zbMATH — the first resource for mathematics

Pacemakers in a reaction-diffusion mechanics system. (English) Zbl 1115.92004
Summary: Nonlinear waves of excitation are found in various biological, physical and chemical systems and are often accompanied by deformations of the medium. We numerically study wave propagation in a deforming excitable medium using a two-variable reaction-diffusion system coupled with equations of continuum mechanics. We study the appearance and dynamics of different excitation patterns organized by pacemakers that occur in the medium as a result of deformation. We also study the interaction of several pacemakers with each other and the characteristics of pacemakers in the presence of heterogeneities in the medium. We found that mechanical deformation not only induces pacemakers, but also has a pronounced effect on spatial organization of various excitation patterns. We show how these effects are modulated by the size of the medium, the location of the initial stimulus, and the properties of the reaction-diffusion-mechanics feedback.

92C05 Biophysics
35K57 Reaction-diffusion equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
92C10 Biomechanics
Full Text: DOI
[1] R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation. Chaos, Solitons Fractals 7:293–301 (1996). · doi:10.1016/0960-0779(95)00089-5
[2] M. A. Allessie, F. I. M. Bonke, and F. J. G. Schopman, Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circ. Res. 33:54–62 (1973).
[3] G. W. Beeler and H. J. Reuter, Reconstruction of the action potential of ventricular myocardial fibers. J. Physiol. 268:177–210 (1977).
[4] M. G. Chang, L. Tung, R. Sekar, J. Cysyk, Y. Qi, L. Xu, E. Marban, and R. Abraham, Calcium overload induces tachyarrhythmias in a 2D ventricular myocyte experimental model. Heart Rhythm 3(5):S109–S110 (2006). · doi:10.1016/j.hrthm.2006.02.331
[5] M. Courtemanche, R. Ramirez, and S. Nattel, Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am. J. Physiol. 275:H301–H321 (1998).
[6] J. M. Davidenko, A. M. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife, Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355:349–351 (1991). · doi:10.1038/355349a0
[7] F. Fenton, E. Cherry, H. Hastings, and S. Evans, Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12:852–892 (2002). · doi:10.1063/1.1504242
[8] F. Fenton and A. Karma, Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos 8:20–47 (1998). · Zbl 1069.92503 · doi:10.1063/1.166311
[9] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1:445–465 (1961). · doi:10.1016/S0006-3495(61)86902-6
[10] M. R. Franz, R. Cima, D. Wang, D. Profitt, and R. Kurz, Electrophysiological effects of myocardial stretch and mechanical determinants of stretch-activated arryhthmias. Circulation 86:968–978 (1992).
[11] G. Gerish, Standienpezifische aggregationsmuster bei dictyostelium discoideum. Wihelm. Roux. Arch. Entwick. Org. 156:127–144 (1965). · doi:10.1007/BF00573870
[12] N. A. Gorelova and J. J. Bures, Spiral waves of spreading depression in the isolated chicken retina. J. Neurobiol. 14:353–363 (1983). · doi:10.1002/neu.480140503
[13] R. A. Gray and J. Jalife, Ventricular fibrillation and atrial fibrillation are two different beats. Chaos 8:65–78 (1997). · Zbl 1069.92505 · doi:10.1063/1.166288
[14] A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544 (1952).
[15] P. Hunter, A. McCulloch, and H. ter Keurs, Modelling the mechanical properties of cardiac muscle. Prog. Biophys. Molec. Biol. 69:289–331 (1998). · doi:10.1016/S0079-6107(98)00013-3
[16] P. J. Hunter, M. P. Nash, and G. B. Sands, Computational electromechanics of the heart. In: A.V. Panfilov and A.V. Holden (Eds.), Computational Biology of the Heart, pp. 345–407. Wiley, Chichester (1997). · Zbl 0905.92005
[17] R. Imbihl and G. Ertl, Oscillatory kinetics in heterogeneous catalysis. Chem. Rev. 95:697–733 (1995). · doi:10.1021/cr00035a012
[18] P. Kohl, P. J. Hunter and D. Noble, Stretch-induced changes in heart rate and rhythm: Clinical observations, experiments and mathematical models. Prog. Biophys. Molec. Biol. 71:91–138 (1999). · doi:10.1016/S0079-6107(98)00038-8
[19] J. Lechleiter, S. Girard, E. Peralta, and D. Clapham, Spiral calcium wave propagation and annihilation in xenopus laevis oocytes. Science 252:123–126 (1991). · doi:10.1126/science.2011747
[20] W. Li, P. Kohl, and N. Trayanova, Induction of ventricular arrhythmias following mechanical impact: a simulation study in 3d. J. Mol. Histol. 35(7):679–6 (2004). · doi:10.1007/s10735-004-6206-3
[21] C. Luo and Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ. Res. 68:1501–1526 (1991).
[22] L. E. Malvern, Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1969). · Zbl 0181.53303
[23] A. D. McCulloch, B. H. Smaill, and P. J. Hunter, Left ventricular epicardial deformation in the isolated arrested dog heart. Am. J. Physiol. 252:H233–H241 (1987).
[24] A. P. Muñuzuri, C. Innocenti, J. Flesselles, J. Gilli, K. I. Agladze, and V. I. Krinsky, Elastic excitable medium. Phys. Rev. E 50:R667–R670 (1994). · doi:10.1103/PhysRevE.50.R667
[25] J. Murray, Mathematical Biology. Springer (2002).
[26] M.P. Nash, A. Mourad, R. H. Clayton, P. M. Sutton, C. P. Bradley, M. Hayward, D. J. Paterson, and P. Taggart, Evidence for multiple mechanisms in human ventricular fibrillation. Circulation 114(6):530–542 (2006). · doi:10.1161/CIRCULATIONAHA.105.602870
[27] M. P. Nash and A. V. Panfilov, Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog. Biophys. Mol. Biol. 85:501–522 (2004). · doi:10.1016/j.pbiomolbio.2004.01.016
[28] D. Noble, A. Varghese, P. Kohl, and P. Noble, Improved guinea-pig ventricular model incorporating diadic space, i kr and i ks , length and tension-dependent processes. Can. J. Cardiol. 14:123–134 (1998).
[29] A. V. Panfilov, R. H. Keldermann, and M. P. Nash, Self-organized pacemakers in a coupled reaction-diffusion-mechanics system. Phys. Rev. Lett. 95(25):258104 (2005). · Zbl 1115.92004 · doi:10.1103/PhysRevLett.95.258104
[30] R. Pool, Heart like a wheel. Science 247:1294–1295 (1990).
[31] W. Sigurdson, A. Ruknudin, and F. Sachs, Calcium imaging of mechanically induced fluxes in tissue-cultured chick heart: Role of stretch-activated ion channels. Am. J. Physiol. 262:H1110–H1115 (1992).
[32] M. Spach and J. Heidlage, The stochastic nature of cardiac propagation at a microscopic level. Electrical description of myocardial architecture and its application to conduction. Circ. Res. 76(3):366–380 (1995).
[33] K. Ten Tusscher and A. Panfilov, Influence of nonexcitable cells on spiral breakup in two-dimensional and three-dimensional excitable media. Phys. Rev. E 68:062902 (2003). · doi:10.1103/PhysRevE.68.062902
[34] K. H. W. J. Ten Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov, A model for human ventricular tissue. Am. J. Physiol. Heart Circ. Physiol. 286:H1573–H1589 (2004). · doi:10.1152/ajpheart.00794.2003
[35] N. Trayanova, W. Li, J. Eason, and P. Kohl, Effect of stretch activated channels on defibrillation efficacy. Heart Rhythm 1:67–77 (2004). · doi:10.1016/j.hrthm.2004.01.002
[36] C. Weijer, Dictyostelium morphogenesis. Curr. Opin. Genet. Dev. 14:392–398 (2004). · doi:10.1016/j.gde.2004.06.006
[37] R. Yoshida, T. Takahashi, T. Yamaguchi, and H. Ichijo, Self-oscillating gel. J. Am. Chem. Soc. 118:5134–5135 (1996). · doi:10.1021/ja9602511
[38] A. N. Zaikin and A. M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-organising system. Nature 225:535–537 (1970). · doi:10.1038/225535b0
[39] H. Zhang, A. Holden, I. Kodama, H. Honjo, M. Lei, T. Varghese, and M. Boyett, Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node. Am. J. Physiol. Heart Circ. Physiol. 279(1):H397–421 (2000).
[40] Z. Zheng, J. Croft, W. Giles, and G. Mensah, Sudden cardiac death in the United States, 1989 to 1998. Circulation 104:2158–2163 (2001). · doi:10.1161/hc4301.098254
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.