Scroll Waves in Anisotropic Excitable Media with Application to the Heart. Sima Setayeshgar Department of Physics Indiana University

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1 Scroll Waves in Anisotropic Excitable Media with Application to the Heart Sima Setayeshgar Department of Physics Indiana University KITP Cardiac Dynamics Mini-Program 1

2 Stripes, Spots and Scrolls KITP Cardiac Dynamics Mini-Program 2

3 Overview Common processes underlying natural patterns give rise to model equations capturing generic features of pattern-forming systems Theoretical approaches accessible near onset, but must resort to numerical tools and experiments far from onset Need for high-fidelity scientific computation to describe realistic physical systems as a bridge between theory and experiment KITP Cardiac Dynamics Mini-Program 3

4 KITP Cardiac Dynamics Mini-Program 4

5 The Heart as a Physical System Sudden cardiac failure is the leading cause of death in industrialized nations deaths/day in North America Growing experimental evidence that self-sustained patterns of electrical 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. KITP Cardiac Dynamics Mini-Program 5

6 Big Picture What are the mechanisms for transition from ventricular tachychardia to fibrillation? How can we control it? Tachycardia: Fibrillation: Courtesy of Sasha Panfilov, University of Utrecht Click for animation. Paradigm: Breakdown of single spiral to disordered state resulting from various mechanisms of spiral instability. KITP Cardiac Dynamics Mini-Program 6

7 Focus What is the role of the anisotropy inherent in the fiber architecture of the heart on scroll wave dynamics? Motivated by: A. T. Winfree, in Progress in Biophysics and Molecular Biology, D. Noble et al. eds., (1997). Numerical ``experiments'' In rectangular slab geometries: Panfilov, A. V. and Keener, J. P., Physica D 84, 545 (1995): Scroll wave breakup due to rotating anisotropy. Fenton, F. and Karma, A., Chaos 8, 20 (1998): Rotating anisotropy leading to ``twistons'', eventually destabilizing scroll filament. Analytical work Dynamics of scroll waves in isotropic excitable media, beginning with: Keener, J. P., Physica D 31, 269 (1988). Biktashev, V. N., Physica D 36, 167 (1989). KITP Cardiac Dynamics Mini-Program 7

8 Tissue Structure 3d conduction pathway with uniaxial anisotropy Propagation speeds: c = 0.5 m/s, c = 0.17 m/s From Textbook of Medical Physiology, by Guyton and Hall From Thomas, Am. J. Anatomy (1957). KITP Cardiac Dynamics Mini-Program 8

9 Credits Collaborator Andrew Bernoff, Mathematics Department, Harvey Mudd College Acknowledgements Alain Karma, Physics Department, Northeastern University Herb Keller, Applied Mathematics Department, Caltech Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 9

10 Rotating anisotropy D D D 2 D D D 2 from Streeter, et al., Circ. Res. 24, p. 339 (1969). Diffusion constants: D > D 1 D 2 Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 10

11 Coordinate System Natural coordinate system defined by fiber direction: x ỹ z } {{ } `new' = α }{{} S cos Θ(z) sin Θ(z) 0 sin Θ(z) cos Θ(z) }{{} R x y z } {{ } `old' S : rescaling, according to 2d anisotropy α (D 1 /D ) 1/2 R : rotation, according to fiber direction Θ(z) Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 11

12 Governing Equations Governing reaction-diffusion equation in new coordinates: u t = f( u) + D 2 u + D 2 u zz { [ + D 2 Θ 2 2 θ + 2 (α2 1)x 2 2 y + 2 2Θ [ θ Θ [ θ + (α 1)x y + (α 1)x y ( 1 )y α x 2 ) y ] u ( 1 α 1 ( ) 1 α 1 x ] y x z } u, ] u Only depends on fiber rotation rate, Θ (no explicit dependence on Θ(z)). For FitzHugh-Naguomo (FHN) kinetics: ( ) ( u u =, f u 3 + 3u v = v ɛ(u δ) ), D = ( D ), etc... Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 12

13 Peskin Fiber Distribution Profile Measured Derived from Streeter, et al., Circ. Res. 24, p. 339 (1969) from Peskin, et al., Comm. on Pure and Appl. Math 42, p. 79 (1989) Θ(z) = sin 1 (z/rl) r = cutoff parameter 2L = thickness of ventricular wall Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 13

14 Perturbation Analysis Consider the limit of `small rotating anisotropy' : Non-dimensional small parameter: ɛ 2 = D 2 ω 0 L 2 1 r 2 1 ( γ ) ( D 2 ω 0 ) 1/2 2L r γ = α + 1/α : transverse diffusion length, l : thickness of ventricular wall : cutoff parameter : `anisotropy' Seek a solution in the form of: where U 0 (r, θ ω 0 t) satisfies: u = U 0 (r, θ ω 0 t + Θ(z) + φ(z, t)) + ɛ 2 u 2, O(1) : U 0 t = f( U 0 ) + D 2 U 0 Scaling assumptions: u 2 O(1), φ z O(ɛ), φ t O(ɛ 2 ). Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 14

15 Validity of Perturbation Analysis? What is the value of the small parameter for the human ventricle? Parameter Value D 1.0 cm 2 s 1 D 0.1 cm 2 s 1 ω s 1 Θ 180 2L 1.0 cm r 1.5 ɛ Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 15

16 Phase Equation At O(ɛ 2 ), introducing Φ(z, t) ( c1 c 2 ) [φ(z, t) ( γ 2 1) Θ(z) ] : Φ t Φ 2 z Φ zz = A(γ, r) F (z; r), 1 < z < 1 Burgers' equation, with forcing given by fiber rotation: F (z; r) = 1 1/r2 1 (z/r) 2, ( A(γ, r) = Ã γ 2 ) r 2 1, Ã = ( ) 2 ( ) c1 4a1 c 2 c 1 1 (a i, c i ) given by inner products from the solvability condition, e.g., a 1 = Ỹθ, D 2 U 0θθ Seek asymptotic and numerical solutions, using constant frequency-shift ansatz: Φ(z, t) = z 1 k(z ) }{{} twist dz + λt + Φ 0 Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 16

17 Scroll Twist For a straight scroll: k(z, t) = ( ) ˆN z ˆN ẑ ˆN = u/ u normal to tip trajectory In new coordinates: k(z, t) = φ z (z, t) + Θ (z). In old coordinates: k( z, t) = Θ ( z) 2α ( φ z ( z, t) + Θ ( z) ) (α 2 1) cos [2 (ω 0 t φ( z, t) Θ( z))] + (1 + α 2 ). Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 17

18 Twist Solutions Diffusive Regime: Φ zz Φ 2 z Twist-dominated Regime: Φ zz Φ 2 z A > 0, A < 0: Maximum twist at boundary A > 0: Formation of large twist in boundary layer in bulk A < 0: Expulsion of large twist from bulk to boundaries Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 18

19 Relevance? Henzi, Lugosi, Winfree, Can. J. Phys. 68, 683 (1990): α = 1 Helical buckling (``sproing instability'') for twist > twist Tip Trajectory Scroll Period vs. Scroll Twist With rotating anisotropy, α 1, Θ z 0??? Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 19

20 Summary What has been done: Extension of asymptotics of scroll waves in isotropic media to include rotating anisotropy of cardiac tissue Phase dynamics (forced Burgers equation): nonconstant fiber rotation rate generates twist a Extensions: Coupling between twist and filament dynamics Extension to biodomain description of cardiac tissue a : Setayeshgar and Bernoff, Phys. Rev. Lett. (2002). Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 20

21 Summary (cont'd) Numerical sproing bifurcation diagram with rotating anisotropy Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 21

22 Filament motion Lab Frame: Fiber Frame: Moving Frame: Simplifying assumptions: Constant fiber rotation rate: Θ = constant φ(z, 0)/ z = 0, R c (z, 0)/ z = 0 in fiber frame No-flux or periodic boundary conditions at vertical boundaries Dynamics reduces to two dimensions: ``Helical'' buckling Motion of spiral center. Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 22

23 Additional coordinate transformation: Filament motion (cont'd) x x X c (t), t y y Y c (t), Phase equation: φ T = Θ 2 Z t + dx c dt x + dy c dt y, [ ( )] 1 c 3 (α) + r 1 α 2Y c 2 + α 2 2 X c Dynamics of the center: d dt ( Xc Y c ) = Θ 2 Z ( µ1 µ 2 ) } µ 3 µ 4 {{ } M ( Xc Y c ) where µ i = µ i (α): µ 1 (1) = µ 4 (1) = t 1 µ 2 (1) = µ 3 (1) = t 2. with: t 1 = Ỹx, D 2 x U 0 x = Ỹy, D 2 y U 0 y, etc. r 1 = Ỹ0, D 2 U 0 xx = Ỹ0, D 2 U 0 yy Notes: Symmetry: α 1/α At O(ɛ 2 ) with φ z = 0: Motion of center is decoupled from dynamics of phase. Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 23

24 Filament Tension Dynamics of spiral center governed by eigenvalues of M: R c (T ) = C + v + e λ + T + C v e λ T No anisotropy: α = 1 λ ± = t 1 ± it 2 Stability determined by filament tension, ( t 1 ). At O(ɛ 2 ), Θ Z determines only the scaling of filament dynamics. Weak anisotropy: α 1 δ, δ 1 λ ± t 1 ± t δ2 (B 2 A 2 ) Rotating anisotropy can lead to change in stability! (Necessary cond'n: B > A.) Dependence on reaction kinetics of α = 1: Filament tension (threshold to buckling) α 1: Possible destabilizing role of rotating anisotropy on buckling Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 24

25 Formally valid for: A 1 Twist-dominated Regime: Φ zz Φ 2 z Cole-Hopf transformation: k(z) = ψ z (z)/ψ(z), d 2 ψ/dz 2 + [ λ V (z)]ψ = 0, V (z) = A(γ, r)f (z; r). Ground state (smallest λ ) determined by potential in the vicinity of its minimum: Negative forcing: A < 0 1d harmonic oscillator equation, λ determined by behaviour at the origin: ( ) λ 0 = Ā/r 2, λ 1 = Ā 1/2 /r 2 Ā = Ã γ 2 /4 1 Positive forcing: A > 0 Airy equation, λ determined by behaviour at boundaries: λ 0 = Ā/(r 2 1), λ 1 = η ( 2Ā ) 2/3 /(r 2 1) 4/3 where η is the first zero of Ai (z). Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 25

26 Twist-dominated Regime: Φ zz Φ 2 z (cont'd) A > 0: Formation of large twist in boundary layer in bulk A < 0: Expulsion of large twist from bulk to boundaries Sima Setayeshgar, Indiana University, Bloomington KITP Cardiac Dynamics Mini-Program 26