American Institute of Mathematical Sciences

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

A mathematical model of the Purkinje-Muscle Junctions

Adnane Azzouzi 1, , Yves Coudière 1, , Rodolphe Turpault 1, and  Nejib Zemzemi 2,

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.
Keywords: finite volumes., monodomain equations, Purkinje-Muscle Junction.
Mathematics Subject Classification: Primary: 35Q92; Secondary: 65M08, 92C9.
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

show all references

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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