# American Institute of Mathematical Sciences

2011, 8(4): 915-930. doi: 10.3934/mbe.2011.8.915

## A mathematical model of the Purkinje-Muscle Junctions

 1 Université de Nantes, Laboratoire de Mathématiques Jean Leray, Nantes, France, France, France 2 INRIA, REO team, Rocquencourt, France

Received  October 2010 Revised  May 2011 Published  August 2011

This paper is devoted to the construction of a mathematical model of the His-Purkinje tree and the Purkinje-Muscle Junctions (PMJ). A simple numerical scheme is proposed in order to perform some simple numerical experiments.
Citation: Adnane Azzouzi, Yves Coudière, Rodolphe Turpault, Nejib Zemzemi. A mathematical model of the Purkinje-Muscle Junctions. Mathematical Biosciences & Engineering, 2011, 8 (4) : 915-930. doi: 10.3934/mbe.2011.8.915
##### References:
 [1] A. V. Aslanidi, P. Stewart, M. R. Boyett and H. Zhang, Optimal velocity and safety of discontinuous conduction through the heterogeneous purkinje-ventricular junction,, Biophysical Journal, (2009). Google Scholar [2] G. Beeler and H. Reuter., Reconstruction of the action potential of ventricular myocardial fibres,, J. Physiol. (Lond.), 268 (1977), 177. Google Scholar [3] O. Berenfeld and J. Jalife, Purkinje-muscle rentry as a mechanism of polymorphic ventricular arrhytmias in a 3-dimensional model of the ventricles,, Circ. Res., 82 (1998), 1063. Google Scholar [4] M. Boulakia, S. Cazeau, M. A. Fernández, J. F. Gerbeau and N. Zemzemi, Mathematical modeling of electrocardiograms: A numerical study,, Annals of Biomedical Engineering, 38 (2010), 1071. Google Scholar [5] M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms,, in, (2007), 240. Google Scholar [6] P. M. Boyle, M. Deo, G. Plank and E. J. Vigmond, Purkinje-mdeiated effects in the response of quiescent ventricles to defibrillation shocks,, Annals of Biomedical Engineering, (2009). Google Scholar [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", North-Holland, (1978). Google Scholar [8] D. DiFrancesco and D. Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes,, Phil. Trans. R. Soc. Lond., 307 (1985), 353. doi: 10.1098/rstb.1985.0001. Google Scholar [9] Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization,", Academic Press Inc., (1981). Google Scholar [10] C. S. Henriquez, A. L. Muzikant and C.K. Smoak, Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: Simulations in a three-dimensional bidomain model,, Journal of Cardiovascular Electrophysiology, 7 (1996), 424. Google Scholar [11] B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation,, in, (2001), 39. Google Scholar [12] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998). Google Scholar [13] M. Potse, B. Dube, J. Richer, A. Vinet and R. M. Gulrajani, A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart,, IEEE Transactions on Biomedical Engineering, (2006). Google Scholar [14] J. C. Neu and W. Krassowska, Homogenization of syncytial tissues,, Crit. Rev. Biomed. Eng., 21 (1993), 137. Google Scholar [15] A. E. Pollard and R. C. Barr, The construction of an anatomically based model of the human ventricular conduction system,, IEEE Transaction on Biomedical Engineering, 37 (1990), 1173. doi: 10.1109/10.64460. Google Scholar [16] W. B. Ross, S. M. Mohiuddin, T. Pagano and D. Hughes, Malposition of a transvenous cardiac electrode associated with amaurosis fugax,, Pacing and Clinical Electrophysiology, 6 (1983), 119. doi: 10.1111/j.1540-8159.1983.tb06589.x. Google Scholar [17] W. A. Schiavone, L. O. N. W. Castle, E. Salcedo and R. Graor, Amaurosis fugax in a patient with a left ventricular endocardial pacemaker,, Pacing and Clinical Electrophysiology, 7 (1984), 288. doi: 10.1111/j.1540-8159.1984.tb04901.x. Google Scholar [18] K. H. W. J. Ten Tusscher and A. V. Panfilov, Modelling of the ventricular conduction system,, Progress in Biophysics and Molecular Biology, (2008). Google Scholar [19] E. J. Vigmond and C. Clements, Construction of a computer model to investigate sawtooth efects in the purkinje system,, IEEE Transaction on Biomedical Engineering, 54 (2007). doi: 10.1109/TBME.2006.888817. Google Scholar

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##### References:
 [1] A. V. Aslanidi, P. Stewart, M. R. Boyett and H. Zhang, Optimal velocity and safety of discontinuous conduction through the heterogeneous purkinje-ventricular junction,, Biophysical Journal, (2009). Google Scholar [2] G. Beeler and H. Reuter., Reconstruction of the action potential of ventricular myocardial fibres,, J. Physiol. (Lond.), 268 (1977), 177. Google Scholar [3] O. Berenfeld and J. Jalife, Purkinje-muscle rentry as a mechanism of polymorphic ventricular arrhytmias in a 3-dimensional model of the ventricles,, Circ. Res., 82 (1998), 1063. Google Scholar [4] M. Boulakia, S. Cazeau, M. A. Fernández, J. F. Gerbeau and N. Zemzemi, Mathematical modeling of electrocardiograms: A numerical study,, Annals of Biomedical Engineering, 38 (2010), 1071. Google Scholar [5] M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms,, in, (2007), 240. Google Scholar [6] P. M. Boyle, M. Deo, G. Plank and E. J. Vigmond, Purkinje-mdeiated effects in the response of quiescent ventricles to defibrillation shocks,, Annals of Biomedical Engineering, (2009). Google Scholar [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", North-Holland, (1978). Google Scholar [8] D. DiFrancesco and D. Noble, A model of cardiac electrical activity incorporating ionic pumps and concentration changes,, Phil. Trans. R. Soc. Lond., 307 (1985), 353. doi: 10.1098/rstb.1985.0001. Google Scholar [9] Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization,", Academic Press Inc., (1981). Google Scholar [10] C. S. Henriquez, A. L. Muzikant and C.K. Smoak, Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: Simulations in a three-dimensional bidomain model,, Journal of Cardiovascular Electrophysiology, 7 (1996), 424. Google Scholar [11] B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation,, in, (2001), 39. Google Scholar [12] J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (1998). Google Scholar [13] M. Potse, B. Dube, J. Richer, A. Vinet and R. M. Gulrajani, A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart,, IEEE Transactions on Biomedical Engineering, (2006). Google Scholar [14] J. C. Neu and W. Krassowska, Homogenization of syncytial tissues,, Crit. Rev. Biomed. Eng., 21 (1993), 137. Google Scholar [15] A. E. Pollard and R. C. Barr, The construction of an anatomically based model of the human ventricular conduction system,, IEEE Transaction on Biomedical Engineering, 37 (1990), 1173. doi: 10.1109/10.64460. Google Scholar [16] W. B. Ross, S. M. Mohiuddin, T. Pagano and D. Hughes, Malposition of a transvenous cardiac electrode associated with amaurosis fugax,, Pacing and Clinical Electrophysiology, 6 (1983), 119. doi: 10.1111/j.1540-8159.1983.tb06589.x. Google Scholar [17] W. A. Schiavone, L. O. N. W. Castle, E. Salcedo and R. Graor, Amaurosis fugax in a patient with a left ventricular endocardial pacemaker,, Pacing and Clinical Electrophysiology, 7 (1984), 288. doi: 10.1111/j.1540-8159.1984.tb04901.x. Google Scholar [18] K. H. W. J. Ten Tusscher and A. V. Panfilov, Modelling of the ventricular conduction system,, Progress in Biophysics and Molecular Biology, (2008). Google Scholar [19] E. J. Vigmond and C. Clements, Construction of a computer model to investigate sawtooth efects in the purkinje system,, IEEE Transaction on Biomedical Engineering, 54 (2007). doi: 10.1109/TBME.2006.888817. Google Scholar
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