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A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times

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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

  1. 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.

    Article  Google Scholar 

  2. 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.

    Article  Google Scholar 

  3. Beeler, G. W., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268:177–210, 1977.

    Google Scholar 

  4. Berbari, E. J., P. Lander, B. J. Scherlag, R. Lazara, and D. B. Gesesowitz. Ambiguities of epicardial mapping. J. Electrocardiol. 24:16–20, 1992.

    Article  Google Scholar 

  5. 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.

    Article  Google Scholar 

  6. 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.

    Article  Google Scholar 

  7. 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.

    Article  MathSciNet  Google Scholar 

  8. 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.

    Article  Google Scholar 

  9. 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.

    Article  Google Scholar 

  10. 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.

  11. Fast, V. G., and A. G. Kleber. Role of wavefront curvature in propagation of cardiac impulse. Cardiovasc. Res. 33:258–271, 1997.

    Article  Google Scholar 

  12. 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.

    Article  Google Scholar 

  13. Gray, R. A., and J. Jalife. Mechanisms of cardiac fibrillation. Science 270:1222–1223, 1995.

    Article  Google Scholar 

  14. Gray, R. A., A. M. Pertsov, and J. Jalife. Spatial and temporal organization during cardiac fibrillation. Nature 392:75–78, 1998.

    Article  Google Scholar 

  15. 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.

    Google Scholar 

  16. 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.

    Article  Google Scholar 

  17. 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.

    Article  Google Scholar 

  18. 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.

    Article  Google Scholar 

  19. 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.

    Article  Google Scholar 

  20. 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.

    Article  Google Scholar 

  21. 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.

    Google Scholar 

  22. 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.

    Article  Google Scholar 

  23. Mazeh, N. The upper limit of vulnerability of the heart. PhD dissertation, Oakland University, Rochester, MI, 2008.

  24. Mazeh, N., and B. J. Roth. A mechanism of the upper limit of vulnerability. Heart Rhythm 6:361–367, 2009.

    Article  Google Scholar 

  25. Moe, G. K., W. C. Rheinboldt, and J. A. Abildskov. A computer model of atrial fibrillation. Am. Heart J. 67:200–220, 1964.

    Article  Google Scholar 

  26. 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.

    Article  Google Scholar 

  27. 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.

    Article  Google Scholar 

  28. 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.

    Article  Google Scholar 

  29. 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.

    Article  Google Scholar 

  30. Rosenbaum, D. S., and J. Jalife. Optical Mapping of Cardiac Excitation and Arrhythmias. Armonk, NY: Futura Pub Co, 2001.

    Google Scholar 

  31. 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.

    Article  MATH  Google Scholar 

  32. Roth, B. J. Electrical conductivity values used with the bidomain model of cardiac tissue. IEEE Trans. Biomed. Eng. 44:326–328, 1997.

    Article  Google Scholar 

  33. 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.

    Google Scholar 

  34. 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.

    Article  Google Scholar 

  35. Spooner, P. M., R. W. Joyner, and J. Jalife. Discontinuous Conduction in the Heart. Armonk, NY: Futura Publishing, 1997.

    Google Scholar 

  36. Tung, L. A bi-domain model for describing ischemic myocardial D-C potentials. PhD Thesis, MIT, Cambridge, MA, 1977.

  37. 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.

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Department of Cardiovascular Medicine, Beaumont Health System, Royal Oak, MI, USA

    Nachaat Mazeh

  2. Department of Cardiovascular Medicine, Oakland University William Beaumont School of Medicine, Royal Oak, MI, USA

    David E. Haines

  3. Department of Electrical and Computer Engineering, George Washington University, Washington, DC, USA

    Matthew W. Kay

  4. Department of Physics, Oakland University, Rochester, MI, USA

    Bradley J. Roth

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  1. Nachaat Mazeh
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Correspondence to Nachaat Mazeh.

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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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