Efficient Numerical Solution of the EMI Model Representing the Extracellular Space (E), Cell Membrane (M) and Intracellular Space (I) of a Collection of Cardiac Cells
The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the...
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| Vydané v: | Frontiers in physics Ročník 8 |
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Frontiers Media S.A
13.01.2021
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| Abstract | The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512
×
256 cells, we solved linear systems with about
2.5
×
10
8
unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance. |
|---|---|
| AbstractList | The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512
×
256 cells, we solved linear systems with about
2.5
×
10
8
unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance. The EMI model represents excitable cells in a more accurate manner than traditional homogenized models at the price of increased computational complexity. The increased complexity of solving the EMI model stems from a significant increase in the number of computational nodes and from the form of the linear systems that need to be solved. Here, we will show that the latter problem can be solved by careful use of operator splitting of the spatially coupled equations. By using this method, the linear systems can be broken into sub-problems that are of the classical type of linear, elliptic boundary value problems. Therefore, the vast collection of methods for solving linear, elliptic partial differential equations can be used. We demonstrate that this enables us to solve the systems using shared-memory parallel computers. The computing time scales perfectly with the number of physical cells. For a collection of 512 × 256 cells, we solved linear systems with about 2.5×108 unknows. Since the computational effort scales linearly with the number of physical cells, we believe that larger computers can be used to simulate millions of excitable cells and thus allow careful analysis of physiological systems of great importance. |
| Author | Tveito, Aslak Cai, Xing Jæger, Karoline Horgmo Hustad, Kristian Gregorius |
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| Cites_doi | 10.1103/physreve.67.051925 10.1088/1741-2560/10/2/026019 10.3389/fphys.2012.00345 10.1016/j.pbiomolbio.2007.07.009 10.1529/biophysj.108.137349 10.1152/ajpheart.00760.2013 10.1137/15m1026419 10.1111/j.1540-8159.2006.00382.x 10.1007/s10439-007-9337-3 10.1109/tbme.2006.889172 10.1137/110838844 10.1007/s10439-005-7257-7 10.1063/1.166300 10.1016/j.bpj.2010.12.3716 10.1109/10.184699 10.1007/s10439-009-9883-y 10.1017/s0962492915000021 10.1007/978-3-319-04801-7 10.1016/j.yjmcc.2009.09.019 10.1006/jcph.2002.7176 10.1002/nla.501 10.1016/j.cam.2015.02.011 10.1016/j.pbiomolbio.2015.12.011 10.1109/TBME.2012.2205575 10.1371/journal.pcbi.1007042 10.3389/fncom.2017.00027 10.1016/s0377-0427(00)00516-1 10.3389/fphys.2011.00014 10.1109/tbme.2009.2014739 10.1161/01.res.0000046237.54156.0a 10.1137/17m1137061 10.1016/s0006-3495(94)80971-3 10.1002/nla.716 10.1152/ajpheart.00868.2011 10.1145/1089014.1089017 10.3389/fphy.2017.00048 10.1155/2010/503906 10.1161/01.res.86.3.302 |
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| References | B43 Franzone (B1) 2014 B44 Marsh (B35) 2012; 59 Whiteley (B3) 2007; 35 Tsumoto (B21) 2011; 100 Sundnes (B2) 2006 Niederer (B26) 2011; 2 Roth (B45) 2003; 67 Liu (B28) 2008; 96 Wathen (B32) 2015; 24 Tveito (B8) 2017; 5 Csepe (B27) 2016; 120 Sundnes (B34) 2009; 56 Agudelo-Toro (B14) 2013; 10 Li (B37) 2005; 31 Jæger (B47) 2020 Mardal (B4) 2007; 14 Veeraraghavan (B24) 2012; 302 Krassowska (B12) 1994; 66 Cervi (B6) 2018; 40 Stinstra (B19) 2005; 33 Tveito (B10) 2017; 11 Hogues (B11) 1992; 39 Stinstra (B16) 2009 Stinstra (B17) 2010; 38 Stinstra (B18) 2006 Bangerth (B36) 2020 Veeraraghavan (B25) 2014; 306 Jæger (B48) 2020 Stüben (B38) 2001; 128 Spach (B23) 2000; 86 Mele (B30) 2020 Louch (B22) 2010; 2010 Benzi (B29) 2002; 182 Hake (B41) Jæger (B9) 2019; 15 Grandi (B33) 2010; 48 Ottino (B7) 2015; 285 Mardal (B31) 2011; 18 Rupp (B39) 2016; 38 Beaudoin (B46) 2006; 29 Linge (B5) 1895; 367 Roberts (B15) 2008; 95 Bell (B40) 2012; 34 Ying (B13) 2007; 54 Kucera (B20) 2002; 91 Kaese (B42) 2012; 3 |
| References_xml | – volume: 67 start-page: 051925 year: 2003 ident: B45 article-title: Approximate analytical solutions of the bidomain equations for electrical stimulation of cardiac tissue with curving fibers publication-title: Phys Rev doi: 10.1103/physreve.67.051925 – volume: 10 start-page: 026019 year: 2013 ident: B14 article-title: Computationally efficient simulation of electrical activity at cell membranes interacting with self-generated and externally imposed electric fields publication-title: J Neural Eng doi: 10.1088/1741-2560/10/2/026019 – ident: B43 – start-page: 1 volume-title: Modeling excitable tissue. year: 2020 ident: B47 article-title: Derivation of a Cell-Based Mathematical Model of Excitable Cells – start-page: 41 volume-title: Computers in cardiology year: 2006 ident: B18 article-title: A model of 3D propagation in discrete cardiac tissue – volume: 3 start-page: 345 year: 2012 ident: B42 article-title: Cardiac electrophysiology in mice: a matter of size publication-title: Front Physiol doi: 10.3389/fphys.2012.00345 – volume: 96 start-page: 294 year: 2008 ident: B28 article-title: Role of pacemaking current in cardiac nodes: insights from a comparative study of sinoatrial node and atrioventricular node publication-title: Prog Biophys Mol Biol doi: 10.1016/j.pbiomolbio.2007.07.009 – volume: 95 start-page: 3724 year: 2008 ident: B15 article-title: Effect of nonuniform interstitial space properties on impulse propagation: a discrete multidomain model publication-title: Biophys J doi: 10.1529/biophysj.108.137349 – volume: 306 start-page: H619 year: 2014 ident: B25 article-title: Mechanisms of cardiac conduction: a history of revisions publication-title: Am J Physiol Heart Circ Physiol doi: 10.1152/ajpheart.00760.2013 – volume-title: Preconditioning for linear systems year: 2020 ident: B30 – volume: 38 start-page: S412 year: 2016 ident: B39 article-title: ViennaCL—Linear algebra library for multi- and many-core architectures publication-title: SIAM J Sci Comput doi: 10.1137/15m1026419 – volume: 29 start-page: 496 year: 2006 ident: B46 article-title: The effect of the fiber curvature gradient on break excitation in cardiac tissue publication-title: Pacing clin electrophysiol doi: 10.1111/j.1540-8159.2006.00382.x – volume: 35 start-page: 1510 year: 2007 ident: B3 article-title: Physiology driven adaptivity for the numerical solution of the bidomain equations publication-title: Ann Biomed Eng doi: 10.1007/s10439-007-9337-3 – volume: 54 start-page: 611 year: 2007 ident: B13 article-title: Hybrid finite element method for describing the electrical response of biological cells to applied fields publication-title: IEEE Trans Biomed Eng doi: 10.1109/tbme.2006.889172 – volume: 34 start-page: C123 year: 2012 ident: B40 article-title: Exposing fine-grained parallelism in algebraic multigrid methods publication-title: SIAM J Sci Comput doi: 10.1137/110838844 – volume: 33 start-page: 1743 year: 2005 ident: B19 article-title: On the passive cardiac conductivity publication-title: Ann Biomed Eng doi: 10.1007/s10439-005-7257-7 – volume: 367 start-page: 1931 year: 1895 ident: B5 article-title: Numerical solution of the bidomain equations publication-title: Phil Trans Roy Soc Lond doi: 10.1063/1.166300 – volume: 100 start-page: 554 year: 2011 ident: B21 article-title: Roles of subcellular Na + channel distributions in the mechanism of cardiac conduction publication-title: Biophys J doi: 10.1016/j.bpj.2010.12.3716 – volume: 39 start-page: 1232 year: 1992 ident: B11 article-title: A model study of electric field interactions between cardiac myocytes publication-title: IEEE Trans Biomed Eng doi: 10.1109/10.184699 – volume-title: Finite element methods in scientific computing year: 2020 ident: B36 – volume: 38 start-page: 1399 year: 2010 ident: B17 article-title: Incorporating histology into a 3D microscopic computer model of myocardium to study propagation at a cellular level publication-title: Ann Biomed Eng doi: 10.1007/s10439-009-9883-y – volume: 24 start-page: 329 year: 2015 ident: B32 article-title: Preconditioning publication-title: Acta Numerica doi: 10.1017/s0962492915000021 – start-page: 397 volume-title: Mathematical cardiac electrophysiology. year: 2014 ident: B1 doi: 10.1007/978-3-319-04801-7 – volume: 48 start-page: 112 year: 2010 ident: B33 article-title: A novel computational model of the human ventricular action potential and Ca transient publication-title: J Mol Cell Cardiol doi: 10.1016/j.yjmcc.2009.09.019 – volume: 182 start-page: 418 year: 2002 ident: B29 article-title: Preconditioning techniques for large linear systems: a survey publication-title: J Comput Phys doi: 10.1006/jcph.2002.7176 – volume: 14 start-page: 83 year: 2007 ident: B4 article-title: An order optimal solver for the discretized bidomain equations publication-title: Numer Lin Algebra Appl doi: 10.1002/nla.501 – volume-title: Gotran–general ODE TRANslator ident: B41 – volume: 285 start-page: 151 year: 2015 ident: B7 article-title: Bpx preconditioners for the bidomain model of electrocardiology publication-title: J Comput Appl Math doi: 10.1016/j.cam.2015.02.011 – volume: 120 start-page: 14 year: 2016 ident: B27 article-title: Human sinoatrial node structure: 3D microanatomy of sinoatrial conduction pathways publication-title: Prog Biophys Mol Biol doi: 10.1016/j.pbiomolbio.2015.12.011 – start-page: 318 volume-title: Computing the electrical activity of the heart. year: 2006 ident: B2 – volume: 59 start-page: 2506 year: 2012 ident: B35 article-title: The secrets to the success of the Rush–Larsen method and its generalizations publication-title: IEEE Trans Biomed Eng doi: 10.1109/TBME.2012.2205575 – volume: 15 start-page: e1007042 year: 2019 ident: B9 article-title: Properties of cardiac conduction in a cell-based computationalmodel publication-title: PLoS Comput Biol doi: 10.1371/journal.pcbi.1007042 – ident: B44 – volume: 11 start-page: 27 year: 2017 ident: B10 article-title: An evaluation of the accuracy of classical models for computing the membrane potential and extracellular potential for neurons publication-title: Front Comput Neurosci doi: 10.3389/fncom.2017.00027 – volume: 128 start-page: 281 year: 2001 ident: B38 article-title: A review of algebraic multigrid publication-title: J Comput Appl Math doi: 10.1016/s0377-0427(00)00516-1 – start-page: 657 volume-title: Computers in cardiology year: 2009 ident: B16 article-title: Comparison of microscopic and bidomain models of anisotropic conduction – start-page: 44 volume-title: Modeling Excitable Tissue. year: 2020 ident: B48 article-title: Operator Splitting and Finite Difference Schemes for Solving the EMI Model – volume: 2 start-page: 1 year: 2011 ident: B26 article-title: Simulating human cardiac electrophysiology on clinical time-scales publication-title: Front Physiol doi: 10.3389/fphys.2011.00014 – volume: 56 start-page: 2546 year: 2009 ident: B34 article-title: A second-order algorithm for solving dynamic cell membrane equations publication-title: IEEE Trans Biomed Eng doi: 10.1109/tbme.2009.2014739 – volume: 91 start-page: 1176 year: 2002 ident: B20 article-title: Localization of sodium channels in intercalated disks modulates cardiac conduction publication-title: Circ Res doi: 10.1161/01.res.0000046237.54156.0a – volume: 40 start-page: A769 year: 2018 ident: B6 article-title: High-order operator splitting for the bidomain and monodomain models publication-title: SIAM J Sci Comput doi: 10.1137/17m1137061 – volume: 66 start-page: 1768 year: 1994 ident: B12 article-title: Response of a single cell to an external electric field publication-title: Biophys J doi: 10.1016/s0006-3495(94)80971-3 – volume: 18 start-page: 1 year: 2011 ident: B31 article-title: Preconditioning discretizations of systems of partial differential equations publication-title: Numer Lin Algebra Appl doi: 10.1002/nla.716 – volume: 302 start-page: H278 year: 2012 ident: B24 article-title: Interstitial volume modulates the conduction velocity-gap junction relationship publication-title: Am J Physiol Heart Circ Physiol doi: 10.1152/ajpheart.00868.2011 – volume: 31 start-page: 302 year: 2005 ident: B37 article-title: An overview of SuperLU publication-title: ACM Trans Math Software doi: 10.1145/1089014.1089017 – volume: 5 start-page: 48 year: 2017 ident: B8 article-title: A cell-based framework for numerical modeling of electrical conduction in cardiac tissue publication-title: Front Phys doi: 10.3389/fphy.2017.00048 – volume: 2010 start-page: 503906 year: 2010 ident: B22 article-title: There goes the neighborhood: pathological alterations in t-tubule morphology and consequences for cardiomyocyte Ca2+ handling publication-title: BioMed Res Int doi: 10.1155/2010/503906 – volume: 86 start-page: 302 year: 2000 ident: B23 article-title: Electrophysiological effects of remodeling cardiac gap junctions and cell size publication-title: Circ Res doi: 10.1161/01.res.86.3.302 |
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