Physics of Cardiac Physics of Cardiac ArrhythmiasArrhythmias

Sitabhra SinhaSitabhra Sinha

Institute of Mathematical Sciences Institute of Mathematical Sciences (IMSc)(IMSc)

Chennai 600 113, INDIAChennai 600 113, INDIACollaborators:

R Pandit, A Pande, T K Shajahan, A Sen(IISc, Bangalore)

D J Christini and K M Stein(WMC- Cornell University, NYC)

OutlineOutline MotivationMotivation Cardiac arrhythmias: tachycardia & Cardiac arrhythmias: tachycardia &

fibrillationfibrillation Reentrant waves and spiral turbulence in Reentrant waves and spiral turbulence in

excitable medium : a model for VF & VTexcitable medium : a model for VF & VT Models: Luo-Rudy I & Panfilov modelsModels: Luo-Rudy I & Panfilov models Spiral formation and breakup: VT and VFSpiral formation and breakup: VT and VF Control of reentry and spiral chaosControl of reentry and spiral chaos Implications for cardiac pacing & Implications for cardiac pacing &

defibrillationdefibrillation SummarySummary

Motivation: Why Study Motivation: Why Study Fibrillation ?Fibrillation ?

Sudden cardiac death due to Sudden cardiac death due to ventricular ventricular fibrillationfibrillation (VF) is the leading cause of death in (VF) is the leading cause of death in the industrialized world.the industrialized world.

One-third of all deaths in the USA are due to One-third of all deaths in the USA are due to cardiac arrest - one out of six due to VF.cardiac arrest - one out of six due to VF.

Understanding VF is an essential prerequisite for Understanding VF is an essential prerequisite for improving current methods of defibrillation improving current methods of defibrillation (massive electrical shocks ~ 600 Volts).(massive electrical shocks ~ 600 Volts).

Possible alternativePossible alternative: Controlling spatio-temporal : Controlling spatio-temporal chaos of VF through low-amplitude chaos of VF through low-amplitude perturbations. perturbations.

What is Ventricular What is Ventricular Fibrillation ?Fibrillation ?

Ventricular Fibrillation (VF): a disorganized Ventricular Fibrillation (VF): a disorganized electrical wave activity that destroys the electrical wave activity that destroys the coherent contraction of ventricular muscle.coherent contraction of ventricular muscle.

Underlying cause of VF: formation of Underlying cause of VF: formation of “electrical vortices” - 2-D (spiral)/3-D (scroll) “electrical vortices” - 2-D (spiral)/3-D (scroll) waves of action potential - creation of re-waves of action potential - creation of re-entrant pathways of electrical activity.entrant pathways of electrical activity.

Spiral/ scroll waves lead to abnormally rapid Spiral/ scroll waves lead to abnormally rapid heart beat (heart beat (tt ~ 100-200 ms) ( ~ 100-200 ms) (TachycardiaTachycardia).).

Ventricular tachycardia (if untreated) leads to Ventricular tachycardia (if untreated) leads to VF in a few seconds through spiral/scroll wave VF in a few seconds through spiral/scroll wave break-up.break-up.

Characterizing Ventricular Characterizing Ventricular FibrillationFibrillation

Normal sinus rhythmNormal sinus rhythm

TachycardiaTachycardia

TachycardiaTachycardia

Ventricular Ventricular FibrillationFibrillation

Spiral wave on the Spiral wave on the surface of a canine surface of a canine ventricle. Color ventricle. Color proportional to proportional to transmembrane transmembrane potential. Image obtained potential. Image obtained with voltage sensitive with voltage sensitive dyes and CCD camera. dyes and CCD camera. http://www.physics.gatech.edu/chaoshttp://www.physics.gatech.edu/chaos

A Brief History of VFA Brief History of VF 1874: “Fibrillation” (Alfred Vulpian)1874: “Fibrillation” (Alfred Vulpian)

“...individual fibers contracting independently... a wad of writhing “...individual fibers contracting independently... a wad of writhing worms” (Tacker and Geddes, 1980)worms” (Tacker and Geddes, 1980)

1888: “Sudden Cardiac Death” (J. A. MacWilliam) 1888: “Sudden Cardiac Death” (J. A. MacWilliam) “...the cardiac pump is thrown out of gear, and the last of “...the cardiac pump is thrown out of gear, and the last of its vital energy is dissipated in a violent and prolonged turmoil of its vital energy is dissipated in a violent and prolonged turmoil of fruitless activity in the ventricular wall.”fruitless activity in the ventricular wall.”

1899: Electrical defibrillation in animals (Prevost and 1899: Electrical defibrillation in animals (Prevost and Batelli).Batelli).

1914: Fibrillation induced through precisely timed 1914: Fibrillation induced through precisely timed electrical stimulus (G. R. Mines).electrical stimulus (G. R. Mines).

1947: Defibrillation of human heart (Claude Beck)1947: Defibrillation of human heart (Claude Beck) 1960s: Initial work in Internal Cardiac Defibrillator 1960s: Initial work in Internal Cardiac Defibrillator

(Mirowski).(Mirowski).

Anatomy of the HeartAnatomy of the Heart

Anatomical vs. Functional Anatomical vs. Functional Reentry Reentry

Anatomically determined Anatomically determined (Mines, 1913)(Mines, 1913)

1. Fixed length of circuit 1. Fixed length of circuit (determined by anatomical (determined by anatomical obstacle).obstacle).

2. Usually excitable gap 2. Usually excitable gap between head and tail of between head and tail of impulse.impulse.

3. Inverse relation between 3. Inverse relation between revolution time and conduction revolution time and conduction velocity.velocity.

Functionally determined Functionally determined (Allessie et al., 1977)(Allessie et al., 1977)

1. Circuit length dependent 1. Circuit length dependent upon electrophysiological upon electrophysiological properties. (“Spiral waves”)properties. (“Spiral waves”)

2. No gap of full excitability.2. No gap of full excitability.

3. Revolution time proportional 3. Revolution time proportional to length of refractory period.to length of refractory period.

George Ralph Mines (1886-George Ralph Mines (1886-1914)1914)

Diagram from Mines(1913) demonstrating circulating rhythms in closed Diagram from Mines(1913) demonstrating circulating rhythms in closed circuits in myocardial tissue: (a) Normal tissue; (b) Abnormal tissue with circuits in myocardial tissue: (a) Normal tissue; (b) Abnormal tissue with delayed conduction.delayed conduction.

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

Implantable Cardioverter- Implantable Cardioverter- Defibrillator (ICD) Defibrillator (ICD)

An ICD consists of a pulse generator An ICD consists of a pulse generator and electrical leads. Endocardial leads and electrical leads. Endocardial leads are inserted through a vein and are inserted through a vein and advanced to the right ventricle and/or advanced to the right ventricle and/or atrium. The pulse generator is placed atrium. The pulse generator is placed subcutaneously or submuscularly and subcutaneously or submuscularly and connected to the leads.connected to the leads.

• Pacing - deliver a sequence of low-amplitude Pacing - deliver a sequence of low-amplitude pulses. pulses. • Cardioversion - a mild shock (if pacing fails in Cardioversion - a mild shock (if pacing fails in terminating VT).terminating VT).• Defibrillation - a large shock to terminate VF.Defibrillation - a large shock to terminate VF.• Pacemaker - for slow heartbeat, can act as Pacemaker - for slow heartbeat, can act as pacemaker.pacemaker.

The ICD constantly monitors heart rhythm.The ICD constantly monitors heart rhythm.Upon detection Upon detection of VT/VF delivers a of VT/VF delivers a programmedprogrammed treatment. treatment.Capable of applying variety of possible treatments:Capable of applying variety of possible treatments:

Internal Internal DefibrillationDefibrillation

Example of an implantable Example of an implantable cardiac defibrillator (ICD)cardiac defibrillator (ICD)

Vol 54 ccMass 97 gmThickness 16mmLongevity 9 yrsBOL Voltage 6.4 VBOL Charge time 6.0 sec

Pacing AlgorithmsPacing AlgorithmsIs there an optimal anti-tachycardia pacing Is there an optimal anti-tachycardia pacing

algorithm ?algorithm ?

Spiral Waves in the HeartSpiral Waves in the Heart(Left) Spiral wave in (Left) Spiral wave in anatomically correct anatomically correct model of the dog model of the dog heart. Color code heart. Color code indicates calculated indicates calculated activation times (in activation times (in milliseconds) of milliseconds) of various regions of the various regions of the heart muscle.heart muscle.

(Right) Reentrant spiral wave (Right) Reentrant spiral wave excitation in a rabbit heart excitation in a rabbit heart observed with a voltage-observed with a voltage-sensitive dye. Color code sensitive dye. Color code indicates measured activation indicates measured activation time in milliseconds.time in milliseconds.

The Cardiac CellThe Cardiac Cell

Voltage-gated Voltage-gated ion channels ion channels are pathways are pathways for charge for charge movement movement (Na(Na++, K, K++, Ca, Ca++ ++

ions).ions).

Myocardial fibers: contractile strand of Myocardial fibers: contractile strand of cardiac muscle composed of many cells.cardiac muscle composed of many cells.

Gap Gap junctionjunction

The Luo-Rudy (L-R) modelThe Luo-Rudy (L-R) model

The L-R I model has 8 coupled The L-R I model has 8 coupled ODEs describing the activity of ODEs describing the activity of each cardiac cell: the each cardiac cell: the transmembrane potential (transmembrane potential (V V ), ), the intracellular Calcium the intracellular Calcium concentration (Caconcentration (Caii) and six ion-) and six ion-

channel gating variables (channel gating variables (mm, , hh, , jj, , xx, , xxii, , dd, , f f ).).Action potential and ionic currents of a Action potential and ionic currents of a ventricular myocyte simulated with L-ventricular myocyte simulated with L-R model.R model.

Biologically realistic model Biologically realistic model for ventricular action for ventricular action potential proposed by Luo & potential proposed by Luo & Rudy (I: 1991, II: 1995): Rudy (I: 1991, II: 1995): incorporates details of ionic incorporates details of ionic currents.currents.

Luo-Rudy I model: Luo-Rudy I model: Membrane Membrane potential equationpotential equation

The transmembrane potential The transmembrane potential VV follows the reaction-diffusion follows the reaction-diffusion equationequation

where where CCmm = 1 = 1 F/cmF/cm22 is the membrane capacitance, is the membrane capacitance, DD (k (k-1-1) is the ) is the conductivity constant and conductivity constant and IILR LR ((A/cmA/cm22) is the instantaneous total ) is the instantaneous total ionic current through the cell : ionic current through the cell :

Inward currents:Inward currents:

IINaNa : Fast sodium current (Na : Fast sodium current (Na++))

IIsisi : Slow inward current (Ca : Slow inward current (Ca++++))

Outward currents :Outward currents :

IKIK : Time-dependent potassium : Time-dependent potassium currentcurrent

IIK1K1 : Time-independent potassium : Time-independent potassium currentcurrent

IKpIKp: Plateau potassium current: Plateau potassium currentIbIb : Background current : Background current

Luo-Rudy I model: Luo-Rudy I model: Formulation Formulation of Ionic Currentsof Ionic Currents

Fast sodium current (E Fast sodium current (E NaNa = 54.4 mV): = 54.4 mV):

Slow inward current (E Slow inward current (E SiSi = 7.7-13.0287 ln ([Ca] = 7.7-13.0287 ln ([Ca] ii)):)):

Time-dependent potassium current (E Time-dependent potassium current (E KK = -77 mV): = -77 mV):

Time-independent potassium current:Time-independent potassium current:

Plateau potassium current:Plateau potassium current:

Background current:Background current:

Luo-Rudy I model: Luo-Rudy I model: CaCa++++ concentration and gating variable concentration and gating variable equationsequations

The intracellular calcium concentration (Ca The intracellular calcium concentration (Ca ii) satisfies the ODE:) satisfies the ODE:

Each ion channel gating variable Each ion channel gating variable ( = ( = mm,,hh,,jj,,xx,,xxii,,dd,,ff ) )

is governed by ODEs of the formis governed by ODEs of the form::

The parameters The parameters and and are functions of the rate constants are functions of the rate constants and and ::

andand

: normalized fraction of the population of ion channels in open : normalized fraction of the population of ion channels in open state, state, : rate at which channels open; : rate at which channels open; : rate at which the channels : rate at which the channels close.close.The rate constants The rate constants and and are complicated are complicated functions of the membrane potential functions of the membrane potential VV..

The Beeler-Reuter (B-R) The Beeler-Reuter (B-R) model model

The B-R model has 8 coupled ODEs describing the activity of The B-R model has 8 coupled ODEs describing the activity of each myocardial cell - corresponding to the transmembrane each myocardial cell - corresponding to the transmembrane potential (potential (V V ), the intracellular Calcium concentration (), the intracellular Calcium concentration (cc) and six ) and six ion-channel gating variables (ion-channel gating variables (mm, , hh, , jj, , xx, , dd, , f f ).).

Variation of Variation of the trans-the trans-membrane membrane potential potential during an during an action action potential in potential in the B-R the B-R model.model.

Biologically realistic model for ventricular action Biologically realistic model for ventricular action potential proposed by Beeler & Reuter (1977) - potential proposed by Beeler & Reuter (1977) - incorporating details of ionic currents.incorporating details of ionic currents.

Beeler-Reuter model: Beeler-Reuter model: Membrane potential equationMembrane potential equationThe transmembrane potential follows the reaction-diffusion The transmembrane potential follows the reaction-diffusion equationequation

where where IIBRBR is the instantaneous total ionic current through the is the instantaneous total ionic current through the cell :cell :

IIKK : transient outward potassium current : transient outward potassium currentIINaNa : fast sodium inward current : fast sodium inward currentIIx x : time-activated outward current (mostly K: time-activated outward current (mostly K++ ions) ions)IIss : slow inward calcium current: slow inward calcium current

Beeler-Reuter model: Beeler-Reuter model: Calcium Calcium concentration and gating variable concentration and gating variable equationsequations

The calcium concentration (The calcium concentration (cc ) satisfies the ODE: ) satisfies the ODE:

Each ion channel gating variable Each ion channel gating variable ( = ( = mm, , hh, , jj, , xx, , dd, , ff ) )is governed by ODEs of the formis governed by ODEs of the form::

The parameters The parameters and and are functions of the rate constants are functions of the rate constants and and ::

andand

Note: Note: represents the normalized fraction of the population of ion represents the normalized fraction of the population of ion channels that is in an open state, channels that is in an open state, is the rate at which channels open is the rate at which channels open and and is the rate at which they close. is the rate at which they close.

Beeler-Reuter model: Beeler-Reuter model: Rate Rate constant equationsconstant equations

Excitable MediaExcitable Media

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

The Heart as an Excitable The Heart as an Excitable MediumMedium

Excitable medium Excitable medium : a small but finite perturbation : a small but finite perturbation from equilibrium can lead to from equilibrium can lead to excitation excitation (a(a large large excursion away from equilibrium) before excursion away from equilibrium) before equilibrium is restored.equilibrium is restored.

Excitation in the Heart Excitation in the Heart : the electromechanical : the electromechanical wave inducing cardiac-muscle contractions which wave inducing cardiac-muscle contractions which pump blood.pump blood.

Refractory period Refractory period : once excited, the medium : once excited, the medium remains quiescent for a certain duration. remains quiescent for a certain duration.

Spiral waves associated with abnormal cardiac Spiral waves associated with abnormal cardiac activity (Winfree).activity (Winfree).

The Panfilov Model for The Panfilov Model for Ventricular FibrillationVentricular Fibrillation

Fitzhugh-Nagumo model for excitable media with Fitzhugh-Nagumo model for excitable media with `Puschino’ kinetics`Puschino’ kinetics

Describes an excitable medium with an absolute and a Describes an excitable medium with an absolute and a relative refractory period - the period after relative refractory period - the period after repolarization during which the membrane recovers repolarization during which the membrane recovers its resting properties.its resting properties.

The simplest model that shows spiral breakup The simplest model that shows spiral breakup qualitatively similar to that seen during VF.qualitatively similar to that seen during VF.

`Puschino’ kinetics shortens the relative refractory `Puschino’ kinetics shortens the relative refractory period.period.

Shows a long chaotic transient, whose duration Shows a long chaotic transient, whose duration increases sharply with system size - agrees with the increases sharply with system size - agrees with the observation that the hearts of larger animals are more observation that the hearts of larger animals are more likely to show VF.likely to show VF.

Panfilov Model: EquationsPanfilov Model: Equations Described by two coupled partial differential equations.Described by two coupled partial differential equations. Variables: membrane potential, Variables: membrane potential, e e ((fast variablefast variable) and ) and

effective membrane conductance, effective membrane conductance, g g ((slow variableslow variable).). e e // t t == i i ddijij j j ee - - f(e)f(e) - - g, g,

g g // t t == ((e, g)e, g) [ [k ek e - - gg]]..

ddijij : conductivity tensor - for isotropic medium, replaced : conductivity tensor - for isotropic medium, replaced by Laplacian. by Laplacian.

Diffusive term describes the coupling among cells.Diffusive term describes the coupling among cells. f(e) f(e) :: nonlinear function - piecewise linear nonlinear function - piecewise linear

nature.nature. ((e, g) e, g) : information about refractory : information about refractory

periods.periods. Conductivity: 2 cmConductivity: 2 cm22 / s. / s.

Panfilov Model: Panfilov Model: ParametersParameters

((e, g) = e, g) = 11, , e e << e e22

((e, g) = e, g) = 22, , e e >> e e22

((e, g) = e, g) = 33, , e e << e e1 1 andand g g << g g1 1

Variables: membrane potential, Variables: membrane potential, e e ((fast variablefast variable) and effective ) and effective membrane conductance, membrane conductance, g g ((slow variableslow variable).).

e e // t t == i i ddijij j j ee - - f(e)f(e) - - g, g,

g g // t t == ((e, g)e, g) [ [k ek e - - gg]]..

ddijij : conductivity tensor - for isotropic medium, replaced by : conductivity tensor - for isotropic medium, replaced by Laplacian. Laplacian.

Diffusive term describes the coupling among cells.Diffusive term describes the coupling among cells. f(e) f(e) :: nonlinear function - piecewise linear nature.nonlinear function - piecewise linear nature. ((e, g) e, g) : information about refractory periods.: information about refractory periods. Conductivity: 2 cmConductivity: 2 cm22 / s. / s.

Panfilov Model: DynamicsPanfilov Model: Dynamics

Dynamics in the Dynamics in the absence of the absence of the diffusion term.diffusion term.

ee changes at a fast changes at a fast rate compared to rate compared to gg..

0 50 100 150 200 250 300 350-0.5

0

0.5

1

1.5

2

2.5

T ( ms )

e , g

e

g

Spiral Turbulence in the 2-Spiral Turbulence in the 2-D Panfilov modelD Panfilov model

Pseudo-color plots of Pseudo-color plots of the the ee field at various field at various values of values of 1 1 (( 33 = 0.3). As = 0.3). As 1 1

decreases, the pitch decreases, the pitch of the spiral of the spiral decreases - ultimately decreases - ultimately leading to spiral leading to spiral breakup.breakup.

Local phase Local phase portraits at various portraits at various values of values of 11. The . The increasing scatter of increasing scatter of points indicates the points indicates the onset of onset of spatiotemporal spatiotemporal chaos.chaos.

Spatiotemporal Chaos in the Spatiotemporal Chaos in the 2-D Panfilov model2-D Panfilov model

The maximum value The maximum value attained by the Kaplan-attained by the Kaplan-Yorke dimension Yorke dimension DDKYKY during spatiotemporal during spatiotemporal chaos - plotted as a chaos - plotted as a function of linear function of linear system size system size L L . The . The lifetime of the chaotic lifetime of the chaotic transient increases with transient increases with LL..

The maximum The maximum Lyapunov exponent Lyapunov exponent ((maxmax) plotted as a ) plotted as a function of time function of time tt : : max max approaches a approaches a positive constant (~ positive constant (~ 0.2) and then decays 0.2) and then decays at large times to at large times to negative values.negative values.

Panfilov Model: Controlling Panfilov Model: Controlling SpatiotemporalSpatiotemporal Chaos Chaos

The model has non-conducting boundaries (The model has non-conducting boundaries (no-fluxno-flux or or NeumannNeumann boundary conditions)boundary conditions) - as the ventricles are - as the ventricles are electrically insulated from the atria.electrically insulated from the atria.

ObservationObservation : : Non-conducting boundaries absorb spiral defects.Non-conducting boundaries absorb spiral defects. Spirals do not last for appreciable periods in small systems.Spirals do not last for appreciable periods in small systems.

Operating Principle for the Control SchemeOperating Principle for the Control Scheme : : To divide the system ( To divide the system ( LL LL ) into ) into K K 22 smaller blocks. smaller blocks. Isolate the blocks ( of size Isolate the blocks ( of size L / K L / K ) by stimulating the system along the ) by stimulating the system along the

block boundaries - driving them to refractory state.block boundaries - driving them to refractory state. Each block is too small to sustain spiral activity - spirals absorbed by Each block is too small to sustain spiral activity - spirals absorbed by

block boundaries. block boundaries. After the system is driven to the quiescent state, controlling After the system is driven to the quiescent state, controlling

stimulation is withdrawn - block boundaries recover from refractory stimulation is withdrawn - block boundaries recover from refractory state.state.

Control parameters in 2-DControl parameters in 2-D

Panfilov modelPanfilov modelL = 128L = 128Pulse amplitude Pulse amplitude 57.3 mV / msec 57.3 mV / mseckept on for kept on for = 41.2 msec. = 41.2 msec.This implies a defibrillating current density of 57 This implies a defibrillating current density of 57 A/cmA/cm22..

Beeler-Reuter modelBeeler-Reuter model

L = 200L = 200Pulse amplitude Pulse amplitude 20 mV / msec 20 mV / mseckept on for kept on for = 120 msec suffices. = 120 msec suffices.This implies a defibrillating current density of 20 This implies a defibrillating current density of 20 A/cmA/cm22..

Panfilov Model: Control in Panfilov Model: Control in 3-D3-DControl algorithm as in 2-D with the following modifications:Control algorithm as in 2-D with the following modifications:

•Control mesh only on one free face of a 3-D domain ( L x Control mesh only on one free face of a 3-D domain ( L x L x L L x L zz).).

• With L = 256 control obtained for 4 With L = 256 control obtained for 4 L L z z ..

• For L For L z z > 4, pulse control is necessary:> 4, pulse control is necessary:• activate control mesh after activate control mesh after msec msec• keep it on for keep it on for ONON msec msec• turn it off for turn it off for OFFOFF msec msec• keep it on for keep it on for ONON msec msec• continue continue n n timestimesWe find We find ON ON = 0.11 msec, = 0.11 msec, OFFOFF = 22 msec and = 22 msec and nn = 30 suffices. = 30 suffices.

Note that Note that OFFOFF is is the duration of one action potential. the duration of one action potential.

Implications for Implications for DefibrillationDefibrillation

Current defibrillation techniques involve applying Current defibrillation techniques involve applying electrical shock to the fibrillating heart .electrical shock to the fibrillating heart .

Principle of operation:Principle of operation: Simultaneous depolarization Simultaneous depolarization of all cells - so that the cardiac pacemaker can take of all cells - so that the cardiac pacemaker can take over. over.

External defibrillation ~ 5 kV.External defibrillation ~ 5 kV. Internal Cardiac Defibrillator (ICD) ~ 600 V.Internal Cardiac Defibrillator (ICD) ~ 600 V. We propose using very-low-amplitude pulse (~ mV) We propose using very-low-amplitude pulse (~ mV)

applied for a brief duration ( ~ 100 ms).applied for a brief duration ( ~ 100 ms). Control over 2-D surface is effective even for 3-D Control over 2-D surface is effective even for 3-D

control.control.

SummarySummary Cardiac arrest due to VF is a spatio-Cardiac arrest due to VF is a spatio-

temporally chaotic phenomenon.temporally chaotic phenomenon. VF arises due to break-up of spiral/scroll VF arises due to break-up of spiral/scroll

waves induced by re-entrant activity.waves induced by re-entrant activity. Panfilov model is the simplest one that Panfilov model is the simplest one that

shows spiral breakup similar to that in VF.shows spiral breakup similar to that in VF. We have controlled spiral break-up in 2 We have controlled spiral break-up in 2

and 3-D in the Panfilov model and the and 3-D in the Panfilov model and the more realistic Luo-Rudy model of the more realistic Luo-Rudy model of the heart.heart.

OutlineOutline MotivationMotivation Heart as excitable mediumHeart as excitable medium Reentrant activity Reentrant activity Fitzhugh-Nagumo model for Ventricular Fibrillation (VF)Fitzhugh-Nagumo model for Ventricular Fibrillation (VF) Spiral formationSpiral formation Spiral breakup and VFSpiral breakup and VF Control of spatio-temporal chaotic activityControl of spatio-temporal chaotic activity Implications for defibrillationImplications for defibrillation SummarySummary

Motivation: Why Study Anti-Motivation: Why Study Anti-Tachycardia Pacing ?Tachycardia Pacing ?

Sudden cardiac death is the leading cause of Sudden cardiac death is the leading cause of death in the industrialized world.death in the industrialized world.

One-third of all deaths in the USA are due to One-third of all deaths in the USA are due to cardiac arrest.cardiac arrest.

One out of six due to VF.One out of six due to VF. Understanding VF is an essential prerequisite for Understanding VF is an essential prerequisite for

improving current methods of defibrillation improving current methods of defibrillation (massive electrical shocks ~ 600 Volts).(massive electrical shocks ~ 600 Volts).

Possible alternativePossible alternative: Controlling spatio-temporal : Controlling spatio-temporal chaos of VF through low-amplitude perturbations. chaos of VF through low-amplitude perturbations.

What is Ventricular What is Ventricular Tachycardia ?Tachycardia ?

Ventricular Tachycardia (VT): Abnormally rapid heart beat (t ~ 100-200 ms).Ventricular Tachycardia (VT): Abnormally rapid heart beat (t ~ 100-200 ms). The heart is unable to pump blood efficiently for such rapid beating.The heart is unable to pump blood efficiently for such rapid beating. Underlying cause of VT: creation of re-entrant pathways of electrical activity.Underlying cause of VT: creation of re-entrant pathways of electrical activity. VT (if untreated) may degenerate to Ventricular Fibrillation (VF) - leading to VT (if untreated) may degenerate to Ventricular Fibrillation (VF) - leading to

death in minutes.death in minutes.

Normal sinus rhythmNormal sinus rhythm

TachycardiaTachycardia

TachycardiaTachycardia

Ventricular Ventricular FibrillationFibrillation

Termination of Reentry by Termination of Reentry by PacingPacing

Pacing can result in : 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 Resetting of the reentrant wave (the retrograde wave collides with the reentrant wave - the anterograde wave becomes the new reentrant the reentrant wave - the anterograde wave becomes the new reentrant wave).wave).• Termination of reentry.Termination of reentry.Termination of reentry occurs by Termination of reentry occurs by block in the anterograde block in the anterograde directiondirection - since the retrograde branch of the wave will collide - since the retrograde branch of the wave will collide with the reentrant wave and annihilate each other.with the reentrant wave and annihilate each other.

Each pacing wave splits into two Each pacing wave splits into two branches while traveling around the branches while traveling around the reentry circuit :reentry circuit :

• Anterograde (along the direction of Anterograde (along the direction of the rentrant wave)the rentrant wave)

• Retrograde (against the direction of Retrograde (against the direction of the reentrant wave).the reentrant wave).

Pacing Termination of Pacing Termination of Ventricular TachycardiaVentricular Tachycardia

Several factors influence the ability of pacing to interact with Several factors influence the ability of pacing to interact with VT: VT: • VT cycle length. VT cycle length. • The refractory period at the stimulation site and at the VT The refractory period at the stimulation site and at the VT circuit. circuit. • The conduction time from the site of stimulation to the VT The conduction time from the site of stimulation to the VT circuit. circuit. • The duration of the excitable gap. The duration of the excitable gap.

The response pattern of VT to the delivery of The response pattern of VT to the delivery of single & double pacing stimuli (Josephson, single & double pacing stimuli (Josephson, 1993)1993)

Why multiple stimuli ?Why multiple stimuli ? Large number of conditions Large number of conditions for reentry to be terminated for reentry to be terminated single stimulus rarely single stimulus rarely sufficient. sufficient. Double stimuli often used: Double stimuli often used: first stimulus used only to first stimulus used only to “peel back” refractoriness, “peel back” refractoriness, allowing the second allowing the second stimulus to interact with the stimulus to interact with the circuit more prematurely. circuit more prematurely.

Pacing Termination of Pacing Termination of Reentry in the 1-D RingReentry in the 1-D Ring

Pacing termination of reentry in 1-D ring - Pacing termination of reentry in 1-D ring - a well-studied problem. Termination occurs a well-studied problem. Termination occurs when the anterogarde branch of the when the anterogarde branch of the reentrant wave is blocked in a region reentrant wave is blocked in a region which is still refractory after the passage of which is still refractory after the passage of the reentrant wave. the reentrant wave.

When the pacing site is not located on the 1-D ring itself (as is generally When the pacing site is not located on the 1-D ring itself (as is generally the case in any the case in any realisticrealistic pacing arrangement) this method of pacing pacing arrangement) this method of pacing termination of reentry fails.termination of reentry fails.

Proper timing of the pacing Proper timing of the pacing wave is crucial.wave is crucial. Glass (1995): From Glass (1995): From continuity arguments, there continuity arguments, there exists a range of stimuli exists a range of stimuli phases and amplitudes that phases and amplitudes that lead to successful reentry lead to successful reentry termination.termination.

Off-circuit Pacing of Off-circuit Pacing of Reentry in 1-D Ring Reentry in 1-D Ring

Consider a homogeneous reentrant circuit Consider a homogeneous reentrant circuit (length= L).(length= L).Pacing site located on ENTRY sidebranch - Pacing site located on ENTRY sidebranch - distance z from circuit. 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 t=z/2c: Stimulus collides with reentrant wave branch propagating out through wave branch propagating out through ENTRY sidebranch. ENTRY sidebranch. • t=r: Pacing site recovers. t=r: Pacing site recovers. • t=r+(z/c): 2nd stimulus (applied at t=r) t=r+(z/c): 2nd stimulus (applied at t=r) reaches the circuit but refractory tail of reaches the circuit but refractory tail of reentrant wave at distance x=z away from reentrant wave at distance x=z away from the ENTRY sidebranch - anterograde the ENTRY sidebranch - anterograde branch of the stimulus will not be blocked. branch of the stimulus will not be blocked.

When z > 0, it is impossible for the When z > 0, it is impossible for the stimulus to catch up with the refractory stimulus to catch up with the refractory tail in a homogeneous medium. tail in a homogeneous medium.

The Critical Role of The Critical Role of InhomogeneityInhomogeneity

Assuming inhomogeneity - e.g., longer refractory period or slower Assuming inhomogeneity - e.g., longer refractory period or slower conduction in narrow channel between non-conducting obstacles may conduction in narrow channel between non-conducting obstacles may lead to successful block of the anterograde branch of the pacing wave. lead to successful block of the anterograde branch of the pacing wave.

(Abildskov & Lux, 1995)(Abildskov & Lux, 1995)

If an inhomogeneity (e.g., a zone of slow If an inhomogeneity (e.g., a zone of slow conduction) exists in the circuit, the conduction) exists in the circuit, the waves travel faster or slower depending waves travel faster or slower depending on location in the circuit. As a result, on location in the circuit. As a result, stimuli may arrive at the circuit from the stimuli may arrive at the circuit from the pacing site and encounter a region that is pacing site and encounter a region that is still refractory - leads to block of the still refractory - leads to block of the anterograde branch of the stimulus anterograde branch of the stimulus

successful terminationsuccessful termination..

L-R model simulation results:L-R model simulation results: One-One-dimensional ringdimensional ring

Spatiotemporal Spatiotemporal propagation of a propagation of a reentrant wave in a reentrant wave in a ring (L=250 ring (L=250 mm)successfully mm)successfully terminated by two terminated by two pacing stimuli pacing stimuli applied at x=0 mm applied at x=0 mm (at T1 = 2600 ms (at T1 = 2600 ms and T2=3200 ms). and T2=3200 ms). Zone of slow Zone of slow conduction has step conduction has step boundary.boundary.Parameter space dgm. of Parameter space dgm. of

Coupling Interval (CI) vs. Coupling Interval (CI) vs. Pacing Interval (PI) at which Pacing Interval (PI) at which termination occurs for 1-D L-R termination occurs for 1-D L-R ring of length 250 mm with a ring of length 250 mm with a zone of slow conduction (25 zone of slow conduction (25 mm) and VT period = 1303.07 mm) and VT period = 1303.07 ms. (For top figure, CI = 899 ms. (For top figure, CI = 899 ms & PI = 600 ms.)ms & PI = 600 ms.)

Two-dimensional Excitable Two-dimensional Excitable Media ModelMedia Model

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

Panfilov model simulation results:Panfilov model simulation results: 2-D 2-D Homogeneous MediaHomogeneous Media

Panfilov model simulation results:Panfilov model simulation results: 2-D 2-D Inhomogeneous MediaInhomogeneous Media

Effect of AnisotropyEffect of AnisotropyCardiac tissue shows anisotropic propagation - the action potential Cardiac tissue shows anisotropic propagation - the action potential propagates faster along the direction of the myocardial fibers than propagates faster along the direction of the myocardial fibers than transverse to it. Axis of anisotropy rotates along the thickness of the transverse to it. Axis of anisotropy rotates along the thickness of the myocardium (from the endocardial to the epicardial layer). myocardium (from the endocardial to the epicardial layer).

Our simulations of pacing in anisotropic models showed no qualitative Our simulations of pacing in anisotropic models showed no qualitative difference from results in isotropic models. difference from results in isotropic models.

For human ventricular myocardium,longitudinal conduction velocity ~ 50 For human ventricular myocardium,longitudinal conduction velocity ~ 50 cm/sec and transverse conduction velocity ~ 14 cm/sec. We used anisotropy cm/sec and transverse conduction velocity ~ 14 cm/sec. We used anisotropy ratio = 1:0.3.ratio = 1:0.3.

OutlookOutlook

Implications for pacing algorithmsImplications for pacing algorithmsLimitations:Limitations:• 1- and 2-Dimensional - instead of 3-Dimensional (the anisotropy 1- and 2-Dimensional - instead of 3-Dimensional (the anisotropy axis rotates along the thickness). axis rotates along the thickness). • Another method of simulating ischemia - increasing K+ Another method of simulating ischemia - increasing K+ concentration (But this does concentration (But this does notnot provide a chronic arrhythmogenic provide a chronic arrhythmogenic substrate).substrate).• Monodomain assumed instead of Bidomain (this is justified for Monodomain assumed instead of Bidomain (this is justified for the low-amplitude stimulus used in pacing). the low-amplitude stimulus used in pacing).

SummarySummary Cardiac arrest due to VF is a spatio-Cardiac arrest due to VF is a spatio-

temporally chaotic phenomenon.temporally chaotic phenomenon. VF arises due to break-up of spiral/scroll VF arises due to break-up of spiral/scroll

waves induced by re-entrant activity.waves induced by re-entrant activity. Fitzhugh-Nagumo model with Puschino Fitzhugh-Nagumo model with Puschino

kinetics (Panfilov) is the simplest one that kinetics (Panfilov) is the simplest one that shows spiral breakup similar to that in VF.shows spiral breakup similar to that in VF.

We have controlled spiral break-up in 2 and We have controlled spiral break-up in 2 and 3-D in the Panfilov model (we are 3-D in the Panfilov model (we are extending the control technique to more extending the control technique to more complex models of the heart).complex models of the heart).

AcknowledgementsAcknowledgements

Collaborators:• Ashwin Pande• Prof. Rahul Pandit Computational Facilities: SERC, IISc

Financial Support:JNCASR, Bangalore & CSIR

Discussion:• Alan Pumir • N. I. Subramanya