Physics of Cardiac Arrhythmias Sitabhra Sinha Institute of Mathematical Sciences (IMSc) Chennai , INDIA Collaborators: R Pandit, A Pande, T K Shajahan, A Sen (IISc, Bangalore) D J Christini and K M Stein (WMC- Cornell University, NYC)

Outline n Motivation n Cardiac arrhythmias: tachycardia & fibrillation n Reentrant waves and spiral turbulence in excitable medium : a model for VF & VT n Models: Luo-Rudy I & Panfilov models n Spiral formation and breakup: VT and VF n Control of reentry and spiral chaos n Implications for cardiac pacing & defibrillation n Summary

Motivation: Why Study Fibrillation ? n Sudden cardiac death due to ventricular fibrillation (VF) is the leading cause of death in the industrialized world. n One-third of all deaths in the USA are due to cardiac arrest - one out of six due to VF. n Understanding VF is an essential prerequisite for improving current methods of defibrillation (massive electrical shocks ~ 600 Volts). n Possible alternative: Controlling spatio-temporal chaos of VF through low-amplitude perturbations.

What is Ventricular Fibrillation ? n Ventricular Fibrillation (VF): a disorganized electrical wave activity that destroys the coherent contraction of ventricular muscle. n Underlying cause of VF: formation of “electrical vortices” - 2-D (spiral)/3-D (scroll) waves of action potential - creation of re-entrant pathways of electrical activity. n Spiral/ scroll waves lead to abnormally rapid heart beat (t ~ ms) (Tachycardia). n Ventricular tachycardia (if untreated) leads to VF in a few seconds through spiral/scroll wave break-up.

Characterizing Ventricular Fibrillation Normal sinus rhythm Tachycardia Tachycardia Ventricular Fibrillation Spiral wave on the surface of a canine ventricle. Color proportional to transmembrane potential. Image obtained with voltage sensitive dyes and CCD camera.

A Brief History of VF n 1874: “Fibrillation” (Alfred Vulpian) “...individual fibers contracting independently... a wad of writhing worms” (Tacker and Geddes, 1980) n 1888: “Sudden Cardiac Death” (J. A. MacWilliam) “...the cardiac pump is thrown out of gear, and the last of its vital energy is dissipated in a violent and prolonged turmoil of fruitless activity in the ventricular wall.” n 1899: Electrical defibrillation in animals (Prevost and Batelli). n 1914: Fibrillation induced through precisely timed electrical stimulus (G. R. Mines). n 1947: Defibrillation of human heart (Claude Beck) n 1960s: Initial work in Internal Cardiac Defibrillator (Mirowski).

Anatomy of the Heart

Anatomical vs. Functional Reentry Anatomically determined (Mines, 1913) 1. Fixed length of circuit (determined by anatomical obstacle). 2. Usually excitable gap between head and tail of impulse. 3. Inverse relation between revolution time and conduction velocity. Functionally determined (Allessie et al., 1977) 1. Circuit length dependent upon electrophysiological properties. (“Spiral waves”) 2. No gap of full excitability. 3. Revolution time proportional to length of refractory period.

George Ralph Mines ( ) Diagram from Mines(1913) demonstrating circulating rhythms in closed circuits in myocardial tissue: (a) Normal tissue; (b) Abnormal tissue with delayed conduction. Proposed the theoretical basis for occurrence of reentrant arrhythmias. 1913: proposed a model for generatiing reentrant rhythms -a dual pathway with differing electrophysiologic properties; suggested that the twin conditions of unidirectional block and slow conduction may occur in abnormal myocardial tissue - allowing a circulating wavefront to be sustained as conductive tissue is always available for excitation. Cambridge University, 1912

Implantable Cardioverter- Defibrillator (ICD) An ICD consists of a pulse generator and electrical leads. Endocardial leads are inserted through a vein and advanced to the right ventricle and/or atrium. The pulse generator is placed subcutaneously or submuscularly and connected to the leads. Pacing - deliver a sequence of low-amplitude pulses. Pacing - deliver a sequence of low-amplitude pulses. Cardioversion - a mild shock (if pacing fails in terminating VT). Cardioversion - a mild shock (if pacing fails in terminating VT). Defibrillation - a large shock to terminate VF. Defibrillation - a large shock to terminate VF. Pacemaker - for slow heartbeat, can act as pacemaker. Pacemaker - for slow heartbeat, can act as pacemaker. The ICD constantly monitors heart rhythm. Upon detection of VT/VF delivers a programmed treatment. Capable of applying variety of possible treatments:

Internal Defibrillation Example of an implantable cardiac defibrillator (ICD) Vol 54 cc Mass 97 gm Thickness 16mm Longevity 9 yrs BOL Voltage 6.4 V BOL Charge time 6.0 sec

Pacing Algorithms Is there an optimal anti-tachycardia pacing algorithm ?

Spiral Waves in the Heart (Left) Spiral wave in anatomically correct model of the dog heart. Color code indicates calculated activation times (in milliseconds) of various regions of the heart muscle. (Right) Reentrant spiral wave excitation in a rabbit heart observed with a voltage-sensitive dye. Color code indicates measured activation time in milliseconds.

The Cardiac Cell Voltage-gated ion channels are pathways for charge movement (Na +, K +, Ca ++ ions). Myocardial fibers: contractile strand of cardiac muscle composed of many cells. Gap junction

The Luo-Rudy (L-R) model The L-R I model has 8 coupled ODEs describing the activity of each cardiac cell: the transmembrane potential (V ), the intracellular Calcium concentration (Ca i ) and six ion- channel gating variables (m, h, j, x, x i, d, f ). Action potential and ionic currents of a ventricular myocyte simulated with L-R model. Biologically realistic model for ventricular action potential proposed by Luo & Rudy (I: 1991, II: 1995): incorporates details of ionic currents.

Luo-Rudy I model: Membrane potential equation The transmembrane potential V follows the reaction-diffusion equation where C m = 1  F/cm 2 is the membrane capacitance, D (k  -1 ) is the conductivity constant and I LR (  A/cm 2 ) is the instantaneous total ionic current through the cell : Inward currents: I Na : Fast sodium current (Na + ) I si : Slow inward current (Ca ++ ) Outward currents : IK : Time-dependent potassium current IK1 : Time-independent potassium current IKp: Plateau potassium current Ib : Background current

Luo-Rudy I model: Formulation of Ionic Currents Fast sodium current (E Na = 54.4 mV): Slow inward current (E Si = ln ([Ca] i )): Time-dependent potassium current (E K = -77 mV): Time-independent potassium current: Plateau potassium current: Background current:

Luo-Rudy I model: Ca ++ concentration and gating variable equations The intracellular calcium concentration (Ca i ) satisfies the ODE: Each ion channel gating variable  ( = m,h,j,x,x i,d,f ) is governed by ODEs of the form : The parameters   and   are functions of the rate constants  and  : and  : normalized fraction of the population of ion channels in open state,  : rate at which channels open;  : rate at which the channels close. The rate constants  and  are complicated functions of the membrane potential V.

The Beeler-Reuter (B-R) model The B-R model has 8 coupled ODEs describing the activity of each myocardial cell - corresponding to the transmembrane potential (V ), the intracellular Calcium concentration (c) and six ion-channel gating variables (m, h, j, x, d, f ). Variation of the trans- membrane potential during an action potential in the B-R model. Biologically realistic model for ventricular action potential proposed by Beeler & Reuter (1977) - incorporating details of ionic currents.

Beeler-Reuter model: Membrane potential equation The transmembrane potential follows the reaction-diffusion equation where I BR is the instantaneous total ionic current through the cell : I K : transient outward potassium current I Na : fast sodium inward current I x : time-activated outward current (mostly K + ions) I s : slow inward calcium current

Beeler-Reuter model: Calcium concentration and gating variable equations The calcium concentration (c ) satisfies the ODE: Each ion channel gating variable  ( = m, h, j, x, d, f ) is governed by ODEs of the form : The parameters   and   are functions of the rate constants  and  : and Note:  represents the normalized fraction of the population of ion channels that is in an open state,  is the rate at which channels open and  is the rate at which they close.

Beeler-Reuter model: Rate constant equations

Excitable Media Excitable Media Subthreshold stimulation  perturbation decaysSubthreshold stimulation  perturbation decays Suprathreshold stimulation Suprathreshold stimulation  Feature: Conduction of propagating waves.Feature: Conduction of propagating waves. Pattern formation: wave of excitation can change the properties of excitable media and cause the formation of spatial patterns.Pattern formation: wave of excitation can change the properties of excitable media and cause the formation of spatial patterns. Examples: Aggregation of Dictyostelium Discoideum amoebae - the monolayer of the starving amoebae is an excitable medium which conducts excitation waves of the intra-cellular mediator, cAMP.Aggregation of Dictyostelium Discoideum amoebae - the monolayer of the starving amoebae is an excitable medium which conducts excitation waves of the intra-cellular mediator, cAMP.

The Heart as an Excitable Medium n Excitable medium : a small but finite perturbation from equilibrium can lead to excitation (a large excursion away from equilibrium) before equilibrium is restored. n Excitation in the Heart : the electromechanical wave inducing cardiac-muscle contractions which pump blood. n Refractory period : once excited, the medium remains quiescent for a certain duration. n Spiral waves associated with abnormal cardiac activity (Winfree).

The Panfilov Model for Ventricular Fibrillation n Fitzhugh-Nagumo model for excitable media with `Puschino’ kinetics n Describes an excitable medium with an absolute and a relative refractory period - the period after repolarization during which the membrane recovers its resting properties. n The simplest model that shows spiral breakup qualitatively similar to that seen during VF. n `Puschino’ kinetics shortens the relative refractory period. n Shows a long chaotic transient, whose duration increases sharply with system size - agrees with the observation that the hearts of larger animals are more likely to show VF.

Panfilov Model: Equations n Described by two coupled partial differential equations. n Variables: membrane potential, e (fast variable) and effective membrane conductance, g (slow variable). n  e /  t =  i d ij  j e - f(e) - g, n  g /  t =  (e, g) [k e - g]. n d ij : conductivity tensor - for isotropic medium, replaced by Laplacian. n Diffusive term describes the coupling among cells. n f(e) : nonlinear function - piecewise linear nature. n  (e, g) : information about refractory periods. n Conductivity: 2 cm 2 / s.

Panfilov Model: Parameters n  (e, g) =  1, e < e 2 n  (e, g) =  2, e > e 2 n  (e, g) =  3, e < e 1 and g < g 1 n Variables: membrane potential, e (fast variable) and effective membrane conductance, g (slow variable). n  e /  t =  i d ij  j e - f(e) - g, n  g /  t =  (e, g) [k e - g]. n d ij : conductivity tensor - for isotropic medium, replaced by Laplacian. n Diffusive term describes the coupling among cells. n f(e) : nonlinear function - piecewise linear nature. n  (e, g) : information about refractory periods. n Conductivity: 2 cm 2 / s.

Panfilov Model: Dynamics n Dynamics in the absence of the diffusion term. n e changes at a fast rate compared to g.

Spiral Turbulence in the 2-D Panfilov model Pseudo-color plots of the e field at various values of  1 (  3 = 0.3). As  1 decreases, the pitch of the spiral decreases - ultimately leading to spiral breakup. Local phase portraits at various values of  1. The increasing scatter of points indicates the onset of spatiotemporal chaos.

Spatiotemporal Chaos in the 2-D Panfilov model The maximum value attained by the Kaplan- Yorke dimension D KY during spatiotemporal chaos - plotted as a function of linear system size L. The lifetime of the chaotic transient increases with L. The maximum Lyapunov exponent ( max ) plotted as a function of time t : max approaches a positive constant (~ 0.2) and then decays at large times to negative values.

Panfilov Model: Controlling Spatiotemporal Chaos n The model has non-conducting boundaries (no-flux or Neumann boundary conditions) - as the ventricles are electrically insulated from the atria. Observation : n Non-conducting boundaries absorb spiral defects. n Spirals do not last for appreciable periods in small systems. Operating Principle for the Control Scheme : n To divide the system ( L  L ) into K 2 smaller blocks. n Isolate the blocks ( of size L / K ) by stimulating the system along the block boundaries - driving them to refractory state. n Each block is too small to sustain spiral activity - spirals absorbed by block boundaries. n After the system is driven to the quiescent state, controlling stimulation is withdrawn - block boundaries recover from refractory state.

Control parameters in 2-D Panfilov model L = 128 Pulse amplitude  57.3 mV / msec kept on for  = 41.2 msec. This implies a defibrillating current density of 57  A/cm 2. Beeler-Reuter model L = 200 Pulse amplitude  20 mV / msec kept on for  = 120 msec suffices. This implies a defibrillating current density of 20  A/cm 2.

Panfilov Model: Control in 3-D Control algorithm as in 2-D with the following modifications: Control mesh only on one free face of a 3-D domain ( L x L x L z ).Control mesh only on one free face of a 3-D domain ( L x L x L z ). With L = 256 control obtained for 4  L z. With L = 256 control obtained for 4  L z. For L z > 4, pulse control is necessary: For L z > 4, pulse control is necessary: activate control mesh after  msec activate control mesh after  msec keep it on for  ON msec keep it on for  ON msec turn it off for  OFF msec turn it off for  OFF msec keep it on for  ON msec keep it on for  ON msec continue n times continue n times We find  ON = 0.11 msec,  OFF = 22 msec and n = 30 suffices. Note that  OFF is  the duration of one action potential.

Implications for Defibrillation n Current defibrillation techniques involve applying electrical shock to the fibrillating heart. n Principle of operation: Simultaneous depolarization of all cells - so that the cardiac pacemaker can take over. n External defibrillation ~ 5 kV. n Internal Cardiac Defibrillator (ICD) ~ 600 V. n We propose using very-low-amplitude pulse (~ mV) applied for a brief duration ( ~ 100 ms). n Control over 2-D surface is effective even for 3-D control.

Summary n Cardiac arrest due to VF is a spatio-temporally chaotic phenomenon. n VF arises due to break-up of spiral/scroll waves induced by re-entrant activity. n Panfilov model is the simplest one that shows spiral breakup similar to that in VF. n We have controlled spiral break-up in 2 and 3-D in the Panfilov model and the more realistic Luo-Rudy model of the heart.

Outline n Motivation n Heart as excitable medium n Reentrant activity n Fitzhugh-Nagumo model for Ventricular Fibrillation (VF) n Spiral formation n Spiral breakup and VF n Control of spatio-temporal chaotic activity n Implications for defibrillation n Summary

Motivation: Why Study Anti- Tachycardia Pacing ? n Sudden cardiac death is the leading cause of death in the industrialized world. n One-third of all deaths in the USA are due to cardiac arrest. n One out of six due to VF. n Understanding VF is an essential prerequisite for improving current methods of defibrillation (massive electrical shocks ~ 600 Volts). n Possible alternative: Controlling spatio-temporal chaos of VF through low-amplitude perturbations.

What is Ventricular Tachycardia ? n Ventricular Tachycardia (VT): Abnormally rapid heart beat (t ~ ms). n The heart is unable to pump blood efficiently for such rapid beating. n Underlying cause of VT: creation of re-entrant pathways of electrical activity. n VT (if untreated) may degenerate to Ventricular Fibrillation (VF) - leading to death in minutes. Normal sinus rhythm Tachycardia Tachycardia Ventricular Fibrillation

Termination of Reentry by Pacing Pacing can result in : No effect on the reentrant wave. No effect on the reentrant wave. Resetting of the reentrant wave (the retrograde wave collides with the reentrant wave - the anterograde wave becomes the new reentrant wave). Resetting of the reentrant wave (the retrograde wave collides with the reentrant wave - the anterograde wave becomes the new reentrant wave). Termination of reentry. Termination of reentry. Termination of reentry occurs by block in the anterograde direction - since the retrograde branch of the wave will collide with the reentrant wave and annihilate each other. Each pacing wave splits into two branches while traveling around the reentry circuit : Anterograde (along the direction of the rentrant wave) Anterograde (along the direction of the rentrant wave) Retrograde (against the direction of the reentrant wave). Retrograde (against the direction of the reentrant wave).

Pacing Termination of Ventricular Tachycardia Several factors influence the ability of pacing to interact with VT: VT cycle length. VT cycle length. The refractory period at the stimulation site and at the VT circuit. The refractory period at the stimulation site and at the VT circuit. The conduction time from the site of stimulation to the VT circuit. The conduction time from the site of stimulation to the VT circuit. The duration of the excitable gap. The duration of the excitable gap. The response pattern of VT to the delivery of single & double pacing stimuli (Josephson, 1993) Why multiple stimuli ? Large number of conditions for reentry to be terminated  single stimulus rarely sufficient. Double stimuli often used: first stimulus used only to “peel back” refractoriness, allowing the second stimulus to interact with the circuit more prematurely.

Pacing Termination of Reentry in the 1-D Ring Pacing termination of reentry in 1-D ring - a well- studied problem. Termination occurs when the anterogarde branch of the reentrant wave is blocked in a region which is still refractory after the passage of the reentrant wave. When the pacing site is not located on the 1-D ring itself (as is generally the case in any realistic pacing arrangement) this method of pacing termination of reentry fails. Proper timing of the pacing wave is crucial. Glass (1995): From continuity arguments, there exists a range of stimuli phases and amplitudes that lead to successful reentry termination.

Off-circuit Pacing of Reentry in 1-D Ring Consider a homogeneous reentrant circuit (length= L). Pacing site located on ENTRY sidebranch - distance z from circuit. ENTRY sidebranch: x = 0. ENTRY sidebranch: x = 0. Conduction velocity = c. Conduction velocity = c. Refractory period = r. Refractory period = r. t=0: Pacing stimulus applied. t=0: Pacing stimulus applied. t=z/2c: Stimulus collides with reentrant wave branch propagating out through ENTRY sidebranch. t=z/2c: Stimulus collides with reentrant wave branch propagating out through ENTRY sidebranch. t=r: Pacing site recovers. t=r: Pacing site recovers. t=r+(z/c): 2nd stimulus (applied at t=r) reaches the circuit but refractory tail of reentrant wave at distance x=z away from the ENTRY sidebranch - anterograde branch of the stimulus will not be blocked. t=r+(z/c): 2nd stimulus (applied at t=r) reaches the circuit but refractory tail of reentrant wave at distance x=z away from the ENTRY sidebranch - anterograde branch of the stimulus will not be blocked. When z > 0, it is impossible for the stimulus to catch up with the refractory tail in a homogeneous medium.

The Critical Role of Inhomogeneity Assuming inhomogeneity - e.g., longer refractory period or slower conduction in narrow channel between non-conducting obstacles may lead to successful block of the anterograde branch of the pacing wave. (Abildskov & Lux, 1995) If an inhomogeneity (e.g., a zone of slow conduction) exists in the circuit, the waves travel faster or slower depending on location in the circuit. As a result, stimuli may arrive at the circuit from the pacing site and encounter a region that is still refractory - leads to block of the anterograde branch of the stimulus  successful termination.

L-R model simulation results: One- dimensional ring Spatiotemporal propagation of a reentrant wave in a ring (L=250 mm)successfully terminated by two pacing stimuli applied at x=0 mm (at T1 = 2600 ms and T2=3200 ms). Zone of slow conduction has step boundary. Parameter space dgm. of Coupling Interval (CI) vs. Pacing Interval (PI) at which termination occurs for 1-D L-R ring of length 250 mm with a zone of slow conduction (25 mm) and VT period = ms. (For top figure, CI = 899 ms & PI = 600 ms.)

Two-dimensional Excitable Media Model Schematic diagram of anatomical figure-of-eight reentry - paced from stimulation site at the apex of the ventricles. Inset: Model used for 2-D simulations. Square patches represent non-conducting scar tissue. Pacing site is at the ventricular apex - pacing waves propagating upward from the site represented as plane waves from the bottom. No-flux boundary conditions at top and bottom; periodic boundary conditions at the sides.

Panfilov model simulation results: 2-D Homogeneous Media

Panfilov model simulation results: 2-D Inhomogeneous Media

Effect of Anisotropy Cardiac tissue shows anisotropic propagation - the action potential propagates faster along the direction of the myocardial fibers than transverse to it. Axis of anisotropy rotates along the thickness of the myocardium (from the endocardial to the epicardial layer). Our simulations of pacing in anisotropic models showed no qualitative difference from results in isotropic models. For human ventricular myocardium,longitudinal conduction velocity ~ 50 cm/sec and transverse conduction velocity ~ 14 cm/sec. We used anisotropy ratio = 1:0.3.

Outlook Implications for pacing algorithms Limitations: 1- and 2-Dimensional - instead of 3-Dimensional (the anisotropy axis rotates along the thickness). 1- and 2-Dimensional - instead of 3-Dimensional (the anisotropy axis rotates along the thickness). Another method of simulating ischemia - increasing K+ concentration (But this does not provide a chronic arrhythmogenic substrate). Another method of simulating ischemia - increasing K+ concentration (But this does not provide a chronic arrhythmogenic substrate). Monodomain assumed instead of Bidomain (this is justified for the low- amplitude stimulus used in pacing). Monodomain assumed instead of Bidomain (this is justified for the low- amplitude stimulus used in pacing).

Summary n Cardiac arrest due to VF is a spatio-temporally chaotic phenomenon. n VF arises due to break-up of spiral/scroll waves induced by re-entrant activity. n Fitzhugh-Nagumo model with Puschino kinetics (Panfilov) is the simplest one that shows spiral breakup similar to that in VF. n We have controlled spiral break-up in 2 and 3-D in the Panfilov model (we are extending the control technique to more complex models of the heart).

Acknowledgements Collaborators: Ashwin Pande Prof. Rahul Pandit Computational Facilities: SERC, IISc Financial Support: JNCASR, Bangalore & CSIR Discussion: Alan Pumir N. I. Subramanya