1	Vacuum Pair Production/Annihilation and Cardiac String Dynamics John P. Wikswo Living State Physics Group Departments of Physics and Astronomy, Molecular Physiology & Biophysics, and Biomedical Engineering Vanderbilt Institute for Integrative Biosystems Research and Education Vanderbilt University Aspen Center for Physics, August 22, 2002
2	Acknowledgements Rubin Aliev Mark Bray Elizabeth Cherry Deborah Echt Flavio Fenton Rick Gray Peter Hunter Alain Karma Mark Lin Neils Otani Arkardy Pertsov Nathalie Virag Jim Weiss And many others
3	Where are the heart strings, and who is pulling them? The normal heart has none The presence of one string is serious The presence of several for a very few minutes is fatal
4	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
5	The Heart is a… Self-assembling, Biochemically powered, Electrically activated, Electrically non-linear, Pressure- and volume-regulated, Two-stage, Tandem, Mechanical pump With a mean time-to-failure of approximately two billion cycles.
6	The heart is ... electrically activated …
7	The heart is an … Electrically activated, Mechanical pump
8	The Normal Heart Beat Courtesy of Rick Gray and CRML, U. Alabama Birmingham
9	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
10	Normal Tachycardia Fibrillation Defibrillation The heart is an electrically activated mechanical pump …with a mean time-to-failure of approximately two billion cycles….
11	Induction of Fibrillation Courtesy of Rick Gray and CRML, U. Alabama Birmingham
12	Termination of Fibrillation Courtesy of Rick Gray and CRML, U. Alabama Birmingham
13	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
14	Spiral and Scroll Waves in Nature A generic property of excitable media Have been shown to occur in Circulating waves of bioelectric activity in cardiac and retinal tissue Autocatalytic chemical reactions, such as Belousov-Zhabotinsky reaction (BZ) cAMP waves in slime mold Dictyostelium discoideum Intracellular calcium release in oocytes Oxidation of CO on crystal surfaces in ultrahigh vacuum conditions Cardiac fibrillation involves multiple scroll waves in 3-D
15	Cardiac fibrillation occurs at the spatial scale of the entire heart, and involves multiple, interacting spiral and/or scroll waves!
16	Transmural versus intramural scroll waves in reentrant arrhythmias and fibrillation Transmural waves can exist in 2-D (thin) or 3-D (thick) Intramural waves require ~1 cm wall thickness
17	Transition from Normal Rhythm to Ventricular Tachycardia to Ventricular Fibrillation
18	Initiation of Spiral Wave Reentry S1-S2 crossed- field stimulation
19	A “Simple” Spiral Wave The nature of the spiral is set by the non-linear properties of the excitable medium Linear core Epicycloidal meander Circular core
20	Nonlinear Properties Determine the Trajectories Six Phenotypes Circular Epicycloidal Cycloid Hypercycloidal Hypermeander Linear core Winfree, Krinsky, Barkley, Efimov, Jalife, Pertsov, Gray, Roth, Fenton, Garfinkel, Chen …
21	Non-linear dynamics of reentry, fibrillation, and defibrillation Reentry -- Self-sustained excitation due to propagating activation wave fronts in the heart that continue to re-excite different regions of tissue rather than terminating after a single excitation Anatomical reentry -- activation wave fronts that travel in one direction around an anatomical obstacle Functional reentry -- activation circulate around a dynamical phase singularity
22	Spiral Wave and Figure-of-Eight Reentry Spiral Wave: S1 vert line S2 horiz line Figure-of-Eight S1 vert line S2 point
23	Spiral Wave, Figure-of-Eight, and Quatrefoil Reentry Spiral Wave (A) S1 vertical line S2 horizontal line One singularity (plus boundary) Figure-of-Eight (B) S1 vertical line S2 point Two singularities Quatrefoil (C & D) Anisotropic cable S1 point S2 point Cathodal (C) or anodal (D) have opposite rotations Four singularities
24	Optical Imaging of Quatrefoil Reentry
25	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
26	Transform into Phase Space The problem: a given voltage can either be rising or falling The solution: represent the cardiac action potential in terms of “phase” in the cardiac cycle: 0, 1, 2, 3 … 1%, 2%, 3%, 3%, 5%, … 0^o, 5^o, 10^o, 15^o, 20^o, 25^o, … One definition of phase (of many):
27	From Voltage to Phase Space Four singularities of indeterminate phase, i.e.,points surrounded by all colors
28	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
29	Why look for strings? Movies of the surface potentials are complicated It is not clear how much of the information is needed Model based upon R.R. Aliev and A.V. Panfilov, A. V., Chaos, Solitons, & Fractals, 7(3): 293-301 (1996) Gray, R. A. and Jalife, J., Chaos, 8(1): 65-78 (1998) Movies by Mark Bray
30	Wavefronts are Better The wavefronts are better Require description of the dynamics of the entire system
31	Strings Alone May Be Best Surface singularities are simpler Filaments (strings) are the best Do they interact in a manner that can allow us to ignore the rest of the problem? HOW DO WE FIND THEM??
32	Local Phase and the Wave Vector
33	Phase and Topological Charge Curl k is proportional to the topological charge! It can be shown that the differential curl evaluates as exactly zero, except at the singularity, where it is undefined. At the singularity, the line integral around the singularity must be used directly to find the topological charge. “Use of Topological Charge to Determine Filament Location in a Numerical Model of Scroll Wave Activity,” M.-A. Bray and J.P. Wikswo, Jr., IEEE Trans BME, in press
34	Phase Singularities in Cardiac Reentry
35	Topological Charge Topological charge n_t is zero about any closed path that does not encircle a phase singularity n_t is +1 or -1 for a path that encircles a singularity with a single arm Topological charge is conserved, i.e., singularities are created and destroyed in pairs.
36	Singularity Motion During Spiral Wave Breakup Voltage Curl of Phase
37	Filaments in Three Dimensions Filaments are the 3-D analogue of the 2-D phase singularity
38	Topological charge Curl k may be approximated by 1) a differential operator, or 2) as a discretized contour interval that is in fact a convolution operation of an image with two Nabla windows
39	Filaments in Three Dimensions Filaments are the 3-D analogue of the 2-D phase singularity
40	3-D Filaments Because curl is a three-dimensional vector operator, this convolution approach can can be extended readily to 3-D in order to visualize scroll wave filaments
41	String Dynamics Strings with positive line tension shrink (Paniflov, Rudenko and Krinsky, Biophysics, 31: 926 (1986))
42	String Dynamics Strings with negative line tension grow and buckle (see V.N. Biktashev, A.V. Holden, and H. Zhang. Phil. Trans. Royal Soc. London, Series A 347: 611-630, 1994) If they touch a surface, a pair of singularities is produced Topological charge is conserved
43	A Little Negative Line Tension
44	A Lot of Negative Line Tension fhnplus_scroll_ring_k40_(filament_plus_wavefront).avi fhnplus_scroll_ring_k40_(filament).avi
45	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
46	Quatrefoil Reentry Follows repeated stimuli applied at a single site Has been used to demonstrate the importance of unequal bidomain anisotropies in cardiac electrodynamics Provides a reproducible, controlled system for study of the interactions of phase singularities and their accompanying filaments
47	Quatrefoil Reentry
48	Quatrefoil Reentry We replicate the experimentally observed quatrefoil reentry configuration using a simulated pair of adjacent circular filaments (scroll rings) oriented along their symmetry axes with varying initial radii and separation distances
49	Reaction-Diffusion System We use a two-variable model of the Belousov-Zhabotinsky (BZ) reaction using the Field-Koros-Noyes formulation where v is the bromous acid concentration, w is the relative ferroin concentration, and d = D_w/D_v (d = 1 in this case) For d = 1, With this BZ formulation, a single ring shrinks with a relative absence of translational drift; permits us to observe interaction without large single ring dynamics
50	Methodology Modeled 3-D system using an axisymmetric cylindrical coordinate system (z,r,q ), such that all results are independent of angle q ® Need only to examine 2-D (z,r) plane Started rings at initial separation (Z₀) and initial radius (R₀) and examined life-time (T_L) and motion in (z,r ) plane Simulated cathode and anode break with appropriate initial conditions
51	Initial Conditions
52	Simulated Singularity Interactions Start with a pair of vortex rings of fixed diameter and positive line tension Measure decay time as a function of separation and initial size
53	Cathodal Break 5: Free decay and self-annihilation per Paniflov, Rudenko and Krinsky, Biophysics, 31: 926 (1986) 4: Repulsion per Elphick and Meron, Physica D, 53: 385 (1991) 1: Enhanced decay, attraction, and mutual annihilation per Elphick and Meron, Physica D, 53: 385 (1991)
54	Cathode break movie
55	Cathodal Break Trajectories
56	Anodal Break 4: Free decay and self-annihilation Paniflov, Rudenko and Krinsky, Biophysics, 31: 926 (1986) 1: Enhanced decay, attraction, and mutual annihilation per per Elphick and Meron, Physica D, 53: 385 (1991) 2: Extended lifetime 3: Repulsion per per Elphick and Meron, Physica D, 53: 385 (1991)
57	Anode break movie
58	Anodal Break Trajectories
59	Initial Velocity = Force
60	Outline The heart is a … Cardiac fibrillation Spiral waves in the heart Two dimensions – Spiral waves Three dimensions – Scroll waves Phase plane analysis Singularity identification Simple reentry Fibrillation Singularity interactions Attraction vs repulsion versus oscillation Annihilation Creation What is needed? Interaction potential String creation operator
61	String Creation and Annihilation: Positive Line Tension with Fiber Rotation Loop pinch-off
62	Wavebreak = Vacuum Creation Wave break occurs when the leading edge of a wave runs into the tail of a preceding wave Wavebreaks create filaments which create reentrant activation
63	Future Questions For both cases, what parameters determine attractive versus repulsive behavior? Parameter gradients? Can a kinematic relationship be derived for the scroll ring interactions? Is the effective mass constant or not, since it is a dissipative system? Can the ring interaction be described by a point-to-point potential, and if so, are there obvious centers of action? In a field model, how do you introduce string creation from the vacuum?
64	Acknowledgements Rubin Aliev Mark Bray Elizabeth Cherry Deborah Echt Flavio Fenton Rick Gray Peter Hunter Alain Karma Mark Lin Neils Otani Arkardy Pertsov Nathalie Virag Jim Weiss And many others
65	The End
66	How do you defibrillate the heart? The Neils Otani Cyber Cardiologist