DAD characterization in electromechanical cardiac models

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July 2013, 33(7): 2651-2665. doi: 10.3934/dcds.2013.33.2651

DAD characterization in electromechanical cardiac models

Paolo Biscari ^1, and Chiara Lelli ^1,

1.	Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy, Italy

Received April 2012 Revised November 2012 Published January 2013

Abstract
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We investigate the possibility of modeling the delayed afterdepolarization (DAD) occurrence in the framework of the classical FitzHugh-Nagumo (FN) dynamical system, as well as in more recent electromechanically-coupled cardiac models. Within the FN model, we identify the domain in the constitutive parameters' space for which orbits exist which exhibit a sufficiently strong secondary impulse. We then address the question whether a locally-induced secondary pulse succeeds or not in originating a self-propagating traveling impulse. Our results evidence that, in the range where secondary impulses exceed the physiological threshold for DAD onset, a local impulse almost certainly causes a traveling impulse (mechanism known as all-or-none). We then consider a recently proposed electromechanically-coupled generalization of the FN model, and show that the mechanical coupling stabilizes the system, in the sense that the more strong the coupling, the less likely is DAD to occur.

Keywords: Electromechanical cardiac models, spike sequence..

Mathematics Subject Classification: Primary: 37N25; Secondary: 92C10, 92C2.

Citation: Paolo Biscari, Chiara Lelli. DAD characterization in electromechanical cardiac models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2651-2665. doi: 10.3934/dcds.2013.33.2651

References:

[1]	C. Antzelevitch and S. Sicouri, Clinical relevance of cardiac arrhythmias generated by afterdepolarizations: Role of M cells in the generation of U waves, triggered activity and torsade de pointes, J. Am. Coll. Cardiol., 23 (1994), 259-277.
[2]	J. Mészáros, D. Khananshvili and G. Hart, Mechanisms underlying delayed afterdepolarizations in hypertrophied left ventricular myocytes of rats, Am. J. Physiol.-Heart. C., 281 (2001), H903-H914.
[3]	D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model, Biophys. J., 68 (1995), 1752-1766.
[4]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
[5]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
[6]	J. Keener and J. Sneyd, "Mathematical Physiology," Ed. Springer Verlag, Berlin, 1998.
[7]	P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model, Eur. Phys. J. Plus, 126 (2011), 1-9.
[8]	A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system, SIAM J. Appl. Math., 63 (2002), 459-484. doi: 10.1137/S0036139901393500.
[9]	K. Schlotthauer and D. M. Bers, Sarcoplasmic reticulum $Ca^{2+}$ release causes myocyte depolarization: Underlying mechanism and threshold for triggered action potentials, Circ. Res., 87 (2000), 774-780.
[10]	Y. Xie, D. Sato, A. Garfinkel, Z. Qu and J. Weiss, So little source, so much sink: Requirements for afterdepolarizations to propagate in tissue, Biophys. J., 99 (2010), 1408-1415.
[11]	C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential II. Afterdepolarization, triggered activity, and potentiation, Circ. Res., 74 (1994), 1097-1113.
[12]	N. A. Wedge, M. S. Branicky and M. C. Cavusoglu, Proc. $26^{th}$ Int. Conf. IEEE engineering in medicine and biology society, Ed. IEEE, New York, (2004), 3027-3030.
[13]	M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Prog. Biophys. Mol. Bio., 85 (2004), 501-522.
[14]	C. Lelli, "Attraction Basin of the Equilibrium Configuration in the FitzHugh-Nagumo Model," Acta Appl. Math., 2012. doi: 10.1007/s10440-012-9744-9.
[15]	M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations, J. Math. Biol., 1 (1974), 153-163.
[16]	C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models," Ph.D. Thesis, Politecnico di Milano, 2012.
[17]	S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260. doi: 10.1137/0142018.
[18]	Y. M. Usachev and S. A. Thayer, All-or-None $Ca^{2+}$ release from intracellular stores triggered by $Ca^{2+}$ influx through voltage-gated $Ca^{2+}$ channels in rat sensory neurons, J. Neuosci., 17 (1997), 7404-7414.
[19]	C. Cherubini, S. Filippi, P. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects, Progr. Biophys. Molec. Biol., 97 (2008), 562-573.
[20]	D. Ambrosi, G. Arioli, F. 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.
[21]	J. M. Rogers and A. D. McCulloch, A Collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng., 41 (1994), 743-757.
[22]	R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos Sol. Fract., 7 (1996), 293-301.
[23]	J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophys. J., 13 (1973), 1313-1337.
[24]	M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua," Ed. Cambridge University Press, 2010.
[25]	S. M. Pogwizd, K. Schlotthauer, L. Li, W. Yuan and D. M. Bers, Arrhythmogenesis and contractile dysfunction in heart failure: Roles of sodium-calcium exchange, inward rectifier potassium current, and residual $\beta$-adrenergic responsiveness, Circ. Res., 88 (2001), 1159-1167.
[26]	N. P. Smith, D. P. Nickerson, E. J. Crampin and P. J. Hunter, Multiscale computational modelling of the heart, Acta Numer., 13 (2004), 371-431. doi: 10.1017/S0962492904000200.

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References:

[1]	C. Antzelevitch and S. Sicouri, Clinical relevance of cardiac arrhythmias generated by afterdepolarizations: Role of M cells in the generation of U waves, triggered activity and torsade de pointes, J. Am. Coll. Cardiol., 23 (1994), 259-277.
[2]	J. Mészáros, D. Khananshvili and G. Hart, Mechanisms underlying delayed afterdepolarizations in hypertrophied left ventricular myocytes of rats, Am. J. Physiol.-Heart. C., 281 (2001), H903-H914.
[3]	D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model, Biophys. J., 68 (1995), 1752-1766.
[4]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
[5]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.
[6]	J. Keener and J. Sneyd, "Mathematical Physiology," Ed. Springer Verlag, Berlin, 1998.
[7]	P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model, Eur. Phys. J. Plus, 126 (2011), 1-9.
[8]	A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system, SIAM J. Appl. Math., 63 (2002), 459-484. doi: 10.1137/S0036139901393500.
[9]	K. Schlotthauer and D. M. Bers, Sarcoplasmic reticulum $Ca^{2+}$ release causes myocyte depolarization: Underlying mechanism and threshold for triggered action potentials, Circ. Res., 87 (2000), 774-780.
[10]	Y. Xie, D. Sato, A. Garfinkel, Z. Qu and J. Weiss, So little source, so much sink: Requirements for afterdepolarizations to propagate in tissue, Biophys. J., 99 (2010), 1408-1415.
[11]	C. H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential II. Afterdepolarization, triggered activity, and potentiation, Circ. Res., 74 (1994), 1097-1113.
[12]	N. A. Wedge, M. S. Branicky and M. C. Cavusoglu, Proc. $26^{th}$ Int. Conf. IEEE engineering in medicine and biology society, Ed. IEEE, New York, (2004), 3027-3030.
[13]	M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Prog. Biophys. Mol. Bio., 85 (2004), 501-522.
[14]	C. Lelli, "Attraction Basin of the Equilibrium Configuration in the FitzHugh-Nagumo Model," Acta Appl. Math., 2012. doi: 10.1007/s10440-012-9744-9.
[15]	M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations, J. Math. Biol., 1 (1974), 153-163.
[16]	C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models," Ph.D. Thesis, Politecnico di Milano, 2012.
[17]	S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247-260. doi: 10.1137/0142018.
[18]	Y. M. Usachev and S. A. Thayer, All-or-None $Ca^{2+}$ release from intracellular stores triggered by $Ca^{2+}$ influx through voltage-gated $Ca^{2+}$ channels in rat sensory neurons, J. Neuosci., 17 (1997), 7404-7414.
[19]	C. Cherubini, S. Filippi, P. Nardinocchi and L. Teresi, An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects, Progr. Biophys. Molec. Biol., 97 (2008), 562-573.
[20]	D. Ambrosi, G. Arioli, F. 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.
[21]	J. M. Rogers and A. D. McCulloch, A Collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng., 41 (1994), 743-757.
[22]	R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos Sol. Fract., 7 (1996), 293-301.
[23]	J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophys. J., 13 (1973), 1313-1337.
[24]	M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua," Ed. Cambridge University Press, 2010.
[25]	S. M. Pogwizd, K. Schlotthauer, L. Li, W. Yuan and D. M. Bers, Arrhythmogenesis and contractile dysfunction in heart failure: Roles of sodium-calcium exchange, inward rectifier potassium current, and residual $\beta$-adrenergic responsiveness, Circ. Res., 88 (2001), 1159-1167.
[26]	N. P. Smith, D. P. Nickerson, E. J. Crampin and P. J. Hunter, Multiscale computational modelling of the heart, Acta Numer., 13 (2004), 371-431. doi: 10.1017/S0962492904000200.

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