{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,26]],"date-time":"2026-07-26T05:58:01Z","timestamp":1785045481861,"version":"3.55.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2020,6]]},"DOI":"10.1007\/s11538-020-00749-4","type":"journal-article","created":{"date-parts":[[2020,6,13]],"date-time":"2020-06-13T07:02:35Z","timestamp":1592031755000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonreflecting Boundary Conditions for a CSF Model of Fourth Ventricle: Spinal SAS Dynamics"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9810-0602","authenticated-orcid":false,"given":"Donatella","family":"Donatelli","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Licia","family":"Romagnoli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,6,13]]},"reference":[{"key":"749_CR1","doi-asserted-by":"publisher","first-page":"99","DOI":"10.3934\/nhm.2007.2.99","volume":"2","author":"D Amadori","year":"2007","unstructured":"Amadori D, Ferrari S, Formaggia L (2007) Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels. Netw Heterog Media 2:99\u2013125","journal-title":"Netw Heterog Media"},{"key":"749_CR2","unstructured":"Baeck K, Goffin J, Sloten JV (2011) The use of different CSF representations in a numerical head model and their effect on the results of FE head impact analyses. In: $$8^{th}$$ European LS-DYNA Users Conference, Strasbourg"},{"key":"749_CR3","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/0021-9991(82)90046-8","volume":"48","author":"A Bayliss","year":"1982","unstructured":"Bayliss A, Turkel E (1982) Far field boundary conditions for compressible flows. J Comput Phys 48:182\u2013199","journal-title":"J Comput Phys"},{"key":"749_CR4","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1137\/120903385","volume":"74","author":"LC Berselli","year":"2014","unstructured":"Berselli LC, Guerra F, Mazzolai B et al (2014) Pulsatile viscous flows in elliptical vessels and annuli: solution to the inverse problem, with application to blood and cerebrospinal fluid flow. SIAM J Appl Math 74:40\u201359","journal-title":"SIAM J Appl Math"},{"key":"749_CR5","doi-asserted-by":"publisher","first-page":"1189","DOI":"10.1016\/j.jfluidstructs.2009.06.008","volume":"25","author":"C Bertram","year":"2009","unstructured":"Bertram C (2009) A numerical investigation of waves propagating in the spinal cord and subarachnoid space in the presence of a syrinx. J Fluids Struct 25:1189\u20131205","journal-title":"J Fluids Struct"},{"key":"749_CR6","doi-asserted-by":"publisher","first-page":"061009","DOI":"10.1115\/1.4001165","volume":"132","author":"C Bertram","year":"2010","unstructured":"Bertram C (2010) Evaluation by fluid\/structure interaction spinal-cord simulation of the effects of subarachnoid-space stenosis on an adjacent syrinx. J Biomech Eng 132:061009","journal-title":"J Biomech Eng"},{"key":"749_CR7","doi-asserted-by":"publisher","first-page":"1099","DOI":"10.1115\/1.2073607","volume":"127","author":"C Bertram","year":"2005","unstructured":"Bertram C, Brodbelt A, Stoodley M (2005) The origins of syringomyelia: numerical models of fluid\/structure interactions in the spinal cord. J Biomech Eng 127:1099\u20131109","journal-title":"J Biomech Eng"},{"key":"749_CR8","doi-asserted-by":"publisher","first-page":"481","DOI":"10.3325\/cmj.2014.55.481","volume":"55","author":"J Buishas","year":"2014","unstructured":"Buishas J, Gould I, Linninger AA (2014) A computational model of cerebrospinal fluid production and reabsorption driven by starling forces. Croat Med J 55:481\u2013497","journal-title":"Croat Med J"},{"key":"749_CR9","doi-asserted-by":"publisher","first-page":"1161","DOI":"10.1002\/mma.407","volume":"26","author":"S \u010cani\u0107","year":"2003","unstructured":"\u010cani\u0107 S, Kim EH (2003) Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels. Math Methods Appl Sci 26:1161\u20131186","journal-title":"Math Methods Appl Sci"},{"key":"749_CR10","doi-asserted-by":"publisher","first-page":"021008","DOI":"10.1115\/1.3005159","volume":"131","author":"S Cirovic","year":"2009","unstructured":"Cirovic S (2009) A coaxial tube model of the cerebrospinal fluid pulse propagation in the spinal column. J Biomech Eng 131:021008","journal-title":"J Biomech Eng"},{"key":"749_CR11","doi-asserted-by":"crossref","unstructured":"Cirovic S, Kim M (2010) One-dimensional model for ceresbrospinal fluid pulse in the spinal column. In: 6th world congress of biomechanics (WCB (2010) Singapore. Springer 366\u2013369","DOI":"10.1007\/978-3-642-14515-5_94"},{"key":"749_CR12","doi-asserted-by":"publisher","first-page":"1801","DOI":"10.1016\/j.jbiomech.2013.05.013","volume":"46","author":"EC Clarke","year":"2013","unstructured":"Clarke EC, Fletcher DF, Stoodley MA et al (2013) Computational fluid dynamics modelling of cerebrospinal fluid pressure in Chiari malformation and syringomyelia. J Biomech 46:1801\u20131809","journal-title":"J Biomech"},{"key":"749_CR13","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1016\/j.nonrwa.2018.05.009","volume":"44","author":"D Donatelli","year":"2018","unstructured":"Donatelli D, Marcati P, Romagnoli L (2018) Analysis of solutions for a cerebrospinal fluid model. Nonlinear Anal Real World Appl 44:417\u2013448","journal-title":"Nonlinear Anal Real World Appl"},{"key":"749_CR14","doi-asserted-by":"publisher","first-page":"2811","DOI":"10.3934\/mbe.2019140","volume":"16","author":"D Donatelli","year":"2019","unstructured":"Donatelli D, Marcati P, Romagnoli L (2019) A comparison of two mathematical models of the cerebrospinal fluid dynamics. Math Biosci Eng 16:2811\u20132851","journal-title":"Math Biosci Eng"},{"key":"749_CR15","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/0045-7825(84)90153-1","volume":"45","author":"J Donea","year":"1984","unstructured":"Donea J, Giuliani S, Laval H, Quartapelle L (1984) Time-accurate solutions of advection\u2013diffusion problems by finite elements. Comput Methods Appl Mech Eng 45:123\u2013145","journal-title":"Comput Methods Appl Mech Eng"},{"key":"749_CR16","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1090\/S0025-5718-1977-0436612-4","volume":"31","author":"B Engquist","year":"1977","unstructured":"Engquist B, Majda A (1977) Absorbing boundary conditions for the numerical simulation of waves. Math Comput 31:629\u2013651","journal-title":"Math Comput"},{"key":"749_CR17","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1148\/radiology.185.3.1438741","volume":"185","author":"DR Enzmann","year":"1992","unstructured":"Enzmann DR, Pelc NJ (1992) Brain motion: measurement with phase-contrast MR imaging. Radiology 185:653\u2013660","journal-title":"Radiology"},{"key":"749_CR18","doi-asserted-by":"publisher","first-page":"5685","DOI":"10.1016\/j.cma.2005.11.011","volume":"195","author":"CA Figueroa","year":"2006","unstructured":"Figueroa CA, Vignon-Clementel IE, Jansen KE et al (2006) A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput Methods Appl Mech Eng 195:5685\u20135706","journal-title":"Comput Methods Appl Mech Eng"},{"key":"749_CR19","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/S0045-7825(01)00302-4","volume":"191","author":"L Formaggia","year":"2001","unstructured":"Formaggia L, Gerbeau JF, Nobile F, Quarteroni A (2001) On the coupling of 3D and 1D Navier\u2013Stokes equations for flow problems in compliant vessels. Comput Methods Appl Mech Eng 191:561\u2013582","journal-title":"Comput Methods Appl Mech Eng"},{"key":"749_CR20","first-page":"145","volume":"27","author":"D Greitz","year":"2004","unstructured":"Greitz D (2004) Radiological assessment of hydrocephalus: new theories and implications for therapy. Neurosurg Rev 27:145\u2013165","journal-title":"Neurosurg Rev"},{"key":"749_CR21","first-page":"021010","volume":"131","author":"S Gupta","year":"2009","unstructured":"Gupta S, Soellinger M, Boesiger P et al (2009) Three-dimensional computational modeling of subjectspecific cerebrospinal fluid flow in the subarachnoid space. J Biomed Eng 131:021010","journal-title":"J Biomed Eng"},{"key":"749_CR22","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1016\/0021-9991(79)90100-1","volume":"30","author":"GW Hedstrom","year":"1979","unstructured":"Hedstrom GW (1979) Nonreflecting boundary conditions for nonlinear hyperbolic systems. J Comput Phys 30:222\u2013237","journal-title":"J Comput Phys"},{"key":"749_CR23","doi-asserted-by":"crossref","unstructured":"LeVeque R (1990) Numerical methods for conservation laws. Birkhauser, Basel","DOI":"10.1007\/978-3-0348-5116-9"},{"key":"749_CR24","doi-asserted-by":"publisher","first-page":"1474","DOI":"10.3174\/ajnr.A2496","volume":"32","author":"SO Linge","year":"2011","unstructured":"Linge SO, Haughton V, L\u00f8vgren AE et al (2011) Effect of tonsillar herniation on cyclic CSF flow studied with computational flow analysis. Am J Neuroradiol (AJNR) 32:1474\u20131481","journal-title":"Am J Neuroradiol (AJNR)"},{"key":"749_CR25","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1146\/annurev-fluid-122414-034321","volume":"48","author":"AA Linninger","year":"2016","unstructured":"Linninger AA, Tangen K, Hsu CY, Frim D (2016) Cerebrospinal fluid mechanics and its coupling to cerebrovascular dynamics. Annu Rev Fluid Mech 48:219\u2013257","journal-title":"Annu Rev Fluid Mech"},{"key":"749_CR26","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1109\/TBME.2005.844021","volume":"52","author":"AA Linninger","year":"2005","unstructured":"Linninger AA, Tsakiris C, Zhu DC et al (2005) Pulsatile cerebrospinal fluid dynamics in the human brain. IEEE T Bio-Med Eng 52:557\u2013565","journal-title":"IEEE T Bio-Med Eng"},{"key":"749_CR27","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s00285-009-0250-2","volume":"59","author":"AA Linninger","year":"2009","unstructured":"Linninger AA, Xenos M, Sweetman B et al (2009) A mathematical model of blood, cerebrospinal fluid and brain dynamics. J Math Biol 59:729\u2013759","journal-title":"J Math Biol"},{"key":"749_CR28","doi-asserted-by":"publisher","first-page":"H1492","DOI":"10.1152\/ajpheart.00658.2011","volume":"302","author":"BA Martin","year":"2012","unstructured":"Martin BA, Reymond P, Novy J et al (2012) A coupled hydrodynamic model of the cardiovascular and cerebrospinal fluid system. Am J Physiol Heart Circul Physiol 302:H1492\u2013H1509","journal-title":"Am J Physiol Heart Circul Physiol"},{"key":"749_CR29","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1002\/cnm.2622","volume":"30","author":"LO M\u00fcller","year":"2014","unstructured":"M\u00fcller LO, Toro EF (2014) A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int J Numer Methods Biomed Eng 30:681\u2013725","journal-title":"Int J Numer Methods Biomed Eng"},{"key":"749_CR30","unstructured":"Nobile F (2001) Numerical approximation of fluid-structure interaction problems with applications to hemodynamics. PhD thesis, Ecole Polytechnique Federeale Lausanne"},{"key":"749_CR31","first-page":"499","volume":"10","author":"Z Ori","year":"1992","unstructured":"Ori Z, Monir G, Weiss J et al (1992) Heart rate variability. Frequency Domain Anal Cardiol Clin 10:499\u2013537","journal-title":"Frequency Domain Anal Cardiol Clin"},{"key":"749_CR32","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1002\/cnm.1212","volume":"25","author":"G Papadakis","year":"2009","unstructured":"Papadakis G (2009) Coupling 3D and 1D fluid structure-interaction models for wave propagation in flexible vessels using a finite volume pressure-correction scheme. Commun Numer Methods Eng 25:533\u2013551","journal-title":"Commun Numer Methods Eng"},{"key":"749_CR33","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511896996","volume-title":"The fluid mechanics of large blood vessels","author":"T Pedley","year":"1980","unstructured":"Pedley T (1980) The fluid mechanics of large blood vessels. Cambridge University Press, Cambridge"},{"key":"749_CR34","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.3171\/jns.2005.102.6.1069","volume":"102","author":"RD Penn","year":"2005","unstructured":"Penn RD, Lee MC, Linninger AA et al (2005) Pressure gradients in the brain in an experimental model of hydrocephalus. J Neurosurg 102:1069\u20131075","journal-title":"J Neurosurg"},{"key":"749_CR35","unstructured":"Pietrabissa R, Quarteroni A, Dubini G et al (2000) From the global cardiovascular system down to the local blood motion: preliminary applications of a multiscale approach. In: Proceedings of ECCOMAS2000 Conference, Barcelona, (CDRom)"},{"key":"749_CR36","doi-asserted-by":"crossref","unstructured":"Quarteroni A, Formaggia L (2004) Mathematical modelling and numerical simulation of the cardiovascular system. In: Handbook of numerical analysis, Vol XII, North Holland, Amsterdam, 3\u2013127","DOI":"10.1016\/S1570-8659(03)12001-7"},{"key":"749_CR37","doi-asserted-by":"crossref","unstructured":"Quarteroni A, Valli A (1994) Numerical approximation of partial differential equations. Springer Verlag Series in computational Mathematics 23","DOI":"10.1007\/978-3-540-85268-1"},{"key":"749_CR38","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.cma.2016.01.007","volume":"302","author":"A Quarteroni","year":"2016","unstructured":"Quarteroni A, Veneziani A, Vergara C (2016) Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput Methods Appl Mech Eng 302:193\u2013252","journal-title":"Comput Methods Appl Mech Eng"},{"key":"749_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3171\/foc.2003.14.4.4","volume":"14","author":"PB Raksin","year":"2003","unstructured":"Raksin PB, Alperin N, Sivaramakrishnan A et al (2003) Noninvasive intracranial compliance and pressure based on dynamic magnetic resonance imaging of blood flow and cerebrospinal fluid flow: Reviewof principles, implementation, and other noninvasive approaches. Neurosurg Focus 14:1\u20138","journal-title":"Neurosurg Focus"},{"key":"749_CR40","doi-asserted-by":"crossref","unstructured":"Romagnoli L (2019) Numerical simulations of different models describing cerebrospinal fluid dynamics. Rend Ma Appl (7). Online first, 28 p","DOI":"10.1101\/573345"},{"key":"749_CR41","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1186\/s12987-017-0085-y","volume":"16","author":"LR Sass","year":"2017","unstructured":"Sass LR, Khani M, Natividad CG et al (2017) A 3D subject-specific model of the spinal subarachnoid space with anatomically realistic ventral and dorsal spinal cord nerve rootlets. Fluids Barriers CNS 16:36","journal-title":"Fluids Barriers CNS"},{"key":"749_CR42","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1023\/B:ENGI.0000007979.32871.e2","volume":"47","author":"SJ Sherwin","year":"2003","unstructured":"Sherwin SJ, Franke V, Peir\u00f2 J, Parker K (2003) One dimensional modelling of a vascular network in space-time variables. J Eng Math 47:217\u2013250","journal-title":"J Eng Math"},{"key":"749_CR43","doi-asserted-by":"publisher","first-page":"2144","DOI":"10.1016\/j.jbiomech.2015.02.018","volume":"48","author":"K Tangen","year":"2015","unstructured":"Tangen K, Hsu Y, Zhu D, Linninger AA (2015) CNS wide simulation of flow resistance and drug transport due to spinal microanatomy. J Biomech 48:2144\u20132154","journal-title":"J Biomech"},{"key":"749_CR44","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2019.2930451","author":"K Tangen","year":"2019","unstructured":"Tangen K, Sullivan J, Holt RW et al (2019) In-vivo intrathecal tracer dispersion in cynomolgus monkey validates wide biodistribution along neuraxis. IEEE Trans Biomed Eng. https:\/\/doi.org\/10.1109\/TBME.2019.2930451","journal-title":"IEEE Trans Biomed Eng"},{"key":"749_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0021-9991(87)90041-6","volume":"68","author":"KW Thompson","year":"1987","unstructured":"Thompson KW (1987) Time dependent boundary conditions for hyperbolic systems. J Comput Phys 68:1\u201324","journal-title":"J Comput Phys"},{"key":"749_CR46","doi-asserted-by":"publisher","first-page":"011004","DOI":"10.1115\/1.4041551","volume":"141","author":"EF Toro","year":"2018","unstructured":"Toro EF, Thornber B, Zhang Q et al (2018) computational model for the dynamics of cerebrospinal fluid in the spinal subarachnoid space. J Biomech Eng 141:011004","journal-title":"J Biomech Eng"},{"key":"749_CR47","doi-asserted-by":"publisher","first-page":"4002","DOI":"10.1016\/j.cma.2005.07.014","volume":"195","author":"SA Urquiza","year":"2006","unstructured":"Urquiza SA, Blanco PJ, V\u00e8nere MJ et al (2006) Multidimensional modelling for the carotid artery blood flow. Comput Methods Appl Mech Eng 195:4002\u20134017","journal-title":"Comput Methods Appl Mech Eng"},{"key":"749_CR48","doi-asserted-by":"publisher","first-page":"3776","DOI":"10.1016\/j.cma.2005.04.014","volume":"195","author":"IE Vignon-Clementel","year":"2006","unstructured":"Vignon-Clementel IE, Figueroa CA, Jensen KE et al (2006) Outflow boundary conditions for three dimensional finite element modelling of blood flow and pressure in arteries. Comput Methods Appl Mech Eng 195:3776\u20133796","journal-title":"Comput Methods Appl Mech Eng"}],"container-title":["Bulletin of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00749-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11538-020-00749-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00749-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,12]],"date-time":"2021-06-12T23:12:06Z","timestamp":1623539526000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11538-020-00749-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6]]},"references-count":48,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["749"],"URL":"https:\/\/doi.org\/10.1007\/s11538-020-00749-4","relation":{},"ISSN":["0092-8240","1522-9602"],"issn-type":[{"value":"0092-8240","type":"print"},{"value":"1522-9602","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6]]},"assertion":[{"value":"6 November 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"77"}}