Skip to main content
SpringerLink
Log in

A finite volume model of cardiac propagation

  • Research Articles
  • Published:
Annals of Biomedical Engineering Aims and scope Submit manuscript

Abstract

This paper describes a two-dimensional cardiac propagation model based on the finite volume method (FVM). This technique, originally derived and applied within the field of computational fluid dynamics, is well suited to the investigation of conduction in cardiac electrophysiology Specifically, the FVM permits the consideration of propagation in a realistic structure, subject to arbitrary fiber orientations and regionally defined properties. In this application of the FVM, an arbitrarily shaped domain is decomposed into a set of constitutive quadrilaterals. Calculations are performed in a computational space, in which the quadrilaterals are all represented simply as squares. Results are related to their physical-space equivalents by means of a transformation matrix. The method is applied to a number of cases. First, large-scale propagation is considered, in which a magnetic resonance-imaged cardiac cross-section serves as the governing geometry. Next, conduction is examined in the presence of an isthmus formed by the microvasculature in a slice of papillary muscle tissue. Under ischemic conditions, the safety factor for propagation is seen to be related to orientation of the fibers within the isthmus. Finally, conduction is studied in the presence of an inexcitable obstacle and a curved fibe field. This example illustrates the dramatic influence of the complex orientation of the fibers on the resulting activation pattern. The FVM provides a means of accurately modeling the cardiac structure and can help bridge the gap between computation and experiment in cardiac electrophysiology.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

Φ i :

intracellular potential (mV)

Φ e :

extracellular potential (mV)

V m :

transmembrane potential (mV)

V max :

maximum transmembrane potential (mV)

V m,j :

V m in a particular point in the distorted grid (mV)

\(\bar V_{m,i} \) :

V m for a particular point in the undistorted grid (mV)

\(\dot V_{max} \) :

maximum rate of rise of the transmembrane potential (mV/sec)

\(\dot V_m \) :

rate of rise of the transmembrane potential (mV/sec)

C :

membrane capacitance (μF/cm2)

β :

ratio of cell surface area to volume (1/cm)

Δt :

increment between successive time steps (msec)

[Na]i :

intracellular sodium (mM)

[Na]o :

extracellular sodium (mM)

[K]i :

intracellular potassium (mM)

[K]o :

extracellular potassium (mM)

[Ca]i :

intracellular calcium (mM)

I ion :

transmembrane ionic current (μA/cm2)

J i :

intracellular current (total) (μA/cm2)

J ix :

intracellular current,x direction (μA/cm2)

J iy :

intracellular current,y direction (μA/cm2)

I v :

volumetric transmembrane current (μA/cm3)

L x :

length of an edge projected on thex axis (cm2)

L y :

length of an edge projected on they axis (cm2)

A :

area of the quadrilateral of interest (cm3)

D i :

intracellular conductivity tensor (mS/cm)

σlong :

conductivity along the fast fiber axis (mS/cm)

σtrans :

conductivity along the slow fiber axis (mS/cm)

σ x :

conductivity in thex direction (mS/cm)

σdiag :

conductivity in they direction (mS/cm)

σ y :

conductivity in the diagonal direction (mS/cm)

α:

fiber angle measured with respect to the positive x axis

ζ:

abscissa in computational space

η:

ordinate in computational space

x :

abscissa in physical space

y :

ordinate in physical space

N j :

shape functions for a four-node plane quadrilateral

T :

Jacobian transformation matrix

T −1 :

inverted Jacobian transformation matrix

k :

summation index

j :

summation index

Det:

determinant

N :

number of elements in the grid

References

  1. Abboud, S., Y. Eshel, S. Levy, and M. Rosenfeld. Numerical calculation of the potential distribution due to dipole sources in a spherical model of the head.Comput. Biomed. Res. 27:441–455, 1994.

    Article  PubMed  CAS  Google Scholar 

  2. Brugada, J., L. Boersma, C. J. H. Kirchhof, V. V. Th. Heynen, and M. A. Allessie, Reentrant excitation around a fixed obstacle in uniform anisotropic ventricular myocardium.Circulation 84:1296–1306, 1991.

    PubMed  CAS  Google Scholar 

  3. Cabo, C., A. M. Pertzov, W. T. Baxter, J. M. Davidenko, R. A. Gray, and J. Jalife. Wave-front curvature as a cause of slow conduction and block in isolated cardiac muscle.Circ. Res. 75:1014–1028, 1994.

    PubMed  CAS  Google Scholar 

  4. Cascio, W. E., T. A. Johnson, and L. S. Gettes. Electrophysiologic changes in ischemic ventricular myocardium. I. Influence of ionic, metabolic, and energetic changes.J. Cardiovasc. Electrophysiol. 6:1–24, 1995.

    Article  Google Scholar 

  5. Clerc, L. Directional differences in impulse spread in trabecular muscle from mammalian heart.J. Physiol. 255:335–346, 1976.

    PubMed  CAS  Google Scholar 

  6. de la Fuente, D., B. Sasyniuk, and G. K. Moe. Conductance through a narrow isthmus in isolated canine atrial tissue: a model of the W-P-W syndrome.Circulation 44:803–809, 1971.

    PubMed  Google Scholar 

  7. Delgado, C., B. Steinhaus, M. Delmar, D. R. Chialvo, and J. Jalife. Directional differences in excitability and margin of safety for propagation in sheep ventricular epicardial muscle.Circ. Res. 67:97–110, 1990.

    PubMed  CAS  Google Scholar 

  8. Demirdžić, I., and S. Muzaferija. Finite volume method for stress analysis in complex domains.Int. J. Numer. Methods Eng. 37:3751–3766, 1994.

    Article  Google Scholar 

  9. Fast, V. G., and A. G. Kléber. Cardiac tissue geometry as a determinant of unidirectional conduction block: assessment of microscopic excitation spread by optical mapping in patterned cell cultures and in a computer model.Cardiovasc. Res. 29:697–707, 1995.

    Article  PubMed  CAS  Google Scholar 

  10. Franzone, P. C., and L. Guerri. Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field.Math. Biosci. 101:155–235, 1990.

    Article  PubMed  CAS  Google Scholar 

  11. Frazier, D. W., W. Krassowska, P.-S. Chen, P. D. Wolf, N. D. Danieley, W. M. Smith, and R. E. Ideker. Transmural activations and stimulus potentials in three-dimensional anisotropic canine myocardium.Circ. Res. 63:135–146, 1988.

    PubMed  CAS  Google Scholar 

  12. Henriquez, C. S., and R. Plonsey. Effects of resistive discontinuities on waveshape and velocity in a single cardiac fibre.Med. Biol. Eng. Comput. 25:428–438, 1987.

    Article  PubMed  CAS  Google Scholar 

  13. Hirsch, C. Numerical Computation of Internal and External Flows, vol. 1. Chichester: John Wiley & Sons, 1988, pp. 1–525.

    Google Scholar 

  14. Idelsohn, S. R., and E. Oñate. Finite volumes and finite elements: two ‘good friends’.Int. J. Numer. Methods Eng. 37:3323–3341, 1994.

    Article  Google Scholar 

  15. Indik, R. A., and J. H. Indik. A new computer method to quickly and accurately compute steady-state temperatures from ferromagnetic seed heating.Med. Phys. 21:1135–1144, 1994.

    Article  PubMed  CAS  Google Scholar 

  16. Janse, M. J., and A. G. Kléber. Propagation of electrical activity in ischemic and infarcted myocardium as the basis of ventricular arrhythmias.J. Cardiovasc. Electrophysiol. 3:77–87, 1992.

    Article  Google Scholar 

  17. Janse, M. J., and A. L. Wit. Electrophysiological mechanisms of ventricular arrhythmias resulting from myocardial ischemia and infarction.Physiol. Rev. 69:1049–1069, 1989.

    PubMed  CAS  Google Scholar 

  18. Joyner, R. W. Mechanisms of unidirectional block in cardiac tissue.Biophys. J. 35:113–125, 1981.

    PubMed  CAS  Google Scholar 

  19. Joyner, R. W., and F. J. L. van Capelle. Propagation through electrically coupled cells. How a small SA node drives a large atrium.Biophys. J. 50:1150–1164, 1986.

    Article  Google Scholar 

  20. Lee, D., and J. J. Chiu. Computation of flow fields and shear rates in an aortic bifurcation.Front. Med. Biol. Eng. 5:23–29, 1993.

    PubMed  CAS  Google Scholar 

  21. Leon, L. J., and F. A. Roberge. Directional characteristics of action potential propagation in cardiac muscle. A model study.Circ. Res. 69:378–395, 1991.

    PubMed  CAS  Google Scholar 

  22. Luo, C.-H., and Y. Rudy. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction.Circ. Res. 68:1501–1526, 1991.

    PubMed  CAS  Google Scholar 

  23. Maglaverase, N., F. Offner, F. J. L. van Capelle, M. A. Allessie, and A. V. Sahakian. Effects of barriers on propagation of action potential in two-dimensional cardiac tissue.J. Electrocardiol. 28:17–31, 1995.

    Article  Google Scholar 

  24. McCulloch, A., L. Waldman, J. Rogers, and J. Guccione. Large-scale finite element analysis of the beating heart.Crit. Rev. Biomed. Eng. 20:427–449, 1992.

    PubMed  CAS  Google Scholar 

  25. Mendez, C., W. J. Muller, J. Merideth, and G. K. Moe. Interaction of transmembrane potentials in canine Purkinje fibers and at Purkinje fiber-muscle junctions.Circ. Res. 24:361–372, 1969.

    PubMed  CAS  Google Scholar 

  26. Miller, C., and C. Henriquez. Finite element analysis of bioelectric phenomena.Crit. Rev. Biomed. Eng. 18:181–205, 1990.

    Google Scholar 

  27. Overholt, E. D., R. W. Joyner, R. D. Veenstra, D. Rawling, and R. Weidmann. Unidirectional block between Purkinje and ventricular layers of papillary muscles.Am. J. Physiol. 247:H584-H595, 1984.

    PubMed  CAS  Google Scholar 

  28. Quan, W., and Y. Rudy. Unidirectional block and reentry of cardiac excitation: a model study.Circ. Res. 66: 367–382, 1990.

    PubMed  CAS  Google Scholar 

  29. Rogers, J. M., and A. D. McCulloch. A collocation-Galerkin finite element model of cardiac action potential propagation.IEEE Trans. Biomed. Eng. 41:743–757, 1994.

    Article  PubMed  CAS  Google Scholar 

  30. Rohr, S., D. M. Schölly, and A. Kléber. Patterned growth of neonatal rat heart cells in culture.Circ. Res. 68:114–130, 1991.

    PubMed  CAS  Google Scholar 

  31. Spach, M. S., W. T. Miller III, D. B. Geselowitz, R. C. Barr, J. M. Kootsey, and E. A. Johnson. The discontinuous nature of propagation in normal canine cardiac muscle.Circ. Res. 48:39–54, 1981.

    PubMed  CAS  Google Scholar 

  32. Ursell, P. C., P. I. Gardner, A. Albala, J. J. Fenolio, Jr., and A. L. Wit. Structural and electrophysiological changes in the epicardial border zone of canine myocardial infarcts during infarct healing.Circ. Res. 5:436–451, 1985.

    Google Scholar 

  33. Wikswo, J. P., Jr., T. A. Wisialowski, W. A. Altemeier, J. R. Balser, H. A. Kopelman, and D. M. Roden. Virtual cathode effects during stimulation of cardiac muscle. Two-dimensionalin vivo experiments.Circ. Res. 68:513–530, 1991.

    PubMed  Google Scholar 

  34. Yamaguchi, T. A computational fluid mechanical study of blood flow in a variety of asymmetric arterial bifurcations.Front. Med. Biol. Eng. 5:135–141, 1993.

    PubMed  CAS  Google Scholar 

  35. Zienkiewicz, O. C., and E. Oñate. Finite volumes versus finite elements. Is there really a choice? In: Nonlinear computations mechanics. State of the art, edited by P. Wriggers and W. Wagner. Berlin: Springer, 1991, pp. 240–254.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, Duke University, Box 90281, 27708-0281, Durham, NC, U.S.A.

    David M. Harrild & Craig S. Henriquez

Authors
  1. David M. Harrild
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Craig S. Henriquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harrild, D.M., Henriquez, C.S. A finite volume model of cardiac propagation. Ann Biomed Eng 25, 315–334 (1997). https://doi.org/10.1007/BF02648046

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02648046

Keywords

Search

Navigation