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A Model of Excitation Wavefronts Spreading in the Anisotropic Cardiac Tissue

  • P. Colli Franzone
  • L. Guerri
  • S. Rovida
  • S. Tentoni
Chapter
Part of the NATO ASI Series book series (NSSB, volume 244)

Abstract

The bioelectric activity of the myocardium during the excitation process, also called depolarization phase, is associated with a propagating front-like variation of the transmembrane potential which is the jump of the potential across the cellular membrane. In this phase the transmembrane potential undergoes an abrupt change from a resting to an excited value occurring in a moving layer about 1 mm thick and lasting about 2 msec.

Keywords

Transmembrane Potential Excitable Medium Eikonal Equation Excitation Process Singular Perturbation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Brebbia, C.A., Telles, J., Wrobel, L. (1984). Boundary Element Techniques — Theory and Applications in Engineering. Springer-Verlag: Berlin, New York.zbMATHGoogle Scholar
  2. [2]
    Casten, R., Cohen, H. & Lagerstrom, P. (1975). Perturbation analysis of approximation to Hodgkin-Huxley theory. Quart. Appl. Math. 32, 365–402.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Colli Franzone, P., Guerri, L., Viganotti, C., Macchi, E., Baruffi, S., Spaggiari, S. & Taccardi, B. (1982). Potential fields generated by oblique dipole layer modelling excitation wavefronts in the anisotropic myocardium. Comparison with potential fields elicited by paced dog hearts in a volume conductor. Circ. Res. 51, 330–346.CrossRefGoogle Scholar
  4. [4]
    Colli Franzone, P., Guerri, L., Taccardi, B., Tentoni, S. & Viganotti, C. (1982). Cardiac fibers orientation and potential fields: model studies. Japanese Heart J. 23(1), 279–281.Google Scholar
  5. [5]
    Colli Franzone, P., Guerri, L. & Rovida, S. (1988). Macroscopic cardiac sources model and propagation wave, in excitable media. In Biomathematics and Related Computational Problems, pp. 615–627, Ricciardi, L.M. (ed.). Kluwer Academic Publishers Group: Dordrecht.CrossRefGoogle Scholar
  6. [6]
    Colli Franzone, P., Guerri, L. & Rovida, S. (1988). Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations. Preprint of I.A.N.-C.N.R. 625, 1–69 (to appear J. Math. Biol., 1989).Google Scholar
  7. [7]
    Colli Franzone, P., Guerri, L. & Tentoni, S. (1990). Mathematical modelling of the excitation process in the myocardial tissue: influence of the fiber rotation on the wavefront propagation and the potential field. Submitted to Math. Biosci.[S] Google Scholar
  8. [8]
    Courant, R. & Hilbert, D. (1966). Methods of Mathematical Physics, Vol. II. Interscience Publishers: New York, London.Google Scholar
  9. [9]
    Di Francesco, D. & Noble, D. (1985). A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Phil. Trans. R. Soc. Lond. B 307, 353–398.ADSCrossRefGoogle Scholar
  10. [10]
    Ebihara, L. & Johnson, E.A. (1980). Fast sodium current in cardiac muscle. Biophys. J. 32, 779–790.CrossRefGoogle Scholar
  11. [11]
    Fife, P.C. (1976). Singular perturbation and wavefront techniques in reaction-diffusion problems. Proceedings SIAM-AMS vol. 10, Symposium on Asymptotic Methods and Singular Perturbations, 23–49, New York.Google Scholar
  12. [12]
    Fife, P.C. (1979). Mathematical aspects of reacting and diffusing systems. In Lecture Notes in Biomathematics 28. Springer Verlag.Google Scholar
  13. [13]
    Fozzard, H.A. (1979). Conduction of the action potential. In Handbook of Physiology, Vol. 1, the Heart, Sect. 2, the Cardiovascular System, pp. 335–356, Berne, R.M., Sperelakis, N., & Geiger, S.R. (eds.). Am. Physiol. Soc., Bethesda, MD.Google Scholar
  14. [14]
    Frazier, D.W., Krassowska, W., Chen, P.S., Wolf, P.D., Danieley, N.D., Smith, W.M. & Ideker, R.E. (1988). Transmural activations and stimulus potentials in three- dimensional anisotropic canine myocardium. Circ. Res. 63(1), 135–146.CrossRefGoogle Scholar
  15. [15]
    Gomatam, J. & Grindrod, P. (1987). Three-dimensional waves in excitable reaction-diffusion systems. J. Math. Biol. 25, 611–622.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544.Google Scholar
  17. [17]
    Jack, J.J.B., Noble, D. & Tsien, R.W. (1983). Electric Current Flow in Excitable Cells. Clarendon Press: Oxford.Google Scholar
  18. [18]
    Keener, J.P. (1980). Waves in excitable media. S.I.A.M. J. Appl. Math. 39(3), 528–548.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Keener, J.P. (1986). A geometrical theory for spiral waves in excitable media. S.I. A.M. J. Appl. Math. 46(6), 1039–1056.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Spaggiari, S., Baruffi, S., Arisi, G., Macchi, E., Taccardi, B. (1987). Effect of intramural fiber direction on epicardial isochrone and potential maps. (Abstract) Circulation 76 11–961.Google Scholar
  21. [21]
    Streeter, D. (1979). Gross Morphology and Fiber Geometry of the Heart. In Handbook of Physiology, Section 2, the Cardiovascular System, Geiger, S.R. (ed.).Google Scholar
  22. [22]
    Zykov, V.S. & Petrov, A. (1977). Role of the inhomogeneity of an excitable medium in the mechanism of self-sustained activity. Biofizika 22(2), 300–306. (English translation Biophysics 22, 307–314.)Google Scholar
  23. [23]
    Zykov, V.S. (1980). Analytical evaluation of the dependence of the speed of an excitation wave in a two-dimensional excitable medium on the curvature of the its front. Biofizika 25(5), 888–892. (English translation Biophysics 25, 906–911.)Google Scholar
  24. [24]
    Zykov, V.S. (1987). Simulation of Wave Processes in Excitable Media. Manchester University Press: Manchester and New York.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • P. Colli Franzone
    • 1
  • L. Guerri
    • 1
  • S. Rovida
    • 2
  • S. Tentoni
    • 2
  1. 1.Dipartimento di Informatica e SistemisticaDell’Università di PaviaPaviaItaly
  2. 2.Istituto di Analisi Numerica del C.N.R.PaviaItaly

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