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• The 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
Numerical Results
Conductivity Tensors
Spiral Waves and Cardiac Arrhymias

• Results of computational experiments with
  different parameters of cardiac tissue

• In Bidomain

W.F. Witkowksi, et al., Nature 392, 78 (1998)

• In Monodomain

• Activation map in cardiac tissue using Bidomain
  model

• 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.

Epicardium
Midmyocardium
Endocardium
Epicardium
Midmyocardium
Endocardium
Filament-finding result for Bidomain Model with
Twist 120 Thickness 10mm
t 0 s
t 100 s
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
conductivities are treated proportionally
Filament number
t 5 s
t 150 s
Patch size 5 cm x 5 cm Time spacing 5 msec
t 10 s
t 200 s
Time (s)
column is for Bidomain model
only Spatial step dx0.5mm, dy0.5mm, dz0.5mm,
dt0.01s
Rotating Anisotropy
Monodomain results are from monodomain code.

Bidomain Model of Cardiac Tissue
Numerical Results

• The orientation of fibers in succe-ssive layers
  of cardiac tissue rotates through the thickness.

t 50 s
t 250 s

• A time sequence of a typical action potential

In Bidomain model, we view cardiac tissue as a
two-phase medium, as if every point in space is
composed of a certain fraction of intracellular
space and a fraction of extracellular space. 1
The convergence result in three-dimension
Bidomain model
Thickness10mm Linear twist 120o dx0.5mm,
dy0.5mm, dz0.5mm dt0.01s Rectangular grid
60 x 60 x 9
Transmembrane potential u (mv)
Transmembrane potential u (mv)

• The comparison of Bidomain model and Monodomain
  model

Bidomain
Reduced Bidomain
t 0 s
Live state physics, Vanderbilt University
1 J. Keener, J. Sneyd, Mathematical
Physiology,
The monodomain result is obtained from reduced
Bidomain model by allowing the conductivities
tensors in the intra- and extra-cellular
proportional. We use filament-finding algorithm
to determine the break-up behavior of the spiral
wave. If there are more than 2 filaments, the
spiral wave breaks up.
Numerical Implementation
Big Picture
Time step (s)
Time (s)
Rectangular grid 60 x 60 x 9 Spatial steps dx
0.5mm, dy0.5mm, dz0.5mm
t 5 s

• Numerical simulation for the parabolic PDE

• Transition from ventricular tachychardia to
  fibrillation

Conclusion
Forward Euler scheme
Tachychardia
Fibrillation
Paradigm Breakdown of a single spiral wave into
disordered state, resulting from various
mechanisms of spiral wave instability

• We developed various numerical methods to solve
  the Bidomain equations in both 2D and 3D models
  with modified Fitz-Nagumo models as an ionic
  model.

t 10 s
Crank-Nicolson scheme
is approximated by the finite difference matrix
operators

• Numerical simulation for the elliptic PDE

• We studied the break-up of the spiral wave in
  both Monodomain and Bidomain models with fiber
  rotation incorporated.

t 50 s
Courtesty of Sasha Panfilov, University of Utrecht

• Focus of our work

• In our Bidomain model, the anisotropy of
  coupling plays an important in the break-up of
  spiral wave, the fiber rotation has a less
  prominent role. While fiber rotation is important
  in Monodomain model.

Computational study of the role of the rotating
anisotropy of cardiac tissue within the Bidomain
model.
Direct solving the system of linear algebraic
equations by LU decomposition
Reduced Bidomain
Bidomain
In both models, Thickness 10mm Twist 120o
The fiber direction is 0o at the epicardium
and 120o at the endocardium. The conductivity
tensors using in the Reduced Monodomain is
t 100 s
Governing Equations
Future Work

• In our rectangular model, we have
  , by re-ordering the indices, we reduce
  the size of the compactly stored band-diagonal
  matrix

• The coupled governing equations describing the
  intra- and extracellular potentials are

t 150 s

• Develop Semi-implicit Algorithm to eliminate
  time step limitation.

• Reduce the computational cost of the linear
  solves by developing more efficient numerical
  methods. The linear system Ax y could benefit
  from applying multigrid methods.

• Transmembrane potential propagation

Transmembrane current, , described by a
neurophysiological model adopted for the
FitzHugh-Nagumo system 1
t 200 s

• Conservative of total current

The conductivity tensors using in the Bidomain
model is
Acknowledgements
t 250 s
I thank my advisor Sima for her suggestions on
the models, algorithms and her encouragement
during the reseach. I thank Xianfeng Song for
helpful suggestions on the boundary conditions
and filament-finding algorithm.
capacitance per unit area of membrane
transmembrane potential intra- (extra-)
cellular potential transmembrane current
conductivity tensor in intra- (extra-) cellular
space
The elements a, b, c .. in the matrix depend are
coefficients depending on discretized equations
1 A. V. Panfilov and J. P. Keener Physica D 1995
1 Roth, B.J. IEEE transactions on Biomedical
Engineering, 1997