American Institute of Mathematical Sciences

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2016, 13(6): 1143-1158. doi: 10.3934/mbe.2016035

Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology

Beata Jackowska-Zduniak 1, and  Urszula Foryś 2,

1. 

Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences, Nowoursynowska 159, 02-776 Warsaw, Poland
2. 

Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw

Received  October 2015 Revised  May 2016 Published  August 2016

A proposed model consisting of two coupled models (Hodgkin-Huxley and Yanagihara-Noma-Irisawa model) is considered as a description of the heart's action potential. System of ordinary differential equations is used to recreate pathological behaviour in the conducting heart's system such as double fire and the most common tachycardia: atrioventricular nodal reentrant tachycardia (AVNRT). Part of the population has an abnormal accessory pathways: fast and slow (Fujiki, 2008). These pathways in the atrioventricular node (AV node) are anatomical and functional contributions of supraventricular tachycardia. However, the appearance of two pathways in the AV node may be a contribution of arrhythmia, which is caused by coexistent conduction by two pathways (called double fire). The difference in the conduction time between these pathways is the most important factor. This is the reason to introduce three types of couplings and delay to our system in order to reproduce various types of the AVNRT. In our research, introducing the feedback loops and couplings entails the creation of waves which can correspond to the re-entry waves occurring in the AVNRT. Our main aim is to study solutions of the given equations and take into consideration the influence of feedback and delays which occur in these pathological modes. We also present stability analysis for both components, that is Hodgkin-Huxley and Yanagihara-Noma-Irisawa models, as well as for the final double-fire model.
Keywords: couplings, ordinary differential equations with delay, feedback, AVNRT., action potential, Hodgkin-Huxley model, double-fire pathology.
Mathematics Subject Classification: Primary: 37N25, 92C30; Secondary: 70K20, 70K5.
Citation: Beata Jackowska-Zduniak, Urszula Foryś. Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1143-1158. doi: 10.3934/mbe.2016035
References:
[1]

D. A. Aabby, Comparatitative Study of Numerical Methods for the Hodgkin-Huxley model of Nerve Cell Action Potentials, U.o. Dayton, Editor, 2009.

[2]

M. A. Akbarzadeh, A. F. Fazelifar and N. B. Bafruee, A case of dual atrioventricular nodal nonreentrant tachycardia: An unusual cause of tachycardia-induced cardiomyopathy, Journal of Arrhythmia, 31 (2015), 328-330. doi: 10.1016/j.joa.2015.04.008.

[3]

A. Borisyuk and J. Rinzel, Understanding neuronal dynamics by geometrical dissection of minimal models, Models and Methods in Neurophysics, Proc Les Houches Summer School, 80 (2005), 17-19, 21-72. doi: 10.1016/S0924-8099(05)80008-3.

[4]

B. Dąbrowska and P. Gajewski, Postępowanie u chorych z nadkomorowymi zaburzeniami rytmu Wytyczne American College of Cardiology, American Heart Association European Society of Cardiology, Medycyna Praktyczna, 6 (2004) (in Polish).

[5]

S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, Springer, Tokyo, 2010. doi: 10.1007/978-4-431-53862-2.

[6]

R. Evertz, F. Merschon, A. Berruezo and L. Mont, Dual ventricular response: Another road to supraventricular tachycardia in dual atrioventricular nodal physiology, Rev Esp Cardiol., 66 (2013), 145-156. doi: 10.1016/j.rec.2012.05.016.

[7]

U. Foryś, Biological delay systems and the Mikhailov Criterion of stability, J. Biological Systems, 12 (2004), 45-60.

[8]

R. A. Freedman and J. W. Mason, Sustained ventricular tachycardia, clinical aspects, Cardiac pacing and electrophysiology, 1991.

[9]

A. Fujiki et al., Junctional rhythm associated with ventriculoatrial block during slow pathway ablation in atypical atrioventricular nodal re-entrant tachycardia, Europace, 10 (2008), 928-987.

[10]

A. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952).

[11]

M. Jastrzębski and P. Kukla, Tachycardia caused by a double fire- simultaneous double atrioventricular nodal conduction: a rare or underdiagnosed arrhythmia. Spectrum of electrocardiographic pictures in three patients, Kardiologia Polska, 67 (2009), 77-84.

[12]

A. Karnik, K. Hematpour, A. Bhatt and M. Mazzini, Dual AV nodal nonreentrant tachycardia resulting in inappropriate icd therapy in a patient with cardiac sarcoidosis, Indian Pacing Electrophysiol. J., 14 (2014), 44-48. doi: 10.1016/S0972-6292(16)30715-X.

[13]

D. G. Katritsis and M. E. Josephson, Classification of electrophysiological types of atrioventricular nodal re-entrant tachycardia: A reappraisal, Europace, 15 (2013), 1231-1240. doi: 10.1093/europace/eut100.

[14]

J. Keener and J. Sneyd, Mathematical Physiology. Systems Physiology, $2^{nd}$ edition, Springer, New York, 2009. doi: 10.1007/978-0-387-75847-3.

[15]

S. Konturek, Fizjologia człowieka. Układ krążenia, Wydawnictwo Uniwersytetu Jagielońskiego, 2001, (in Polish).

[16]

K. Małaczyńska and K. Błaszczyk, Atrioventricular nodal reentrant tachycardia, Polski Przegląd Kardiologiczny, 14 (2012), 196-203.

[17]

S. Masoli, S. Solinas and E. D'Angelo, Action potential processing in a detailed Purkinje cell model reveals a critical role for axonal compartmentalization, Front. Cell. Neurosci., 9 (2015), 1-21. doi: 10.3389/fncel.2015.00047.

[18]

P. Podziemski and J. J. .Zebrowski, A simple model of the right atrium of the human heart with the sinoatrial and atrioventricular nodes included, J Clin Monit Comput., 27 (2013), 481-498. doi: 10.1007/s10877-013-9429-6.

[19]

W. G. Stevenson, Exploring postinforction reentrant ventrivular tachycardia with entertainment mapping, J. Am. Coll. Cardiol., 29 (1997).

[20]

J. Wang, L. Chen and X. Fei, Analysis and control of the bifurcation Hodgkin-Huxley model, Chaos, Solitons and Fractals, 31 (2007), 247-256. doi: 10.1016/j.chaos.2005.09.060.

[21]

D. Wu, P. Denes, R. Dhingra and R. Pietras, New manifestation of dual AV nodal pathways, Eur J Cardiol., 2 (1975), 459-466.

[22]

K. Yanagihara, A. Noma and H. Irisawa, Reconstruction of sino-atrial node pacemaker potential based on the voltage clamp experiments, Japanese Journal of Physiology, 30 (1980), 841-857. doi: 10.2170/jjphysiol.30.841.

[23]

B. Zduniak, M. Bodnar and U. Foryś, A modified van der Pol equation with delay in a description of the heart action, Int. J. Appl. Math. Comput. Sci., 24 (2014), 853-863. doi: 10.2478/amcs-2014-0063.

[24]

Y. Zhang, K. Wang, Y. Yuan, D. Sui and H. Zhang, Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley model, Computational and Mathematical Methods in Medicine, (2014), Art. ID 761907, 9 pp. doi: 10.1155/2014/761907.

show all references

References:
[1]

D. A. Aabby, Comparatitative Study of Numerical Methods for the Hodgkin-Huxley model of Nerve Cell Action Potentials, U.o. Dayton, Editor, 2009.

[2]

M. A. Akbarzadeh, A. F. Fazelifar and N. B. Bafruee, A case of dual atrioventricular nodal nonreentrant tachycardia: An unusual cause of tachycardia-induced cardiomyopathy, Journal of Arrhythmia, 31 (2015), 328-330. doi: 10.1016/j.joa.2015.04.008.

[3]

A. Borisyuk and J. Rinzel, Understanding neuronal dynamics by geometrical dissection of minimal models, Models and Methods in Neurophysics, Proc Les Houches Summer School, 80 (2005), 17-19, 21-72. doi: 10.1016/S0924-8099(05)80008-3.

[4]

B. Dąbrowska and P. Gajewski, Postępowanie u chorych z nadkomorowymi zaburzeniami rytmu Wytyczne American College of Cardiology, American Heart Association European Society of Cardiology, Medycyna Praktyczna, 6 (2004) (in Polish).

[5]

S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, Springer, Tokyo, 2010. doi: 10.1007/978-4-431-53862-2.

[6]

R. Evertz, F. Merschon, A. Berruezo and L. Mont, Dual ventricular response: Another road to supraventricular tachycardia in dual atrioventricular nodal physiology, Rev Esp Cardiol., 66 (2013), 145-156. doi: 10.1016/j.rec.2012.05.016.

[7]

U. Foryś, Biological delay systems and the Mikhailov Criterion of stability, J. Biological Systems, 12 (2004), 45-60.

[8]

R. A. Freedman and J. W. Mason, Sustained ventricular tachycardia, clinical aspects, Cardiac pacing and electrophysiology, 1991.

[9]

A. Fujiki et al., Junctional rhythm associated with ventriculoatrial block during slow pathway ablation in atypical atrioventricular nodal re-entrant tachycardia, Europace, 10 (2008), 928-987.

[10]

A. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952).

[11]

M. Jastrzębski and P. Kukla, Tachycardia caused by a double fire- simultaneous double atrioventricular nodal conduction: a rare or underdiagnosed arrhythmia. Spectrum of electrocardiographic pictures in three patients, Kardiologia Polska, 67 (2009), 77-84.

[12]

A. Karnik, K. Hematpour, A. Bhatt and M. Mazzini, Dual AV nodal nonreentrant tachycardia resulting in inappropriate icd therapy in a patient with cardiac sarcoidosis, Indian Pacing Electrophysiol. J., 14 (2014), 44-48. doi: 10.1016/S0972-6292(16)30715-X.

[13]

D. G. Katritsis and M. E. Josephson, Classification of electrophysiological types of atrioventricular nodal re-entrant tachycardia: A reappraisal, Europace, 15 (2013), 1231-1240. doi: 10.1093/europace/eut100.

[14]

J. Keener and J. Sneyd, Mathematical Physiology. Systems Physiology, $2^{nd}$ edition, Springer, New York, 2009. doi: 10.1007/978-0-387-75847-3.

[15]

S. Konturek, Fizjologia człowieka. Układ krążenia, Wydawnictwo Uniwersytetu Jagielońskiego, 2001, (in Polish).

[16]

K. Małaczyńska and K. Błaszczyk, Atrioventricular nodal reentrant tachycardia, Polski Przegląd Kardiologiczny, 14 (2012), 196-203.

[17]

S. Masoli, S. Solinas and E. D'Angelo, Action potential processing in a detailed Purkinje cell model reveals a critical role for axonal compartmentalization, Front. Cell. Neurosci., 9 (2015), 1-21. doi: 10.3389/fncel.2015.00047.

[18]

P. Podziemski and J. J. .Zebrowski, A simple model of the right atrium of the human heart with the sinoatrial and atrioventricular nodes included, J Clin Monit Comput., 27 (2013), 481-498. doi: 10.1007/s10877-013-9429-6.

[19]

W. G. Stevenson, Exploring postinforction reentrant ventrivular tachycardia with entertainment mapping, J. Am. Coll. Cardiol., 29 (1997).

[20]

J. Wang, L. Chen and X. Fei, Analysis and control of the bifurcation Hodgkin-Huxley model, Chaos, Solitons and Fractals, 31 (2007), 247-256. doi: 10.1016/j.chaos.2005.09.060.

[21]

D. Wu, P. Denes, R. Dhingra and R. Pietras, New manifestation of dual AV nodal pathways, Eur J Cardiol., 2 (1975), 459-466.

[22]

K. Yanagihara, A. Noma and H. Irisawa, Reconstruction of sino-atrial node pacemaker potential based on the voltage clamp experiments, Japanese Journal of Physiology, 30 (1980), 841-857. doi: 10.2170/jjphysiol.30.841.

[23]

B. Zduniak, M. Bodnar and U. Foryś, A modified van der Pol equation with delay in a description of the heart action, Int. J. Appl. Math. Comput. Sci., 24 (2014), 853-863. doi: 10.2478/amcs-2014-0063.

[24]

Y. Zhang, K. Wang, Y. Yuan, D. Sui and H. Zhang, Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley model, Computational and Mathematical Methods in Medicine, (2014), Art. ID 761907, 9 pp. doi: 10.1155/2014/761907.

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