A finite element method for an eikonal equation model of myocardial excitation wavefront propagation

SIAM Journal on Applied Mathematics

Volumn 63, Issue 1, 2002, Pages 324-350

  Tomlinson, Karl A ^a   Hunter, Peter J ^a   Pullan, Andrew J ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

Author keywords

Eikonal equation; Hermite interpolation; Myocardial excitation; Numerical continuation; Petrov Galerkin method; Wavefront propagation

Indexed keywords

APPROXIMATION THEORY; BIOELECTRIC POTENTIALS; BOUNDARY CONDITIONS; CARDIOLOGY; COMPUTER SIMULATION; FINITE DIFFERENCE METHOD; GALERKIN METHODS; INTEGRAL EQUATIONS; INTERPOLATION; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; TISSUE;

EIKONAL EQUATION; HERMITE INTERPOLATION; MYOCARDIAL EXCITATION; NUMERICAL CONTINUATION METHOD; PETROV-GALERKIN METHOD; WAVEFRONT PROPAGATION; WEIGHTED RESIDUAL METHOD;

FINITE ELEMENT METHOD;

EID: 0036706241     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901389513     Document Type: Article

References (24)

1. E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, 1990.
2. J. W. Barrett and K. W. Morton, Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 97-122.
3. A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), pp. 199-259.
4. I. Christie, D. F. Giffiths, A. R. Mitchell, and O. C. Zienkiewicz, Finite element methods for second order differential equations with significant first derivatives, Internat. J. Numer. Methods Engrg., 10 (1976), pp. 1389-1396.
5. P. Colli Franzone and L. Guerri, Models of the spreading of excitation in myocardial tissue, in High-Performance Computing in Biomedical Research, T. C. Pilkington, B. Loftis, J. F. Thompson, S. L.-Y. Woo, T. C. Plamer, and T. F. Budinger, eds., CRC Press, Boca Raton, FL, 1993, pp. 359-401.
6. P. Colli Franzone and L. Guerri, Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I. Validation of the eikonal model, Math. Biosci., 113 (1993), pp. 145-209.
7. P. Colli Franzone, L. Guerri, M. Pennacchio, and B. Taccardi, Spread of excitation in 3-D models of the anisotropic cardiac tissue. II. Effects of fiber architecture and ventricular geometry, Math. Biosci., 147 (1998), pp. 131-171.
8. 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), pp. 121-176.
9. P. Colli Franzone, L. Guerri, and S. Tentoni, Mathematical modelling of the excitation process in the myocardial tissue: Influence of the fibre rotation on the wavefront propagation and the potential field, Math. Biosci., 101 (1990), pp. 155-235.
10. M. Falcone, T. Giorgi, and P. Loreti, Level sets of viscosity solutions: Some applications to fronts and rendez-vous problems, SIAM J. Appl. Math., 54 (1994), pp. 1335-1354.
11. L. S. Green, B. Taccardi, P. R. Ershler, and R. L. Lux, Epicardial potential mapping. Effects of conducting media on isopotential and isochrone distributions, Circulation, 84 (1991), pp. 2513-2521.
12. C. S. Henriquez, Simulating the electrical behaviour of cardiac tissue using the bidomain model, Crit. Rev. Biomed. Engrg., 21 (1993), pp. 1-77.
13. J. P. Keener, An eikonal-curvature equation for action potential propagation in myocardium, J. Math. Biol., 29 (1991), pp. 629-651.
14. J. P. Keener and A. V. Panfilov, Three-dimensional propagation in the heart: The effects of geometry and fiber orientation on propagation in myocardium, in Cardiac Electrophysiology: From Cell to Bedside, 2nd ed., D. P. Zipes and J. Jalife, eds., Saunders, Philadelphia, 1995, pp. 335-347.
15. W. Krassowska and J. C. Neu, Effective boundary conditions for syncytial tissue, IEEE Trans. Biomed. Engrg., 41 (1994), pp. 143-150.
16. I. J. Le Grice, A Finite Element Model of Myocardial Structure: Implications for Electrical Activation in the Heart, Ph.D. thesis, The University of Auckland, Auckland, New Zealand, 1992.
17. K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Appl. Math. Math. Comput. 12, Chapman & Hall/CRC, Boca Raton, FL, 1996.
18. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
19. R. Plonsey and R. C. Barr, Mathematical modeling of electrical activity of the heart, J. Electrocardial., 20 (1987), pp. 219-226.
20. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
21. O. H. Schmitt, Biological information processing using the concept of interpenetrating domains, in Information Processing in the Nervous System, K. N. Leibovic, ed., Springer-Verlag, New York, 1969, pp. 325-331.
22. J. A. Sethian, Theory, algorithms, and applications of level set methods for propagating interfaces, Acta Numer., 1996 (1996), pp. 309-395.
23. K. A. Tomlinson, Finite Element Solution of an Eikonol Equation for Excitation Wavefront Propagation in Ventricular Myocardium, Ph.D. thesis, The University of Auckland, Auckland, New Zealand, 2000.
24. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th ed., McGraw-Hill, Maidenhead, England, 1994.