Role of the Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities

1

• Role of the Bidomain Model of Cardiac Tissue in
  the Dynamics of Phase Singularities
• Jianfeng Lv and Sima SetayeshgarDepartment of
  Physics, Indiana University, Bloomington, Indiana
  47405

Numerical Results
Motivation
Rotating Anisotropy
Comparison of break-up in bidomain and
monodomain models
1 W.F. Witkowksi, et al., Nature 392, 78
(1998)

• Ventricular fibrillation (VF) is the main cause
  of sudden cardiac death in industrialized
  nations, accounting for 1 out of 10 deaths.
• Strong experimental evidence suggests that
  self-sustained waves of electrical wave activity
  in cardiac tissue are related to fatal
  arrhythmias.
• Goal is to use analytical and numerical tools to
  study the dynamics of reentrant waves in the
  heart on physiologically realistic domains.
• And the heart is an interesting arena for
  applying the ideas of pattern formation.

Dissection results indicate that cardiac fibers
are arranged in surfaces, where fibers are
approximately parallel in each surface while the
mean fiber angle rotates from the outer
(epicardium) to inner (endocardium) wall.
Focus of This Work
Patch size 5 cm x 5 cm Time spacing 5 msec
Computational study of the role of the rotating
anisotropy of cardiac tissue on the dynamics of
phase singularities in the bidomain model of
cardiac tissue.
Rectangular grid 60 x 60 x 9 dx0.5 mm,
dy0.5 mm, dz0.5 mm dt0.01s
Spiral Waves and Cardiac Arrhythmias
Governing Equations
Example of filament-finding results used to
characterize breakup (D Q 120)
Transition from ventricular tachychardia to
fibrillation is conjectured to occur as a result
of breakdown of a single spiral (scroll) into a
spatiotemporally disordered state, resulting from
various mechanisms of spiral (scroll) wave
instability.
Governing equations describing the intra- and
extracellular potentials
Filament length(grid points)

• Ionic current, , described by a
  FitzHugh-Nagumo-like kinetics 1

• Transmembrane potential propagation

Filament number
Fibrillation
Tachychardia

• Conservation of total current

Time (s)
Time (s)
Courtesty of Sasha Panfilov, University of Utrecht
Insert refs.
capacitance per unit area of membrane
transmembrane potential intra- (extra-)
cellular potential transmembrane current
conductivity tensor in intra- (extra-) cellular
space
Filament length(grid points)
Filament number
Bidomain Model of Cardiac Tissue
1 A. V. Panfilov and J. P. Keener Physica D
(1995).
Numerical Implementation
The bidomain model treats the complex
microstructure of cardiac tissue is as a
two-phase conducting medium, where every point in
space is composed of both intra- and
extracellular spaces and both conductivity
tensors are specified at each point.

• Numerical solution of parabolic PDE (for um )

Time (s)
Time (s)
Conclusions
Forward Euler scheme

• We have numerically implemented electrical wave
  propagation in the bidomain model of cardiac
  tissue in the presence of rotating anisotropy
  using FHN-like reaction kinetics.
• Preliminary numerical results indicate that in
  the bidomain model, scroll wave breakup is more
  sensitive to the anisotropy ratio than the fiber
  rotation rate, in contrast with the monodomain
  model.

From Laboortatory of Living State Physics,
Vanderbilt University
Crank-Nicolson scheme
Conductivity Tensors
is approximated by the finite difference matrix
operator,

• Numerical solution of elliptic PDE (for ue )

Direct solution of the resulting systems of
linear algebraic equations by LU decomposition.
Bidomain
Monodomain
The ratios of the diffusion constants along and
perpendicular to the fiber direction in the
intra- and extra-cellular spaces are different.
The intracellular and extracellular conductivity
tensors are proportional.
Future Work
Index re-ordering to reduce size of
band-diagonal system
Insert text refs.
Acknowledgements
We acknowledge support from the National Science
Foundation and Indiana University. We thank
Xianfeng Song in our group for helpful advice on
various aspects of the numerical implementation.
2 J. P. Keener and J. Sneyd, Mathematical
Physiology 3 C. S. Henriquez, Critical Reviews
in Biomedical Engiineering 21, 1-77 (1993)
Elements ai, bi, ci are constants obtained in
finite difference approximation to the elliptic
equation.