Carleman estimate for a linearized bidomain model in electrocardiology and its applications

Article
  • 24 Downloads

Abstract

This paper concerns Carleman estimate and its applications for a linearized bidomain model in electrocardiology, which describes the electrical activity in the cardiac tissue. We first establish a new Carleman estimate for this reaction–diffusion system. By means of this Carleman estimate, we study two problems for the linearized bidomian model, a Cauchy problem and an inverse conductivities problem. We prove a conditional stability result for the Cauchy problem and a Hölder stability result for the inverse conductivities problem.

Keywords

Electrocardiology Cauchy problem Conditional stability 

Mathematics Subject Classification

35L05 35L10 35R09 35R30 

Notes

Acknowledgements

This work is supported by NSFC (Nos. 11661004, 11601240)

References

  1. 1.
    Ainseba, B., Bendahmane, M., He, Y.: Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Netw. Heterog. Media 10, 369–385 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellassoued, M., Yamamoto, M.: Lipschitz stability in determining density and two Lamé coefficients. J. Math. Anal. Appl. 329, 1240–1259 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bellassoued, M., Yamamoto, M.: Carleman estimates and an inverse heat source problem for the thermoelasticity system. Inverse Probl. 27(1), 015006 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bendahmane, M., Chamakuri, N., Comte, E., Aïnseba, B.: A 3D boundary optimal control for the bidomain-bath system modeling the thoracic shock therapy for cardiac defibrillation. J. Math. Anal. Appl. 437, 972–998 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bendahmane, M., Chaves-Silva, F.W.: Controllability of a degenerating reactiondiffusion system in electrocardiology. SIAM J. Control Optim. 53, 3483–3502 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boulakia, M., Schenone, E.: Stability estimates for some parameters of a reaction-diffusion equation coupled with an ODE. Appl. Anal. 96, 1138–1145 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bourgault, Y., Coudière, Y., Pierre, C.: Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal Real World Appl 10, 458–482 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bukhgeim, A., Klibanov, M.V.: Uniqueness in the large of a class of multidimensional inverse problems. Sov. Math. Dokl. 17, 244–247 (1981)Google Scholar
  9. 9.
    Franzone, P.C., Savaré, G.: 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: In memory of Brunello Terreni, vol. 50, pp. 49–78. Birkhauser, Basel (2002)Google Scholar
  10. 10.
    Corrado, C., Lassoued, J., Mahjoub, M., Zemzemi, N.: Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology. Math. Biosci. 272, 81–91 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dal, H., Göktepe, S., Kaliske, M., Kuhl, E.: A fully implicit finite element method for bidomain models of cardiac electromechanics. Comput. Methods Appl. Mech. Eng. 253, 323–336 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fan, J., Cristo, M.D., Jiang, Y., Nakamura, G.: Inverse viscosity problem for the Navier-Stokes equation. J. Math. Anal. Appl. 365, 750–757 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Henriquez, C.S.: Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21, 1–77 (1993)MathSciNetGoogle Scholar
  14. 14.
    Imanuvilov, O.Y.: Controllability of parabolic equations. Sbornik Math. 186, 879–900 (1995)Google Scholar
  15. 15.
    Imanuvilov, O.Y., Yamamoto, M.: Carleman estimates for the non-stationary Lemé system and application to an inverse problem, ESAIM: control. Optim. Calc. Var. 11, 1–56 (2005)CrossRefGoogle Scholar
  16. 16.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998)CrossRefMATHGoogle Scholar
  17. 17.
    Klibanov, M.V., Timonov, A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Kunisch, K., Wagner, M.: Optimal control of the bidomain system (I): the monodomain approximation with the Rogers-McCulloch model. Nonlinear Anal. Real World Appl. 13, 1525–1550 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lassoued, J., Mahjoub, M., Zemzemi, N.: Stability results for the parameter identification inverse problem in cardiac electrophysiology. Inverse Probl. 32, 115002 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lopez-Rincon, A., Bendahmane, M., Ainseba, B.: On 3d numerical inverse problems for the bidomain model in electrocardiology. Comput. Math. Appl. 69, 255–274 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lü, Q., Yin, Z.: Unique continuation for stochastic heat equations. Esaim Control Optim. Calc. Var. 21, 378–398 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Puel, J.P., Yamamoto, M.: On a global estimate in a linear inverse hyperbolic problem. Inverse Probl. 12, 995–1002 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Seo, I.: Global unique continuation from a half space for the Schrödinger equation. J. Funct. Anal. 266, 85–98 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Uesaka, M., Yamamoto, M.: Carleman estimate and unique continuation for a structured population model. Appl. Anal. 95, 599–614 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Veneroni, M.: Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10, 849–868 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vigmond, E.J., Santos, R.W., Prassl, A.J., Deo, M., Plank, G.: Solvers for the cardiac bidomain equations. Progr. Biophys. Mol. Biol. 96(1–3), 3–18 (2008)CrossRefGoogle Scholar
  27. 27.
    Wu, B., Yu, J.: Hölder stability of an inverse problem for a strongly coupled reaction-diffusion system. IMA J. Appl. Math. 82, 424–444 (2017)MathSciNetGoogle Scholar
  28. 28.
    Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Probl. 25, 123013 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Yuan, G., Yamamoto, M.: Lipshitz stability in the determination of the principal part of a parabolic equation. ESAIM Control Optim. Calc. Var. 15, 525–554 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina

Personalised recommendations