Physics of Cardiac Arrhythmias

1
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)
2
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

3
Motivation Why Study Fibrillation ?

• 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.

4
What is Ventricular Fibrillation ?

• 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.

5
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
6
A Brief History of VF

• 1874 Fibrillation (Alfred Vulpian)
  ...individual fibers
  contracting independently... a wad of writhing
  worms (Tacker and Geddes, 1980)
• 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.
• 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).

7
Anatomy of the Heart
8
Anatomical vs. Functional Reentry
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.
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.
9
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.
10
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.
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.

11
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
12
Pacing AlgorithmsIs there an optimal
anti-tachycardia pacing algorithm ?
13
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.
14
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).
15
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 ion-channel gating
variables (m, h, j, x, xi, d, f ).
Action potential and ionic currents of a
ventricular myocyte simulated with L-R model.
16
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
Outward currents IK Time-dependent potassium
current IK1 Time-independent potassium
current IKp Plateau potassium current Ib
Background current
Inward currents INa Fast sodium current
(Na) Isi Slow inward current (Ca)
17
Luo-Rudy I model Formulation of Ionic Currents
Fast sodium current (E Na 54.4 mV)
Slow inward current (E Si 7.7-13.0287 ln (Ca
i))
Time-dependent potassium current (E K -77 mV)
Time-independent potassium current
Plateau potassium current
Background current
18
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.
19
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 trans-membrane potential during
an action potential in the B-R model.
20
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
21
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.
22
Beeler-Reuter model Rate constant equations
23
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.

24
The Heart as an Excitable Medium

• 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).

25
The Panfilov Model for Ventricular Fibrillation

• 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 likely to show VF.

26
Panfilov Model Equations

• 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.
• Conductivity 2 cm2 / s.

27
Panfilov Model Parameters

• ? (e, g) ?1, e lt e2
• ? (e, g) ?2, e gt e2
• ? (e, g) ?3, e lt e1 and g lt 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.
• ? (e, g) information
  about refractory periods.
• Conductivity 2 cm2 / s.

28
Panfilov Model Dynamics

• Dynamics in the absence of the diffusion term.
• e changes at a fast rate compared to g.

29
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.
30
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 Kaplan-Yorke
dimension DKY during spatiotemporal chaos -
plotted as a function of linear system size L .
The lifetime of the chaotic transient increases
with L.
31
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
• Non-conducting boundaries absorb spiral defects.
• Spirals do not last for appreciable periods in
  small systems.
• Operating Principle for the Control Scheme
• 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 state.

32
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.
33
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 .
• For L z gt 4, pulse control is necessary

• activate control mesh after ? msec
• 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.
34
Implications for Defibrillation

• 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.

35
Summary

• 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.

36
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
• Summary

37
Motivation Why Study Anti-Tachycardia Pacing ?

• 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.

38
What is Ventricular Tachycardia ?

• Ventricular Tachycardia (VT) Abnormally rapid
  heart beat (t 100-200 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
39
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.
40
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 prematurely.
The response pattern of VT to the delivery of
single double pacing stimuli (Josephson, 1993)
41
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.
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.
42
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.
• t0 Pacing stimulus applied.
• tz/2c Stimulus collides with reentrant wave
  branch propagating out through ENTRY sidebranch.
• tr Pacing site recovers.
• tr(z/c) 2nd stimulus (applied at tr) reaches
  the circuit but refractory tail of reentrant wave
  at distance xz away from the ENTRY sidebranch -
  anterograde branch of the stimulus will not be
  blocked.

When z gt 0, it is impossible for the stimulus to
catch up with the refractory tail in a
homogeneous medium.
43
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.
44
L-R model simulation results One-dimensional ring
Spatiotemporal propagation of a reentrant wave in
a ring (L250 mm)successfully terminated by two
pacing stimuli applied at x0 mm (at T1 2600 ms
and T23200 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
1303.07 ms. (For top figure, CI 899 ms PI
600 ms.)
45
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.
46
Panfilov model simulation results 2-D
Homogeneous Media
47
Panfilov model simulation results 2-D
Inhomogeneous Media
48
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 10.3.
Our simulations of pacing in anisotropic models
showed no qualitative difference from results in
isotropic models.
49
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 low-amplitude stimulus used in
  pacing).

50
Summary

• 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).

51
Acknowledgements

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