1. Advisor: Sima Setayeshgar
Role of Bidomain Model of Cardiac Tissue in the Dynamics of Phase Singularities
Jianfeng Lv
Advisor: Sima Setayeshgar
May 15, 2009

2. Outline Motivation Numerical Implementation Numerical Results
Conclusions and Future Work

3. [1] W.F. Witkowski, et al., Nature 392, 78 (1998)
Motivation:
Patch size: 5 cm x 5 cm Time spacing: 5 msec
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.
And … the heart is an interesting arena for applying the ideas of pattern formation.
[1] W.F. Witkowski, et al., Nature 392, 78 (1998)

4. Spiral Waves and Cardiac Arrhythmias
Transition from ventricular tachycardia 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. [1]
Tachychardia
Fibrillation
Courtesy of Sasha Panfilov, University of Utrecht
Goal is to use analytical and numerical tools to study the dynamics of reentrant waves in the heart on physiologically realistic domains.
[1] A. V. Panfilov, Chaos 8, (1998)

5. Cardiac Tissue Structure
Cells are typically
30 – 100 µm long
8 – 20 µm wide
Propagation Speeds
= 0.5 m / s
= 0.17 m / s
Guyton and Hall, “Textbook of Medical Physiology”
Nigel F. Hooke, “Efficient simulation of action potential propagation in a bidomain”, 1992

6. Cable Equation and Monodomain Model
Early studies used the 1-D cable equation to describe the electrical behavior of a cylindrical fiber.
transmembrane potential:
intra- (extra-) cellular potential:
transmembrane current (per unit length):
axial currents:
ionic current:
conductivity tensor:
capacitance per unit area of membrane:
resistances (per unit length):
Adapted from J. P. Keener and J. Sneyd, Mathematical Physiology

7. Bidomain Model of Cardiac Tissue
The bidomain model treats the complex microstructure of cardiac tissue 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.[1-3]
From Laboratory of Living State Physics, Vanderbilt University
[1] J. P. Keener and J. Sneyd, Mathematical Physiology
[2] C. S. Henriquez, Critical Reviews in Biomedical Engineering 21, 1-77 (1993)
[3] J. C. Neu and W. Krassowska, Critical Reviews in Biomedical Engineering 21, (1993)

8. Bidomain Model Transmembrane current: Ohmic axial currents:
Conservation of total currents:

9. Conductivity Tensors Bidomain: Monodomain:
The ratio of the intracellular and extracellular conductivity tensors;
Bidomain:
Monodomain:
Cardiac tissue is more accurately described as a three-dimensional anisotropic bidomain, especially under conditions of applied external current such as in defibrillation studies.[1-2]
[1] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, (1994)
[2] J. P. Wikswo, et al., Biophysical Journal 69, (1995)

10. Monodomain Reduction
By setting the intra- and extra-cellular conductivity matrices proportional to each other, the bidomain model can be reduced to monodomain model.
(1)
Substitute (1) into
If , then we obtain the monodomain model.

11. From Streeter, et al., Circ. Res. 24, p.339 (1969)
Rotating Anisotropy
Local Coordinate
Lab Coordinate
From Streeter, et al., Circ. Res. 24, p.339 (1969)

12. Coordinate System

13. Governing Equations

14. Perturbation Analysis

15. Scroll Twist Solutions
Scroll Twist, Fz
Twist
Twist
Rotating anisotropy generated scroll twist,
either at the boundaries or in the bulk.

16. Significance?
In isotropic excitable media (a = 1), for twist > twistcritical,
straight filament undergoes buckling (“sproing”) instability [1]
What happens in the presence of rotating anisotropy (a > 1)??
Henzi, Lugosi and Winfree, Can. J. Phys. (1990).

17. Filament Motion

18. Filament motion (cont’d)

19. Filament Tension
Destabilizing or restabilizing role of rotating anisotropy!!

20. Phase Singularity
Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively.

21. Focus of this work
Analytical and numerical works[1-5] have been done on studying the dynamic of scroll waves in monodomain in the presence of rotating anisotropy .
Rotating anisotropy can induce the breakdown of scroll wave;
Rotating anisotropy leads to “twistons”, eventually destabilizing scroll filament;
The focus of this work is computational study of the role of rotating anisotropy on the dynamics of phase singularities in bidomain model of cardiac tissue as a conducting medium.
[1] Biktashev, V. N. and Holden, A. V. Physica D 347, 611(1994)
[2] Keener, J. P. Physica D 31, 269 (1988)
[3] S. Setayeshgar and A. J. Bernoff, PRL 88, (2002)
[4] A. V. Panfilov and J. P. Keener, Physica D 84, 545 (1995)
[5] Fenton, F. and Karma, A. Chaos 8, 20 (1998):

22. Numerical Implementation
of the Bidomain Equations with Rotating Anisotropy

23. Governing Equations
Governing equations describing the intra- and extracellular potentials:
Transmembrane potential propagation
Conservation of total current
: transmembrane potential
: intra- (extra-) cellular potential
: ionic current
: conductivity tensor in intra- (extra-) cellular space

24. Ionic current models
Ionic current, , described by a FitzHugh-Nagumo-like kinetics [1]
These parameters specify the fast processes such as initiation of the action potential. The refractoriness of the model is determined by the function
[1] A. V. Panfilov and J. P. Keener, Physica D 84, (1995)

25. Boundary conditions No-flux boundary conditions: Let
For a rectangular,
Normal vector to the domain boundary:
Conductivity tensors in natural frame:

26. Numerical Implementation
Numerical solution of parabolic PDE (for Vm )
Forward Euler scheme:
Crank-Nicolson scheme:
The spacial operator is approximated by the finite difference matrix operator

27. Numerical Implementation cont’d
Numerical solution of elliptic PDE (for Ve )
Direct solution of the resulting systems of linear algebraic equations by LU decomposition.
ai , bi , ci , mi are coefficients of terms after discretization of LHS.
denotes the extracellular potential Ve on node (x=i, y=j, z=k).
denotes the corresponding RHS after discretization.

28. Numerical Implementation cont’d
Index re-ordering to reduce size of band-diagonal system
Elements ai, bi, ci … are constants obtained in finite difference approximation to the elliptic equation.

29. Numerical Convergence
A time sequence of a typical action potential with various time-steps.
The figures show that time step δt = 0.01 is suitable taking both efficiency and accuracy of computation into account.

30. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Search for the closest tip

31. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Make connection

32. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Continue doing search

33. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Continue

34. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Continue

35. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Continue

36. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
The closest tip is too far

37. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Reverse the search direction

38. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Continue

39. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Complete the filament

40. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Start a new filament

41. Filament-finding algorithm
“Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface
Repeat until all tips are consumed

42. Numerical Results

43. Numerical Results Filament dynamics of Bidomain
Examples of filament-finding results used to characterize breakup.
Total filament length(grid points)
Filament number
Time (s)
Time (s)
Filament number
Total filament length(grid points)
Time (s)
Time (s)

44. Numerical Results of previous work in Monodomain
Previous study has shown rotating anisotropy can induce the breakdown of scroll wave.[1]
Model size : 60x60x9 for 10mm thickness
No break-up while the fiber rotation is less then 60o or total thickness is less than 3.3mm.
Iso surfaces of 3D view of scroll wave in the medium with =
[1] A. V. Panfilov and J. P. Keener, Physica D 84, (1995)

45. Numerical Results Bidomain/Monodomain Comparison
Results of computational experiments with different parameters of cardiac tissue.
Twist
Thickness
(layer)
Irregular behavior
Monodomain[1]
Monodomain
Bidomain
∆x=0.5
∆x=0.2
0.3
120o 9 No
0.1
Yes
0.06
60o
40o 5 3
For ∆x=0.5, the size of rectangular grid is 60x60x9 points
For ∆x=0.2, the size of rectangular grid is 150x150x23 points
[1] A. V. Panfilov and J. P. Keener Physica D 1995

46. Numerical Results: Larger Domain Size Result
Contour plots of transmembrane potential selected tissue layers at t = 750 time units. Scroll wave breakup is evident in the middle layers.
Filament number
Time (s)
Total filament length(grid points)
Model size: 140x294x48;
∆x = ∆y = ∆z = 0.25 (space units) Time step: ∆t = 0.01 (time units) ;
Time (s)

47. Conclusions so far …
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.
In the finer monodomain model and bidomain model, the boundaries of irregular behavior shift;
Numerical Limitation:
Large space step in previous study causes mesh effect;
Model size is too small. Increasing model size in bidomain model is limited by the physical memory;

48. Multigrid Techniques: Multigrid Hierarchy
Relax
Restrict
Interpolate
Relax
Relax
Relax
Relax
Dragica Vasileska, “Multi-Grid Method”

49. Multigrid Techniques:
Multigrid method
Coarse-grid correction
Structure of multigrid cycles
Compute the defect on the fine grid;
Restrict the defect;
Solve exactly on the coarse grid for the correction;
Interpolate the correction to the fine grid;
Compute the next approximation
Relaxation
“Numerical Recipes in C”, 2nd Editoin
S denotes smoothing; E denotes exact solution on the finest grid.
Descending line \ denotes restriction, each ascending line / denotes prolongation.
William L. Briggs, “A Multigrid Tutorial”

50. Multigrid Techniques:
Full Multigrid Algorithm
Multigrid method starts with some initial guess on the finest grid and carries out enough cycles to achieve convergence. Efficiency can be improved by using the Full Multigrid Algorithm (FMG)
FMG with the exact solution at the coarsest level. It uses V-cycles (W-cycles) as the solver on each grid level.
“Numerical Recipes in C”, 2nd Editoin

51. Multigrid Techniques:
Interpolation
Trilinear interpolation between the grids
3D interpolation
2D interpolation
The arrows denote the coarse grid points to be used for interpolating the fine grid point. The numbers attached to the arrows denote the contribution of the specific coarse grid point.
Dragica Vasileska, “Multi-Grid Method”

52. Multigrid Techniques:
Restriction
3D Restriction
2D Restriction
In 3D, A 27-point full weighting scheme is used. The number in front of each grid point denotes its weight in this operation.
Dragica Vasileska, “Multi-Grid Method”

53. Multigrid Results Convergence in 2D
Typical action potential with various Pre and Post Relaxation-steps.
The figures show that in 2D relaxation step 200 is suitable taking both efficiency and accuracy of computation into account.
The domain is 127x127

54. Multigrid Results Convergence in 3D
Typical action potential with various Pre and Post Relaxation-steps.
In the case of 3D, relaxation step 200 is also an appropriate number taken both efficiency and accuracy into account.
The domain is 127x127x7, the convergence plot and density plot are taken at Z=4.

55. Future Work
Improve numerical efficiency, optimize the multigrid code to reduce the computation time;
Systematic exploration of the role of cell electrophysiology in rotating anisotropy-induced scroll break-up in the Bidomain model;

56. Thank you

57. Ionic current models cont.
Ionic current described by a FitzHugh-Nagumo-like kinetics[1]
It is a simple model to study the dynamics of spiral wave in the excitable media.
The e is chosen small so that the time scale of Vm is much faster than that of w. The local kinetics in the absence of spatial derivatives, has a stable but excitable fixed point at the intersection of the nullclines Im = 0 and dv/dt = 0; At any instant in time, almost all spatial points are within the dash line boundary layer. Uth is the excitability threshold for the fixed point. I.C. near the fixed point and to the left of the threshold decay to the fixed point. I.C. to the right of the threshold undergo a large excursion before returning to the fixed point.
[1] Barkley D. (1991) "A model for fast computer simulation of waves in excitable media". Physica 49D, 61–70.