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

Jianfeng Lv

Advisor: Sima Setayeshgar

May 15, 2009

Outline

Motivation

Numerical Implementation

Numerical Results

Conclusions and Future Work

Motivation:

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.

Patch size: 5 cm x 5 cm Time spacing: 5 msec

[1] W.F. Witkowski, et al., Nature 392, 78 (1998)

Spiral Waves and Cardiac ArrhythmiasTransition 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, 57-64 (1998)

Cardiac Tissue Structure

Cells are typically30 – 100 µm long8 – 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

Cable Equation and Monodomain Model Early studies used the 1-D cable equation to describe the electrical behavior of a cylindrical fiber.

mm m m

VC V I

��

Adapted from J. P. Keener and J. Sneyd, Mathematical Physiology

transmembrane potential: intra- (extra-) cellular potential:

capacitance per unit area of membrane:conductivity tensor:

transmembrane current (per unit length):

tI( )i eV V

axial currents:

resistances (per unit length):

ionic current:, i eI I

, i er r

Bidomain Model of Cardiac Tissue

From Laboratory of Living State Physics, Vanderbilt University

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-

[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, 137-1999 (1993)

Bidomain Model

Ohmic axial currents:

Conservation of total currents: 0i i e eV V D D��

, i i i e e eI V I V D D��

, 0a i e aI I I I ��

Transmembrane current:

t i i e eI V V D D��

( )mt m m e e

VI C I V

��

mt m m

VI C I

|| 0 0

Bidomain:

Conductivity Tensors

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]

The ratio of the intracellular and extracellular conductivity tensors;

Monodomain:

[1] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, 232-240 (1994)[2] J. P. Wikswo, et al., Biophysical Journal 69, 2195-2210 (1995)

Monodomain ReductionBy setting the intra- and extra-cellular conductivity matrices proportional to each other, the bidomain model can be reduced to monodomain model.

i i e e m aV V I D D D

��

1 1mm m i i e e m i i e a

VC I V I

D D D D D D D��

, a i i e e m i eI V V V V V D D��

If , then we obtain the monodomain model.i eD D

mm m m

VC I V

��

Substitute (1) into ( )mm m i i

VC I V

��

1( )i i e e D D D D D

Rotating AnisotropyLocal Coordinate Lab Coordinate

1lab localR RD D

cos sin

sin cos

From Streeter, et al., Circ. Res. 24, p.339 (1969)

Coordinate System

Governing Equations

Perturbation Analysis

Scroll Twist Solutions

Scroll Twist, z

Rotating anisotropy generated scroll twist, either at the boundaries or in the bulk.

Significance?

In isotropic excitable media ( = 1), for twist > twistcritical, straight filament undergoes buckling (“sproing”) instability [1]

Henzi, Lugosi and Winfree, Can. J. Phys. (1990).

What happens in the presence of rotating anisotropy ( > 1)??

Filament Motion

Filament motion (cont’d)

Filament Tension

Destabilizing or restabilizing role of rotating anisotropy!!

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

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 .

[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, 028101 (2002) [4] A. V. Panfilov and J. P. Keener, Physica D 84, 545 (1995)[5] Fenton, F. and Karma, A. Chaos 8, 20 (1998):

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.

• Rotating anisotropy can induce the breakdown of scroll wave;• Rotating anisotropy leads to “twistons”, eventually destabilizing scroll filament;

Numerical Implementationof the Bidomain Equations with Rotating Anisotropy

Transmembrane potential propagation

: transmembrane potential: intra- (extra-) cellular potential: ionic current: conductivity tensor in intra- (extra-) cellular space

Governing equations describing the intra- and extracellular potentials:

( ) (( ) ) 0i m i e eV V D D D��

Governing Equations

Conservation of total current

mI( )i eD D

( )i eV V

( )me e m

��

Ionic current models Ionic current, , described by a FitzHugh-Nagumo-like kinetics [1]

( )( )

I f V w

dwV kV w

2 2 1 2

( ) , ( ) , when V

( ) , ( ) , when e

( ) ( 1), ( ) , when V

where 0.0065, 0.841, 0.15, 3

20, 3, 15;

0.14; 0

m m m m

f V c V V e

f V c V a V V e

f V c V V e

e e a k

3.0589; 2.5

[1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)

These parameters specify the fast processes such as initiation of the action potential. The refractoriness of the model is determined by the function . ( )mV

Boundary conditions No-flux boundary conditions:

Normal vector to the domain boundary: Conductivity tensors in natural frame:

��

,i eD D

or , or ( )i e e e mV V V V D D D

D D V x

n D D V y

(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1) and (0,0,-1)n For a rectangular,

Numerical Implementation

1 ( )n

m mn e e m

V VD V I V

1( ) ( )

m me e e e m

V VD V D V I V

Numerical solution of parabolic PDE (for Vm )

Forward Euler scheme:

Crank-Nicolson scheme:

( )me e m

��

The spacial operator is approximated by the finite difference matrix operator ( )e eV D��

Numerical solution of elliptic PDE (for Ve )

Direct solution of the resulting systems of linear algebraic equations by LU decomposition.

(( ) ) ( )i e e i mV V D D D��

1 1 1 111 1

2 2 2 2 211 2

1 3 3 3 311 3

m a b V f V

c m a b V f V

d c m a V f V

Numerical Implementationcont’d

ai , bi , ci , mi are coefficients of terms after discretization of LHS.

i j kV denotes the extracellular potential Ve on node (x=i, y=j, z=k).

( )mif V denotes the corresponding RHS after discretization.

Index re-ordering to reduce size of band-diagonal system

1 1 11

2 2 2 22

3 3 33

2 2 2 2

111 211 311 11 112 212 312 1 121 221 321

x x x x

N N N N

N N jx z

m a b cd m a b c

d m b c

e d m a

Elements ai, bi, ci … are constants obtained in finite difference approximation to the elliptic equation.

Numerical Implementationcont’d

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.

Filament-finding algorithm

Search for the closest tip

“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

Continue doing search

Continue

The closest tip is too far

Reverse the search direction

Continue

Complete the filament

Start a new filament

Repeat until all tips are consumed

Numerical Results

Numerical ResultsFilament dynamics of Bidomain

Examples of filament-finding results used to characterize breakup.

Time (s)

|| ||/ 0.06, / 0.4i i e eD D D D

Time (s)

Time (s) Time (s)

|| ||/ 0.3, / 0.4i i e eD D D D

Numerical Resultsof previous work in Monodomain

Previous study has shown rotating anisotropy can induce the breakdown of scroll wave.[1]

[1] A. V. Panfilov and J. P. Keener, Physica D 84, 545-552 (1995)

Iso surfaces of 3D view of scroll wave in the medium with = 0.1111||/D D

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.

||/D D

Results of computational experiments with different parameters of cardiac tissue.

TwistThickness

(layer)

Irregular behavior

Monodomain[1] Monodomain Bidomain

∆x=0.5 ∆x=0.2 ∆x=0.5

0.3 120o 9 No No No

0.1 120o 9 Yes No Yes

0.06 120o 9 Yes Yes Yes

0.1 40o 9 No No Yes

0.1 40o 3 No No No

[1] A. V. Panfilov and J. P. Keener Physica D 1995

Numerical ResultsBidomain/Monodomain Comparison

||/D D

For ∆x=0.5, the size of rectangular grid is 60x60x9 pointsFor ∆x=0.2, the size of rectangular grid is 150x150x23 points

Numerical Results:Larger Domain Size Result

Time (s)

Contour plots of transmembrane potential selected tissue layers at t = 750 time units. Scroll wave breakup is evident in the middle layers.

Model size: 140x294x48; ∆x = ∆y = ∆z = 0.25 (space units) Time step: ∆t = 0.01 (time units) ;

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;

Multigrid Techniques:Multigrid Hierarchy

InterpolateRestrict

RelaxDragica Vasileska, “Multi-Grid Method”

Multigrid Techniques:Multigrid method

Coarse-grid correction•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

Structure of multigrid cycles

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”

“Numerical Recipes in C”, 2nd Editoin

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

Multigrid Techniques:Interpolation

Trilinear interpolation between the grids

2D interpolation

4 2 41 1

12 21 1 1

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.

3D interpolation

Dragica Vasileska, “Multi-Grid Method”

Multigrid Techniques:Restriction

2D Restriction 3D 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”

Multigrid ResultsConvergence 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

Multigrid ResultsConvergence 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.

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;

Thank you

Ionic current models cont.

Ionic current described by a FitzHugh-Nagumo-like kinetics[1]

1(1 )[ ( )]m m m m

I V V V f w

( ) ( ) f w w b a

[1] Barkley D. (1991) "A model for fast computer simulation of waves in excitable media". Physica 49D, 61–70.

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

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