1. W.F. Witkowksi, et al., Nature 392, 78 (1998)
The role of the bidomain model of cardiac tissue in the dynamics of phase singularities
Jianfeng Lv and Sima Setayeshgar Department of Physics, Indiana University, Bloomington, Indiana 47405
Numerical Results
Spiral Waves and Cardiac Arrhymias
Conductivity Tensors
Numerical Results
Results of computational experiments with different parameters of cardiac tissue
W.F. Witkowksi, et al., Nature 392, 78 (1998)
In Monodomain:
In Bidomain:
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.
Activation map in cardiac tissue using Bidomain model
Epicardium
Midmyocardium
Endocardium
Twist
Thickness
Break-up
Monodomain
Bidomain
0.7
0.4
120o
10mm No
0.5
Yes
0.3
0.1
0.06
60o
40o
5 mm
3.3 mm
Epicardium
Midmyocardium
Endocardium
Filament-finding result for Bidomain Model with
Twist = 120; Thickness = 10mm;
t = 0 s
t = 100 s
The intracellular and extracellular conductivities are treated proportionally
The ratios of the diffusion constants along and perpendicular to the fiber direction in the intra- and extra- cellular spaces are different.
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 dx=0.5mm, dy=0.5mm, dz=0.5mm, dt=0.01s
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
Thickness=10mm; Linear twist = 120o;
dx=0.5mm, dy=0.5mm, dz=0.5mm; dt=0.01s;
Rectangular grid 60 x 60 x 9
Transmembrane potential u (mv)
Transmembrane potential u (mv)
The comparison of Bidomain model and Monodomain model
Reduced Bidomain
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 (s)
Time step (s)
Rectangular grid 60 x 60 x 9; Spatial steps dx = 0.5mm, dy=0.5mm, dz=0.5mm;
t = 5 s
Numerical simulation for the parabolic PDE
Transition from ventricular tachychardia to fibrillation
Conclusion
Tachychardia
Forward Euler scheme:
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
[1] A. V. Panfilov and J. P. Keener Physica D 1995
The elements a, b, c .. in the matrix depend are coefficients depending on discretized equations
[1] Roth, B.J. IEEE transactions on Biomedical Engineering, 1997