Skip to main content
SpringerLink
Log in

Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro- and Macroscopic Level

  • Chapter
Evolution Equations, Semigroups and Functional Analysis

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 50))

Abstract

The aim of this paper is to study the reaction-diffusion systems arising from the mathematical models of the cardiac electric activity at the micro- and macroscopic level.

Dedicated to the memory of Brunello Terreni.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Attouch, Variational convergence for functions and operators, Pitman (Advanced Publishing Program), Boston, MA, 1984.

    MATH  Google Scholar 

  2. C. Baiocchi, Discretization of evolution variational inequalities, Partial differential equations and the calculus of variations, Vol. I (F. Colombini, A. Marino, L. Modica, and S. Spagnolo, eds.), Birkhäuser Boston, Boston, MA, 1989, pp. 59–92.

    Chapter  Google Scholar 

  3. N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Kluwer Academic Publishers Group, Dordrecht, 1989, Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. Leĭtes.

    MATH  Google Scholar 

  4. F. Bassetti, Variable time-step discretization of degenerate evolution equations in Banach spaces, Tech. report, IAN-CNR, Pavia, 2002.

    Google Scholar 

  5. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, North-Holland Publishing Co., Amsterdam, 1978.

    MATH  Google Scholar 

  6. H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contribution to Nonlinear Functional Analysis, Proc. Sympos. Math. Res. Center, Univ. Wisconsin, Madison, 1971, Academic Press, New York, 1971, pp. 101–156.

    Google Scholar 

  7. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).

    MATH  Google Scholar 

  8. H. Brezis, On some degenerate nonlinear parabolic equations, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, 111., 1968), Amer. Math. Soc, Providence, R.I., 1970, pp. 28-38.

    Google Scholar 

  9. H. Brézis, Intégrales convexes dans les espaces de Sobolev, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), vol. 13, 1972, pp. 9–23 (1973).

    Google Scholar 

  10. H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1–168.

    MathSciNet  Google Scholar 

  11. H. Brézis C. Bardos, Sur une classe de problèmes d’évolution non linéaires, J. Differential Equations 6 (1969), 345–394.

    Article  MathSciNet  MATH  Google Scholar 

  12. N. F. Britton, Reaction-diffusion equations and their applications to biology, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1986.

    MATH  Google Scholar 

  13. R. W. Carroll and R. E. Showalter, Singular and degenerate Cauchy problems, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976, Mathematics in Science and Engineering, Vol. 127.

    Google Scholar 

  14. R. G. Casten, H. Cohen, and P. A. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory, Quart. Appl. Math. 32 (1974/75), 365–402.

    MathSciNet  Google Scholar 

  15. P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), no. 5, 737–756.

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Colli Franzone and L. Guerri, Spreading of excitation in 3-d models of the anisotropic cardiac tissue, Math. Biosc. 113 (1993), 145–209.

    Article  MATH  Google Scholar 

  17. P. Colli Franzone, L. Guerri, and S. Rovida, Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations, J. Math. Biol. 28 (1990), no. 2, 121–176.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Cronin, Mathematics of cell electrophysiology, Marcel Dekker Inc., New York, 1981.

    MATH  Google Scholar 

  19. E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal. 12 (1981), no. 5, 731–751.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Ebihara and E. A. Johnson, Fast sodium current in cardia muscle, Biophys. J. 32 (1980), 779–790.

    Article  Google Scholar 

  21. A. Favini and A. Yagi, Degenerate differential equations in Banach spaces, Marcel Dekker Inc., New York, 1999.

    MATH  Google Scholar 

  22. R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical analysis of variational inequalities, North-Holland Publishing Co., Amsterdam, 1981, Translated from the French.

    MATH  Google Scholar 

  23. C. S. Henriquez, A. L. Muzikant, and C. K. Smoak, Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: simulations in a three dimensional bidomain model, J. Cardiovasc. Electrophysiol. 7 (1996), 424–444.

    Article  Google Scholar 

  24. C. S. Henriquez, Simulating the electrical behavior of cardiac tissue using the bidomain model, Crit. Rev. Biomed. Engr. 21 (1993), 1–77.

    MathSciNet  Google Scholar 

  25. A. L. Hodkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952), 500–544.

    Google Scholar 

  26. D. Hoff, Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1161–1177.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. J. B. Jack, D. Noble, and R. W. Tsien, Electric current flow in excitable cells, Clarendon Press, Oxford, 1983.

    Google Scholar 

  28. J. W. Jerome, Convergence of successive iterative semidiscretizations for FitzHugh-Nagumo reaction diffusion systems, SIAM J. Numer. Anal. 17 (1980), no. 2, 192–206.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994, Translated from the Russian by G. A. Yosifian [G. A. Iosifyan].

    Book  Google Scholar 

  30. J. Keener and J. Sneyd, Mathematical physiology, Springer-Verlag, New York, 1998.

    MATH  Google Scholar 

  31. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.

    MATH  Google Scholar 

  32. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications.Vol. I-II, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.

    Book  Google Scholar 

  33. J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appi. Math. 20 (1967), 493–519.

    Article  MathSciNet  MATH  Google Scholar 

  34. CH. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential, i. simulations of ionic currents and concentration changes, Circ. Res. 74 (1994), 1071–1096.

    Article  Google Scholar 

  35. M. Mascagni, The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve conduction, SIAM J. Numer. Anal. 27 (1990), no. 4, 941–962.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. M. Miura, Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations, J. Math. Biol. 13 (1981/82), no. 3, 247–269.

    Article  MathSciNet  Google Scholar 

  37. J. S. Neu and W. Krassowska, Homogenization of syncitial tissues, Crit. Rev. Biom. Engr. 21 (1993), 137–199.

    Google Scholar 

  38. R. H. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), no. 5, 525–589.

    Article  MathSciNet  MATH  Google Scholar 

  39. O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, North-Holland Publishing Co., Amsterdam, 1992.

    MATH  Google Scholar 

  40. O. A. Oleĭnik and T. Shaposhnikova, On homogenization problems for the Laplace operator in partially perforated domains with Neumann’s condition on the boundary of cavities, Atti Accad. Naz. Lincei Cl. Sei. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), no. 3, 133–142.

    MATH  Google Scholar 

  41. B. J. Roth, How the anisotropy of the intracellular and extracellular conductivities influence stimulation of cardiac muscle, J. Math. Biol. 30 (1992), 633–646.

    Article  MATH  Google Scholar 

  42. B. J. Roth and W. Krassowska, The induction of reentry in cardiac tissue. The missing link: how electric fields alter transmembrane potential, Chaos 8 (1998), 204–219.

    Article  Google Scholar 

  43. J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. Numer. Anal. 33 (1996), 68–87.

    Article  MathSciNet  MATH  Google Scholar 

  44. E. Sánchez-Palencia and A. Zaoui (eds.), Homogenization techniques for composite media, Springer-Verlag, Berlin, 1987, Papers from the course held in Udine, July 1-5, 1985.

    MATH  Google Scholar 

  45. E. Sánchez-Palencia, Nonhomogeneous media and vibration theory, Springer-Verlag, Berlin, 1980.

    Google Scholar 

  46. S. Sanfelici, Convergence of the galerkin approximation of a degenerate evolution problem in electrocardiology, Numer. Methods for Partial Differential Equations 18 (2002), 218–240.

    Article  MathSciNet  MATH  Google Scholar 

  47. G. Savaré, Approximation and regularity of evolution variational inequalities, Rend. Acc. Naz. Sei. XL Mem. Mat. XVII (1993), 83–111.

    Google Scholar 

  48. G. Savaré, Weak solutions and maximal regularity for abstract evolution inequalities, Adv. Math. Sci. Appl. 6 (1996), 377–418.

    MathSciNet  MATH  Google Scholar 

  49. R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  50. J. Smoller, Shock waves and reaction-diffusion equations, second ed., Springer-Verlag, New York, 1994.

    Book  MATH  Google Scholar 

  51. N. Trayanova, K. Skouibine, F. Aguel, The role of cardiac tissue structure in defibrillation, Chaos 8 (1998), 221–253.

    Article  Google Scholar 

  52. J. P. Wikswo, Tissue anisotropy, the cardiac bidomain, and the virtual cathod effect, Cardiac Electrophysiology: From Cell to Beside (D. P. Zipes and J. Jalife, eds.), W. B. Saunders Co., Philadelphia, 1994, pp. 348–361.

    Google Scholar 

  53. A. L. Wit, S. M. Dillon, and J. Coromilas, Anisotropy reentry as a cause of ventricular tachyarhythmias, Cardiac Electrophysiology: From Cell to Beside (D. P. Zipes and J. Jalife, eds.), W. B. Saunders Co., Philadelphia, 1994, pp. 511–526.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata, 1, I-27100, Pavia, Italy

    Piero Colli Franzone & Giuseppe Savaré

Authors
  1. Piero Colli Franzone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Savaré
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, 20133, Milano, Italy

    Alfredo Lorenzi  & Bernhard Ruf  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Basel AG

About this chapter

Cite this chapter

Franzone, P.C., Savaré, G. (2002). Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro- and Macroscopic Level. In: Lorenzi, A., Ruf, B. (eds) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8221-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8221-7_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9480-7

  • Online ISBN: 978-3-0348-8221-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation