Observability and Inverse Problems Arising in Electrocardiography

  • I. Iakovidis
  • C. F. Martin
  • S. Xie
Part of the Progress in Systems and Control Theory book series (PSCT, volume 1)

Abstract

The problem we consider arises in electrocardiography. It is one of a general class of inverse problems in electrocardiography and its goal is the reconstruction of cardiac electrical events from information obtained non-invasively at the body surface. In this paper we formulate the problem as a problem in observability by considering the heart-torso system as a dynamical system governed by the Laplace’s equation with input the electrical potential distribution on the surface of the heart and output the potential distribution on the torso. The problem then becomes the reconstruction of as much information as possible about the state of the system using only a finite number of measurements on the torso. We will present two analytical approaches to the inverse problem under the assumptions of spherical and cylindrical geometry of the heart-torso system. Also, we will point out the dependence of the solution to the problem on the selection of the location of the measurements. We note that the aim of this paper is to describe the problem and suggest reasonable modes of attack.

Keywords

Inverse Problem Potential Distribution Cubature Formula Body Surface Potential Epicardial Potential 
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]
    R.V CHURCHIll, “Fourier series and boundary value problems”, Mc-Graw Hill, (1963).Google Scholar
  2. [2]
    J.A. Cochran, “Remarks on the zeros of cross-product Bessel functions”, SIAM Jour, of Numer. Analysis, vol. 12, 1964, pp. 580–587.MathSciNetMATHGoogle Scholar
  3. [3]
    P. Colli Franzone, L. Guerri, B. Taccardi and C. Viganotti, “The direct and inverse potential problem in electro-cardiology. Numerical aspects of some regularization methods and appLication to data collected in isolated dog heart experiments”, Publ. No 222, IAN-CNR, Pavia, 1979, pp. 1–82.Google Scholar
  4. [4]
    D. Gilliam, Z. Li and C. Martin, “The observabiLity of the heat equation with sampLing in time and space”, International Journal of Control, 48, 1988, pp. 755–780.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    V.K. Ivanov , “On Linear problems which are not well-posed” Dokl Akad. Nauk SSSR, 145, no.2,1963, pp. 270–272.Google Scholar
  6. [6]
    N.S. KOSHLYAKOV, M.M. SMIRNOV, E.B. GLiNEV, “Differential Equations of Mathematical Physics”, North-Holland, (1964).Google Scholar
  7. [7]
    C.Martin, S.XIE, “Reconstruction of the solution of Laplace’s equation from point measurements on the boundary of the disk”, to appear in The International Journal of Control.Google Scholar
  8. [8]
    B. J. MESSINGER-RAPPORT, “The inverse problem in electrocardiography: A model study of geometry, conductivity, and sampLing parameters”, M. S. thesis, Case Western Reserve Univ., Cleveland, OH, 1985.Google Scholar
  9. [9]
    S.G. MIKHLiN, “Mathematical Physics, An advanced course”, North-Holland, (1964).Google Scholar
  10. [10]
    R.H. Okada, “The image surface of a circular cyLinder”, American Heart Journal, v. 51,1956, pp. 489–500.CrossRefGoogle Scholar
  11. [11]
    D.L. Ragozin,“Uniform convergence of spherical harmonic expansions” Math. Ann., 195,1972, pp. 87–94.Google Scholar
  12. [12]
    Y. Rudy and R. Plonsey, “The eccentric spheres model as the basis for the study of the role of geometry and inhomogeneities in electrocardiography”, IEEE Trans. Biomed. Eng., v. BME-26,1979, pp. 392–399.CrossRefGoogle Scholar
  13. [13]
    A.H. STROUD, “Approximate calculation of multiple integrals”, Prentice Hall, (1971).Google Scholar
  14. [14]
    A. TIKHONOV, V. ARSENIN, “Solutions to ill posed problems”, Halsted Press, (1977).Google Scholar
  15. [15]
    G.P. TOLSTOV, “Fourier Series”, Prentice-Hall, Inc., 1965.Google Scholar
  16. [16]
    G.N. Watson, “A Treaties on the Theory of Bessel Functions”, 2nd ed.,London: Cambridge University Press, 1944.Google Scholar
  17. [17]
    J. A. WOLF, “ObservabiLity and group representation theory, this volume”.Google Scholar
  18. [18]
    Y. Yamashita , “Inverse solution in electrocardiography: Determining epicardial from body surface maps by using the finite element method”, Japan Circ. J., v. 45, November, 1981, pp. 1312–1322.Google Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • I. Iakovidis
    • 1
  • C. F. Martin
    • 1
  • S. Xie
    • 1
  1. 1.Department of MathematicsTexas UniversityLubbockUSA

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