# 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. [2] G. Beeler and H. Reuter., Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol. (Lond.), 268 (1977), 177-210. [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), 1063-1077. [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), 1071-1097. [5] M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms, in "Functional Imaging and Modeling of the Heart" (eds. F. B. Sachse and G. Seemann), Lecture Notes in Computer Science, Springer-Verlag, (2007), 240-249. [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. [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," North-Holland, 1978. [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), 353-398. doi: 10.1098/rstb.1985.0001. [9] Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization," Academic Press Inc., Harcourt Brace Jovanovich Publishers, London, 1981. [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), 424-44. [11] B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation, in "Functional Imaging and Modeling of the Heart," LNCS, (2001), 39-46. [12] J. Keener and J. Sneyd, "Mathematical Physiology," Springer, New York, 1998. [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. [14] J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199. [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), 1173-1185. doi: 10.1109/10.64460. [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), 119-123. doi: 10.1111/j.1540-8159.1983.tb06589.x. [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), 288-292. doi: 10.1111/j.1540-8159.1984.tb04901.x. [18] K. H. W. J. Ten Tusscher and A. V. Panfilov, Modelling of the ventricular conduction system, Progress in Biophysics and Molecular Biology, 2008. [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.

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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. [2] G. Beeler and H. Reuter., Reconstruction of the action potential of ventricular myocardial fibres, J. Physiol. (Lond.), 268 (1977), 177-210. [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), 1063-1077. [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), 1071-1097. [5] M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms, in "Functional Imaging and Modeling of the Heart" (eds. F. B. Sachse and G. Seemann), Lecture Notes in Computer Science, Springer-Verlag, (2007), 240-249. [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. [7] P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," North-Holland, 1978. [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), 353-398. doi: 10.1098/rstb.1985.0001. [9] Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization," Academic Press Inc., Harcourt Brace Jovanovich Publishers, London, 1981. [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), 424-44. [11] B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation, in "Functional Imaging and Modeling of the Heart," LNCS, (2001), 39-46. [12] J. Keener and J. Sneyd, "Mathematical Physiology," Springer, New York, 1998. [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. [14] J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199. [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), 1173-1185. doi: 10.1109/10.64460. [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), 119-123. doi: 10.1111/j.1540-8159.1983.tb06589.x. [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), 288-292. doi: 10.1111/j.1540-8159.1984.tb04901.x. [18] K. H. W. J. Ten Tusscher and A. V. Panfilov, Modelling of the ventricular conduction system, Progress in Biophysics and Molecular Biology, 2008. [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.
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