The Physics of the Heart. Sima Setayeshgar

Transcription

1 The Physics of the Heart Sima Setayeshgar Department of Physics Indiana University Indiana Unversity Physics REU Seminar: August 1,

2 Stripes, Spots and Scrolls Indiana Unversity Physics REU Seminar: August 1,

3 Indiana Unversity Physics REU Seminar: August 1,

4 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. Indiana Unversity Physics REU Seminar: August 1,

5 The heart is an electrically activated mechanical pump Electrical Activity = Mechanical Function Seconds From Textbook of Medical Physiology, by Guyton and Hall Indiana Unversity Physics REU Seminar: August 1,

6 Electric potential propagation in a biological cable Cable Equation: Hodgkin and Rushton (1946), Rall (1957,...) τ m V / t + R m I ion = λ 2 m 2 V / x 2 Adapted from Mathematical Physiology, by Keener & Sneyd (1998). Axial current : I a = I i + I e T ransmembrane current : I t = I ionic + I capacitive + I applied T ransmembrane potential : V = V i V e P hysical properties : C m, R m, r i, r e, p, d Kirchoff's law: I i (x + dx) I i (x) = I e (x) I e (x + dx) = I t dx Conservation of charge: I a / x = 0 Ohmic axial currents: V i,e (x + dx) V i,e (x) = I i,e (x)r i,e dx Indiana Unversity Physics REU Seminar: August 1,

7 Ionic currents in the cardiac cell Ionic Currents (I Na +(V ), I K +(V ), I Ca ++(V ),...): I ion V/R ion!! Electrophysiologically realistic models: Hodgkin-Huxley (1952): Squid giant axon Noble (1960), Beeler-Reuter (1977), Luo-Rudy (1991, 1994),... Reduced models: FitzHugh-Nagumo (1960),... C.-H. Choi and Y. Rudy, Circ. Res. 74, 1071 (1994). Indiana Unversity Physics REU Seminar: August 1,

8 The heart is a three-dimensional anisotropic medium 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 Streeter, et al., Circ. Res. 24, p. 339 (1969). Indiana Unversity Physics REU Seminar: August 1,

9 Excitability and Nonlinear Traveling Waves (a) Propagating band Ω + : Excited Ω : Rest speed c = c(v) c(v front ) = c(v back ) Click for movie. (b) Spiral wave c(v) varies continuously through zero: c(v front ) > 0 and c(v back ) < 0 Existence of pivot point: c(v ) = 0 Click for movie. Reviews: Fife (1984), Keener and Tyson (1988), Cross and Hohenberg (1993) Indiana Unversity Physics REU Seminar: August 1,

10 Experimental Evidence of Spiral Waves From W. F. Witkowski, et al., Nature 392, p. 78. Time spacing between frames 5 ms Image size 5 cm Click for movie. Indiana Unversity Physics REU Seminar: August 1,

11 Big Picture What are the basic mechanisms of the onset of fibrillation? How can we control them? Tachycardia: Fibrillation: Courtesy of Sasha Panfilov, University of Utrecht Click for animation. Indiana Unversity Physics REU Seminar: August 1,

12 Postgenomic Systems Biology/Physiology......allows linking genetic information to cellular phenotypes that lead to cardiac arrhythmias C. Clancy et al., Nature 400, 566 (1999). Indiana Unversity Physics REU Seminar: August 1,

13 Calculating the heart's function from first principles requires data-driven modeling over orders of magnitude in space and time scales! Ion channel: 10 nm 1 channel/µm 2 Cardiac cell: 150 µm 15 µm 15 µm 500 to 30,000 channels per cell depending upon cell type Heart: 10 cm cells channels Ratio of spatial scales: 10 8 Channels change in 1 10 ns, fibrillation time scale 10 s Ratio of temporal scales: 10 9 Indiana Unversity Physics REU Seminar: August 1,

14 The Problem of Scales: Numerical Approaches Fibrillation occurs on the scale of the entire heart! Divide each cardiac cell into 10 segments: segments/heart 50 currents/variables per segment: variables/heart 5 µs/timestep: timesteps/10 s of fibrillation 1 year on a TFLOPS workstation! Indiana Unversity Physics REU Seminar: August 1,

15 Ongoing Research Approaches: Use combination of analytical and computational tools Use combination of idealized and realistic modeling Topics: Electrophysiology Electro-mechanical coupling Solid and fluid mechanics What is my group doing: Paradigm: Breakdown of single spiral to disordered state resulting from various mechanisms of spiral instability. Questions: Role of passive versus active properties of heart tissue as a conducting medium on the generation of electrical wave instabilities Members: Jianfeng Lv (G), Xianfeng Song (G), Le Mai Nguyen (UG) Indiana Unversity Physics REU Seminar: August 1,

16 A snapshot of what we do... with Prof. Andrew Bernoff, Harvey Mudd College Indiana Unversity Physics REU Seminar: August 1,

17 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 Indiana Unversity Physics REU Seminar: August 1,

18 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) Indiana Unversity Physics REU Seminar: August 1,

19 Governing Equations Governing equations 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: ( ) ( V u =, f V 3 + 3V v = v ɛ(v δ) ), D = ( D ), etc... Indiana Unversity Physics REU Seminar: August 1,

20 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 Indiana Unversity Physics REU Seminar: August 1,

21 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 ). Indiana Unversity Physics REU Seminar: August 1,

22 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 Indiana Unversity Physics REU Seminar: August 1,

23 Scroll Twist Twist and... Twist Solution A > 0: Formation of large twist in boundary layer in bulk A < 0: Expulsion of large twist from bulk to boundaries Setayeshgar and Bernoff, Phys. Rev. Lett. 88, Rotating anisotropy of medium generates scroll twist. Indiana Unversity Physics REU Seminar: August 1,

24 ... Sproing Filament buckling instability above twist threshold. Buckled filament Tip trajectory Destabilizing (``sproing instability'') role of cardiac tissue structure as a conducting medium on dynamics of scroll waves depends on membrane ion kinetics. Setayeshgar and Bernoff, preprint. Indiana Unversity Physics REU Seminar: August 1,

25 Concluding Remarks The heart is an important physiological system that is amenable to physical analysis. From 1920's: first clinical EKG machine... Indiana Unversity Physics REU Seminar: August 1,

26 To 2020's: engineered heart on a scaffold?? Polymer Scaffold Disordered Culture Ordered Culture from Ratner Group, University of Washington, Bioengineering Indiana Unversity Physics REU Seminar: August 1,

27 What else do we do... Indiana Unversity Physics REU Seminar: August 1,

28 Overview Indiana Unversity Physics REU Seminar: August 1,

29 Physics of Diffusion I: Limits to Biochemical Signaling Intra-/Extracelluar Signaling Physical Analogy RNAP CheY CheY-P TF Biological receptors as ``particle detectors'' perform at limit set by diffusive counting noise. W. Bialek and S. Setayeshgar, PNAS 102, (2005). Indiana Unversity Physics REU Seminar: August 1,

30 Physics of Diffusion II: Optimal Strategy in Nutrient Uptake I cell (N)/I cell Add 10,000 absorbers to cell body: Increase rate of uptake by only 5%!! OR Number of absorbers, N I stalk (N)/I cell L = 10 mm L = 5 mm L = 1 mm Number of absorbers, N Put 10,000 absorbers on stalk of length L: Increase rate of uptake by approximately 2 times 1.25 times 0.5 times maximum rate of uptake by cell body! J. Wagner, S. Setayeshgar, L. Sharon, J. Reilly and Y. Brun, PNAS (2006). Indiana Unversity Physics REU Seminar: August 1,