# Dynamics of Pivoting Electrical Waves in a Cardiac Tissue Model

- 2 Downloads

## Abstract

Through a detailed mathematical analysis we seek to advance our understanding of how cardiac tissue conductances govern pivoting (spiral, scroll, rotor, functional reentry) wave dynamics. This is an important problem in cardiology since pivoting waves likely underlie most reentrant tachycardias. The problem is complex, and to advance our methods of analysis we introduce two new tools: a ray tracing method and a moving-interface model. When used in combination with an ionic model, they permit us to elucidate the role played by tissue conductances on pivoting wave dynamics. Specifically we simulate traveling electrical waves with an ionic model that can reproduce the characteristics of plane and pivoting waves in small patches of cardiac tissue. Then ray tracing is applied to the simulated pivoting waves in a manner to expose their real displacement. In this exercise we find loci with special characteristics, as well as zones where a part of a pivoting wave quickly transitions from a regenerative to a non-regenerative propagation mode. The loci themselves and the monitoring of the ionic model state variables in this zone permit to elucidate several aspects of pivoting wave dynamics. We then formulate the moving-interface model based on the information gathered with the above-mentioned analysis. Equipped with a velocity profile *v*(*s*), *s*: distance along of the pivoting wave contour and the steady- state action potential duration (APD) of a plane wave during entrainment, *APDss*(*T*), at period *T*, this simple model can predict: shape, orbit of revolution, rotation period, whether a pivoting wave will break up or not, and whether the tissue will admit pivoting waves or not. Because *v*(*s*) and *APDss*(*T*) are linked to the ionic model, dynamical analysis with the moving-interface model conveys information on the role played by tissue conductances on pivoting wave dynamics. The analysis conducted here enables us to better understand previous results on the termination of pivoting waves. We surmise the method put forth here could become a means to discover how to alter tissue conductances in a manner to terminate pivoting waves at the origin of reentrant tachycardias.

## Keywords

Electrical waves Pivoting waves Traveling waves Vortices of electrical waves Finite element method Dynamical analysis## Notes

### Acknowledgements

Funding from the Whitaker Foundation. Support provided by the Center for Computational Research from the University at Buffalo. Computational resources from the Texas Advanced Computer Center, Grant: TG-BCS110013. Funding from Complex Biosystems Inc. We thank an anonymous reviewer to: help us improve our model formulation, and make recommendations to clearly introduce our findings to analysts in the field.

## Supplementary material

## References

- Allessie M, Bonke F, Schopman F (1973) Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circ Res 33(1):54–62CrossRefGoogle Scholar
- Allessie M, Bonke F, Schopman F (1976) Circus movement in rabbit atrial muscle as a mechanism of tachycardia. ii. the role of nonuniform recovery of excitability in the occurrence of unidirectional block, as studied with multiple microelectrodes. Circ Res 39(2):168–177CrossRefGoogle Scholar
- Allessie M, Bonke F, Schopman F (1977) Circus movement in rabbit atrial muscle as a mechanism of tachycardia. iii. the ’leading circle’ concept: a ne model of circus movement in cardiac tissue without the involvement of an anatomical obstacle. Circ Res 41:9–18CrossRefGoogle Scholar
- Beaumont J, Davidenko N, Davidenko J, Jalife J (1995) Self-sustaining spiral wave activity in a two-dimensional ionic model of cardiac ventricular muscle. In: Power H, Hart RT (eds) Computational mechanics, chap. 10, Southampton, pp 75–86Google Scholar
- Beaumont J, Davidenko N, Davidenko J, Jalife J (1998) Spiral waves in two-dimensional models of ventricular muscle: formation of a stationary core. Biophys J 75:1–75CrossRefGoogle Scholar
- Beaumont J, Jalife J (2000) Cardiac electrophysiology. From cell to bedside. W.B., In: Zipes D, Jalife J (eds) Rotors and spiral waves in two dimensions, 3rd edn, chap. 39, PhiladelphiaGoogle Scholar
- Bursac N, Parker K, Iravanian S, Tung L (2002) Cardiomyocyte cultures with controlled cacroscopic anisotropy: a model for functional electrophysiological studies of cardiac muscle. Circ Res 91:e45–e54CrossRefGoogle Scholar
- Connolly S, Hallstrom A, Cappato R, Schron E, Kuck KH, Zipes D, Greene H, Boczor S, Domanski M, Follmann D, Gent M, Roberts R (2000) On behalf of the investigators of the AVID, C., studies, C.: Meta-analysis of the implantable cardioverter defibrillator secondary prevention trials. Eur Heart J 21:2071–2078CrossRefGoogle Scholar
- Costantino A, Hyatt CJ, Kollisch-Singule M, Beaumont J, Roth B, Pertsov A (2017) Determining the light scattering and absorbtion parameters from forward directed flux measurements in cardiac tissue. J Biomed Optics 22(7):076009CrossRefGoogle Scholar
- Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings ACM National conference, pp 157–192Google Scholar
- Davidenko J, Pertsov A, Salomonsz R, Baxter W, Jalife J (1991) Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355:349–351CrossRefGoogle Scholar
- Duff I, Meurant G (1989) The effect of ordering on preconditioned conjugate gradients. BIT Numerical Mathematics 29(4):635–657MathSciNetCrossRefzbMATHGoogle Scholar
- Ebihara L, Johnson E (1980) Fast sodium current kinetic in cardiac muscle: a quantitative description. Biophys J 32:779–790CrossRefGoogle Scholar
- Fast V, Darrow B, Saffitz J, Kléber A (1996) Anisotropic activation spread in heart cell monolayers assessed by high-resolution optical mapping. Circ Res 79(1):115–127CrossRefGoogle Scholar
- Fast V, Kléber A (1993) Microscopic conduction in cultured strands of neonatal rat heart cells measured with voltage-sensitive dyes. Circ Res 73:914–925CrossRefGoogle Scholar
- Fast V, Kléber A (1994) Anisotropic conduction in monolayers of neonatal rat heart cells cultured on collagen substrate. Circ Res 75(3):591–595CrossRefGoogle Scholar
- Fitzhugh R (1961) Impulse and physiological states in models of nerve membrane. Biophys J 1:445–466CrossRefGoogle Scholar
- Golub G, Van Loan C (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
- Gray R, Pertsov A, Jalife J (1998) Spatial and temporal organization during cardiac fibrillation. Nature 392:75–78CrossRefGoogle Scholar
- Beeler Go W, Reuter H (1977) Reconstruction of the action potential of ventricular myocardial fibers. J Physiol 268:177–210CrossRefGoogle Scholar
- Matsuura H, Ehara T, Imoto Y (1987) An analysis of the delayed outward current in single ventricular cells of the guinea-pig. Pflugers Arch 410:596–603CrossRefGoogle Scholar
- Hakim V, Karma A (1999) Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications. Phys Rev E 60(5):5073–5104MathSciNetCrossRefGoogle Scholar
- Hohnloser S (1997) Proarrhythmia with class iii antiarrhythmic drugs: types, risks, and management. Am J Cardiol 23:82G–89GCrossRefGoogle Scholar
- Huikuri HV, Castellanos A, Myerburg RJ (2001) Sudden death due to cardiac arrhythmias. N Engl J Med 345(20):1473–1482CrossRefGoogle Scholar
- Ikeda T, Uchida L, Hough T, Lee J, Fishbein M, Mandel W, Chen P, Karageuzian H (1996) Mechanism of spontaneous termination of functional reentry in isolated canine right atrium. Circulation 94:1962–1973CrossRefGoogle Scholar
- Jack J, Noble D, Tsien R (1983) Electric current flow in excitable cells. Oxford University Press, OxfordGoogle Scholar
- Keener J (1986) A geometrical theory for spiral waves in excitable media. SIAM J Appl Math 46:1039–1056MathSciNetCrossRefzbMATHGoogle Scholar
- Keener J (1992) The core of the spiral. SIAM J Appl Math 52(5):1370–1390MathSciNetCrossRefzbMATHGoogle Scholar
- Kirsten H, Tusscher T, Hren R, Panfilov A (2007) Organization of ventricular gibrillation in the human heart. Circ Res 100:e87–e101Google Scholar
- Kleber A, Riegger C (1987) Electrical constants of arterially perfused rabbit papillary muscle. J Physiol Lond 385:307–324CrossRefGoogle Scholar
- Kreyszig E (1973) Advanced engineering mathematics. Wiley, HobokenzbMATHGoogle Scholar
- Krinsky V (1984) Self-organization. Autowaves and structures far from equilibrium. In: Proceedings of an international symposium held under the auspices of the Institute of Biological Physics, USSR academy of Sciences, Springer-VerlagGoogle Scholar
- Krummen D, Hayase J, Morris D, Ho J, SmetaK M, Clopton P, Narayan SM (2014) Rotor stability separates sustained ventricular fibrillation from self-terminating episodes in humans. J Am Coll Cardiol 63(24):2712–2721CrossRefGoogle Scholar
- Kucera J, Kléber A, Rohr S (1998) Slow conduction in cardiac tissue, ii effects of branching tissue geometry. Circ Res 83:795–805CrossRefGoogle Scholar
- Lafuente-Lafuente C, Mouly S, Longás-Tejero M (2006) Antiarrhythmic drugs for maintaining sinus rhythm after cardioversion of atrial fibrillation. a systematic review of randomized controlled trials. Arch Intern Med 166(7):719–728CrossRefGoogle Scholar
- Lancaster P, Salkauskas K (1990) Curve and surface fitting. Academic Press, CambridgezbMATHGoogle Scholar
- Lee J, Kamjoo K, Hough D, Hwand C, Fan W, Fishbein M, Bonometti C, Ikeda C, Karageuzian H, Chen P (1996) Reentrant wave fronts in wigger’s stage ii ventricular fibrillation. Circ Res 78:660–675CrossRefGoogle Scholar
- Liu WH, Sherman AH (1976) Comparative analysis of the cuthill-mckee and the reverse cuthill-mckee ordering algorithms for sparse matrices. SIAM J Numer Anal 13(2):198–213MathSciNetCrossRefzbMATHGoogle Scholar
- Luo C, Rudy Y (1991) A model of ventricular cardiac action potential: depolarization, repolarization and their interaction. Circ Res 68:1501–1526CrossRefGoogle Scholar
- Luther S, Fenton H, Kornreich B, Squires A, Bittihn P, Hornung D, Zabel M, Flanders J, Gladuli A, Campoy L, Cherry L, Luther G, Hasenfuss G, Krinsky V, Pumir A, Gilmour R (2011) Low-energy control of electrical turbulence in the heart. Nature 475:235–239CrossRefGoogle Scholar
- Mandapati R, Asano Y, Baxter W, Gray R, Davidenko J, Jalife J (1998) Quantification of effects of global ischemia on dynamics of ventricular fibrillation in isolated rabbit heart. Circulation 98:1688–1696CrossRefGoogle Scholar
- Mandapati R, Skanes A, Chen J, Berenfeld O, Jalife J (2000) Stable microreentrant sources as a mechanism of atrial fibrillation in the isolated sheep heart. Circulation 101:194–199CrossRefGoogle Scholar
- Massé S, Downar E, Chauhan V, Sevaptsidis E, Nanthakumar K (2006) Ventricular fibrillation in myopathic human hearts: mechanistic insights from in vivo global endocardial and epicardial mapping. Am J Physiol Heart Circ Physiol 292:H2589–H2597CrossRefGoogle Scholar
- Mines G (1914) On circulating excitations in heart muscles and their possible relation to tachycardia fibrillation. Trans R Soc Can 4:43–53Google Scholar
- Nair G, Nery P, Diwakaramenon S, Healy J, Connoly S, Morillo C (2009) A systematic review of randomized trials comparing radiofrequency ablation with antiarrhythmic medications in patients with atrial fibrillation. J Cardiovascul Electrophysiol 20(2):127–240CrossRefGoogle Scholar
- Nanthakumar K, Walcott G, Melnick S, Rogers J, Kay MW, Smith W, Ideker R, Holman W (2004) Epicardial organization of human ventricular fibrillation. Heart Rhythm 1:14–23CrossRefGoogle Scholar
- Nash M, Mourad A, Clayton R, Sutton PM, Bradley C, Hayward M, Paterson D, Taggart P (2006) Evidence for multiple mechanisms in human ventricular fibrillation. Circulation 114:536–542CrossRefGoogle Scholar
- Pandit S, Jalife J (2013) Rotors and the dynamics of cardiac fibrillation. Circ Res 112(5):849–862CrossRefGoogle Scholar
- Pelce P, Sun S (1991) Wave front interaction in steadily rotating spirals. Physica D 48:353–366CrossRefzbMATHGoogle Scholar
- Pertsov A, Davidenko J, Salomonsz R, Baxter R, Jalife J (1993) Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ Res 72:631–650CrossRefGoogle Scholar
- Pertsov A, Ermakova E, Panfivlov A (1984) Rotating spiral waves in a modified Fitx-Hugh-Nagumo model. Physica D 14:117–124MathSciNetCrossRefzbMATHGoogle Scholar
- Raba A, Cordeiro J, Antzelevitch A, Beaumont J (2013) Extending the conditions of application of an inversion of the Hodgkin-Huxley gating model. J Math Biol 75(5):752–773MathSciNetCrossRefzbMATHGoogle Scholar
- Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, New DelhiCrossRefzbMATHGoogle Scholar
- Samie F, Berenfeld O, Anumonwo J, Mironov S, Udassi S, Beaumont J, Taffet S, Pertsov A, Jalife J (2001) Rectification of the background potassium current. a determinant of rotor dynamics in ventricular fibrillation. Circ Res 89:1216–1223CrossRefGoogle Scholar
- Samie F, Mandapati R, Gray R, Watanabe Y, Zuur C, Beaumont J, Jalife J (2000) A mechanism of transition from ventricular fibrillation to tachycardia: effect of calcium channel blockade on the dynamics of rotating waves. Circ Res 86:684–691CrossRefGoogle Scholar
- Skanes A, Mandapati R, Berenfeld O, Davidenko J, Jalife J (1998) Spatiotemporal periodicity during atrial fibrillation in isolated sheep heart. Circulation 98(12):1236–1248CrossRefGoogle Scholar
- Tse G (2016) Mechanisms of cardiac arrhythmias. J Arrhythm 32:75–81CrossRefGoogle Scholar
- Walcott G, Kay N, Plumb V, Smith W, Rogers J, Epstein A, Ideker R (2002) Endocardial wave front organization during ventricular fibrillation in humans. J Am College Cardiol 39(1):109–115CrossRefGoogle Scholar
- Weidmann S (1970) Electrical constants of trabeculear muscle from mammalian heart. J Physiol 210:1041–1054CrossRefGoogle Scholar
- Westfall M, Pasyk K, Yule D, Samuelson L, Metzger J (1997) Ultrastructure and cell-cell coupling of cardiac myocytes differentiating in embryonic stem cell culture. Cell Motil Cytoskelet 36:43–54CrossRefGoogle Scholar
- Wilber D, Pappone C, Neuzil P Marchlinski F, Natale A, Macle L, Daoud E, Calkins H, Hall B, Reddy V, Augello G, Reynolds M, Vinekar C, Liu C, Berry S, Berry D (2010) Comparison of antiarrhythmic drug therapy and radiofrequency catheter ablation in patients with paroxysmal atrial fibrillation. A randomized controlled trial. JAMA 303(4):333–340CrossRefGoogle Scholar
- Wolbrette D (2003) Risk of proarrhythmia with class iii antiarrhythmic agents: sex-based differences and other issues. Am J Cardiol 9(6):39–44CrossRefGoogle Scholar
- Yue D, Marban E (1988) A novel cardiac potassium channel that is active and conductive at depolarized potentials. Pflugers Arch 413:127–133CrossRefGoogle Scholar
- Zykov V, Winfree A (1992) Simulation of wave processes in excitable media. Wiley, New YorkGoogle Scholar