{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:31Z","timestamp":1787337271738,"version":"build-2736575974"},"reference-count":59,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>The paper introduces and studies numerical methods that are fully adaptive in both three-dimensional (3D) space and time to challenging multiscale cardiac reaction-diffusion models. In these methods, temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration, while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators. In contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time (so-called method of lines), our method discretizes first in time and then in space (so-called Rothe method)---an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology (KARDOS code library). With this method, the evolution of a complete heartbeat, from the excitation to the recovery phase, is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models, coupled with either a variant of the simple FitzHugh--Nagumo model or the more complex phase-I Luo--Rudy ionic model. The numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction-diffusion models on 3D domains up to moderate sizes. In particular, the method accurately resolves the evolution of the intra- and extracellular potentials, gating variables, and ion concentrations during the excitation, plateau, and recovery phases.<\/jats:p>","DOI":"10.1137\/050634785","type":"journal-article","created":{"date-parts":[[2006,8,24]],"date-time":"2006-08-24T15:29:50Z","timestamp":1156433390000},"page":"942-962","source":"Crossref","is-referenced-by-count":97,"title":["Adaptivity in Space and Time for Reaction-Diffusion Systems in Electrocardiology"],"prefix":"10.1137","volume":"28","author":[{"given":"Piero Colli","family":"Franzone","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Deuflhard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bodo","family":"Erdmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jens","family":"Lang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luca F.","family":"Pavarino","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,4]]},"reference":[{"key":"R1","unstructured":"Amira homepage\n                      , http:\/\/amira.zib.de\/."},{"key":"R2","unstructured":"KARDOS homepage\n                      , http:\/\/www.zib.de\/Numerik\/numsoft\/kardos."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1046\/j.1540-8167.2004.03381.x"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0715049"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0730048"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0899-8248(92)90015-Z"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0733059"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.84.1343"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(93)90001-Q"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF00163143"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202504003489"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2005.04.003"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Piero Colli Franzone, Savar\u00e9, Degenerate evolution systems modeling the cardiac electric field at micro\u2010 and macroscopic level, Progr. Nonlinear Differential Equations Appl., Vol. 50, Birkh\u00e4user, Basel, 2002, 49\u2013782003m:35121","DOI":"10.1007\/978-3-0348-8221-7_4"},{"key":"R14","unstructured":"P. Deuflhard, Uniqueness theorems for stiff ODE initial value problems, Pitman Res. Notes Math. Ser., Vol. 228, Longman Sci. Tech., Harlow, 1990, 74\u2013881124455"},{"key":"R15","volume-title":"Newton methods for nonlinear problems","author":"Deuflhard Peter","year":"2004"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21582-2"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"P. Deuflhard, J. Lang, and U. Nowak,\n                      Adaptive algorithms in dynamical process simulation\n                      , in Progress in Industrial Mathematics at ECMI\u201994, H. Neunzert, ed., Wiley\u2013Teubner, Stuttgart, 1996, pp. 122\u2013137.","DOI":"10.1007\/978-3-322-82967-2_16"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0899-8248(89)90018-9"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/PL00013546"},{"key":"R20","unstructured":"B. Erdmann, J. Lang, and R. Roitzsch,\n                      Kardos User\u2019s Guide\n                      , Technical Report ZR\u201002\u201042, Zuse Institute Berlin (ZIB), Berlin\u2010Dahlem, Germany, 2002."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02344627"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1145\/198429.198437"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934091"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05221-7"},{"key":"R25","first-page":"1","volume":"21","author":"Henriquez C.","year":"1993","journal-title":"Crit. Rev. Biomed. Eng.","ISSN":"https:\/\/id.crossref.org\/issn\/0278-940X","issn-type":"print"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-8167.1996.tb00548.x"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1113\/jphysiol.1952.sp004764"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(94)90049-3"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1161\/01.CIR.0000147231.69595.D3"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/b98841"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1152\/physrev.00025.2003"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04484-1"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1109\/10.784145"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021900219772"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1016\/0899-8248(92)90004-R"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1152\/ajpheart.1997.272.5.H2466"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-003-0100-5"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1007\/BF02241653"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1161\/01.RES.68.6.1501"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598349197"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1002\/nla.381"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1114\/1.1509453"},{"key":"R43","unstructured":"A. Panfilov and A. Holden, eds.\n                      Computational Biology of the Heart\n                      , Wiley, Chichester 1997."},{"key":"R44","doi-asserted-by":"crossref","unstructured":"L. Pavarino and P. Colli Franzone,\n                      Parallel solution of cardiac reaction\u2010diffusion models\n                      , in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 40, R. Kornhuber, R. Hoppe, J. P\u00e9riaux, O. Pironneau, O. Widlund, and J. Xu, eds., Springer\u2010Verlag, Berlin, 2004, pp. 669\u2013676.","DOI":"10.1007\/3-540-26825-1_72"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(01)00535-0"},{"key":"R46","unstructured":"J. Pormann,\n                      A Simulation System for the Bidomain Equations\n                      , Ph.D. thesis, Duke University, Durham, NC, 1999."},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1109\/10.310090"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1007\/BF00948895"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1152\/ajpheart.01108.2003"},{"key":"R50","doi-asserted-by":"crossref","unstructured":"D. Stalling, M. Westerhoff, and H.\u2010C. Hege,\n                      Amira: A highly interactive system for visual data analysis\n                      , in The Visualization Handbook, C. Hansen and C. Johnson, eds., Elsevier, Amsterdam, 2005, pp. 749\u2013767.","DOI":"10.1016\/B978-012387582-2\/50040-X"},{"key":"R51","unstructured":"D. Streeter,\n                      Gross morphology and fiber geometry in the heart\n                      , in Handbook of Physiology, Vol. 1, R. Berne, ed., Williams & Wilkins, Baltimore, MD, 1979, pp. 61\u2013112."},{"key":"R52","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18237-2"},{"key":"R53","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.11.014"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1137\/0913035"},{"key":"R55","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2002.804597"},{"key":"R56","doi-asserted-by":"crossref","unstructured":"R. W. Dos Santos, G. Plank, S. Bauer, and E. Vigmond,\n                      Preconditioning techniques for the Bidomain equations\n                      , in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 40, R. Kornhuber, R. Hoppe, J. P\u00e9riaux, O. Pironneau, O. Widlund, and J. Xu, eds., Springer\u2010Verlag, Berlin, 2004, pp. 571\u2013580.","DOI":"10.1007\/3-540-26825-1_60"},{"key":"R57","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500315360"},{"key":"R58","unstructured":"S. Zachow, M. Weiser, H.\u2010C. Hege, and P. Deuflhard,\n                      Soft tissue prediction in computer assisted maxillofacial surgery planning: A quantitative evaluation of histomechanical modeling using pre\u2010 and postoperative CT\u2010Data\n                      , in Biomechanics Applied to Computer Assisted Surgery, Research Signpost, Trivandrum, Kerala, India 2005, pp. 277\u2013298."},{"key":"R59","unstructured":"D. Zipes and J. Jalife,\n                      Cardiac Electrophysiology\n                      , 4th ed., W. B. Saunders, Philadelphia, 2004."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050634785","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:53:50Z","timestamp":1787334830000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050634785"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":59,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/050634785"],"URL":"https:\/\/doi.org\/10.1137\/050634785","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}