{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:27:24Z","timestamp":1762918044017,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"5-6","license":[{"start":{"date-parts":[[2019,12,1]],"date-time":"2019-12-01T00:00:00Z","timestamp":1575158400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,12,1]],"date-time":"2019-12-01T00:00:00Z","timestamp":1575158400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1446139"],"award-info":[{"award-number":["DMS-1446139"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["U54CA132383"],"award-info":[{"award-number":["U54CA132383"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2019,12]]},"DOI":"10.1007\/s10444-019-09727-4","type":"journal-article","created":{"date-parts":[[2019,12,3]],"date-time":"2019-12-03T05:02:49Z","timestamp":1575349369000},"page":"3057-3094","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On finite volume discretization of infiltration dynamics in tumor growth models"],"prefix":"10.1007","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8295-7743","authenticated-orcid":false,"given":"Xianyi","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Mashriq Ahmed","family":"Saleh","sequence":"additional","affiliation":[]},{"given":"Jianjun Paul","family":"Tian","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,3]]},"reference":[{"key":"9727_CR1","doi-asserted-by":"crossref","unstructured":"Ascher, U.M., Petzold, L.R.: Computer methods for ordinary differential equations and differential-algebraic equations. Society for Industrial and Applied Mathematics (1998)","DOI":"10.1137\/1.9781611971392"},{"issue":"4","key":"9727_CR2","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1137\/050637923","volume":"38","author":"M Burger","year":"2006","unstructured":"Burger, M., Francesco, M.D., Dolak-Struss, Y.: The Keller\u2013Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38(4), 1288\u20131315 (2006)","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"9727_CR3","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.matpur.2006.04.002","volume":"86","author":"V Calvez","year":"2006","unstructured":"Calvez, V., Carrillo, J.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pure. Appl. 86(2), 155\u2013175 (2006)","journal-title":"J. Math. Pure. Appl."},{"issue":"1","key":"9727_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/j.1365-2184.1992.tb01433.x","volume":"25","author":"JJ Casciari","year":"1992","unstructured":"Casciari, J.J., Sotirchos, S.V., Sutherland, R.M.: Mathematical modelling of microenvironment and growth in EMT6\/Ro multicellular tumour spheroids. Cell Proliferat. 25(1), 1\u201322 (1992)","journal-title":"Cell Proliferat."},{"issue":"4","key":"9727_CR5","doi-asserted-by":"publisher","first-page":"974","DOI":"10.1137\/S0036141002418388","volume":"35","author":"X Chen","year":"2003","unstructured":"Chen, X., Friedman, A.: A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth. SIAM J. Math. Anal. 35(4), 974\u2013986 (2003)","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"9727_CR6","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s10444-017-9545-9","volume":"44","author":"A Chertock","year":"2018","unstructured":"Chertock, A., Epshteyn, Y., Hu, H., Kurganov, A.: High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems. Adv. Comput. Math. 44(1), 327\u2013350 (2018)","journal-title":"Adv. Comput. Math."},{"issue":"1","key":"9727_CR7","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/0021-9991(84)90143-8","volume":"54","author":"P Colella","year":"1984","unstructured":"Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54(1), 174\u2013201 (1984)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"9727_CR8","doi-asserted-by":"publisher","first-page":"159","DOI":"10.4171\/IFB\/76","volume":"5","author":"S Cui","year":"2003","unstructured":"Cui, S., Friedman, A.: A hyperbolic free boundary problem modeling tumor growth. Interface Free Bound. 5(2), 159\u2013182 (2003)","journal-title":"Interface Free Bound."},{"issue":"2","key":"9727_CR9","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1017\/S0956792512000411","volume":"24","author":"E Espejo","year":"2013","unstructured":"Espejo, E., Vilches, K., Conca, C.: Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\\mathbb {R}^{2}$. Eur. J. Appl. Math. 24(2), 297\u2013313 (2013)","journal-title":"Eur. J. Appl. Math."},{"issue":"2","key":"9727_CR10","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1006\/jcph.2001.6932","volume":"174","author":"C Farhat","year":"2001","unstructured":"Farhat, C., Geuzaine, P., Grandmont, C.: The discrete geometric conservation law and the nonlinear stability of ale schemes for the solution of flow problems on moving grids. J. Comput. Phys. 174(2), 669\u2013694 (2001)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"9727_CR11","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/s00211-006-0024-3","volume":"104","author":"F Filbet","year":"2006","unstructured":"Filbet, F.: A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104(4), 457\u2013488 (2006)","journal-title":"Numer. Math."},{"issue":"4","key":"9727_CR12","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1016\/j.jde.2015.02.002","volume":"259","author":"A Friedman","year":"2015","unstructured":"Friedman, A., Hao, W., Hu, B.: A free boundary problem for steady small plaques in the artery and their stability. J. Differ. Equations 259(4), 1227\u20131255 (2015)","journal-title":"J. Differ. Equations"},{"issue":"5","key":"9727_CR13","doi-asserted-by":"publisher","first-page":"2013","DOI":"10.1137\/090772630","volume":"42","author":"A Friedman","year":"2010","unstructured":"Friedman, A., Hu, B., Xue, C.: Analysis of a mathematical model of ischemic cutaneous wounds. SIAM J. Math. Anal. 42(5), 2013\u20132040 (2010)","journal-title":"SIAM J. Math. Anal."},{"issue":"3","key":"9727_CR14","doi-asserted-by":"publisher","first-page":"e90,497","DOI":"10.1371\/journal.pone.0090497","volume":"9","author":"W Hao","year":"2014","unstructured":"Hao, W., Friedman, A.: The LDL\u2013HDL profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE 9(3), e90,497 (2014)","journal-title":"PLoS ONE"},{"issue":"3","key":"9727_CR15","first-page":"e014,738","volume":"11","author":"W Hao","year":"2016","unstructured":"Hao, W., Schlesinger, L.S., Friedman, A.: Modeling granulomas in response to infection in the lung. PLoS ONE 11(3), e014,738 (2016)","journal-title":"PLoS ONE"},{"issue":"2","key":"9727_CR16","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/0022-5193(71)90050-6","volume":"30","author":"EF Keller","year":"1971","unstructured":"Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30(2), 225\u2013234 (1971)","journal-title":"J. Theor. Biol."},{"issue":"2","key":"9727_CR17","doi-asserted-by":"publisher","first-page":"568","DOI":"10.1137\/110823584","volume":"44","author":"I Kim","year":"2012","unstructured":"Kim, I., Yao, Y.: The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle. SIAM J. Math. Anal. 44(2), 568\u2013602 (2012)","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"9727_CR18","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/0021-9991(79)90145-1","volume":"32","author":"B van Leer","year":"1979","unstructured":"van Leer, B.: Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov\u2019s method. J. Comput. Phys. 32(1), 101\u2013136 (1979)","journal-title":"J. Comput. Phys."},{"key":"9727_CR19","doi-asserted-by":"crossref","unstructured":"LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press (2002)","DOI":"10.1017\/CBO9780511791253"},{"key":"9727_CR20","unstructured":"Niu, B., Zeng, X., Szulzewsky, F., Holte, S., Maini, P.K., Holland, E.C., Tian, J.P.: Mathematical modeling of PDGF-driven glioma reveals the infiltrating dynamics of immune cells into tumors. Submitted (2018)"},{"issue":"17","key":"9727_CR21","doi-asserted-by":"publisher","first-page":"6709","DOI":"10.1016\/j.jcp.2011.05.005","volume":"230","author":"AL Ortega","year":"2011","unstructured":"Ortega, A.L., Scovazzi, G.: A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements. J. Comput. Phys. 230(17), 6709\u20136741 (2011)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9727_CR22","doi-asserted-by":"publisher","first-page":"2319","DOI":"10.1090\/S0002-9947-08-04656-4","volume":"361","author":"B Perthame","year":"2009","unstructured":"Perthame, B., Dalibard, A.L.: Existence of solutions of the hyperbolic Keller-Segel model. T. Am. Math. Soc. 361(5), 2319\u20132335 (2009)","journal-title":"T. Am. Math. Soc."},{"issue":"2","key":"9727_CR23","doi-asserted-by":"publisher","first-page":"A789","DOI":"10.1137\/140970185","volume":"38","author":"X Zeng","year":"2016","unstructured":"Zeng, X.: A general approach to enhance slope limiters in MUSCL schemes on nonuniform rectilinear grids. SIAM J. Sci. Comput. 38(2), A789\u2013A813 (2016)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-019-09727-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-019-09727-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-019-09727-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,1]],"date-time":"2020-12-01T19:50:12Z","timestamp":1606852212000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-019-09727-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12]]},"references-count":23,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["9727"],"URL":"https:\/\/doi.org\/10.1007\/s10444-019-09727-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2019,12]]},"assertion":[{"value":"14 September 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 September 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}