Abstract
The velocity and curvature of a wave front are important factors governing the propagation of electrical activity through cardiac tissue, particularly during heart arrhythmias of clinical importance such as fibrillation. Presently, no simple computational model exists to determine these values simultaneously. The proposed model uses the arrival times at four or five sites to determine the wave front speed (v), direction (θ), and radius of curvature (ROC) (r 0). If the arrival times are measured, then v, θ, and r 0 can be found from differences in arrival times and the distance between these sites. During isotropic conduction, we found good correlation between measured values of the ROC r 0 and the distance from the unipolar stimulus (r = 0.9043 and p < 0.0001). The conduction velocity (m/s) was correlated (r = 0.998, p < 0.0001) using our method (mean = 0.2403, SD = 0.0533) and an empirical method (mean = 0.2352, SD = 0.0560). The model was applied to a condition of anisotropy and a complex case of reentry with a high voltage extra stimulus. Again, results show good correlation between our simplified approach and established methods for multiple wavefront morphologies. In conclusion, insignificant measurement errors were observed between this simplified approach and an approach that was more computationally demanding. Accuracy was maintained when the requirement that ε (ε = b/r 0, ratio of recording site spacing over wave fronts ROC) was between 0.001 and 0.5. The present simplified model can be applied to a variety of clinical conditions to predict behavior of planar, elliptical, and reentrant wave fronts. It may be used to study the genesis and propagation of rotors in human arrhythmias and could lead to rotor mapping using low density endocardial recording electrodes.
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Abbreviations
- v :
-
Wave front speed
- θ :
-
Angle specifying wave front velocity direction
- r 0 :
-
Radius of curvature
- b :
-
Recording sites spacing (shortest distance)
- ε :
-
Ratio of electrode spacing over radius of curvature
- g ix :
-
Intracellular conductivity in the x-direction
- g iy :
-
Intracellular conductivity in the y-direction
- g ex :
-
Extracellular conductivity in the x-direction
- g ey :
-
Extracellular conductivity in the y-direction
- t n :
-
Activation time at electrode n, where n = 1, 2, 3, or 4
- Δt ij :
-
Difference of activation times between the ith and jth electrodes
- Δτ i :
-
Time for wave front to travel segment i
- S1–S2 :
-
Stimulation protocol using stimulus of strength S1 and at a later time stimulus S2
- D :
-
Side of square inside the tissue where fibers curve
- POI:
-
Point of interest
- DF:
-
Distance formula
- LS:
-
Line segments method
- 4E:
-
Our computational method
- VF:
-
Ventricular fibrillation
- Vm:
-
Membrane potential
References
Allessie, M. A., F. J. M. Bonke, and F. J. G. Schopman. Circus movement in rabbit atrial muscle as a mechanism of tachycardia, III: the “leading circle” concept. A new model of circus movement in cardiac tissue without involvement of an anatomical obstacle. Circ. Res. 41:9–18, 1977.
Bayly, P. V., B. H. KenKnight, J. M. Rogers, R. E. Hillsley, R. E. Ideker, and W. M. Smith. Estimation of conduction velocity vector fields from epicardial mapping data. IEEE Trans. Biomed. Eng. 45:563–571, 1998.
Beeler, G. W., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268:177–210, 1977.
Berbari, E. J., P. Lander, B. J. Scherlag, R. Lazara, and D. B. Gesesowitz. Ambiguities of epicardial mapping. J. Electrocardiol. 24:16–20, 1992.
Berenfeld, O., and A. M. Pertsov. Dynamics of intramural scroll waves in three dimensional continuous myocardium with rotational anisotropy. J. Theor. Biol. 199:383–394, 1999.
Cabo, C., A. M. Pertsov, W. T. Baxter, J. M. Davidenko, R. A. Gray, and J. Jalife. Wave-front curvature as a cause of slow conduction and block in isolated cardiac muscle. Circ. Res. 75:1014–1028, 1994.
Charteris, N., and B. J. Roth. How hyperpolarization and recovery of excitability affect propagation through a virtual anode in the heart. Comput. Math. Methods Med. 2011:375059, 2011.
Davidenko, J. M., A. V. Pertsov, R. Salomonsz, W. Baxter, and J. Jalife. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355:349–351, 1992.
El-Sherif, N., E. B. Caref, H. Yin, and M. Restivo. The electrophysiological mechanism of ventricular arrhythmias in the long QT syndrome: tridimensional mapping of activation and recovery times. Circ. Res. 79:474–492, 1996.
Ershler, P. R., and R. L. Lux. Derivative mapping in the study of activation sequence during ventricular arrhythmias. In: Proceedings of Computers in Cardiology, edited by K. L. Ripley. New York: IEEE Computer Society Press, 1986, pp. 623–624.
Fast, V. G., and A. G. Kleber. Role of wavefront curvature in propagation of cardiac impulse. Cardiovasc. Res. 33:258–271, 1997.
Girouard, S. D., J. M. Pastore, K. R. Laurita, K. W. Gregory, and D. S. Rosenbaum. Optical mapping in a new guinea pig model of ventricular tachycardia reveals mechanisms for multiple wavelengths in a single reentrant circuit. Circulation 93:603–613, 1996.
Gray, R. A., and J. Jalife. Mechanisms of cardiac fibrillation. Science 270:1222–1223, 1995.
Gray, R. A., A. M. Pertsov, and J. Jalife. Spatial and temporal organization during cardiac fibrillation. Nature 392:75–78, 1998.
Horner, S. M., Z. Vespalcova, and M. J. Lab. Electrode for recording direction of activation, conduction velocity, and monophasic action potential of myocardium. Am. J. Physiol. 272:H-1917–H-1927, 1997.
Ideker, R. E., W. M. Smith, S. M. Blanchard, S. L. Reiser, and E. V. Simpson. The assumptions of isochronal cardiac mapping. PACE 12:456–478, 1989.
Janse, M. J., F. J. L. van Capelle, H. Morsink, A. G. Kléber, F. Wilms-Schopman, R. Cardinal, C. N. d’Alnoncourt, and D. Durrer. Flow of “injury” current and patterns of excitation during early ventricular arrhythmias in acute regional myocardial ischemia in isolated porcine and canine hearts: evidence for two different arrhythmic mechanisms. Circ. Res. 47:151–165, 1980.
Kadish, A. H., J. F. Spear, J. H. Levine, R. F. Hanich, C. Prood, and E. N. Moore. Vector mapping of myocardial activation. Circulation 74:603–615, 1986.
Kay, M. W., and R. A. Gray. Measuring curvature and velocity vector fields for waves of cardiac excitation in 2-D media. IEEE Trans. Biomed. Eng. 52:50–63, 2005.
KenKnight, B. H., P. V. Bayly, R. J. Gerstle, D. L. Rollins, P. D. Wolf, W. M. Smith, and R. E. Ideker. Regional capture of fibrillating ventricular myocardium. Evidence of an excitable gap. Circ. Res. 77:849–855, 1995.
Laxer, C., C. Alferness, W. M. Smith, and R. E. Ideker. The use of computer animation of mapped cardiac potentials in studying electrical conduction properties of arrhythmias. In: Proceedings of Computers in Cardiology, edited by A. Murray, and K. L. Ripley. Chicago, IL: IEEE Computer Society Press, 1990, pp. 23–26.
Lin, S.-F., R. A. Abbas, and J. P. Wikswo, Jr. High-resolution high-speed synchronous epifluorescence imaging of cardiac activation. Rev. Sci. Instrum. 68:213–217, 1997.
Mazeh, N. The upper limit of vulnerability of the heart. PhD dissertation, Oakland University, Rochester, MI, 2008.
Mazeh, N., and B. J. Roth. A mechanism of the upper limit of vulnerability. Heart Rhythm 6:361–367, 2009.
Moe, G. K., W. C. Rheinboldt, and J. A. Abildskov. A computer model of atrial fibrillation. Am. Heart J. 67:200–220, 1964.
Pertsov, A. M., J. M. Davidenko, R. Salomonsz, W. T. Baxter, and J. Jalife. Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ. Res. 72:631–650, 1993.
Pogwizd, S. M., and P. B. Corr. Reentrant and nonreentrant mechanisms contribute to arrhythmogenesis during early myocardial ischemia: results using three-dimensional mapping. Circ. Res. 61:352–371, 1987.
Punske, B. B., Q. Ni, R. L. Lux, R. S. MacLeod, P. R. Ershler, T. J. Dustman, M. J. Allison, and B. Taccardi. Spatial methods of epicardial activation time determination in normal hearts. Ann. Biomed. Eng. 31:781–792, 2003.
Rogers, J. M., M. Usui, B. H. KenKnight, R. E. Ideker, and W. M. Smith. A quantitative framework for analyzing epicardial activation patterns during ventricular fibrillation. Ann. Biomed. Eng. 25:749–760, 1997.
Rosenbaum, D. S., and J. Jalife. Optical Mapping of Cardiac Excitation and Arrhythmias. Armonk, NY: Futura Pub Co, 2001.
Roth, B. J. How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle. J. Math. Biol. 30(6):633–646, 1992.
Roth, B. J. Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng. 44:326–328, 1997.
Smith, W. M., P. D. Wolf, E. V. Simpson, N. D. Danieley, and R. E. Ideker. Mapping ventricular fibrillation and defibrillation. In: Cardiac Mapping, edited by M. Shenesa, M. Borggrefe, and G. Breithard. Mount Kisco, NY: Futura Publishing Co, 1993, pp. 251–260.
Spach, M. S., W. T. Miller, III, D. B. Geselowitz, R. C. Barr, J. Kootsey, and E. A. Johnson. The discontinuous nature of propagation in normal canine cardiac muscle: evidence for recurrent discontinuities of intracellular resistance that affect the membrane currents. Circ. Res. 48:39–54, 1981.
Spooner, P. M., R. W. Joyner, and J. Jalife. Discontinuous Conduction in the Heart. Armonk, NY: Futura Publishing, 1997.
Tung, L. A bi-domain model for describing ischemic myocardial D-C potentials. PhD Thesis, MIT, Cambridge, MA, 1977.
Winfree, A. T. Heart muscle as a reaction-diffusion medium: the roles of electric potential diffusion, activation front curvature, and anisotropy. Int. J. Bifurcation Chaos 7:487–526, 1997.
Acknowledgments
This research was funded by the Department of Cardiovascular Medicine at Beaumont Health System, Royal Oak, Michigan. Dr. M. W. Kay received support from the NIH Grant (HL095828). We wish to thank Drs. R. A. Gray and J. M. Rogers for their helpful discussions and insights.
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Associate Editor Ajit P. Yoganathan oversaw the review of this article.
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Mazeh, N., Haines, D.E., Kay, M.W. et al. A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times. Cardiovasc Eng Tech 4, 520–534 (2013). https://doi.org/10.1007/s13239-013-0158-2
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DOI: https://doi.org/10.1007/s13239-013-0158-2