Role of the Bidomain Model of Cardiac Tissue in the Dynamics of Phase SingularitiesJianfeng Lv and Sima Setayeshgar

Department of Physics, Indiana University, Bloomington, Indiana 47405

Motivation

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

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

Bidomain Model of Cardiac Tissue

Spiral Waves and Cardiac ArrhythmiasTransition 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. [2]

Tachychardia Fibrillation

Courtesty of Sasha Panfilov, University of Utrecht

Transmembrane potential propagation

: capacitance per unit area of membrane: transmembrane potential: intra- (extra-) cellular potential: transmembrane current: conductivity tensor in intra- (extra-) cellular space

Governing equations describing the intra- and extracellular potentials:

mCmu

mI( )i eD D

( )i eu u

Ionic current, , described by a FitzHugh-Nagumo-like kinetics [9]

( )

( )( )

m m

m

I f u w

dwu ke w

dtε

=− −

= −

1 1 1

2 2 1 2

3 3 2

1 2

1 2 3

( ) , ( ) , when

( ) 1, ( ) , when e

( ) ( 1), ( ) , when

where 0.0065, 0.841, 0.15, 3

20, 3, 15;

m m m m

m m m m

m m m m

f u c u u u e

f u c u u u e

f u c u u u e

e e a k

c c c

ε εε εε ε

= = <= + = ≤ ≤= − = >

= = = == = =

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

mI

( ) (( ) ) 0i i em eD u D D u∇⋅ ∇ +∇⋅ + ∇ =sr sr sr sr sr sr sr

1( )m

e e mm

uD u I

t Cχ∂

=− ∇⋅ ∇ +∂

sr sr sr

0 0

0 0

0 0

i

ii

i

D

D D

D⊥

⊥

⎛ ⎞⎜ ⎟

=⎜ ⎟⎜ ⎟⎝ ⎠

Psr

The ratios of the diffusion constants along and perpendicular to the fiber direction in the intra- and extra-cellular spaces are different.

0 0

0 0

0 0

e

ee

e

D

D D

D⊥

⊥

⎛ ⎞⎜ ⎟

=⎜ ⎟⎜ ⎟⎝ ⎠

Psr

i i

e e

D D

D D⊥

⊥

≠P

P

Bidomain:

Numerical Implementation

1 ( )n

n

m mn e e m

n m

u uD u I u

t+ −

= +Δ

1

1

1( ) ( )

2n n

n n n

m me e e e m

m

u uD u D u I u

t+

+

−= + +

Δ

Numerical solution of parabolic PDE (for um )

Forward Euler scheme:

Crank-Nicolson scheme:

1( )m

e e mm

uD u I

t Cχ∂

=− ∇⋅ ∇ +∂

sr sr sr

is approximated by the finite difference matrix operator, 1

( )e em

D uCχ

− ∇⋅ ∇sr sr sr

eD

Numerical solution of elliptic PDE (for ue )

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

( ) (( ) ) 0i i em eD u D D u∇⋅ ∇ +∇⋅ + ∇ =sr sr sr sr sr sr sr

Numerical Results

Monodomain Bidomain Fiber Rotation

ThicknessBreak-up (>= 2 filaments)

Monodomain Bidomain

1.0 1.0 0.4 120o 10mm No No

0.9 0.9 0.4 120o 10mm No No

0.8 0.8 0.4 120o 10mm No No

0.7 0.7 0.4 120o 10mm No No

0.6 0.6 0.4 120o 10mm No No

0.5 0.5 0.4 120o 10mm No Yes

0.3 0.3 0.4 120o 10 mm No Yes

0.1 0.1 0.4 120o 10 mm Yes Yes

0.06 0.06 0.4 120o 10 mm Yes Yes

0.3 0.3 0.4 60o 10mm No Yes

0.1 0.1 0.4 60o 10 mm Yes Yes

0.3 0.3 0.4 40o 10mm No Yes

0.1 0.1 0.4 40o 10 mm Yes Yes

0.1 0.1 0.4 60o 5 mm Yes Yes

Conclusions

From Laboortatory 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.[3-5]

[3] J. P. Keener and J. Sneyd, Mathematical Physiology[4] C. S. Henriquez, Critical Reviews in Biomedical Engineering 21, 1-77 (1993)[5] J. C. Neu and W. Krassowska, Critical Reviews in Biomedical Engineering 21, 137-1999 (1993)[6] B. J. Roth and J. P. Wikswo, IEEE Transactions on Biomedical Engineering 41, 232-240 (1994)[7] J. P. Wikswo, et al., Biophysical Journal 69, 2195-2210 (1995)[8] J. P. Keener and K. Bogar, Chaos 8, 234-241 (1998)

Governing Equations

Conservation of total current

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

1 1 1 111 1

2 2 2 2 211 2

1 3 3 3 311 3

( )

( )

( )

e m

e m

e m

m a b u f u

c m a b u f u

d c m a u f u

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

O

O O O O M MO O O M M

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

Comparison of break-up in bidomain and monodomain models:

Future Work

Acknowledgements

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.

/e eD D⊥ P

Example of filament-finding results used to characterize breakup ( ):

Rectangular grid: 60 x 60 x 9; dx=0.5 mm, dy=0.5 mm, dz=0.5 mm; dt=0.01s

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.

Conductivity Tensors

Focus of This WorkComputational study of the role of the rotating anisotropy of cardiac tissue on the dynamics of phase singularities in the bidomain model of cardiac tissue.

/D D⊥ P /i iD D⊥ P

Time (s)

/ 0.06, / 0.4i i e eD D D D⊥ ⊥= =P P

Time (s)

Goal is to use analytical and numerical tools to study the dynamics of reentrant

waves in the heart on physiologically realistic domains.

Rotating Anisotropy

Time (s) Time (s)

/ 0.7, / 0.4i i e eD D D D⊥ ⊥= =P P

1 1 11

2 2 2 22

3 3 33

1 1 1

2 2 2 2

3 3 3

, 1

111 211 311 11 112 212 312 1 121 221 321

x

x x x

x x x x

x x x

x x z

N

N N N

N N N N

N N N

N N jx z

N N N

m a b cd m a b c

d m b c

m

e m a

e d m a

e d m

+ + +

+ + + +

+ + +

× =6 4 4 4 4 4 4 4 4 4 4 4 4 447 4 4 4 4 4 4 4 4 4 4 4 4 4 48

L L L L L L

O

O O O OO O

O

O

O O O

O

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.

Cardiac tissue is more accurately described as a three-dimensional anisotropic bidomain, especially under conditions of applied external current such as in defibrillation studies.[6-7]

However, unlike the monodomain, analytical and numerical studies based on the bidomain model remain technically challenging. [8]

120QΔ = o

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.

ii

e e

DD

D Dα⊥

⊥

= =P

P

The intracellular and extracellular conductivity tensors are proportional.

Monodomain:

[4] A. V. Panfilov, Chaos 8, 57-64 (1998)