Resolution Strategies for the Finite-Element-Based Solution of the ECG Inverse Problem

IEEE Transactions on Biomedical Engineering

Volumn 57, Issue 2, 2010, Pages 220-237

  Wang, Dafang ^a   Kirby, Robert M ^a   Johnson, Chris R ^a  

^a Department of Electrical and Computer Engineering   (United States)

Author keywords

Adaptive refinement; ECG forward problem; ECG inverse problem; finite element method (FEM); resolution studies

Indexed keywords

ALGORITHM; ARTICLE; BIOLOGICAL MODEL; COMPUTER SIMULATION; ELECTROCARDIOGRAPHY; FINITE ELEMENT ANALYSIS; HUMAN; IMAGE PROCESSING; METHODOLOGY; SIGNAL PROCESSING;

ALGORITHMS; COMPUTER SIMULATION; ELECTROCARDIOGRAPHY; FINITE ELEMENT ANALYSIS; HUMANS; IMAGE PROCESSING, COMPUTER-ASSISTED; MODELS, BIOLOGICAL; SIGNAL PROCESSING, COMPUTER-ASSISTED;

EID: 77950359471     PISSN: 00189294     EISSN: 15582531     Source Type: Journal    
DOI: 10.1109/TBME.2009.2024928     Document Type: Article

References (40)

1. I. Babuska and J. T. Oden, “Verification and validation in computational engineering and science: Basic concepts,” Comput. Methods Appl. Mech. Eng., vol. 193, no. 36–38, pp. 4057–4066, 2004.
2. J. A. Schmidt, C. R. Johnson, J. C. Eason, and R. S. MacLeod, “Applications of automatic mesh generation and adaptive methods in computational medicine,” in Modeling, Mesh Generation, and Adaptive Methods for Partial Differential Equations. Berlin, Germany: Springer-Verlag, 1995, pp. 367–390.
3. C. R. Johnson, “Computational and numerical methods for bioelectric field problems,” Crit. Rev. Biomed. Eng., vol. 25, no. 1, pp. 1–81, 1997.
4. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. Washington, DC: V. H. Winston & Sons, 1977.
5. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems. Dordrecht, The Netherlands: Kluwer, 1995.
6. V. Isakov, Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer-Verlag, 2006.
7. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems. Norwell, MA: Kluwer, 2000.
8. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996.
9. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems. New York: Springer-Verlag, 2004.
10. F. Natterer, “Error bounds for Tikhonov regularization in Hilbert scales,” Appl. Anal., vol. 18, pp. 29–37, 1984.
11. P. Mathe and S. V. Pereverzev, “Optimal discretization of inverse problems in Hilbert scales, regularization and self-regularization of projection methods,” SIAM J. Numer. Anal., vol. 38, no. 6, pp. 1999–2021, 2001.
12. H. Engl and A. Neubauer, “Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales,” Proc. Amer. Math. Soc., vol. 102, pp. 587–592, 1988.
13. J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math., vol. 198, no. 2, pp. 493–504, 2007.
14. S. Lasanen and L. Roininen, “Statistical inversion with Green's priors,” presented at the 5th Int. Conf. Inverse Problems Eng.: Theory Pract., Cambridge, U.K., 2005.
15. M. Seger, G. Fischer, R. Modre, B. Messnarz, F. Hanser, and B. Tilg, “Lead field computation for the electrocardiographic inverse problem finite elements versus boundary elements,” Comput. Methods Programs Biomed., vol. 77, pp. 241–252, 2005.
16. D. M. Weinstein, C. R. Johnson, and J. A. Schmidt, “Effects of adaptive refinement on the inverse EEG solution,” Proc. SPIE, vol. 2570, pp. 2–11, 1995.
17. C. R. Johnson and R. S. MacLeod, “Nonuniform spatial mesh adaptation using a posteriori error estimates: Applications to forward and inverse problems,” Appl. Numer. Math., vol. 14, pp. 311–326, 1994.
18. G. Shou, L. Xia, M. Jiang, F. Liu, and S. Crozier, “Forward and inverse solutions of electrocardiography problem using an adaptive BEM method,” in Functional Imaging and Modeling of the Heart (Lecture Notes in Computer Science), vol. 4466. Berlin, Germany: Springer-Verlag, 2007, pp. 290–299.
19. C. R. Johnson, “Adaptive finite element local regularization methods for the inverse ECG problem,” in Inverse Problems in Electrocardiology (Advances in Computational Biomedicine). Southampton, U.K.: WIT Press, 2001.
20. A. J. Pullan and C. P. Bradley, “A coupled cubic Hermite finite element/boundary element procedure for electrocardiographic problems,” Comput. Mech., vol. 18, pp. 356–368, 1996.
21. G. Fischera, B. Tilga, P. Wacha, R. Modrea, U. Lederb, and H. Nowak, “Application of high-order boundary elements to the electrocardiographic inverse problem,” Comput. Methods Programs Biomed., vol. 58, pp. 119–131, 1999.
22. B. Kaltenbacher, “V-cycle convergence of some multigrid methods for ill-posed problems,” Math. Comput., vol. 72, no. 244, pp. 1711–1730, 2003.
23. M. Mohr and U. Rude, “Multilevel techniques for the solution of the inverse problem of electrocardiography,” presented at the Multigrid Methods VI, 6th Eur. Multigrid Conf., Gent, Belgium, vol. 14, 1999.
24. C. R. Johnson, M. Mohr, U. Rude, A. Samsonov, and K. Zyp, “Multi-level methods for inverse bioelelectric field problems,” in Lecture Notes in Computational Science and Engineering—Multiscale and Multiresolution Methods: Theory and Applications. Heidelberg, Germany: Springer-Verlag, 2001.
25. P. R. Johnston, S. J. Walker, J. A. K. Hyttinen, and D. Kilpatrick, “Inverse electrocardiographic transformations: Dependence on the number of epicardial regions and body surface data points,” Math. Biosci., vol. 120, pp. 165–187, 1994.
26. R. M. Gulrajani, F. A. Roberge, and G. E. Mailloux, “The forward problem of electrocardiography,” in Comprehensive Electrocardiology, P. W. Macfarlane and T. D. V. Lawrie, Eds. Oxford, U.K.: Pergamon, 1989, pp. 197–236.
27. T. J. R. Hughes, The Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall, 1987.
28. O. Axelsson, Iterative Solution Methods. New York: Cambridge Univ. Press, 1994.
29. G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD. New York: Oxford Univ. Press, 1999.
30. Y. Yamashita, “Theoretical studies on the inverse problem in electrocardiography and the uniqueness of the solution,” IEEE Trans. Biomed. Eng., vol. BME-29, no. 11, pp. 719–725, Nov. 1982.
31. T. Ohe and K. Ohnaka, “A precise estimation method for locations in an inverse logarithmic potential problem for point mass models,” Appl. Math. Model., vol. 18, pp. 446–452, 1994.
32. T. Ohe and K. Ohnaka, “An estimation method for the number of point masses in an inverse logarithmic potential problem using discrete Fourier transform,” Appl. Math. Model., vol. 19, pp. 429–436, 1995.
33. T. Nara and S. Ando, “A projective method for an inverse source problem of the Poisson equation,” Inverse Problems, vol. 19, pp. 355–369, 2003.
34. B. Messnarz, M. Seger, R. Modre, G. Fischer, F. Hanser, and B. Tilg, “A comparison of noninvasive reconstruction of epicardial versus transmembrane potentials in consideration of the null space,” IEEE Trans. Biomed. Eng., vol. 51, no. 9, pp. 1609–1618, Sep. 2004.
35. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the 1-curve,” SIAM Rev., vol. 34, no. 4, pp. 561–580, 1992.
36. B. Punske, Q. Ni, R. F. Fux, R. S. MacFeod, 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., vol. 31, pp. 781–792, 2003.
37. A. Ghodrati, D. Brooks, G. Tadmor, and R. Macleod, “Wavefront-based models for inverse electrocardiography,” IEEE Trans. Biomed. Eng., vol. 53, no. 9, pp. 1821–1831, Sep. 2006.
38. I. Babuska, T. Strouboulis, C. S. Upadhyay, and S. K. Gangaraj, “A model study of the quality of a posteriori estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles,” Comput. Methods Appl. Mech. Eng., vol. 114, pp. 307–378, 1994.
39. R. S. MacFeod, C. R. Johnson, and P. R. Ershler, “Construction of an inhomogeneous model of the human torso for use in computational electrocardiography,” in Proc. C-EMBS91. Piscataway, NJ: IEEE Press, 1991, pp. 688–689.
40. C. R. Johnson, R. S. MacLeod, and P. R. Ershler, “A computer model for the study of electrical current flow in the human thorax,” Comput. Biol. Med., vol. 22, no. 3, pp. 305–323, 1992.