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International Journal of Bifurcation and Chaos, Vol. 12, No. 9 (2002) 1953–1968c© World Scientific Publishing Company
HOW ANTIARRHYTHMIC DRUGS INCREASETHE RATE OF SUDDEN CARDIAC DEATH
C. FRANK STARMERMedical University of South Carolina, Charleston, SC 29425, USA
Received January 10, 1998; Revised October 18, 2001
Two large clinical trials of drugs that exhibited significant antiarrhythmic properties in singlecells were found to increase the rate of sudden cardiac death in patients by two to three foldover untreated patients. We hypothesized that premature excitation within the drug-alteredvulnerable region, a region that trails each excitation wave, might be one mechanism for initi-ating re-entrant tachyarrhythmias that could lead to ventricular fibrillation and sudden cardiacdeath. With numerical studies of a cardiac cell model, we probed the determinants of the vul-nerable period. We found that antiarrhythmic drugs that block the sodium channel can increasethe duration of the cardiac vulnerable period by both slowing conduction velocity and reducingthe gradient of excitability. Coupling the dynamics of drug binding to ion channels with waveformation in a nonlinear excitable medium provides new insights into possible arrhythmogenicmechanisms.
Keywords: Nonlinear dynamics; antiarrhythmic drugs; vulnerable window; use-dependent; car-diac model; re-entry; proarrhythmic.
1. Introduction
“In few specialties of medicine are new promis-ing drugs shown to be so much inferior to placeboand, even worse, to increase mortality,” wroteProf. John Sanderson in an editorial discussing theunexpected and disappointing results of yet anotherlarge scale clinical trial of cardiac antiarrhythmicdrugs [Sanderson, 1996]. Until recently, cliniciansthought that depressing the response to an unex-pected excitation of the heart (PVC or prematureventricular contraction) would reduce the incidenceof arrhythmias leading to sudden cardiac death.Sanderson’s paradox was that new antiarrhythmicdrugs, able to extend the refractory period in iso-lated cells, actually increased the rate of suddencardiac death observed in clinical trials, in spite ofthe fact that > 80% of the responses to unexpectedendogenous excitations of the heart were suppressed[CAST Investigators, 1989; CAST II Investigators,
1992]. Is there a specific mechanism that links cel-lular antiarrhythmic effects (suppression of a highpercentage of unexpected excitatory events) withmulticellular proarrhythmic effects (increased inci-dence of sustained spiral wave activity)?
Potentially fatal cardiac arrhythmias arise fromunexpected stimuli or PVCs [Allessie et al., 1973;Krinsky, 1966]. For example, stress, vigorous exer-cise or a heart attack can alter the cellular environ-ment in the heart, leading to spontaneous oscilla-tion of otherwise quiescent cells. If the amplitudeof the oscillation is sufficient to excite neighboringcells, then either a continuous wave or a wave frag-ment will form depending on the state of neighbor-ing cells (some can be in the rest state while oth-ers might be in the refractory state). A continuouswave propagates in all directions away from the ex-citation site and eventually is extinguished by col-liding with itself. On the other hand, a wave frag-ment or wavelet, formed by successful propagation
1953
1954 C. F. Starmer
in some directions and failed propagation in otherdirections, can evolve into a spiral wave. Since a spi-ral wave is self-sustaining, a pattern of re-excitationevolves that can lead to loss of mechanical pumping,reduced blood pressure and possibly death.
A standard clinical paradigm for managingpatients with a predisposition to serious cardiacrhythm disturbances has been to reduce the likeli-hood of aberrant excitation. This was accomplishedby treating patients with drugs (ion channel block-ers) that either amplified the requirements to exciteindividual cardiac cells (by blocking Na channels) orprolonged the period of time a cell was inexcitable(by blocking K channels).
The paradox articulated by Sanderson was thatall drugs tested in two major recent clinical trials[CAST Investigators, 1989; CAST II Investigators,1992] significantly prolonged refractoriness in stud-ies of isolated cells. During clinical trials, though,treated patients died from sudden cardiac death at arate two to three times that of the untreated groups.The purpose of this paper is to explore how antiar-rhythmic drugs might facilitate initiation of poten-tially life threatening arrhythmias. As stated above,wavelets can be initiated by excitation within a vul-nerable region of the heart. Consequently, we ex-plored the determinants of the vulnerable region(VR), first articulated by Wiener and Rosenblueth[1946] and the role Na channel availability playedin the dynamics of front formation. The extent ofthe VR was sensitive to both the velocity of theparent wave and the gradient of excitability at thepoint of excitation. More important, we found thatslowly unbinding antiarrhythmic drugs reduced thegradient of excitability, thus extending the VR.
2. Numerical Methods
Although there are many models available to ex-plore numerically the behavior of an excitable cell,we chose the Beeler–Reuter (BR) [1977] model inorder to avoid the complexities introduced by acti-vation of numerous ion channels included in morerealistic cardiac models [Luo & Rudy, 1991]. Thebehaviors that we explore in this paper (thresh-old of excitation, front formation, front collapse,unidirectional conduction) are generic properties ofall excitable media and are determined by net di-rectional flow of transmembrane currents. Conse-quently, even the two current model of FitzHughand Nagumo [1961] is suitable for exploring mech-
anisms associated with refractory and vulnerableregions.
The excitable cell model used in our studies wasbased on a parallel circuit composed of a membranecapacitance and several gated ion channels that per-mit flow of ions between the extracellular and intra-cellular spaces [Hodgkin & Huxley, 1952; Beeler &Reuter, 1977] and is expressed as:
Cm∂V
∂t=
1
Ri∇2V − Iion (1)
where Ri is the internal resistivity of the cell, andthe ionic currents, Iion are defined in terms of gatedconductances of the form
Iion = gionagatedgate(V − Vion) (2)
where gion is the maximum ensemble conductance,agate and dgate represent the fractions of open acti-vation and inactivation gates and Vion is the reversalpotential for the channel charge carrier. The chan-nel gating variables vary between 0 (closed) and 1(open) and are modeled in terms of a first-ordertransition between closed and open states [Hodgkin& Huxley, 1952]:
Closedα⇐⇒β
Open (3)
The dynamics are described by
da
dt= αa(V )(1− a)− βa(V )a (4)
where α(V ) and β(V ) are voltage sensitive transi-tion rates and a is the fraction of open channels.
Propagation depends on the relationship be-tween the source of charge available to diffuse intoneighboring cells and the charge required by neigh-boring cells to exceed the threshold of excitation.Here we will focus on the Na current and fac-tors that modulate its magnitude and write thereaction–diffusion equation as:
Cm∂V
∂t=
1
Ri∇2V − (gNam
3hj(V −VNa)+ Iother ion)
(5)
where gNa is the maximum Na conductance, m3 isthe fraction of open activation gates and h and jare the fractions of open fast and slow inactivationgates. For our analysis, we approximate excitabil-ity in terms of the density of Na channels availableto open upon excitation, i.e. Availability = A =gNahj.
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1955
A cable was constructed by linking excitablesegments with an axial resistivity of 250 ohm-cmand a cell radius of 7 microns. The cable equa-tion was discretized with space steps of 1/512 cmand time steps 1/128 ms. The discretized equationswere solved using an implicit method. The bound-ary conditions at the cable ends were nonconduct-ing, i.e. ∂V/∂x = 0.
We measured the vulnerable period (VP) in a10 cm 1D cable using a conditioning wave followedby a test stimulus. We initiated a conditioning waveat the left boundary of the cable (see Fig. 6 top)with current injection (−10 µA/cm2). We probedthe post-refractory interval with test stimuli at asite 2.5 cm from the left end of the cable by in-jecting −10 µA/cm2 for 0.25 ms. For studies of theVP dependence on electrode length, the test stimuliwere produced by injecting a near-threshold current(−1 µA/cm2 for 0.25 ms) into a test electrode withlengths varying from 1.5 to 3.9 mm. The VP waslocated by using a binary search of the time window
bounded by the refractory period and the return tofull excitability following the passage of the condi-tioning wave.
3. Cardiac Cell Model
Under normal conditions, excitable cells exhibit astable resting state and an unstable excited state.With low amplitude stimuli, there is no response,but with stimuli that exceed a certain threshold,the cell’s electrical potential makes a major depar-ture from equilibrium generating an action poten-tial shown in Fig. 1.
To visualize the dynamics of channel gating, wecomputed a typical cardiac action potential (Fig. 2)and plotted the time course of the sodium chan-nel activation gate (m), the fast inactivation gate(h) and the slow inactivation gate (j). At therest potential, the activation gate, m = 0, and thefast and slow inactivation gates (h and j) are open(h = j = 1). Immediately after stimulation the
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–50.0�
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Cardiac Membrane Action PotentialSub- and Supra-threshold Stimulation�
Sub-thresholdexcitation
Suprathresholdexcitation
Unexcitable (refractory)
* Na channels do not open
** Na channels open, permitting an inward
current to flow
*** Na channel inactivate (close)
K channels permit an outward current to flow
*
***
**
Fig. 1. The computed action potential following subthreshold and suprathreshold stimulation. A subthreshold current wasapplied at t = 10 ms and was insufficient to raise the transmembrane potential above the threshold for opening Na channelactivation (m) gates. Following suprathreshold stimulation at t = 200 ms a sufficient number of Na channels opened such thatthe influx of Na current continued to depolarize the membrane. Inactivation of Na channels terminates current flow within1 ms. On a slower time scale, voltage sensitive K channels open (permitting a small outward current) and Ca channels open(permitting a small inward current). The result is a net outward current which gradually repolarizes the membrane, returningthe potential to the rest value.
1956 C. F. Starmer
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Fig. 2. A computed cardiac action potential and channel gating. Shown in the lower panels are the time course of theactivation gate (m) and the fast and slow inactivation gates (h and j). At the rest potential, the m gate is closed (m = 0)while the inactivation gates are open (h = j = 1). Following excitation, the membrane depolarizes and the activation gaterapidly changes from a closed to an open configuration (m > 0). On a slower time course, the inactivation gates change fromopen to closed (h, j < 1). The composite gating behavior creates a time varying conductance: gNa = m3hj which rapidlyincreases and then more slowly returns to zero. The two lower panels display gating dynamics during the excitatory phase ofthe action potential (at t = 10 ms) and during repolarization (at t = 300 ms).
fraction of open m gates increases rapidly while thefraction of open inactivation gates (h and j) de-crease on a slower rate. In cardiac cells, the rel-atively slow inactivation process, which terminatesconduction, is responsible for the refractory period,an interval of time during which the cell remainsunresponsive to stimulation. Due to the voltagesensitivity of the inactivation rate constants, recov-ery from inactivation typically does not occur untilthe membrane potential is within 10–15 mV of therest potential.
Action potentials are initiated by stimulationthat facilitates opening of channels carrying a de-polarizing current. As channel gates open, ioniccurrents flow down their concentration gradient.Following excitation, membrane depolarization pro-ceeds by the inward flow of Na (or Ca) ions throughNa (or Ca) channels. On a slower time course,potassium channels begin to open, initiating mem-brane repolarization by the outward flow of K ions.As the membrane potential repolarizes, the Nachannel gates revert to their rest configuration: theactivation gates close while the inactivation gates
open (but on a time scale roughly ten times slowerthan the activation time scale). Recovery fromchannel inactivation terminates the period of re-fractoriness. At the rest potential (about −84 mV),the fraction of inactivated Na channels is essentiallyzero, i.e. h = j = 1, and the cell is maximallyexcitable.
4. Modeling the Drug-ChannelInteraction
Many interactions between drugs and ion chan-nels are interesting because the chemistry of theseinteractions departs from traditional “continuous”chemistry. In continuous chemical reactions, whenone mixes two chemical reagents, each componenthas continuous access to each other and the reac-tion proceeds at a constant rate and is schematicallyrepresented by
Channel + Drugk⇐⇒l
Blocked (6)
Let b(t) represent the fraction of drug-complexed channels, then the time-dependent
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1957
fraction of drug-complexed channels is described by:
db
dt= k[Drug](1− b)− lb (7)
where k and l are forward and reverse rate con-stants. The time course of binding is described by:
b(t) = b(∞) + [b(0) − b(∞)]e−(k[Drug]+l)t (8)
Early studies of antiarrhythmic drugs, though,revealed that in the presence of an antiarrhyth-mic drug, the rate of rise of the cellular actionpotential was reduced as the rate of stimulationwas increased. This observation was consistentwith a transient reduction in Na conductance andsuggested that membrane excitability was in somemanner dependent on the rate of stimulation [John-son & McKinnon, 1957; Heistracher, 1971]. Morerecent voltage-clamp studies of cardiac cells re-vealed that the altered excitability was due totransient blockade of the Na channel by the drugmolecule [Gilliam et al., 1989; Starmer et al., 1991;Zilberter et al., 1994].
The frequency-dependent pattern of channelblockade was modeled by postulating that the drugmust selectively bind to one of the states of a chan-nel: either rest, open or inactivated [Starmer &
Grant, 1985; Starmer, 1988] as shown in Fig. 3. Wecaptured the state dependence of channel blockadeby adding a blocked state to the channel gate tran-sition model. Blockade of an activated channel wasthus described by:
Closedα⇐⇒β
OpenkD⇐⇒l
Blocked (9)
Let b represent the fraction of blocked channels,m represent the fraction of open channels, c repre-sent the fraction of closed channels and D representthe drug concentration, k and l represent the ratesof block and unblock. From conservation of chan-nels: c+m+ b = 1 and we can write the differentialequations characterizing the blocking process as
db
dt= k[D]m− lb (10)
dm
dt= α(1−m− b) + lb− (β + kD)m (11)
For most drug-channel reactions, the rates of drugbinding and unbinding are slow compared with thevoltage-dependent transition rates between openand closed channel conformations. Assuming rapid
Na+
Nagate
Na+
drug
gate
mem
bra
ne
mem
bra
ne
mem
bra
ne
gat
e +
BlockedAccessibleInaccessible(V)α
(V)β
kD
1
drug
drug
Fig. 3. A simple model of ion channel blockade. Shown is the cell membrane and a single sodium ion channel. The channel isin one of three conformations: either the binding site is inaccessible (left), accessible but unblocked (middle) or accessible andblocked (right). The channel gate is sensitive to the transmembrane potential, controls the flow of ions through the channeland is either closed (left) or open (middle and right). We hypothesize that the gating process also controls access to the drugbinding site. As shown, when the gate is in the closed configuration the diffusion path to the binding site is blocked. Whenthe gate is in the open configuration, the diffusion path is unrestricted and the drug is free to migrate to the binding siteand possibly form a drug-complexed (blocked) channel (right). Drug-complexed channels do not conduct. When the gatingtransitions are rapid compared with the time constant of drug binding, then Eq. (12) accurately describes the time course ofthe blockade process as a function of the rates of binding and unbinding as well as the voltage sensitive gate transition rates,α and β.
1958 C. F. Starmer
equilibrium between the closed and open states,Eq. (4) reduces to α(1−m) = βm or m = α/(α+β)so that the blockade equation can be rewritten as
db
dt=
α
α+ βk[D](1− b)− lb (12)
The voltage-sensitive gating rate constants, α andβ, thus modulate the rate of formation of drug com-plexed channels, k[D]. Because the channel gatingtransition rates α and β are sensitive to membranepotential, the diffusion path of drug to the bindingsite is modulated by membrane potential. At depo-larized potentials (i.e. during the action potential)α � β, thus promoting drug-channel interactions.At the rest potential, α � β, thus inhibiting bind-ing. During periodic stimulation, when the intervalbetween successive action potentials is greater than4/l, there is no net accumulation of blocked chan-nels. However, when the recovery interval is lessthan 4/l, there is a net increase in blocked channels[see Figs. 4 and 5(b)].
To explore the dynamic effects of use-dependentdrug-channel interactions, we augmented the mem-brane models [Beeler & Reuter, 1977] with equa-tions similar to Eq. (12) that describe the gate con-trol of the drug diffusion path. When the inacti-vation gate (h or j) is assumed to control bind-ing site access, the blockade equation (inactivationstate blockade) was written as:
db
dt= k[D](1 − h)(1− b)− lb (13)
Similarly, when the fraction of open channels is as-sumed to control binding site access, the kineticsare described by:
db
dt= k[D]m3hj(1 − b)− lb (14)
With either blockade model, the sodium conduc-tance and hence, excitability, was reduced by thefraction of blocked channels. The macroscopic Naconductance in the presence of drug is representedby gNahj(1 − b). The Na current is now describedby:
INa = gNam3hj(1 − b)(V − VNa) (15)
We approximated cellular excitability in terms ofthe Na channels available for transition from therest state to the activated state: Availability =gNa(1− b)hj.
Use- and frequency-dependent blockade arereadily demonstrated with this model. With pulse
Use- and Frequency-Dependent Channel Blockade
Stimulation Interval: 250 ms Stimulation Interval: 800 ms
Mem
bran
e C
urre
ntM
embr
ane
Pot
entia
l
Fig. 4. Repetitive stimulation of cardiac displays “use- andfrequency-dependence”. The top trace is a sequence of com-puted action potentials using the Beeler–Reuter model aug-mented by Eqs. (12) and (14) and stimulated with a 4 Hzpulse train (left) and a 1.25 Hz pulse train (right). The cal-cium conductance (gCa) was reduced to 0.045 (from the stan-dard BR value of 0.09) in order to reduce the action potentialduration. The rates of binding (kD) and unbinding (l) were4/s and 1/s respectively. The bottom trace displays the totalmembrane current. During an action potential, the channelsswitch between different configurations that alter accessibil-ity of the drug binding site. Here we depict open channelblock where the open channel permits diffusion of drug tothe binding site. Consequently, formation of drug-complexedchannels can only take place when the channel is open. Sim-ilarly, some drugs can only block inactivated channels as ifthe inactivation gate controls access to the binding site.
train stimulation, we computed cellular action po-tentials and total membrane current for two stim-ulation frequencies. That reduction in the recov-ery time enhances the accumulated block is readilyseen when comparing results associated with 250 msstimulus intervals (Fig. 3, left) with 800 ms stimulusintervals (Fig. 3, right).
In summary, we extended the BR model equa-tions with Eqs. (13) or (14), depending on the block-ade process, and substituted Eq. (15) for the Nacurrent. The validity of this approach has beentested with cellular studies of both open channeland inactivated blockade. We found good agree-ment, over a wide range of stimulus frequenciesand drug concentrations, between observed block-ade and model-based predictions of the time andfrequency-dependent interaction of drugs with ei-ther inactivated or the open Na channels [Gilliamet al., 1989; Starmer et al., 1991a; Whitcomb et al.,1989; Zilberter et al., 1994].
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1959
5. Drug Alteration of Single CellResponses to Stimulation
Until recently, the clinical paradigm for managingcardiac rhythm disturbances was to reduce cellu-lar excitability either by using Na channel block-ing drugs to prolong the refractory period or by us-ing K channel blocking drugs to prolong the actionpotential duration. It had been thought that byprolonging the refractory period of a cardiac cell,either with Na or K channel blockade, the num-ber of unexpected extra stimuli (PVCs arising fromcells that decide to suddenly switch from a sta-ble mode to an oscillatory mode) would be dimin-ished thereby reducing the number of episodes of
potentially fatal rhythm disturbances. As clini-cal trials have demonstrated, therapy based on ionchannel blockade was successful in reducing thenumber of unexpected extra stimuli but actuallyincreased the number of fatal rhythm disturbances.To explore the nature of this paradox, we first inves-tigated single cell responses to stimulation (wherethese drugs display their antiarrhythmic properties)and then probed multicellular responses to stimu-lation and subsequent front formation (where thedrugs display their proarrhythmic properties).
The refractory period and its modulation bydrugs were readily demonstrated with paired stim-uli (a conditioning stimulus, s1, followed after somedelay by a test stimulus, s2). Figure 5 illustrates
(a)
Fig. 5. Periodic stimulation and the refractory period of a single cell. Shown in (a) is a train of action potentials (uppertrace) initiated in response to a 2 Hz train of 5 s1 pulses followed by a test pulse (s2) under drug free conditions. Thelower trace displays the fraction of available (noninactivated) Na channels (h ∗ j) where h and j are the BR fast and slowinactivation variables. The cell was refractory when the s1–s2 delay was less than 300 msec. (b) illustrates the effect of a drug.For these studies we assumed blockade of inactivated channels with kD = 4/sec and l = 1/sec. The upper trace shows theaction potentials associated with the 2 Hz train of stimuli while the lower trace shows the use-dependent fraction of unblockedchannels. Since the fraction of available channels is proportional to h ∗ j ∗ (1 − b), and h ∗ j is approximately 1, the bottomtrace indirectly reflects excitability, i.e. a large fractional block results in a poorly excitable preparation. The large fraction ofblocked channels after the fifth stimulus pulse extended the refractory period to 450 msec.
1960 C. F. Starmer
(b)
Fig. 5. (Continued )
the refractory period for a single cell in the absenceof drug (a) and in the presence of a use-dependentdrug that blocks inactivated channels (b). In (a),a train of five conditioning (s1) stimuli (2 Hz) wasused to establish a steady state. Following the fifthpulse, a test (s2) stimulus (1.5×threshold) was usedto assess the excitability of the cell. When the de-lay between the last conditioning pulse (s1) and thetest pulse (s2) was < 300 ms, the membrane wasunable to initiate a new action potential. For s1–s2 delays > 300 ms, there was greater Na channelavailability since more channels had recovered frominactivation. Consequently action potentials of in-creasing duration (in proportion to the s1–s2 delay)were initiated.
The refractory period was extended beyond theend of the action potential by periodic blockade ofthe Na channels as shown in Fig. 5(b). In the pres-ence of a use-dependent drug, the refractory periodwas determined by the time required for Na chan-nel availability (hj(1 − b)) to recover sufficientlyto support initiation of a new action potential. A2 Hz train of five pulses was used to establish a
steady-state pattern of channel blockade as seen inthe lower panel. Using the s1–s2 stimulation proto-col, the refractory period was extended to 450 msthereby reducing the period of susceptibility to pre-mature stimulation to 50 ms.
6. Vulnerability: A MulticellularResponse
Ferris et al. [1936] and later, Wiggers and Wegria[1939] observed that ventricular fibrillation couldbe readily induced by stimulating a heart duringan interval they called the vulnerable phase. Asshown above [Fig. 5(b)], use-dependent Na block-ade displayed antiarrhythmic properties by extend-ing the refractory period secondary to a periodicreduction of the number of Na channels availableto generate a new action potential [Johnson &McKinnon, 1957; Heistracher, 1971; Gilliam et al.,1989]. In patients, though, use-dependent Na block-ade increased the probability of spontaneous initi-ation of re-entrant arrhythmias that led to suddencardiac death [CAST Investigators, 1989; CAST IIInvestigators, 1992].
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1961
Krinsky [1966] proposed that ventricular fib-rillation could be linked to spiral waves of cardiacactivation which could be initiated by stimulationwithin the vulnerable region. Utilizing the trav-eling boundary separating excitable and refractorycells that follows a propagating wave, Wiener andRosenblueth [1946] proposed a model for the vul-nerable period, an interval within which stimula-tion would lead to wavelet formation (a wave frag-ment). The W–R model of the vulnerable perioddepended on the velocity of the wave leading us[Starmer et al., 1991b; Starmer et al., 1994] to hy-pothesize that Na channel blockade would increasethe cardiac vulnerable period and increase the like-lihood of initiating a unidirectionally propagatingwave (in 1D) or wavelets capable of evolving intospiral waves (in 2D).
The combined effect of an increase in the re-fractory interval and an increase in the vulnera-ble period will increase the probability that a ran-domly timed stimulus can trigger events leading toa re-entrant arrhythmia. Let RR be the intervalbetween successive heart beats, RP be the refrac-tory period and VP be the vulnerable period. ThenP(arrhythmia) = VP/(RR − RP). This ratio in-creases with either a prolonged refractory period orwith a prolonged vulnerable period, or both. Giventhis direct link between likelihoods for triggeringre-entrant arrhythmias by altering the RP and VP,we probed the vulnerable period to determine itssensitivity to Na channel availability, use-dependentchannel blockade and electrode configuration.
Figure 6 illustrates a typical numerical experi-ment demonstrating formation of a unidirectionallypropagating wave front by timing stimuli to occurwithin the vulnerable period. Shown at the top ofthe figure is the experimental setup: a length of ex-citable cable with two stimulation sites, s1 which isused to initiate a conditioning wave and s2 whichis used to probe for the vulnerable period. For the1D case, the vulnerable period was defined as therange of delays between the s1 and s2 stimuli thatresulted in a unidirectional propagation. Shown aretwo conditions, the top is considered normal, with amaximum Na = conductance of 5 mS/cm2 and thelower is considered diseased where the maximum Naconductance is reduced to 2 mS/cm2. Each of thefour panels shows the fate of a stimulus introducedat the s2 site. The time between each successivetrace is 5 ms.
With a Na conductance of 5 mS/cm2 (top se-quence), a wave was initiated at the left bound-
ary and propagated from left to right. At an s1–s2 delay of 483.3 ms, the stimulus at the s2 siteignited a region that collaped as shown by thetraces at 5 ms intervals [Fig. 6(A)]. Delaying thes2 stimulus by 0.1 ms resulted in an ignited re-gion that expanded in the retrograde direction andcollapsed in the antegrade direction (unidirectionalconduction) [Fig. 6(B)]. With an s1–s2 delay of485.4 ms, the ignited region expanded in both direc-tions [bidirectional conduction, Fig. 6(D)]. The VPwas 1.9 ms and, assuming a uniform distribution ofPVCs, then for a heart rate of 75 (interbeat inter-val of 800 ms), P(arrhythmia) = 1.9/(800−483.3) =0.004. When the Na conductance is reduced by 60%to 2 mS/cm2, the conduction velocity of the s1 frontslowed from 64.3 cm/sec to 31.7 cm/sec [Figs. 6(E)–6(H)], the refractory period increased from 483.3 msto 554.5 ms and the VP increased from 1.9 ms to22.3 ms. Thus under conditions of low excitabilitythe P (arrhythmia) = 22.3/(800 − 554.5) = 0.09, anapproximately 20 fold increase compared with thenormal or control value.
In addition to the demonstrating asymmet-ric front development resulting from stimulationwithin the vulnerable period, the two conditions[Figs. 6(A)–6(D) and Figs. 6(E)–6(H)] display dif-ferences in the transition from unidirectional tobidirectional conduction. Figure 6(C) shows a rapidcollapse of the developing antegrade wave whileFig. 6(D) displays an immediate transition to incre-mental conduction. On the other hand Fig. 6(G)shows a very slow collapse of the developing an-tegrade wave and 6(H) shows a very slow transi-tion from decremental to incremental conduction.The differences in these transition dynamics suggestthat the vulnerable region is sensitive to mediumexcitability.
Wiener and Rosenbleuth [1946] viewed thevulnerable region as determined by the movementof the transition point between refractory and ex-citable mediums. Traveling with each wave ofexcitation is a phase wave of excitability. Withinthis wave is a point that represents the transitionfrom refractory (unexcitable) to excitable medium.Winfree [1987] realized that the transit of this pointacross the stimulus field would define a period ofvulnerability and defined the VP as the ratio ofthe length of the stimulus field (L) to the wavevelocity (v).
We tested the predicted linear relationships(VP = L/v) between the VP and electrode lengthand between VP and 1/velocity and found good
1962 C. F. Starmer
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s2 = 483.3 ms s2 = 483.4 ms s2 = 485.3 ms s2 = 485.4 ms
VP gNa = 5.0 mS/cm**2
s1 wave
(s1 velocity: 64.30 cm/sec VP: 1.9 ms)
A B C D
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Excitable Cable
(a)
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s1 wave
E F G H
(b)
Fig. 6. Demonstration of the vulnerable period. Using a 10 cm cable, a conditioning wave was initiated (s1) at the left endof the cable. A test wave was initiated (s2) at a location 2.5 cm to the right of the conditioning site. Shown are spatialdistributions (expanding away from the stimulation site) of the membrane potential plotted at 5 ms intervals. Varying thes1–s2 delay reveals the boundaries of the VP, as defined by the transition from block (A, E) to unidirectional retrogradeconduction (B, C, F, G) and the transition from retrograde conduction to bidirectional conduction (D, H). Wavelets (or wavefragments) are the 2D and 3D equivalents of unidirectional conduction.
agreement between theory and data. Figure 7 dis-plays the relationship between the VP and elec-trode lengths varying from 1.5 mm to 3.9 mm wheregNa = 4 mS/cm2. Least squares fitting of a lin-ear model yielded a slope of 17.79 ms/cm whichshould correspond to the reciprocal conduction ve-locity. The conduction velocity was 56.87 cm/secand its reciprocal was 17.58 ms/cm, in good agree-ment with the data.
We next explored the dependence of the vulner-able period on reductions in both the maximum Naconductance, gNa, and use-dependent reductions inNa conductance. Shown in Fig. 8 is a plot of the
vulnerable period as a function of the inverse ve-locity of the conditioning (s1) wave. Frequency-dependent Na channel blockade produced a linearincrease in VP as a function of 1/v while reductionsin gNa resulted in a nonlinear relationship. More-over, there was an offset between the frequency-dependent results (+) and reductions in gNa (box)indicating that other factors were significantlyaltering the VP.
The VP model [Wiener & Rosenbleuth, 1946;Winfree, 1987; Starmer, et al., 1994; Starobin et al.,1994] predicted a linear relationship between VPand stimulus field extent and between VP and the
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1963
0�
0.1 0.2 0.3 0.4Electrode Length (cm)
0
2
4
6
8
10V
ulne
rabl
e P
erio
d (m
s)
Slope = 17.79 ms/cmIntercept = 2.86 msr = .999
Conditioning Velocity = 56.87 cm/sec
1/velocity = 17.58 ms/cm
Fig. 7. The sensitivity of the VP — electrode length rela-tionship to the dynamics of Na channel blockade. The stim-ulus region was varied from 1.5 to 3.9 mm. The conditioningwave velocity was 56.87 cm/sec gNa = 4 mS/cm2. The leastsquares slope of the VP — 1/v relationship was 17.79 ms/cm,in good agreement with the reciprocal velocity, 17.58 ms/cm.
15.0�
20.0�
25.0�
30.0�
35.0�
1/velocity (ms/cm)�
0.0
10.0
20.0
30.0
40.0
50.0
Vul
nera
ble
Per
iod
(ms)
Extension of the Vulnerable PeriodConstant and Use−dependent Blockade
�
Use−dependent Blockade:
isi = 4 s
(Gradient effect)
isi = 2 s
isi = 1.5 s
isi = .6 s (2:1 block)
isi = 1 s
isi = .9 s
(kD = .004/ms, l = .001/ms)
Fig. 8. Variations in the vulnerable period associated withfixed reductions in Na channel availability (reducing gNa,box) and with use-dependent blockade (+). As gNa was re-duced, the conduction velocity was similarly reduced and theVP increased in a nonlinear manner. By changing the stim-ulus interval from 4 s to 600 ms we observed a progressiveincrease in the VP, but the magnitude was approximately fiveto seven fold greater than that associated with simply reduc-ing gNa. The dashed line represents the least squares fit ofL/v where 1/v was the independent variable. The dashedline represents the fit of a linear model (VP = L/v) tothe frequency-dependent VP measurements. The solid linerepresents the least squares fit of VP = [L − (δ/∇(A))]/vto the data where ∇(A) and v were the independentvariables.
reciprocal conditioning velocity. Thus, the nonlin-ear relationship between VP and the inverse con-ditioning velocity was unexpected. These modelswere based on the assumption that there was asteep transition between excitable and refractorymediums (large gradient). To test this assumption,we presumed that Na channel availability wouldaccurately approximate membrane excitability andplotted the spatial distribution of Na channel avail-ability at the two VP boundaries: the time of thetransition from block and retrograde propagationand the time of the transition from retrograde prop-agation and bidirectional propagation (Fig. 9). Weplotted the availability curves for values of gNa rang-ing from 2 mS/cm2 (bottom curve) to 5 mS/cm2
(top curve). We identified two critical values of Naavailability: one associated with initiation of sta-ble retrograde propagation, Pr, and a larger valueassociated with initiation of stable antegrade prop-agation, Pa.
These results revealed that the vulnerable re-gion was sensitive not only by the passage of thecritical point over the excitation field, but also tothe time required to establish stable propagationin the antegrade direction [compare Figs. 6(C) and6(G)]. We approximated the extension of the vul-nerable region, a virtual electrode of length, Lv, byestimating the time required for the retrograde crit-ical availability, Pr to recover to the antegrade crit-ical value, Pa. Let δ = Pa − Pr then the distancerequired to recover this increment of availability is
Lv = − δ
∇A (16)
where the availability is A = gNajh. Consequently,the VP can be written as
V P =L− δ
∇Av
(17)
Fitting Eq. (17) (with v and ∇A as independentvariables) was in good agreement with the datashown in Fig. 8 (solid line). Given that there is acritical excitability associated with establishing ret-rograde propagation (unidirectional propagation),then reducing the maximum Na conductance willalways reduce the spatial gradient at the criticalpoint, a consequence of moving up the excitabilitycurve towards the plateau (maximum) value.
To be effective, use- and frequency-dependentantiarrhythmic drugs must be characterized byboth a rapid rate of binding and a long unbinding
1964 C. F. Starmer
0�
1�
2�
3�
4�
5�
Distance (cm)�
0
1
2
3
4
0�
1�
2�
3�
4�
5�
Distance (cm)�
0
1
2
3
4
Na
Ava
ilabi
lity
(mS
/cm
**2)
� AntegradePaPr
RetrogradeLvThreshold
Threshold
Direction of Successful
DirectionofSuccessfulPropagation
Propagation
Availability Aligned at theBlock−Unidirection
Availability Aligned at theUnidirection−BidirectionTransitionTransition
Fig. 9. The spatial distribution of Na channel availability (excitability) trailing the conditioning wave. Shown are superpo-sitions of Na channel availability for values of gNa ranging from 2 to 5 mS/cm2 measured at the s1–s2 delay associated withthe transition from block to retrograde conduction (left) and retrograde conduction to bidirectional conduction (right). Thes2 electrode was located at x = 2.5 cm. Each panel displays an intersection, Pr (retrograde conduction) and Pa (antegradeconduction), representing the critical excitability associated with the transition in front development. The asymmetry inpropagation development is reflected by Pr < Pa since development of stable retrograde propagation occurs in a region ofmore recovered medium while development of stable antegrade propagation occurs in a region of less recovered medium.
time constant. Rapidly unbinding drugs would pro-duce only a small, if any, accumulated effect duringperiodic activation and thus would not significantlyprolong the refractory period. The drugs used in theCAST study [CAST, 1989] had time constants ex-ceeding 1 sec (moricizine, τ = 1.3 sec and flecainide,τ = 15.5 sec) [Starmer et al., 1991]. The link be-tween drug unbinding rate, the spatial gradient andaltering the vulnerable period can be readily shown.
The gradient of availability in the presence of achannel blocking drug is
∇A = gNadb
dx=gNa
db
dtv
(18)
Between heart beats, the Na channel binding sitesare not accessible, and thus
db
dt= −lb (19)
so the gradient at Pr is proportional to the unbind-ing rate constant:
∇A = gNa−lbcrit
v(20)
where bcrit is the fraction of blocked channels as-sociated with the critical availability, Pr. Let δ bethe difference between the critical antegrade andretrograde availabilities, A(Pa) − A(Pr). Then the
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1965
0�
500�
1000 1500 2000Unbinding Time Constant (ms)�
0
5
10
15
20V
P (
ms)
�
(56.57 cm/sec)
(55.51 cm/sec)
(55.82 cm/sec)
(56.30 cm/sec)
(56.53 cm/sec)
(56.30 cm/sec)
l = .0007
l = .001
l = .002
l = .003
l = 0.1
l = .0005
gNa = 4.0 mS/cm**2
Fig. 10. The VPs computed for different rates of unblock ofa use-dependent drug from the Na channel. Drug blockadewas modeled as block of inactivated channels with a bindingrate of k[D](1 − h) and an unbinding rate of l. Numerically,k[D] = 0.002 events/ms and l ranged from 0.0005 events/msto 0.1 events/ms. The test stimulus was introduced after asingle conditioning wave. The conditioning wave velocity wasvirtually constant (∼ 56.5 cm/sec) so that the VP should beconstant if there were no gradient effect. There was a fourfold increase in the VP as the unbinding rate was reducedfrom 0.1 to 0.0005, consistent with Eq. (15).
length of the virtual electrode is −(δ/∇A(x = Pr)).So the virtual electrode extension is proportional tothe unbinding time constant τ = 1/l
Lv = − δ
∇A(x = Pr)
= −
δ
−gNalbcrit
v=
δτ
gNabcritv(21)
To test this hypothesis, we computed the VPfor drugs with unbinding rates varying from 0.01(100 ms time constant) to 0.0005 events/ms (2 stime constant) following the first conditioning stim-ulus (so that the conditioning wave velocity wouldbe relatively constant). As seen in Fig. 10, the VPwas linearly related to the unbinding time constant,which is in qualitative agreement with clinical andtissue studies.
The gradient effect provides a basis forinterpreting the difference between the linearVP-velocity relationship (Fig. 8) for frequency-
0�
10 20Distance (cm)�
Model of the Vulnerable PeriodVP = (L + L*) / v
Test Electrode:
Excitability
Membrane Potential
direction of
propagation
L
exci
tabi
lity
0
1
Transition point
(P)
exitableexcitable unexcitable (refractory)
VulnerableRegion
Virtual Extension L* = −a/grad(availability)
L*
Fig. 11. Model of the vulnerable period. Shown is the spatial distribution of the Na channel availability (excitability) and apropagating action potential. As the critical availability for retrograde propagation, Pr, traverses the s2 electrode, an impulse isformed that extends in the retrograde direction and collapses in the antegrade direction. After Pr passes the end of the physicalelectrode, antegrade propagation continues to fail due to front extension into less recovered medium. Pr must continue foran additional distance until the critical antegrade availability, Pa arrives at the end of the physical electrode. This additionaldistance is inversely proportional to the availability gradient and in a marginally excitable medium, can dramatically extendthe VP.
1966 C. F. Starmer
dependent blockade and the nonlinear VP-velocityrelationship associated with reductions in gNa. Allthe frequency-dependent measurements of the VPused gNa = 4 mS/cm2. Consequently, any alter-ation in the availability gradient would be a resultof the unbinding kinetics of the drug. Since therates of drug binding and unbinding were identicalfor each stimulus rate and the critical excitabilityexists, then the gradient effect would be indepen-dent of the stimulus rate and would lead to a con-stant virtual extension of the electrode. The resultwould be a linear relationship between VP and ve-locity. On the other hand, altering the maximumNa channel availability, gNa, altered the availabil-ity gradient as seen in the lower plot in Fig. 9 re-flecting alteration in the virtual electrode length.Consequently, the virtual electrode length increasedas gNa is reduced, with a concomitant increase inthe VP.
With these observations, we refined the originalW–R model of vulnerability by incorporating a vir-tual electrode reflecting the gradient effect. Shownin Fig. 11 is a model of a cardiac action potential,the phase wave of excitability and their relation-ship to a stimulus field. As the critical point, Pr,passes over the physical electrode field, a retrogradewave is formed that propagates into more excitablemedium (to the left). On the other hand, the an-tegrade wave collapses because the medium to theright has not yet exceeded the threshold for estab-lishing stable antegrade propagation. As Pr con-tinues to pass beyond the right boundary of thephysical electrode and across the virtual extension,unstable antegrade wave motion continues to per-sist until the excitability at the physical electrodeboundary reaches the higher threshold, Pa, neces-sary to support stable propagation into less recov-ered medium.
7. Discussion
Sanderson [1996] expressed disappointment thatdrugs showing antiarrhythmic promise when stud-ied in single cells actually increased the incidence ofsudden cardiac death when used in patients. It is in-deed paradoxical that the same property of a drugthat leads to antiarrhythmic responses in isolatedcells can become proarrhythmic in a multicellular(whole heart) preparation. Within this setting, itis interesting to note that many abused substancesblock Na channels and frequently drug-abusers
appear in emergency rooms with re-entrant car-diac arrhythmias, consistent with this property[Whitcomb et al., 1989]. Perhaps this model, link-ing the cellular antiarrhythmic potential with themulticellular proarrhythmic potential is a candidatefor resolving Sanderson’s paradox.
To probe the possible links between anti- andpro-arrhythmic processes, we exploited a theoret-ical model suggested by Wiener and Rosenblueth[1946] and with numerical studies probed the re-lationship between Na channel availability and thevulnerable period in an homogeneous and contin-uous medium. These studies were based on thegeneric property of vulnerability seen not only incardiac preparations but also in chemical reactions[Gomez-Gesteira et al., 1994].
Earlier in vitro studies with isolated guinea-pigventricular muscle [Nesterenko et al., 1992] and rab-bit isolated left atrium [Starmer et al., 1992] re-vealed drug-induced prolongation of the VP thatwas out of proportion to that expected from a re-duction in the conduction velocity. In both prepa-rations we found that the VP could be readily ex-tended with drugs that blocked the Na channel. Inthe guinea-pig studies, under drug-free conditions,we measured a maximum VP of 5 ms. With 1 µMmoricizine, the VP was extended to 17 ms and in-creasing the concentration to 12 µM extended theVP to 35 ms. Similarly, in the presence of 3 µMflecainide (used in the CAST study [CAST Inves-tigators, 1989; CAST II Investigators, 1992]) wemeasured a VP of 50 ms. Recognizing that manyabused substances exhibited Na blocking proper-ties, we tested propoxyphene and cocaine in rab-bit atrial tissue [Starmer et al., 1992]. In thesestudies, the VP under drug-free conditions couldnot be detected. However, in the presence of 2 µMpropoxyphene or 1 µM cocaine, the VP was greaterthan 30 ms.
The link between the VP, membrane excitabil-ity and its spatial gradients that trail a propagatingwave reveals how ion channel blockade can promotesimultaneously anti- and pro-arrhythmic properties.Na channel blockade (which reduces excitabilityand slows the recovery of excitability in individualcells, an antiarrhythmic property) simultaneously,in multicellular preparations, reduces the excitabil-ity gradient and extends the period of vulnerabilityduring which unexpected stimuli can initiate unidi-rectional propagation (by slowing conduction andaltering the gradient of excitability). These prop-erties promote wavelet formation which can lead to
How Antiarrhythmic Drugs Increase the Rate of Sudden Cardiac Death 1967
spiral waves following premature stimulation. Thatthe gradient of excitability is important has been re-cently highlighted in studies of defibrillation. Chenet al. [1999] and Yamanouchi et al. [2001] demon-strated that defibrillation pulse polarity, which al-ters membrane excitability within the defibrillationfield, was linked to defibrillation failure.
In summary, these studies suggest that manyof the complexities of a real cardiac cell can beignored when studying re-entrant arrhythmogenicprocesses. Altering excitability and the dynamics ofthe recovery of excitability displays both anti- andpro-arrhythmic effects. Both velocity and spatialgradients of excitability alter the VP. The dominantmechanism of the VP appears to be the interactionbetween the front formation process in a nonlinearreaction–diffusion medium and the dynamic alter-ation of the medium’s excitability, in this case, withNa channel availability. The mechanism of vulnera-bility is a generic property of any excitable mediumand as such should be the target of intensiveinvestigation.
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