Stochastic Aspects of Cardiac Arrhythmias

28 Pages·2007·0.74 MB·English

JournalofStatisticalPhysics,Vol.128,Nos.1/2,July2007((cid:1)C 2007)
DOI:10.1007/s10955-006-9191-y
Stochastic Aspects of Cardiac Arrhythmias
ClaudiaLerma,1,2TrineKrogh-Madsen,3MichaelGuevara1andLeonGlass1
ReceivedFebruary23,2006;acceptedJuly26,2006
PublishedOnlineAugust26,2006
Abnormalcardiacrhythms(cardiacarrhythmias)oftendisplaycomplexchangesover
timethatcanhavearandomorhaphazardappearance.Mathematically,thesechanges
canonoccasionbeidentifiedwithbifurcationsindifferenceordifferentialequation
models of the arrhythmias. One source for the variability of these rhythms is the
fluctuatingenvironment.However,intheneighborhoodofbifurcationpoints,thefluc-
tuationsinducedbythestochasticopeningandclosingofindividualionchannelsin
thecellmembrane,whichresultsinmembranenoise,mayleadtorandomnessinthe
observeddynamics.Toillustratethis,weconsidertheeffectsofstochasticproperties
ofionchannelsontheresettingofpacemakeroscillationsandonthegenerationof
earlyafterdepolarizations.Thecomparisonofthestatisticalpropertiesoflongrecords
showingarrhythmiaswiththepredictionsfromtheoreticalmodelsshouldhelpinthe
identificationofdifferentmechanismsunderlyingcardiacarrhythmias.
KEYWORDS: stochasticdifferentialequations,earlyafterdepolarizations,ionicmod-
els,prematureventricularcomplexes,phaseresetting.
1. INTRODUCTION
The heart is an amazing organ. In human beings, the heart beats over two
billiontimesovera70-yearlifetime.Aninterruptionofthisbeatingpatternfora
timeasbriefasafewminutesoftenleadstoseriousneurologicaldamage.Thus,
theheartrhythmmustbeincrediblyrobust,abletosustainitselfdespiteavariety
of changes in the body that arise over the short term as a consequence of one’s
1CentreforNonlinearDynamicsinPhysiologyandMedicineandDepartmentofPhysiology,McGill
University, 3655 Promenade Sir William Osler, Montreal, Quebec, Canada H3G 1Y6; e-mails:
[email protected],[email protected],[email protected].
2Departamento de Instrumentacion Electromecanica, Instituto Nacional de Cardiologa “Ignacio
Chavez,”JuanBadiano1,MexicoD.F.,Mexico14080.
3DepartmentofMedicine,DivisionofCardiology,WeillMedicalCollegeofCornellUniversity,520
E.70thStreet,Starr463,NewYork,NY10021,USA;e-mail:[email protected].
347
0022-4715/07/0700-0347/0(cid:1)C2007SpringerScience+BusinessMedia,Inc.
348 Lerma,Krogh-Madsen,GuevaraandGlass
dailyactivities,aswellasoverthelongtermasaconsequenceofnormalagingand
disease.Viewedfromatheoreticalperspective,onecanthinkoftheheartrhythm
as a stable limit-cycle oscillation, some of whose properties, such as the period,
maybemodifiedtosuitbodilydemandsthatareconveyedtotheheartbyneural
activityandcirculatinghormonesthatregulatecardiacactivity.Inthisarticlewe
arguethatinsomeexperimentalandclinicalsituations,deterministicdifferential
equationsmaygiveresultsthatarequalitativelyincorrectandthatitisessentialto
considerstochasticmathematicalmodels.
InSec.2wegiveabriefintroductiontothekeyconceptsofcardiacelectro-
physiologythatweuseinthisarticle.InSec.3,wegivesomephenomenological
observations about abnormal heart rhythms as recorded from the electrocardio-
gram,focusingespeciallyonthepatternsofatypeofabnormalheartbeatcalled
a premature ventricular complex. In Sec. 4 we introduce a few of the stochas-
tic sources that influence the cardiac rhythm. In Sec. 5, we give two examples
of how the stochastic opening and closing of ion channels in the cell mem-
branecanleadtoimportantqualitativechangesinthedynamicsinmathematical
models of cardiac systems when compared with the dynamics in deterministic
models.
2. APRIMERONCARDIACELECTROPHYSIOLOGY(36)
Inthenormalheart,electricalactivityoriginatesoneachheartbeatinaspe-
cializedpacemakerregioncalledthesinusnode.Theactivitythenspreadsthrough
theupperchambersoftheheart(theatria),thenthroughtheatrioventricularnode
andtheHis-Purkinjesystemtothelowerchambersoftheheart(theventricles).At
thecellularlevel,theheartbeatisassociatedwithcyclicchangesintheelectrical
potential difference across the cell membrane, which separates the intracellular
andextracellularmilieu.Thispotentialdifferencearisesasaconsequenceofcon-
centrationdifferencesofseveralions,chieflyNa+,K+,andCa2+,acrossthecell
membrane.Theseconcentrationdifferencesaremaintainedbyspecializedmolec-
ularcomplexescalledionpumpsthatuseenergytotransportionsacrossthecell
membrane. Further, there are individual channels in the cell membrane which
stochastically open or close. Ions flow through these channels and thus change
thevoltageacrossthecellmembrane.Therateatwhichionicchannelsopenand
close is different for each type of channel and is based largely on the potential
difference across the membrane in which they are embedded. The activity of
channels can also be modulated by neurotransmitters and circulating hormones.
On each heartbeat, there is an action potential, in which there is an increase in
thetransmembranevoltage(depolarization),associatedwithatransientincreased
permeabilityofthecellmembranetoNa+andCa2+,followedbyarepolarization
totherestingmembranepotential,associatedwithanincreasingpermeabilityof
thecellmembranetoK+.Thechangesinmembranepotentialleadtoasequence
StochasticAspectsofCardiacArrhythmias 349
of events that result in the contraction of the heart muscle and the consequent
pumpingofbloodthroughthebody.
A central goal of research over the past 50 years has been to understand
and mathematically model the ionic processes underlying activity in the heart.
The foundation was set in landmark work by Hodgkin and Huxley who devel-
opednonlinearordinarydifferentialandpartialdifferentialequationsfortheionic
processesunderlyingthegenerationandconductionofnerveimpulses.(29)Subse-
quentlyNoble(55) andothersextendedthisapproachtotheheart.Currentmathe-
maticalmodelsofasinglecardiaccellareformulatedassystemsoftensofcoupled
equationswithhundredsofparameters(e.g.,Refs.50,75).
Abnormalcardiacrhythms(i.e.,cardiacarrhythmias)canbeviewedasarising
as a consequence of one of two different mechanisms. There is either abnormal
generation of action potentials or abnormal conduction of the action potential
withintheheart.Abnormalitiesinheartbeatgenerationoccurifthesinusnodebeats
tooquicklyortooslowly,orifotherregionsoftheheartdevelopanintrinsicrhythm
thatisnotentrainedtothenormalsinusrhythm,leadingtoectopicbeats.Abnormal
patterns of conduction can arise as a consequence of blocked conduction. For
example,insomepeoplenotalltheactionpotentialsoriginatinginthesinusnode
areconductedtotheventricles.Inotherpeople,conductionabnormalitiesproduce
reentrantrhythms,inwhichtheperiodofthecardiacrhythmissetbythetimeit
takesforanexcitationtotravelinareentrantpath,ratherthanbytheperiodofthe
sinusrhythm.(36,84)
The recognition of the presence of cardiac arrhythmias musthave arisen in
antiquity when people felt abnormalities in the rhythm of the pulse. However,
theanalysisofarrhythmiashasbeenenormouslyaidedbytheelectrocardiogram,
which measures the potential difference arising between points on the surface
of the body as a consequence of the propagation of the action potential through
the entire heart. The electrocardiographic signal, which is of the order of 0.1–5
millivolts in amplitude, has been recorded and analyzed for about the past 100
years. Examples of electrocardiograms, which we will discuss in more detail as
thepaperprogresses,aregiveninFig.1.Abnormalitiesinthequalitativefeatures
of the electrocardiogram are used to classify cardiac arrhythmias into a number
of different types, based on the nature of the abnormality and the portion of the
heart affected. For example, ventricular tachycardia refers to an abnormally fast
heartbeat originating in the ventricles. But there are several types of ventricular
tachycardia:someoftheseresultintheheartpumpinganadequatebloodflowto
thebodyandsocanbeconsistentwiththecontinuedexistenceoflife,whileothers
do not generate enough blood flow and will lead to death. In most people, the
terminalrhythmisventricularfibrillation,arhythminwhichtherearebelievedto
bemultipleco-existingreentrantspiralwavesofexcitationintheventricles.(84)In
Fig.1,theendofeachrecordshowsventriculartachycardia,whichcandegenerate
350 Lerma,Krogh-Madsen,GuevaraandGlass
Fig.1. ElectrocardiographicHolter-recordingtracesfromtwopatientswhosufferedsustainedventric-
ulartachycardia(arrowsindicateonset),obtainedfromtheSuddenCardiacDeathHolterDatabase.(72)
(A) An 82-year-old woman with heart failure (patient 52). (B) A 68-year-old man with a history
of ventricular arrhythmias who was taking quinidine and digoxin (patient 45). The electrocardio-
graphiccomplexesarelabeledasbeingofnormal(N)orventricular(V)origin.Thenumberofnormal
interveningbeatsbetweentwoventricularbeats(NIB)isindicatedbeloweachtrace.
intoventricularfibrillation,resultingindeath.Infact,thisisexactlywhathappened
subsequenttotheendofthelastrecordshowninFig.1A.
Wenowtrytoplacetheinitiationofarrhythmiasintoanonlineardynamics
context. Clearly, the normal pacemaker oscillation and propagation in the intact
heartareextraordinarilyrobust.Bythiswemeanthatunderawiderangeofcir-
cumstances,therhythmisqualitativelyidentical.Thesinusnodeisthepacemaker
and sets the rate, initiating an orderly spread of excitation over the entire heart.
StochasticAspectsofCardiacArrhythmias 351
However, in some circumstances, parameters describing part or all of the heart
may change from normal values so that qualitatively different dynamics occur.
Mathematically,suchqualitativechangesindynamicsarecalledbifurcations.In
somecaseschangesinparametervaluesareabrupt,takingplaceoveratimescale
ofsecondsorminutes(aheartattack,changesintheactivityofthenervoussys-
temorinthecirculationofhormones,administrationofdrugsthatchangecardiac
properties). In other cases the changes are gradual: e.g., slow changes over the
years as a consequence of a faulty heart valve leading to increased atrial pres-
sure,consequentdevelopmentoffibrotictissue,andremodellingofthemixofion
channels in the atria, resulting in atrial fibrillation.(54) The bifurcation boundary
inparameterspacebetweennormalandabnormaldynamicsmightbetraversed
slowlywithrespecttothetimebetweenheartbeats.Insuchasituation,stochastic
effectswillbecomeprominentsinceaminutechangeinsomeparameterwouldlead
toonebehaviororanother.Thisconceptiscentraltothefollowingdiscussion.
3. ELECTROCARDIOGRAMANALYSIS
The electrocardiogram provides a visualization of the electrical activity of
theheart.InFig.1,weshowexamplesofelectrocardiogramsoftwopatientstaken
fromthePhysionetSuddenCardiacDeathHolterdatabase.(72)AHolterrecording
is an ambulatory recording of the electrocardiogram, usually over a period of
∼24h.Thesetwopatientshadventriculartachycardia(onsetindicatedbyarrows
inFig.1)andthesubjectinFig.1Adiedwhileherelectrocardiogramwasbeing
recorded.Basedonthemorphologyofthedeflectionsontheelectrocardiogram,
we distinguish normal sinus beats (labelled N) and premature ventricular com-
plexes(labelledV).Theprematureventricularcomplexesarisefromasitewithin
theventricles.InthepatientinFig.1Athereisonemorphologyforthepremature
ventricular complex, whereas, in Fig. 1B, there is more than one morphology.
There are two possible mechanisms for different morphologies in the same pa-
tient.Eithertheprematureventricularcomplexesarisefromdifferentsitesinthe
ventricles, or the premature ventricular complexes arise from a single site, but
areconductedthroughtheventriclesdifferentlyondifferentheartbeats.Although
thereareseveraldifferentphysiologicalmechanismsthathavebeenhypothesized
togenerateprematureventricularcomplexes,inmostcasesitisnotknownhowto
identifyamechanismfortheprematureventricularcomplexbasedoninspection
of the electrocardiogram.(36,65) One of the main points of this article is to make
statistical physicists aware of the fascinating problems encountered in trying to
decodethepatternsofprematureventricularcomplexes.
One way to obtain an impression of the pattern of premature ventricular
complexesintheelectrocardiogramistocountthenumberofnormalsinusbeats
betweentwoprematureventricularcomplexes.IntherecordsinFig.1,wedisplay
these numbers under each trace for several different segments of the recording.
352 Lerma,Krogh-Madsen,GuevaraandGlass
Most readers of this will be aware that medical exams often evaluate the elec-
trocardiogramonlyforshorttimeintervalsoftheorderofseveralseconds.Such
short segments do not always give a clear impression of the record over more
extendedtimes.Onewaytocharacterizeelectrocardiogramswithprematureven-
tricularcomplexesoverlongertimesistosimplywriteouttheintegersequenceof
thenumberofsuchcomplexesbetweenconsecutivenormalbeatsoverlongtimes.
InFig.2,weshowthesesequencesfortherecordsfromwhichFig.1wasderived.
The data in Fig. 2A shows a preponderance of low integers. These are not
randomly distributed. There are long sequences of consecutive 1s, but also an
apparentgradualincreaseoftheintegervaluesfollowedbyadecrease.Thereare
alsolongsequencesinwhichtherearenoprematureventricularcomplexes,sothat
theintegersinthetablearethenontheorderofseveralhundred.ThedatainFig.
2Barequitedifferent.Althoughthereareagainlongsequencesofconsecutive1s,
there are many more long stretches in which there are no premature ventricular
complexes.Moreover,thereisalsoastrongpreponderanceofoddnumbersinthe
sequence. The middle trace in Fig. 1B comes from a stretch of 45 numbers of
which6areeven.
Alikelyhypothesisabouttheserecordsisthatoverthelongtimeintervalsof
theserecordings,therearesomesortofchangesintheparametersdescribingthe
state of the heart. Unfortunately, unlike the situation in laboratory experiments,
datacollectedwhilewearingportablemonitorsisnotwellcontrolled,anditisnot
routine to simultaneously document some of the changes that might underly the
changes in rhythm in these subjects (e.g., change in posture, respiration, mental
state,drugs).Worsestill,therearealmostcertainlyphysiologicalchangesofwhich
wearenotawareanddonotthereforecurrentlymonitor.
Consequently, as a means of displaying this information over long times,
wehavedevelopedavisualizationtechniquecalleda“heartprint.”(64-66) Wedes-
ignate the number of intervening N beats between two consecutive V beats as
the NIB value. A pair of two consecutive V beats is termed a couplet, while
a sequence of 3 or more successive V beats that spontaneously terminates is
termed non-sustained ventricular tachycardia. Premature ventricular complexes
that are not part of a couplet or non-sustained ventricular tachycardia are called
isolated.TheNNintervalisthetimebetweentwoconsecutiveNbeats,whilethe
coupling interval (CI) is the time from an N beat to an immediately following
Vbeat.
A heartprint (Fig. 3) is a way to represent dependencies between the NN
intervaland(i)theectopicbeatinterval(betweentwoVbeats,orVVintervals),
(ii)NIBvalues,and(iii)theCI.Theordinateofthe3coloredplotsintheheartprint
istheNNinterval.TheincidenceoftheVVintervals,NIBvalues,andtheCIare
indicated in the three colored plots, respectively, where the relative frequency
of occurrence is indicated by the color (e.g., red is associated with the highest
incidence,anddarkbluewiththelowest).Thehistogramsabovethecoloredplots
StochasticAspectsofCardiacArrhythmias 353
Fig.2. Excerptsofconsecutivenumberofinterveningnormal(N)beatsbetweentwoventricular(V)
beats(NIB)measuredfromsametwopatientsasinFig.1.TheboxedsequencesindicatetheNIB
valuesassociatedwiththeECGsegmentsshowninFig.1.
354 Lerma,Krogh-Madsen,GuevaraandGlass
Fig. 3. (Color online) Heartprints from the same two patients presented in Fig. 1. The heartprint
represents the dependency between the intervals between two normal beats (NN) and three other
intervals:timebetweentwoventricularbeats(VV),numberofinterveningnormalbeats(NIB),and
thecouplinginterval(CI),i.e.thetimefromanNbeattoaVbeat.Theordinateofthethreecolored
plotsistheNNinterval.TheincidenceoftheVVintervals,NIBvalues,andtheCIisindicatedinthe
threecoloredplotsrespectively,wheretherelativefrequencyofoccurrenceisindicatedbythecolor
(e.g.,redisassociatedwithhigherincidence).Thehistogramsabovethecoloredplotsarethoseofthe
VVintervals,theNIBvalues,andtheCI,respectively,whilethehistogramstotheleftgivethatofthe
NNvalues.
StochasticAspectsofCardiacArrhythmias 355
givethehistogramsoftheVVintervals,theNIBvalues,andtheCI,respectively.
ThehistogramtotheleftofthecoloredplotsgivesthehistogramofNNvalues.
Figure3showstheheartprintsforthesubjectsfromwhomtheelectrocardio-
gramsinFig.1weretaken.Therearestrikingdifferences,especiallywithrespect
tothedistributionofthenumbersofsinusbeatsbetweenectopicbeats,thesinus
rates,andthecouplingintervals.InFig.3B,thereisevidencethatthedistribution
of the NIB values depends on the sinus rate, with a larger range of NIB values
occurringatlowerNNintervals.
An underlying goal of our work is to decode the mechanisms of ventricu-
lar arrhythmia by analyzing data such as that in Figs. 1–3. Further, since some
mechanisms may be associated with a high risk, whereas other mechanisms are
associatedwithbenignrhythms,theanalysisofarrhythmiamayhelpguidetherapy.
Foroneclassofarrhythmias,calledparasystole,therearestrikingqualitative
featuresoftheheartprintthatarereproducedintheoreticalmodels.Inparasystole,
there is an independent pacemaker in the ventricle that beats with its own fre-
quencyandcompeteswiththesinusrhythmforcontroloftheventricles.Insome
circumstances, the parasystolic rhythm is only marginally affected by the sinus
rhythm. In an earlier paper we have analyzed and modeled a record of this sort
(Case3inRef.65)byusingastochasticdifferenceequation,obtainingexcellent
agreementbetweenthemodelandtheclinicalrecord.However,thetworecordsin
Fig.1arequalitativelydifferentfromthiscasethatwehaveanalyzedandwedo
nothaveagoodtheoreticalunderstandingofthedynamicsintheserecords.
There are several possible mechanisms for the dynamics in these
records.(64,65) Itispossiblethatthereisaparasystolicfocusthatisstronglyreset
bythesinusrhythm-asituationthatistermedmodulatedparasystole.(35)Itisalso
possible that there are abnormal regions in the heart that initiate an extra action
potential. On the cellular level, one mechanism that can lead to this is called
an early afterdepolarization. An early afterdepolarization is a transient increase
in the membrane potential following an action potential. Although afterdepolar-
izations have been recognized fora long timebased on experimental studies,(12)
theirimportanceinaclinicalcontextisbecomingincreasinglyclear.Forexample,
several drugs that have been associated with premature death also lead to early
afterdepolarizations.(15,36)Further,somegeneticdefectsinNa+andK+channels
havebeenidentifiedwhichleadtoanincreasedrateofearlyafterdepolarizations
and increased risk of sudden death.(53) Evidence for a mechanism implicating
early afterdepolarizations is particularly strong for the record in Fig. 1B, since
thereareseveralelectrocardiographiccharacteristicsthatareconsistentwithearly
afterdepolarizations(abnormallylongQT-interval,presenceofU-waves)andthe
patient was taking a drug, quinidine, that can produce early afterdepolarizations
andventriculartachyarrhythmiasinexperimentalsettings.(1,46,49)
To date, there has not been a thorough theoretical analysis of the expected
dynamicsthatwouldresultifanectopicfocusoranearlyafterdepolarizationfocus
356 Lerma,Krogh-Madsen,GuevaraandGlass
wereembeddedintheventricle.Inourview,amodelwouldhavetoincludeboth
propagation into and out of the focus. Further, in order to understand statistical
aspects of records such as those in Fig. 1, we believe it would be essential to
treatstochasticaspectsincludingthefluctuationofthesinusrhythm.Carryingout
such a computation is a future goal. As a partial step in that direction, in Sec. 5
we will use two ionic models to demonstrate that stochastic effects on the level
of the ion channel can lead togross macroscopic changes indynamics. Firstwe
reviewearlierexperimentalandtheoreticalworkonstochasticdynamicsincardiac
systems.
4. SOURCESOFSTOCHASTICITYOFCARDIACDYNAMICS
4.1. ExternalStochasticInfluencesontheHeartbeat
Therearenumerousinfluencesthatcontroltheheartrhythm.Someofthese
areexternaltotheheart(oreventhebody)whereasothersareintheheartitself.
People exist in a fluctuating environment. During the course of the day, as
activitychanges,theheartreactstothechangingdemands.Forexample,everyone
isfamiliarwiththenotionthatphysicalactivityleadstoamorerapidheartbeat.But
theheartratealsotypicallyincreasessomewhatduringinspirationanddecreases
during expiration. These changes are under the control of a large number of
feedbackcontrolsystemsandaremediatedbythenervoussystemandcirculating
hormones. Activity of a class of neurons called sympathetic neurons tends to
increasetheheartrateandtheforceofcontractionoftheheart,whereasactivity
of another class of neurons, called parasympathetic neurons, tends to decrease
the heart rate. There are stochastic aspects of this influence. The firing (action
potential)ofanervecellleadstothereleaseofneurotransmittersinthevicinityof
heartcells,whichinturninfluencetheheart.Theneurotransmittersarereleased
in discrete quantal packets called vesicles. In experimental systems, the number
ofvesiclesreleasedduetoasingleactionpotentialisnotconstant,butisgenerally
thought to reflect an underlying stochastic process, being often described using
binomialorPoissondistributions.(51)Therehasbeensomemodellingofthecontrol
systemsregulatingheartratethatincludesastochasticcomponent.(38,41,60,69)
Theresultoftheseinfluencesleadstofluctuationsintheheartrate.Analysis
ofthefluctuationsoftheheartrateinnormalpeoplehasbeenintensivelystudied
byanalysisof24-hourHolterrecordings.Thefluctuationsarevariouslydescribed
asbeingchaoticordisplaying1/f noise,multifractality,orlong-rangescaling.(34)
Althoughthereisnostrongevidencefordeterministicchaosinnormalheartrate
variability,complexfluctuationsareobservedevenifenvironmentalfluctuations
are held constant, perhaps reflecting the dynamics of multiple feedback control
circuits.Inthenormalheartthevariabilityisgreatlyreducedwhendrugsaregiven
thatblocktheeffectsofsympatheticandparasympatheticnerveactivity,(86) orin