtbi.2015.07.021","article-title":"Multiscale modelling of palisade formation in gliobastoma multiforme","volume":"383","author":"Caiazzo","year":"2015","journal-title":"J. Theor. Biol."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kumar, P., Li, J., and Surulescu, C. (2020). Multiscale modeling of glioma pseudopalisades: Contributions from the tumor microenvironment. arXiv.","DOI":"10.1007\/s00285-021-01599-x"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"37283","DOI":"10.1038\/srep37283","article-title":"Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: In silico insights","volume":"6","author":"Alfonso","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2875","DOI":"10.1007\/s11538-012-9786-1","article-title":"Hypoxic cell waves around necrotic cores in glioblastoma: A biomathematical model and its therapeutic implications","volume":"74","author":"Calvo","year":"2012","journal-title":"Bull. Math. Biol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Bellomo, N., Bellouquid, A., Gibelli, L., and Outada, N. (2017). A Quest Towards a Mathematical Theory of Living Systems, Birkh\u00e4user.","DOI":"10.1007\/978-3-319-57436-3"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"333","DOI":"10.3934\/nhm.2007.2.333","article-title":"Modeling cell movement in anisotropic and heterogeneous network tissues","volume":"2","author":"Hillen","year":"2007","journal-title":"Netw. Heterog. Media"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Conte, M., and Surulescu, C. (2020). Mathematical modeling of glioma invasion: Acid-and vasculature mediated go-or-grow dichotomy and the influence of tissue anisotropy. arXiv.","DOI":"10.1016\/j.amc.2021.126305"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Corbin, G., Engwer, C., Klar, A., Nieto, J., Soler, J., Surulescu, C., and Wenske, M. (2020). Modeling glioma invasion with anisotropy-and hypoxia-triggered motility enhancement: From subcellular dynamics to macroscopic PDEs with multiple taxis. arXiv.","DOI":"10.1142\/S0218202521500056"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1142\/S0218202518400055","article-title":"Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum","volume":"28","author":"Corbin","year":"2018","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_25","unstructured":"Dietrich, A., Kolbe, N., Sfakianakis, N., and Surulescu, C. (2020). Multiscale modeling of glioma invasion: From receptor binding to flux-limited macroscopic PDEs. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s00285-014-0822-7","article-title":"Glioma follow white matter tracts: A multiscale DTI-based model","volume":"71","author":"Engwer","year":"2015","journal-title":"J. Math. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"443","DOI":"10.3934\/mbe.2015011","article-title":"A multiscale model for glioma spread including cell-tissue interactions and proliferation","volume":"13","author":"Engwer","year":"2016","journal-title":"Math. Biosci. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1093\/imammb\/dqv030","article-title":"Effective equations for anisotropic glioma spread with proliferation: A multiscale approach and comparisons with previous settings","volume":"33","author":"Engwer","year":"2016","journal-title":"Math. Med. Biol. A J. IMA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s00285-006-0017-y","article-title":"M 5 mesoscopic and macroscopic models for mesenchymal motion","volume":"53","author":"Hillen","year":"2006","journal-title":"J. Math. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Hillen, T., and Painter, K.J. (2013). Transport and anisotropic diffusion models for movement in oriented habitats. Dispersal, Individual Movement and Spatial Ecology, Springer.","DOI":"10.1007\/978-3-642-35497-7_7"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s10013-016-0223-x","article-title":"A multiscale modeling approach to glioma invasion with therapy","volume":"45","author":"Hunt","year":"2017","journal-title":"Vietnam. J. Math."},{"key":"ref_32","unstructured":"Zhigun, A., and Surulescu, C. (2020). A novel derivation of rigorous macroscopic limits from a micro-meso description of signal-triggered cell migration in fibrous environments. arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Aylaj, B., Bellomo, N., Chouhad, N., and Knopoff, D. (2020). On the Interaction Between Soft and Hard Sciences: The Role of Mathematical Sciences. Vietnam. J. Math.","DOI":"10.1007\/s10013-019-00381-3"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jtbi.2013.01.014","article-title":"Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion","volume":"323","author":"Painter","year":"2013","journal-title":"J. Theor. Biol."},{"key":"ref_35","unstructured":"(2020, July 17). Available online: https:\/\/bionumbers.hms.harvard.edu\/bionumber.aspx?s=n&v=0&id=108941."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1038\/nrc3110","article-title":"Dysregulated pH: A perfect storm for cancer progression","volume":"11","author":"Webb","year":"2011","journal-title":"Nat. Rev. Cancer"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-36347-7","article-title":"Behaviors of glioblastoma cells in in vitro microenvironments","volume":"9","author":"Diao","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1242\/jcs.115.2.283","article-title":"NCAM regulates cell motility","volume":"115","author":"Prag","year":"2002","journal-title":"J. Cell Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1083\/jcb.200707009","article-title":"Cofilin determines the migration behavior and turning frequency of metastatic cancer cells","volume":"179","author":"Sidani","year":"2007","journal-title":"J. Cell Biol."},{"key":"ref_40","first-page":"5670","article-title":"Noninvasive Measurement of Interstitial pH Profiles in Normal and Neoplastic Tissue Using Fluorescence Ratio Imaging Microscopy","volume":"54","author":"Martin","year":"1994","journal-title":"Cancer Res."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1529\/biophysj.106.093468","article-title":"A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment","volume":"92","author":"Stein","year":"2007","journal-title":"Biophys. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1111\/j.1365-2184.2009.00613.x","article-title":"Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model","volume":"42","author":"Eikenberry","year":"2009","journal-title":"Cell Prolif."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1002\/wsbm.1233","article-title":"Endothelial cell motility, coordination and pattern formation during vasculogenesis","volume":"5","author":"Czirok","year":"2013","journal-title":"Wiley Interdiscip. Rev. Syst. Biol. Med."},{"key":"ref_44","unstructured":"(2020, June 22). Available online: https:\/\/www.lab.anhb.uwa.edu.au\/mb140\/MoreAbout\/Endothel.htm."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.jtbi.2004.04.012","article-title":"A mathematical model of tumour angiogenesis, regulated by vascular endothelial growth factor and the angiopoietins","volume":"229","author":"Plank","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-27303-6","article-title":"Mathematical modeling of the function of Warburg effect in tumor microenvironment","volume":"8","author":"Shamsi","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.jtbi.2006.07.002","article-title":"Modeling the effects of vasculature evolution on early brain tumor growth","volume":"243","author":"Gevertz","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_48","unstructured":"Weickert, J. (1998). Anisotropic Diffusion in Image Processing, Teubner Stuttgart."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1142\/S0218202519400062","article-title":"A multiscale view of nonlinear diffusion in biology: From cells to tissues","volume":"29","author":"Burini","year":"2019","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1142\/S0218202510004568","article-title":"Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems","volume":"20","author":"Bellomo","year":"2010","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Perthame, B., Vauchelet, N., and Wang, Z. (2018). The Flux Limited Keller-Segel System; Properties and Derivation from Kinetic Equations. arXiv.","DOI":"10.4171\/rmi\/1132"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Conte, M., Casas-Tint\u00f2, S., and Soler, J. (2020). Modeling invasion patterns in the glioblastoma battlefield. bioRxiv.","DOI":"10.1101\/2020.06.17.156497"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.jtbi.2009.06.025","article-title":"A mathematical model for pattern formation of glioma cells outside the tumor spheroid core","volume":"260","author":"Kim","year":"2009","journal-title":"J. Theor. Biol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1017\/S0956792512000447","article-title":"Anisotropic diffusion in oriented environments can lead to singularity formation","volume":"24","author":"Hillen","year":"2013","journal-title":"Eur. J. Appl. Math."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.4310\/CMS.2017.v15.n6.a5","article-title":"Global weak solutions to a strongly degenerate haptotaxis model","volume":"15","author":"Winkler","year":"2017","journal-title":"Comm. Math. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.matpur.2017.11.002","article-title":"Singular structure formation in a degenerate haptotaxis model involving myopic diffusion","volume":"112","author":"Winkler","year":"2018","journal-title":"J. Math. Pures Appl."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1090\/btran\/17","article-title":"Finite-time blow-up in a degenerate chemotaxis system with flux limitation","volume":"4","author":"Bellomo","year":"2017","journal-title":"Trans. Am. Math. Soc. Ser. B"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1080\/03605302.2016.1277237","article-title":"A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up","volume":"42","author":"Bellomo","year":"2017","journal-title":"Comm. Partial. Differ. Equ."},{"key":"ref_59","unstructured":"Kolbe, N., Sfakianakis, N., Stinner, C., Surulescu, C., and Lenz, J. (2020). Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discret. Contin. Dyn. Syst. 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