Solvability analysis and numerical approximation of linearized cardiac electromechanics
Abstract
This paper is concerned with the mathematical analysis of a coupled elliptic–parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction–diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction–diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo–Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.
References
- Chaos, Solitons Fractals 7 , 293 ( 1996 ) . Crossref, ISI, Google Scholar
- Math. Z. 183 , 311 ( 1983 ) . Crossref, ISI, Google Scholar
- SIAM J. Appl. Math. 71 , 605 ( 2011 ) . Crossref, ISI, Google Scholar
- J. Elasticity 107 , 199 ( 2012 ) . Crossref, ISI, Google Scholar
- Netw. Heterog. Media 6 , 195 ( 2011 ) . Crossref, ISI, Google Scholar
- Arch. Rational Mech. Anal. 63 , 337 ( 1977 ) . Crossref, ISI, Google Scholar
- Adv. Comput. Math. 39 , 425 ( 2013 ) . Crossref, ISI, Google Scholar
- Netw. Heterog. Media 1 , 185 ( 2006 ) . Crossref, ISI, Google Scholar
- Nature 415 , 198 ( 2002 ) . Crossref, ISI, Google Scholar
- SIAM J. Appl. Math. 72 , 1618 ( 2012 ) . Crossref, ISI, Google Scholar
- Appl. Math. Res. Express AMRX 2 , 28 ( 2008 ) . Google Scholar
- Nonlinear Anal.: Real World Appl. 10 , 458 ( 2009 ) . Crossref, ISI, Google Scholar
- Progr. Biophys. Molec. Biol. 97 , 562 ( 2008 ) . Crossref, ISI, Google Scholar
-
P. G. Ciarlet , Mathematical Elasticity, Vol I: Three-Dimensional Elasticity ( North-Holland , 1978 ) . Google Scholar P. Colli Franzone and G. Savaré , Evolution Equations, Semigroups and Functional Analysis,Progress in Nonlinear Differential Equations Application 50 (Birkhäuser, 2002) pp. 49–78. Crossref, Google Scholar- SIAM J. Numer. Anal. 46 , 2443 ( 2008 ) . Crossref, ISI, Google Scholar
- Math. Biosci. 226 , 58 ( 2010 ) . Crossref, ISI, Google Scholar
- Biophys. J. 1 , 445 ( 1961 ) . Crossref, ISI, Google Scholar
- Comput. Mech. 45 , 227 ( 2010 ) . Crossref, ISI, Google Scholar
- Philos. Trans. Roy. Soc. A 367 , 3445 ( 2009 ) . Crossref, ISI, Google Scholar
- Duke Math. J. 107 , 497 ( 2001 ) . Crossref, ISI, Google Scholar
- Nonlinear Anal. Real World Appl. 7 , 535 ( 2006 ) . Crossref, ISI, Google Scholar
- Int. J. Numer. Methods Engrg. 96 , 712 ( 2013 ) . Crossref, ISI, Google Scholar
- Int. J. Numer. Methods Biomed. Engrg. 28 , 72 ( 2012 ) . Crossref, ISI, Google Scholar
- Nonlinear Anal. 14 , 501 ( 1990 ) . Crossref, ISI, Google Scholar
-
J.-L. Lions , Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires ( Dunod , 1969 ) . Google Scholar - Discrete Contin. Dynam. Syst. 29 , 1573 ( 2011 ) . Crossref, ISI, Google Scholar
- Bull. Math. Biol. 65 , 767 ( 2003 ) . Crossref, ISI, Google Scholar
- Proc. Inst. Radio Engrg. 50 , 2061 ( 1962 ) . Crossref, ISI, Google Scholar
- J. Elasticity 88 , 27 ( 2007 ) . Crossref, ISI, Google Scholar
- J. Elasticity 61 , 113 ( 2000 ) . Crossref, ISI, Google Scholar
- Progr. Biophys. Molec. Biol. 85 , 501 ( 2004 ) . Crossref, ISI, Google Scholar
- Int. J. Numer. Meth. Biomed. Engrg. 28 , 52 ( 2012 ) . Crossref, ISI, Google Scholar
- Progr. Biophys. Molec. Biol. 104 , 77 ( 2011 ) . Crossref, ISI, Google Scholar
- Quart. J. Mech. Appl. Math. 63 , 375 ( 2010 ) . Crossref, ISI, Google Scholar
- P. Pathmanathan, C Ortner and D. Kay, Existence of solutions of partially degenerate visco-elastic problems, and applications to modelling muscular contraction and cardiac electro-mechanical activity, submitted . Google Scholar
-
A. Quarteroni and A. Valli , Numerical Approximation of Partial Differential Equations ,Springer Series in Computational Mathematics 23 ( Springer , 1994 ) . Crossref, Google Scholar - IEEE Trans. Biomed. Engrg. 41 , 743 ( 1994 ) . Crossref, ISI, Google Scholar
- Eur. J. Mech. A Solids 48 , 129 ( 2014 ) . Crossref, ISI, Google Scholar
- Int. J. Numer. Meth. Biomed. Engrg. 28 , 761 ( 2012 ) . Crossref, ISI, Google Scholar
- O. Rousseau, Geometrical modeling of the heart, Ph.D. thesis, Université d'Ottawa (2010) . Google Scholar
- , Computer Models in Biomechanics: From Nano to Macro, eds.
G. A. Holzapfel and E. Kuhl (Springer, 2013) pp. 189–201. Crossref, Google Scholar - Math. Med. Biol. 31 , 259 ( 2014 ) . Crossref, ISI, Google Scholar
- Numer. Methods Partial Differential Equations 18 , 218 ( 2002 ) . Crossref, ISI, Google Scholar
-
J. Sundnes , Computing the Electrical Activity in the Heart ,Monographs in Computational Science and Engineering 1 ( Springer , 2006 ) . Google Scholar - Comput. Meth. Biomech. Biomed. Engrg. 17 , 604 ( 2014 ) . Crossref, ISI, Google Scholar
- Circ. Res. 108 , 113 ( 2011 ) . Crossref, ISI, Google Scholar
- L. Tung, A bidomain model for describing ischemic myocardial D–C potentials, Ph.D. thesis, MIT (1978) . Google Scholar
- Nonlinear Anal. Real World Appl. 10 , 849 ( 2009 ) . Crossref, ISI, Google Scholar
| Remember to check out the Most Cited Articles! |
|---|
|
View our Mathematical Modelling books
|
