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

DAD characterization in electromechanical cardiac models

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

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.
Citation: Paolo Biscari, Chiara Lelli. DAD characterization in electromechanical cardiac models. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[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). Google Scholar

[3]

D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model,, Biophys. J., 68 (1995), 1752. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. Google Scholar

[5]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar

[6]

J. Keener and J. Sneyd, "Mathematical Physiology,", Ed. Springer Verlag, (1998). Google Scholar

[7]

P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model,, Eur. Phys. J. Plus, 126 (2011), 1. Google Scholar

[8]

A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system,, SIAM J. Appl. Math., 63 (2002), 459. doi: 10.1137/S0036139901393500. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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, (2004), 3027. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations,, J. Math. Biol., 1 (1974), 153. Google Scholar

[16]

C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models,", Ph.D. Thesis, (2012). Google Scholar

[17]

S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations,, SIAM J. Appl. Math., 42 (1982), 247. doi: 10.1137/0142018. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/100788379. Google Scholar

[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. Google Scholar

[22]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation,, Chaos Sol. Fract., 7 (1996), 293. Google Scholar

[23]

J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation,, Biophys. J., 13 (1973), 1313. Google Scholar

[24]

M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua,", Ed. Cambridge University Press, (2010). Google Scholar

[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. Google Scholar

[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. doi: 10.1017/S0962492904000200. Google Scholar

show all references

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. Google Scholar

[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). Google Scholar

[3]

D. D. Friel, $[Ca^{2+}]_i$ oscillations in symphathetic neurons: An experimental test of a theoretical model,, Biophys. J., 68 (1995), 1752. Google Scholar

[4]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445. Google Scholar

[5]

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon,, Proc. IRE, 50 (1962), 2061. Google Scholar

[6]

J. Keener and J. Sneyd, "Mathematical Physiology,", Ed. Springer Verlag, (1998). Google Scholar

[7]

P. Biscari and C. Lelli, Spike transitions in the FitzHugh-Nagumo model,, Eur. Phys. J. Plus, 126 (2011), 1. Google Scholar

[8]

A. Tonnelier, The McKean's caricature of the Fitzhugh-Nagumo model I. The space-clamped system,, SIAM J. Appl. Math., 63 (2002), 459. doi: 10.1137/S0036139901393500. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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, (2004), 3027. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

M. W. Green and B. D. Sleeman, On FitzHugh's nerve axon equations,, J. Math. Biol., 1 (1974), 153. Google Scholar

[16]

C. Lelli, "Characterization of Delayed After-Depolarization in Extended FitzHugh-Nagumo Models,", Ph.D. Thesis, (2012). Google Scholar

[17]

S. P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations,, SIAM J. Appl. Math., 42 (1982), 247. doi: 10.1137/0142018. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/100788379. Google Scholar

[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. Google Scholar

[22]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation,, Chaos Sol. Fract., 7 (1996), 293. Google Scholar

[23]

J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation,, Biophys. J., 13 (1973), 1313. Google Scholar

[24]

M. E. Gurtin, E. Fried and L. Anand, "The Mechanics and Thermodynamics of Continua,", Ed. Cambridge University Press, (2010). Google Scholar

[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. Google Scholar

[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. doi: 10.1017/S0962492904000200. Google Scholar

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