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.

[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.

[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.

[5]

M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms,, in, (2007), 240.

[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. doi: 10.1098/rstb.1985.0001.

[9]

Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization,", Academic Press Inc., (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.

[11]

B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation,, in, (2001), 39.

[12]

J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (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.

[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.

[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.

[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.

[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.

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).

[2]

G. Beeler and H. Reuter., Reconstruction of the action potential of ventricular myocardial fibres,, J. Physiol. (Lond.), 268 (1977), 177.

[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.

[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.

[5]

M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms,, in, (2007), 240.

[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. doi: 10.1098/rstb.1985.0001.

[9]

Philip E. Gill, Walter Murray and Margaret H. Wright, "Practical Optimization,", Academic Press Inc., (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.

[11]

B. Milan Horacek, K. Simelius, R. Hren and J. Nenonen, Challenges in modelling human heart's total excitation,, in, (2001), 39.

[12]

J. Keener and J. Sneyd, "Mathematical Physiology,", Springer, (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.

[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.

[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.

[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.

[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.

[1]

Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013

[2]

Anna Kaźmierczak, Jan Sokolowski, Antoni Zochowski. Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes. Mathematical Control & Related Fields, 2018, 8 (1) : 89-115. doi: 10.3934/mcrf.2018004

[3]

Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411

[4]

Daniel Guan. Classification of compact complex homogeneous spaces with invariant volumes. Electronic Research Announcements, 1997, 3: 90-92.

[5]

Vitali Milman, Liran Rotem. $\alpha$-concave functions and a functional extension of mixed volumes. Electronic Research Announcements, 2013, 20: 1-11. doi: 10.3934/era.2013.20.1

[6]

Mauro Garavello. The LWR traffic model at a junction with multibuffers. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 463-482. doi: 10.3934/dcdss.2014.7.463

[7]

Daniela Calvetti, Jenni Heino, Erkki Somersalo, Knarik Tunyan. Bayesian stationary state flux balance analysis for a skeletal muscle metabolic model. Inverse Problems & Imaging, 2007, 1 (2) : 247-263. doi: 10.3934/ipi.2007.1.247

[8]

Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243

[9]

Rinaldo M. Colombo, Mauro Garavello. A Well Posed Riemann Problem for the $p$--System at a Junction. Networks & Heterogeneous Media, 2006, 1 (3) : 495-511. doi: 10.3934/nhm.2006.1.495

[10]

Tomasz Dobrowolski. The dynamics of the kink in curved large area Josephson junction. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1095-1105. doi: 10.3934/dcdss.2011.4.1095

[11]

Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149

[12]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225

[13]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[14]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104

[15]

Rinaldo M. Colombo, Mauro Garavello. Comparison among different notions of solution for the $p$-system at a junction. Conference Publications, 2009, 2009 (Special) : 181-190. doi: 10.3934/proc.2009.2009.181

[16]

Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks & Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111

[17]

Anne-Laure Bessoud. A variational convergence for bifunctionals. Application to a model of strong junction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 399-417. doi: 10.3934/dcdss.2012.5.399

[18]

Gunhild A. Reigstad. Numerical network models and entropy principles for isothermal junction flow. Networks & Heterogeneous Media, 2014, 9 (1) : 65-95. doi: 10.3934/nhm.2014.9.65

[19]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

[20]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (0)

[Back to Top]