Physics of Cardiac Arrhythmias
Sitabhra Sinha
Institute of Mathematical Sciences (IMSc)
Chennai 600 113, 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
Motivation
 Cardiac arrhythmias: tachycardia & fibrillation
 Reentrant waves and spiral turbulence in
excitable medium : a model for VF & VT
 Models: Luo-Rudy I & Panfilov models
 Spiral formation and breakup: VT and VF
 Control of reentry and spiral chaos
 Implications for cardiac pacing & defibrillation
 Summary
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Motivation: Why Study
Fibrillation ?
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Sudden cardiac death due to ventricular fibrillation
(VF) is the leading cause of death in the
industrialized world.
One-third of all deaths in the USA are due to
cardiac arrest - one out of six due to VF.
Understanding VF is an essential prerequisite for
improving current methods of defibrillation
(massive electrical shocks ~ 600 Volts).
Possible alternative: Controlling spatio-temporal
chaos of VF through low-amplitude perturbations.
What is Ventricular Fibrillation ?
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Ventricular Fibrillation (VF): a disorganized
electrical wave activity that destroys the coherent
contraction of ventricular muscle.
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.
Spiral/ scroll waves lead to abnormally rapid heart
beat (t ~ 100-200 ms) (Tachycardia).
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.
http://www.physics.gatech.edu/chaos
A Brief History of VF
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1874: “Fibrillation” (Alfred Vulpian)
“...individual fibers contracting independently... a wad of writhing worms”
(Tacker and Geddes, 1980)
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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.”
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1899: Electrical defibrillation in animals (Prevost and Batelli).
1914: Fibrillation induced through precisely timed electrical
stimulus (G. R. Mines).
1947: Defibrillation of human heart (Claude Beck)
1960s: Initial work in Internal Cardiac Defibrillator (Mirowski).
Anatomy of the Heart
Anatomical vs. Functional
Reentry
Anatomically determined (Mines,
1913)
Functionally determined (Allessie
et al., 1977)
1. Fixed length of circuit (determined
by anatomical obstacle).
1. Circuit length dependent upon
electrophysiological properties.
(“Spiral waves”)
2. Usually excitable gap between
head and tail of impulse.
2. No gap of full excitability.
3. Inverse relation between
revolution time and conduction
3. Revolution time proportional to
George Ralph Mines (1886-1914)
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
Diagram from Mines(1913) demonstrating circulating rhythms in closed circuits in
myocardial tissue: (a) Normal tissue; (b) Abnormal tissue with delayed conduction.
Implantable CardioverterDefibrillator (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.
The ICD constantly monitors heart rhythm.
Upon detection of VT/VF delivers a programmed treatment.
Capable of applying variety of possible treatments:
• Pacing - deliver a sequence of low-amplitude pulses.
• Cardioversion - a mild shock (if pacing fails in
terminating VT).
• Defibrillation - a large shock to terminate VF.
• Pacemaker - for slow heartbeat, can act as
pacemaker.
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
Gap
junction
Myocardial fibers: contractile strand of cardiac
muscle composed of many cells.
Voltage-gated
ion channels are
pathways for
charge
movement (Na+,
K+, Ca++ ions).
The Luo-Rudy (L-R) model
Biologically realistic model for
ventricular action potential
proposed by Luo & Rudy (I:
1991, II: 1995): incorporates
details of ionic currents.
The L-R I model has 8 coupled ODEs
describing the activity of each cardiac
cell: the transmembrane potential (V
), the intracellular Calcium
concentration (Cai) and six ionchannel gating variables (m, h, j, x,
xi, d, f ).
Action potential and ionic currents of a
ventricular myocyte simulated with L-R
model.
Luo-Rudy I model: Membrane
potential equation
The transmembrane potential V follows the reaction-diffusion equation
where Cm = 1 F/cm2 is the membrane capacitance, D (k-1) is the
conductivity constant and ILR (A/cm2) is the instantaneous total ionic
current through the cell :
Inward currents:
INa : Fast sodium current (Na+)
Isi : 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
Slow inward current (E
Na
Si
= 54.4 mV):
= 7.7-13.0287 ln ([Ca] i)):
Time-dependent potassium current (E
Time-independent potassium current:
Plateau potassium current:
Background current:
K
= -77 mV):
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,xi,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
Biologically realistic model for ventricular action potential
proposed by Beeler & Reuter (1977) - incorporating details of
ionic currents.
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 transmembrane
potential
during an
action
potential in the
B-R model.
Beeler-Reuter model: Membrane
potential equation
The transmembrane potential follows the reaction-diffusion equation
where IBR is the instantaneous total ionic current through the cell :
IK : transient outward potassium current
INa : fast sodium inward current
Ix : time-activated outward current (mostly K+ ions)
Is : 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
•Subthreshold stimulation  perturbation decays
•Suprathreshold stimulation 
•Feature: Conduction of propagating waves.
•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.
The Heart as an Excitable
Medium
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Excitable medium : a small but finite
perturbation from equilibrium can lead to
excitation (a large excursion away from
equilibrium) before equilibrium is restored.
Excitation in the Heart : the electromechanical
wave inducing cardiac-muscle contractions which
pump blood.
Refractory period : once excited, the medium
remains quiescent for a certain duration.
Spiral waves associated with abnormal cardiac
activity (Winfree).
The Panfilov Model for
Ventricular Fibrillation
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Fitzhugh-Nagumo model for excitable media with
`Puschino’ kinetics
Describes an excitable medium with an absolute and a
relative refractory period - the period after
repolarization during which the membrane recovers its
resting properties.
The simplest model that shows spiral breakup
qualitatively similar to that seen during VF.
`Puschino’ kinetics shortens the relative refractory
period.
Shows a long chaotic transient, whose duration
increases sharply with system size - agrees with the
observation that the hearts of larger animals are more
Panfilov Model: Equations
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Described by two coupled partial differential equations.
Variables: membrane potential, e (fast variable) and
effective membrane conductance, g (slow variable).
 e / t = i dij j e - f(e) - g,
 g / t =  (e, g) [k e - g].
dij : conductivity tensor - for isotropic medium,
replaced by Laplacian.
Diffusive term describes the coupling among cells.
f(e) : nonlinear function
- piecewise
linear nature.
 (e, g) : information
about refractory
periods.
Panfilov Model: Parameters
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 (e, g) = 1, e < e2
 (e, g) = 2, e > e2
 (e, g) = 3, e < e1 and g < g1
Variables: membrane potential, e (fast variable) and
effective membrane conductance, g (slow variable).
 e / t = i dij j e - f(e) - g,
 g / t =  (e, g) [k e - g].
dij : conductivity tensor - for isotropic medium,
replaced by Laplacian.
Diffusive term describes the coupling among cells.
f(e) : nonlinear function
- piecewise
linear nature.
Panfilov Model: Dynamics
2
.
5
.
5
,
g
2
1
g
e
1
0
.
5
e
0
-
0
0
.
5
5
0
1
0
01
5
02
T
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
Dynamics in the absence
of the diffusion term.
e changes at a fast rate
compared to g.
0
(
02
5
m
03
0
s
03
)
5
0
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
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.
The maximum value
attained by the KaplanYorke dimension DKY
during spatiotemporal
chaos - plotted as a
function of linear system
size L . The lifetime of the
chaotic transient increases
with L.
Panfilov Model: Controlling
Spatiotemporal Chaos
The model has non-conducting boundaries (no-flux or
Neumann boundary conditions) - as the ventricles are
electrically insulated from the atria.
Observation :
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Non-conducting boundaries absorb spiral defects.
Spirals do not last for appreciable periods in small systems.
Operating Principle for the Control Scheme :
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To divide the system ( L  L ) into K 2 smaller blocks.
Isolate the blocks ( of size L / K ) by stimulating the system along the
block boundaries - driving them to refractory state.
Each block is too small to sustain spiral activity - spirals absorbed by
block boundaries.
After the system is driven to the quiescent state, controlling
stimulation is withdrawn - block boundaries recover from refractory
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/cm2.
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/cm2.
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).
• With L = 256 control obtained for 4  L z .
control
meshisafter
 msec
• For•Lactivate
>
4,
pulse
control
necessary:
z
• keep it on for  ON msec
• turn it off for  OFF msec
• keep it on for  ON msec
• 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
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Current defibrillation techniques involve applying
electrical shock to the fibrillating heart .
Principle of operation:
Simultaneous
depolarization of all cells - so that the cardiac
pacemaker can take over.
External defibrillation ~ 5 kV.
Internal Cardiac Defibrillator
(ICD) ~ 600 V.
We propose using very-low-amplitude pulse (~
mV) applied for a brief duration ( ~ 100 ms).
Control over 2-D surface is effective even for 3-D
control.
Summary
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Cardiac arrest due to VF is a spatio-temporally
chaotic phenomenon.
VF arises due to break-up of spiral/scroll waves
induced by re-entrant activity.
Panfilov model is the simplest one that shows
spiral breakup similar to that in VF.
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
Motivation
 Heart as excitable medium
 Reentrant activity
 Fitzhugh-Nagumo model for Ventricular
Fibrillation (VF)
 Spiral formation
 Spiral breakup and VF
 Control of spatio-temporal chaotic activity
 Implications for defibrillation
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Motivation: Why Study AntiTachycardia Pacing ?
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Sudden cardiac death is the leading cause of death
in the industrialized world.
One-third of all deaths in the USA are due to
cardiac arrest.
One out of six due to VF.
Understanding VF is an essential prerequisite for
improving current methods of defibrillation
(massive electrical shocks ~ 600 Volts).
Possible alternative: Controlling spatio-temporal
chaos of VF through low-amplitude perturbations.
What is Ventricular
Tachycardia ?
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Ventricular Tachycardia (VT): Abnormally rapid heart beat (t ~ 100200 ms).
The heart is unable to pump blood efficiently for such rapid beating.
Underlying cause of VT: creation of re-entrant pathways of electrical
activity.
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
Each pacing wave splits into two branches
while traveling around the reentry circuit :
• Anterograde (along the direction of the
rentrant wave)
• Retrograde (against the direction of the
reentrant wave).
Pacing can result in :
• 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).
• 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.
Pacing Termination of
Ventricular Tachycardia
Several factors influence the ability of pacing to interact with VT:
• VT cycle length.
• 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 duration of the excitable gap.
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
The response pattern of VT to the delivery of single prematurely.
& double pacing stimuli (Josephson, 1993)
Pacing Termination of Reentry in
the 1-D Ring
Pacing termination of reentry in 1-D ring - a wellstudied 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.
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.
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.
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.
• Conduction velocity = c.
• Refractory period = r.
• t=0: Pacing stimulus applied.
• t=z/2c: Stimulus collides with reentrant wave
branch propagating out through ENTRY
sidebranch.
• t=r: Pacing site recovers.
• t=r+(z/c): 2nd stimulus (applied at t=r)
reaches the circuit but refractory tail of reentrant
wave
away from
the stimulus
ENTRY
Whenatzdistance
> 0, it isx=z
impossible
for the
sidebranch
anterograde
branchtail
of in
thea stimulus
to catch up- with
the refractory
will
not be blocked.
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
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 =
1303.07 ms. (For top figure, CI =
899 ms & PI = 600 ms.)
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.
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
Panfilov model simulation results:
Homogeneous Media
2-D
Panfilov model simulation results:
Inhomogeneous Media
2-D
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).
For human ventricular myocardium,longitudinal conduction velocity ~ 50 cm/sec and
transverse conduction velocity ~ 14 cm/sec. We used anisotropy ratio = 1:0.3.
Our simulations of pacing in anisotropic models showed no qualitative
difference from results in isotropic models.
Outlook
Implications for pacing algorithms
Limitations:
• 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).
• Monodomain assumed instead of Bidomain (this is justified for the lowamplitude stimulus used in pacing).
Summary

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

Cardiac arrest due to VF is a spatio-temporally
chaotic phenomenon.
VF arises due to break-up of spiral/scroll waves
induced by re-entrant activity.
Fitzhugh-Nagumo model with Puschino kinetics
(Panfilov) is the simplest one that shows spiral
breakup similar to that in VF.
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