September  2019, 24(9): 4863-4897. doi: 10.3934/dcdsb.2019035

Mathematical analysis of cardiac electromechanics with physiological ionic model

1. 

Institut de mathématiques de Bordeaux (IMB), Institut de rythmologie et modélisation cardiaque (Liryc), Université de Bordeaux and INRIA-Carmen Bordeaux Sud-Ouest, France

2. 

Mathematics Laboratory, Doctoral School of Sciences and Technologies, Lebanese University, Hadat, Lebanon

3. 

Département d'informatique et mathématiques, Centrale Nantes, Laboratoire de Mathématiques Jean Leray, France

4. 

Mathematics Laboratory, Doctoral School of Sciences and Technologies, Lebanese University, Hadat, Lebanon

* Corresponding author: Mostafa Bendahmane

Received  March 2018 Revised  September 2018 Published  February 2019

This paper is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential coupled with general physiological ionic models and subsequent deformation of the cardiac tissue. A prototype system belonging to this class is provided by the electromechanical bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. 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. We prove existence of weak solutions to the underlying coupled electromechanical bidomain model under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities. The proof of the existence result, which constitutes the main thrust of this paper, is proved by means of a non-degenerate approximation system, the Faedo-Galerkin method, and the compactness method.

Citation: Mostafa Bendahmane, Fatima Mroue, Mazen Saad, Raafat Talhouk. Mathematical analysis of cardiac electromechanics with physiological ionic model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4863-4897. doi: 10.3934/dcdsb.2019035
References:
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H. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differentail equations, Mathematische Zeitschrift, 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[2]

D. AmbrosiG. ArioliF. Nobile and A. Quarteroni, Electromechanical coupling in cardiac dynamics: The active strain approach, SIAM J. Appl. Math., 71 (2011), 605-621.  doi: 10.1137/100788379.  Google Scholar

[3]

D. Ambrosi and S. Pezzuto, Active stress vs. active strain in mechanobiology: Constitutive issues, J. Elasticity, 107 (2012), 199-212.  doi: 10.1007/s10659-011-9351-4.  Google Scholar

[4]

B. AndreianovM. BendahmaneA. Quarteroni and R. Ruiz-Baier, Solvability analysis and numerical approximation of linearized cardiac electromechanics, Mathematical Models and Methods in Applied Sciences, 25 (2015), 959-993.  doi: 10.1142/S0218202515500244.  Google Scholar

[5]

D. N. Arnold, L. R. Scott and M. Vogelius, Regular inversion of the divergence operator with dirichlet boundary conditions on a polygon. Google Scholar

[6]

J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[7]

D. BaroliA. Quarteroni and R. Ruiz-Baier, Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity, Adv. Comput. Math., 39 (2013), 425-443.  doi: 10.1007/s10444-012-9286-8.  Google Scholar

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G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, The Journal of Physiology, 268 (1977), 177-210.  doi: 10.1113/jphysiol.1977.sp011853.  Google Scholar

[9]

M. Bendahmane and K. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.  doi: 10.3934/nhm.2006.1.185.  Google Scholar

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D. Bers, Cardiac excitation-contraction coupling, Nature, 415 (2002), 198-205.   Google Scholar

[11]

J. Bourgain and H. Brezis, On the equation divy = f and application to control of phases, Journal of the American Mathematical Society, 16 (2003), 393-426.  doi: 10.1090/S0894-0347-02-00411-3.  Google Scholar

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Y. BourgaultY. Coudière and C. Pierre, Existence and uniqeness of the solution for the bidomain model used in cardiac electrophysiology, Nonl. Anal.: Real World Appl., 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.  Google Scholar

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F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

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C. CherubiniS. FilippiP. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects, Progr. Biophys. Molec. Biol., 97 (2008), 562-573.  doi: 10.1016/j.pbiomolbio.2008.02.001.  Google Scholar

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P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, 2002, 49–78.  Google Scholar

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L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998.  Google Scholar

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S. Göktepe and E. Kuhl, Electromechanics of the heart: A unified approach to the strongly coupled excitation–contraction problem, Comput. Mech., 45 (2010), 227-243.  doi: 10.1007/s00466-009-0434-z.  Google Scholar

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P. KrejčíJ. Sainte-MarieM. Sorine and J. Urquiza, Solutions to muscle fiber equations and their long time behaviour, Nonl. Anal.: Real World Appl., 7 (2006), 535-558.  doi: 10.1016/j.nonrwa.2005.03.021.  Google Scholar

[22]

A. LaadhariR. Ruiz-Baier and A. Quarteroni, Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells, Int. J. Numer. Methods Engrg., 96 (2013), 712-738.  doi: 10.1002/nme.4582.  Google Scholar

[23]

P. LafortuneR. ArísM. Vázquez and G. Houzeaux, Coupled electromechanical model of the heart: Parallel finite element formulation, Int. J. Numer. Methods Biomed. Engrg., 28 (2012), 72-86.  doi: 10.1002/cnm.1494.  Google Scholar

[24]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[25]

C.-h. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. simulations of ionic currents and concentration changes, Circulation Research, 74 (1994), 1071-1096.  doi: 10.1161/01.RES.74.6.1071.  Google Scholar

[26]

H. Matano and Y. Mori, Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst., 29 (2011), 1573-1636.  doi: 10.3934/dcds.2011.29.1573.  Google Scholar

[27]

P. Nardinocchi and L. Teresi, On the active response of soft living tissues, J. Elasticity, 88 (2007), 27-39.  doi: 10.1007/s10659-007-9111-7.  Google Scholar

[28]

M. Nash and P. Hunter, Computational mechanics of the heart. From tissue structure to ventricular function, J. Elasticity, 61 (2000), 113-141.  doi: 10.1023/A:1011084330767.  Google Scholar

[29]

M. Nash and A. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progr. Biophys. Molec. Biol., 85 (2004), 501-522.  doi: 10.1016/j.pbiomolbio.2004.01.016.  Google Scholar

[30]

J. Nečas, Sur les normes équivalentes dans $w^k_ p(\omega)$ et sur la coercivité des formes formellement positives, Les presses de l'Université de Montréal. Google Scholar

[31]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2011. Google Scholar

[32]

F. NobileA. Quarteroni and R. Ruiz-Baier, An active strain electromechanical model for cardiac tissue, Int. J. Numer. Meth. Biomed. Engrg., 28 (2012), 52-71.  doi: 10.1002/cnm.1468.  Google Scholar

[33]

D. NordslettenS. NiedererM. NashP. Hunter and N. Smith, Coupling multi-physics models to cardiac mechanics, Progr. Biophys. Molec. Biol., 104 (2011), 77-88.  doi: 10.1016/j.pbiomolbio.2009.11.001.  Google Scholar

[34]

P. PathmanathanS. ChapmanD. Gavaghan and J. Whiteley, Cardiac electromechanics: the effect of contraction model on the mathematical problem and accuracy of the numerical scheme, Quart. J. Mech. Appl. Math., 63 (2010), 375-399.  doi: 10.1093/qjmam/hbq014.  Google Scholar

[35]

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

[36]

P.-A. Raviart, J.-M. Thomas, P. G. Ciarlet and J. L. Lions, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, vol. 2, Dunod Paris, 1998. Google Scholar

[37]

S. RossiT. LassilaR. Ruiz-BaierA. Sequeira and A. Quarteroni, Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics, Eur. J. Mechanics A/Solids, 48 (2014), 129-142.  doi: 10.1016/j.euromechsol.2013.10.009.  Google Scholar

[38]

S. RossiR. Ruiz-BaierL. Pavarino and A. Quarteroni, Orthotropic active strain models for the numerical simulation of cardiac biomechanics, Int. J. Numer. Meth. Biomed. Engrg., 28 (2012), 761-788.  doi: 10.1002/cnm.2473.  Google Scholar

[39]

R. Ruiz-Baier, D. Ambrosi, S. Pezzuto, S. Rossi and A. Quarteroni, Activation models for the numerical simulation of cardiac electromechanical interactions, in Computer Models in Biomechanics: From Nano to Macro (eds. G. Holzapfel and E. Kuhl), Springer-Verlag, Heidelberg, 2013, 189–201. doi: 10.1007/978-94-007-5464-5_14.  Google Scholar

[40]

R. Ruiz-Baier, A. Gizzi, S. Rossi, C. Cherubini, A. Laadhari, S. Filippi and A. Quarteroni, Mathematical modeling of active contraction in isolated cardiomyocytes, Math. Medicine Biol., 31 (2014), 259–283. doi: 10.1093/imammb/dqt009.  Google Scholar

[41]

J. Simon, Démonstration constructive d'un théorème de g. de rham, CR Acad. Sci. Paris Sér. I Math, 316 (1993), 1167–1172.  Google Scholar

[42]

J. Sundnes, G. Lines, X. Cai, B. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart, vol. 1 of Monographs in Computational Science and Engineering, Springer-Verlag, Berlin, 2006.  Google Scholar

[43]

J. SundnesS. WallH. OsnesT. Thorvaldsen and A. McCulloch, Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations, Comput. Meth. Biomech. Biomed. Engrg., 17 (2014), 604-615.  doi: 10.1080/10255842.2012.704368.  Google Scholar

[44]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[45]

N. Trayanova, Whole-heart modeling: Applications to cardiac electrophysiology and electromechanics, Circ. Res., 108 (2011), 113-128.  doi: 10.1161/CIRCRESAHA.110.223610.  Google Scholar

[46]

L. Tung, A Bi-Domain Model for Describing Ischemic Myocardial D–C potentials, PhD thesis, MIT, 1978. Google Scholar

[47]

M. Veneroni, Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonl. Anal.: Real World Appl., 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.  Google Scholar

[48]

X. Wang, A remark on the characterization of the gradient of a distribution, Applicable Analysis, 51 (1993), 35-40.  doi: 10.1080/00036819308840202.  Google Scholar

show all references

References:
[1]

H. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differentail equations, Mathematische Zeitschrift, 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[2]

D. AmbrosiG. ArioliF. Nobile and A. Quarteroni, Electromechanical coupling in cardiac dynamics: The active strain approach, SIAM J. Appl. Math., 71 (2011), 605-621.  doi: 10.1137/100788379.  Google Scholar

[3]

D. Ambrosi and S. Pezzuto, Active stress vs. active strain in mechanobiology: Constitutive issues, J. Elasticity, 107 (2012), 199-212.  doi: 10.1007/s10659-011-9351-4.  Google Scholar

[4]

B. AndreianovM. BendahmaneA. Quarteroni and R. Ruiz-Baier, Solvability analysis and numerical approximation of linearized cardiac electromechanics, Mathematical Models and Methods in Applied Sciences, 25 (2015), 959-993.  doi: 10.1142/S0218202515500244.  Google Scholar

[5]

D. N. Arnold, L. R. Scott and M. Vogelius, Regular inversion of the divergence operator with dirichlet boundary conditions on a polygon. Google Scholar

[6]

J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63 (1976/77), 337-403.  doi: 10.1007/BF00279992.  Google Scholar

[7]

D. BaroliA. Quarteroni and R. Ruiz-Baier, Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity, Adv. Comput. Math., 39 (2013), 425-443.  doi: 10.1007/s10444-012-9286-8.  Google Scholar

[8]

G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, The Journal of Physiology, 268 (1977), 177-210.  doi: 10.1113/jphysiol.1977.sp011853.  Google Scholar

[9]

M. Bendahmane and K. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.  doi: 10.3934/nhm.2006.1.185.  Google Scholar

[10]

D. Bers, Cardiac excitation-contraction coupling, Nature, 415 (2002), 198-205.   Google Scholar

[11]

J. Bourgain and H. Brezis, On the equation divy = f and application to control of phases, Journal of the American Mathematical Society, 16 (2003), 393-426.  doi: 10.1090/S0894-0347-02-00411-3.  Google Scholar

[12]

Y. BourgaultY. Coudière and C. Pierre, Existence and uniqeness of the solution for the bidomain model used in cardiac electrophysiology, Nonl. Anal.: Real World Appl., 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.  Google Scholar

[13]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[14]

C. CherubiniS. FilippiP. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects, Progr. Biophys. Molec. Biol., 97 (2008), 562-573.  doi: 10.1016/j.pbiomolbio.2008.02.001.  Google Scholar

[15]

P. Ciarlet, Mathematical Elasticity, Vol I : Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988.  Google Scholar

[16]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, 2002, 49–78.  Google Scholar

[17]

G. de Rham, Differentiable Manifolds, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-61752-2.  Google Scholar

[18]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 1998.  Google Scholar

[19]

S. Göktepe and E. Kuhl, Electromechanics of the heart: A unified approach to the strongly coupled excitation–contraction problem, Comput. Mech., 45 (2010), 227-243.  doi: 10.1007/s00466-009-0434-z.  Google Scholar

[20]

G. Holzapfel and R. Ogden, Constitutive modelling of passive myocardium: A structurally based framework for material characterization, Phil. Trans. Royal Soc. Lond. A, 367 (2009), 3445-3475.  doi: 10.1098/rsta.2009.0091.  Google Scholar

[21]

P. KrejčíJ. Sainte-MarieM. Sorine and J. Urquiza, Solutions to muscle fiber equations and their long time behaviour, Nonl. Anal.: Real World Appl., 7 (2006), 535-558.  doi: 10.1016/j.nonrwa.2005.03.021.  Google Scholar

[22]

A. LaadhariR. Ruiz-Baier and A. Quarteroni, Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells, Int. J. Numer. Methods Engrg., 96 (2013), 712-738.  doi: 10.1002/nme.4582.  Google Scholar

[23]

P. LafortuneR. ArísM. Vázquez and G. Houzeaux, Coupled electromechanical model of the heart: Parallel finite element formulation, Int. J. Numer. Methods Biomed. Engrg., 28 (2012), 72-86.  doi: 10.1002/cnm.1494.  Google Scholar

[24]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[25]

C.-h. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential. I. simulations of ionic currents and concentration changes, Circulation Research, 74 (1994), 1071-1096.  doi: 10.1161/01.RES.74.6.1071.  Google Scholar

[26]

H. Matano and Y. Mori, Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst., 29 (2011), 1573-1636.  doi: 10.3934/dcds.2011.29.1573.  Google Scholar

[27]

P. Nardinocchi and L. Teresi, On the active response of soft living tissues, J. Elasticity, 88 (2007), 27-39.  doi: 10.1007/s10659-007-9111-7.  Google Scholar

[28]

M. Nash and P. Hunter, Computational mechanics of the heart. From tissue structure to ventricular function, J. Elasticity, 61 (2000), 113-141.  doi: 10.1023/A:1011084330767.  Google Scholar

[29]

M. Nash and A. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progr. Biophys. Molec. Biol., 85 (2004), 501-522.  doi: 10.1016/j.pbiomolbio.2004.01.016.  Google Scholar

[30]

J. Nečas, Sur les normes équivalentes dans $w^k_ p(\omega)$ et sur la coercivité des formes formellement positives, Les presses de l'Université de Montréal. Google Scholar

[31]

J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2011. Google Scholar

[32]

F. NobileA. Quarteroni and R. Ruiz-Baier, An active strain electromechanical model for cardiac tissue, Int. J. Numer. Meth. Biomed. Engrg., 28 (2012), 52-71.  doi: 10.1002/cnm.1468.  Google Scholar

[33]

D. NordslettenS. NiedererM. NashP. Hunter and N. Smith, Coupling multi-physics models to cardiac mechanics, Progr. Biophys. Molec. Biol., 104 (2011), 77-88.  doi: 10.1016/j.pbiomolbio.2009.11.001.  Google Scholar

[34]

P. PathmanathanS. ChapmanD. Gavaghan and J. Whiteley, Cardiac electromechanics: the effect of contraction model on the mathematical problem and accuracy of the numerical scheme, Quart. J. Mech. Appl. Math., 63 (2010), 375-399.  doi: 10.1093/qjmam/hbq014.  Google Scholar

[35]

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

[36]

P.-A. Raviart, J.-M. Thomas, P. G. Ciarlet and J. L. Lions, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, vol. 2, Dunod Paris, 1998. Google Scholar

[37]

S. RossiT. LassilaR. Ruiz-BaierA. Sequeira and A. Quarteroni, Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics, Eur. J. Mechanics A/Solids, 48 (2014), 129-142.  doi: 10.1016/j.euromechsol.2013.10.009.  Google Scholar

[38]

S. RossiR. Ruiz-BaierL. Pavarino and A. Quarteroni, Orthotropic active strain models for the numerical simulation of cardiac biomechanics, Int. J. Numer. Meth. Biomed. Engrg., 28 (2012), 761-788.  doi: 10.1002/cnm.2473.  Google Scholar

[39]

R. Ruiz-Baier, D. Ambrosi, S. Pezzuto, S. Rossi and A. Quarteroni, Activation models for the numerical simulation of cardiac electromechanical interactions, in Computer Models in Biomechanics: From Nano to Macro (eds. G. Holzapfel and E. Kuhl), Springer-Verlag, Heidelberg, 2013, 189–201. doi: 10.1007/978-94-007-5464-5_14.  Google Scholar

[40]

R. Ruiz-Baier, A. Gizzi, S. Rossi, C. Cherubini, A. Laadhari, S. Filippi and A. Quarteroni, Mathematical modeling of active contraction in isolated cardiomyocytes, Math. Medicine Biol., 31 (2014), 259–283. doi: 10.1093/imammb/dqt009.  Google Scholar

[41]

J. Simon, Démonstration constructive d'un théorème de g. de rham, CR Acad. Sci. Paris Sér. I Math, 316 (1993), 1167–1172.  Google Scholar

[42]

J. Sundnes, G. Lines, X. Cai, B. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart, vol. 1 of Monographs in Computational Science and Engineering, Springer-Verlag, Berlin, 2006.  Google Scholar

[43]

J. SundnesS. WallH. OsnesT. Thorvaldsen and A. McCulloch, Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations, Comput. Meth. Biomech. Biomed. Engrg., 17 (2014), 604-615.  doi: 10.1080/10255842.2012.704368.  Google Scholar

[44]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343, American Mathematical Soc., 2001. doi: 10.1090/chel/343.  Google Scholar

[45]

N. Trayanova, Whole-heart modeling: Applications to cardiac electrophysiology and electromechanics, Circ. Res., 108 (2011), 113-128.  doi: 10.1161/CIRCRESAHA.110.223610.  Google Scholar

[46]

L. Tung, A Bi-Domain Model for Describing Ischemic Myocardial D–C potentials, PhD thesis, MIT, 1978. Google Scholar

[47]

M. Veneroni, Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonl. Anal.: Real World Appl., 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.  Google Scholar

[48]

X. Wang, A remark on the characterization of the gradient of a distribution, Applicable Analysis, 51 (1993), 35-40.  doi: 10.1080/00036819308840202.  Google Scholar

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