Toward Prediction of the Local Onset of Alternans in the Heart

868 Biophysical Journal Volume 100 February 2011 868–874

Toward Prediction of the Local Onset of Alternans in the Heart

Alexander R. Cram, Hrishikesh M. Rao, and Elena G. Tolkacheva*Biomedical Engineering Department, University of Minnesota, Minneapolis, Minnesota

ABSTRACT A beat-to-beat variation in the cardiac action potential duration is a phenomenon known as alternans. Alternanshas been linked to ventricular fibrillation, and thus the ability to predict the onset of alternans could be clinically beneficial. Theo-retically, it has been proposed that the slope of a restitution curve, which relates the duration of the action potential to thepreceding diastolic interval, can predict the onset of alternans. Experimentally, however, this hypothesis has not been consis-tently proven, mainly because of the intrinsic complexity of the dynamics of cardiac tissue. It was recently shown that the resti-tution portrait, which combines several restitution curves simultaneously, is associated with the onset of alternans in isolatedmyocytes. Our main purpose in this study was to determine whether the restitution portrait is correlated with the onset of alter-nans in the heart, where the dynamics include a spatial complexity. We performed optical mapping experiments in isolated Lan-gendorff-perfused rabbit hearts in which alternans was induced by periodic pacing at different frequencies, and identified thelocal onset of alternans, Bonset. We identified two regions of the heart: the area that exhibited alternans at Bonset (1:1alt) andthe area that did not (1:1). We constructed two-dimensional restitution portraits for the epicardial surface of the heart andmeasured the spatial distribution of three different slopes (the dynamic restitution slope, SRPdyn, and two local S1-S2 slopes,S12 and Smax

12 ) separately for these two regions. We found that the S12 and Smax12 slopes differed significantly between the

1:1alt and 1:1 regions just before the onset of alternans, and SRPdyn slopes were statistically similar. In addition, we found thatthe slopes of the dynamic restitution curve Sdyn were also statistically similar between these two regions. On the other hand,the quantitative values of all slopes were significantly different from the theoretically predicted value of one. These resultsdemonstrate that the slopes measured in the restitution portrait correlate with the onset of alternans in the heart.

INTRODUCTION

A beat-to-beat variation in the cardiac action potential dura-tion (APD) is a phenomenon known as electrical alternans(Fig. 1, A and B). Alternans desynchronizes depolarization,increases dispersion of refractoriness, and creates a substratefor ventricular fibrillation (1–4). It has been shown that elec-trical alternans in single cells is the primary cause of T-wavealternans in the heart (1). Moreover, T-wave alternans isrecognized as a precursor of ventricular arrhythmias,because it has been observed in a wide variety of clinicaland experimental conditions associated with such arrhyth-mias (1,5–9).

It has been suggested that one can determine the onset ofalternans in cardiac myocytes by analyzing their responsesto periodic stimulation and constructing a restitution curve(10–13) representing the nonlinear relationship betweenthe APD and the preceding diastolic interval (DI). Further-more, it has been proposed theoretically that a slope of therestitution curve equal to one predicts the onset of alternansin cardiac myocytes (13).

However, the actual dynamics of periodically pacedcardiac myocytes are more complex, and usually the APDwill depend on the entire pacing history (14–20), not justthe preceding DI. This phenomenon, known as short-termmemory, is an intrinsic property of cardiac myocytes. One

Submitted December 20, 2010, and accepted for publication January 7,

2011.

*Correspondence: [email protected]

Editor: Randall L. Rasmusson.

� 2011 by the Biophysical Society

0006-3495/11/02/0868/7 $2.00

of the main consequences of short-term memory is a depen-dence of the restitution curve on the pacing protocol used toobtain it. Investigators have used various pacing protocolsexperimentally to construct different restitution curves,with dynamic and S1-S2 restitution curves being the mostcommon. In a dynamic protocol, steady-state (SS) APDand DI are measured as the basic cycle length (BCL)decreases. In an S1-S2 protocol, a premature stimulus (S2)is applied at various times relative to the end of a series ofpaced (S1) beats. Thus, the dynamic and S1-S2 restitutioncurves describe different aspects of cardiac dynamics (SSresponses and responses to perturbations, respectively). Inthe presence of short-term memory, these different restitu-tion curves have different slopes, and none of these havebeen clearly linked to the onset of alternans (15–18). Indeed,several experimental studies demonstrated the existence ofalternans for a shallow restitution, and no alternans wasobserved for a steep restitution (11,12,19). Hence, it iswidely accepted that individual restitution curves fail topredict the onset of alternans correctly.

To enable more accurate determination of the onset of al-ternans in periodically paced myocytes, a downsweeppacing protocol was developed theoretically (20) and imple-mented experimentally in small bullfrog cardiac tissue (21)and isolated guinea pig and rabbit myocytes (22). Thisprotocol allows one to measure the restitution portrait ofcardiac myocytes, which consists of several restitutioncurves measured simultaneously at various BCLs. Thus,the restitution portrait captures several aspects of cardiac

doi: 10.1016/j.bpj.2011.01.009

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FIGURE 1 Response of periodically paced cardiac tissue to stimulation

(arrows) during (A) normal pacing (1:1 response) and (B) alternans. (C)

A segment of the perturbed downsweep pacing protocol for a given value

of B. Corresponding cardiac responses that were measured at each step

are shown in brackets.

Toward Prediction of Alternans 869

dynamics simultaneously, in contrast to individual restitu-tion curves. It was experimentally confirmed that one ofthe slopes measured in the restitution portrait, Smax

12 , is corre-lated with the onset of alternans in isolated rabbit and guineapig myocytes (22).

However, the dynamics of periodically paced wholehearts are more complex, mainly due to the presence ofa spatial component. Therefore, a single restitution curvemeasured in the heart does not reflect its spatiotemporaldynamics, and thus cannot correctly determine the onsetof alternans. Various investigators have attempted to demon-strate the presence of spatial heterogeneity in the restitutionproperties of the heart using both individual restitutioncurves and restitution portraits (23–25), but a direct linkbetween the onset of alternans in the heart and local restitu-tion properties is still lacking.

Our main purpose in this study was to investigate thespatiotemporal formation of alternans in whole hearts andto determine whether the slopes measured in the restitutionportrait are correlated with its onset. We induced alternansin isolated rabbit hearts by periodic pacing at differentfrequencies and identified its temporal and spatial localonset, Bonset. We constructed two-dimensional (2D) restitu-tion portraits and dynamic restitution curves for the epicar-dial surface of the heart and measured the spatialdistribution of several slopes separately for two differentregions of the heart: the area that exhibited alternans atBonset (1:1alt) and the area that did not (1:1). We demon-strated that several slopes measured in the restitutionportrait (S12 and Smax

12 ) were significantly different betweenthe 1:1alt and 1:1 regions just before the onset of alternans,indicating a correlation with the local onset of alternans in

the heart. On the other hand, the dynamical slopes measuredin the restitution portrait (SRPdyn) or restitution curve (Sdyn)were statistically similar between the 1:1 and 1:1alt regions.Moreover, the quantitative values of all slopes were signifi-cantly different from the theoretically predicted value ofone. These results demonstrate that the slopes measured inthe restitution portrait correlate with the onset of alternansin the heart.

MATERIALS AND METHODS

Optical mapping of whole hearts

All experiments were performed according to the guidelines of the National

Institutes of Health and the University of Minnesota. New Zealand White

rabbits of either sex (n ¼ 6, 2.5–3 kg) were injected with heparin sulfate

(300 U) and anesthetized with sodium pentobarbital (75 mg/kg IV). After

a thoracotomy was performed, the hearts were quickly removed and

immersed in cardioplegic solution (in mM: glucose 280, KCl 13.44,

NaHCO3 12.6, mannitol 34). The aorta was quickly cannulated and retro-

gradely perfused with warm (36 5 1�C) oxygenated Tyrode’s solution

(in mM: NaCl 130, CaCl2 1.8, KCl 4, MgCl2 1.0, NaH2PO4 1.2, NaHCO3

24, glucose 5.5) under constant pressure (70 mm Hg). The pH of the

Tyrode’s solution was maintained at 7.4, with adjustments made by addition

of HCl. The hearts were immersed in a chamber and superfused with the

same Tyrode’s solution. Blebbistatin (10 mmol/L) was added to the Tyrode’s

solution to reduce motion artifacts.

A bolus of 5 mL of the voltage-sensitive dye Di-4-ANEPPS (10 mmol/L)

was injected into the heart and excited with the use of a diode-pumped,

continuous-excitation green laser (532 nm, 1 W; Shanghai Dream Lasers

Technology, Shanghai, China). Two CCD cameras (CA-D1-0128T;

DALSA, Waterloo, Ontario, Canada) were used to record simultaneously

the epicardial surfaces of both the left and right ventricles (>80% of total

surface). Movies were acquired at 600 frames per second with a spatial

resolution of 64�64 pixels. The background fluorescence was subtracted

from each frame. In addition, spatial (3�3 pixels) and temporal (5 pixels)

conical convolution filters were used.

Pacing protocol

External stimuli (5 ms duration, twice the threshold) were applied to the

base of the heart by means of dynamic and perturbed downsweep pacing

protocols (21). In the dynamic pacing protocol (n ¼ 5), 100 stimuli were

applied at different BCLs B to achieve SS, starting from B ¼ 300 ms in

20 ms decrements until B¼ 100 ms or until ventricular fibrillation occurred.

In the perturbed downsweep pacing protocol (n ¼ 5), the following steps

were applied at each BCL B, starting with B ¼ 300 ms (Fig. 1 C):

1. One hundred stimuli were applied at BCL B to achieve SS.

2. One additional stimulus (long perturbation (LP)) was applied at a longer

BCL BLP ¼ B þ 10 ms.

3. Ten stimuli were applied at BCL B to return to SS.

4. One additional stimulus (short perturbation (SP)) was applied at a shorter

BCL BSP ¼ B �10 ms.

5. Ten stimuli were applied at BCL B to return to SS.

After steps 1–5 were completed, the BCL B was progressively reduced in

20 ms decrements until B¼ 100 ms or until ventricular fibrillation occurred.

At each BCL, the APDs were measured at 80% repolarization during SS

(ASS) in both protocols, and after SP (ASP) and LP (ALP) in the perturbed

downsweep pacing protocol. The preceding DI DP was calculated at each

pixel according to the following formula:

Dp ¼ Bp � ASS (1)

Biophysical Journal 100(4) 868–874

870 Cram et al.

where p denotes either SS, LP, or SP. Then, 2D APD and DI maps were

created from the (DP, AP) pairs for epicardial surfaces of the heart at each

BCL for SS, LP, and SP responses. These maps were used to construct

2D dynamic restitution curve and restitution portraits of the heart.

At each pixel, the dynamic restitution curve consists of (ASS, DSS)

responses measured at different BCLs (Table 1). We calculated the slope

of the dynamic restitution curve (Sdyn) at each pixel by fitting these

responses from all B-values with a second-degree polynomial function

using custom-made software written in PV-WAVE (Visual Numeric,

Boulder, CO).

At each pixel, the restitution portrait consists of a combination of the

following responses: (ASS,DSS), (ALP,DLP), (ASP,DSP) measured at different

BCLs (Table 1). Thus, as described previously (21,22), the following resti-

tution curves can be visualized in the restitution portrait: 1), a unique

dynamic restitution curve that consists of (ASS,DSS) responses measured

at different B-values; and 2), several local S1-S2 restitution curves that

consist of (ASS,DSS), (ALP,DLP), and (ASP,DSP) responses measured sepa-

rately for each B-value. We calculated the slope of the dynamic restitution

curve measured in the restitution portrait (SRPdyn) at each pixel by fitting

(ASS,DSS) responses from all B-values with a second-degree polynomial

function. The local S1-S2 restitution curves were fitted with a second-

degree polynomial function at each B-value, and two slopes (at the SS

(S12) and the SP (Smax12 )) were measured.

Data analysis

Alternans was calculated at each pixel as a difference in APDs between

odd and even beats, as described previously (25): DAPD ¼ jAPDeven �APDoddj R 5 ms (Fig. 1 B). All APD variations smaller than the temporal

threshold of alternans (5 ms) were defined as 1:1 responses (Fig. 1 A). 2D

DAPD maps were constructed to reveal the spatial distribution and ampli-

tude of alternans for the epicardial surfaces of the heart. The phase of the

alternans was not taken into account, so no distinction between spatially

concordant and discordant alternans was made. The presence of alternans

was denoted as a red color, and the presence of 1:1 behavior (absence of

alternans) was denoted as white. Note that in the case of spatially discor-

dant alternans, the white color also indicates the presence of the nodal

lines (26,27).

The local spatial onset of alternans, Bonset, was defined as the BCL at

which at least 10% of the ventricular surfaces exhibited alternans. In

each experiment, only the ventricle where the onset of alternans appeared

first was considered for further analysis. Two spatial regions of the heart

were defined at the Bonset: the 1:1alt region, where alternans was present,

and the 1:1 region, which exhibited 1:1 behavior. These two regions were

virtually projected to all previous BCLs, and the mean values and standard

errors for all parameters were calculated and averaged separately for these

two regions.

Because in different experiments the onset of alternans occurs at different

BCLs, a normalized BCL BN was introduced as BN ¼ B�Bonset so that the

onset of alternans would always occur at BonsetN ¼ 0 ms.

Group data are presented as the mean 5 SE. We performed statistical

comparisons between the two regions in the heart using a two-sample

t-test (Origin Software, Northampton, MA). Values of p < 0.05 were

considered statistically significant.

TABLE 1 Different responses and corresponding slopes

measured from the pacing protocols

Responses Slopes

LP: (ALP, DLP) S12 Measured at SS at each B

Restitution portrait SS: (ASS, DSS) Smax12 Measured at SP at each B

SP: (ASP, DSP)

SS: (ASS, DSS) SRPdyn Measured at SS from all B

Dynamic restitution

curve

SS: (ASS, DSS) Sdyn Measured at SS from all B


RESULTS

Spatiotemporal evolution of alternans in the heart

We observed that alternans first appeared locally at B¼Bonset

in one of the ventricles and evolved spatially throughout theheart as the BCL decreased. Fig. 2 shows a representativeexample of the spatiotemporal evolution of alternans in theleft ventricle of a rabbit heart at different values of B (top)or the corresponding normalizedBCLBN (bottom). The colorbar represents the amplitude of alternans DAPD. Note thepresence of 1:1 behavior (white) at large B, and the appear-ance of alternans (red) as B decreases. For instance, in thisparticular example, the onset of alternans occurred atBonset ¼ 170 ms (Bonset

N ¼ 0 ms). At this specific BCL, thetwo regions, 1:1alt and 1:1, were identified at the surface ofthe heart and projected back into all prior BCLs (seeFig. 2). Note that alternans spread throughout the heart asthe BCLs became smaller than Bonset (negative BN), andspatially concordant alternans became spatially discordant,as previously described (26,27). Spatially concordant anddiscordant alternans can be distinguished by their phases;however, a corresponding analysis is beyond the scope ofthis study and thus is not shown here. However, note thatthe presence of white color at B¼ 150 ms in Fig. 2 indicatesthe presence of the nodal line during spatially discordant al-ternans. The total area occupied by alternans (both spatiallyconcordant and discordant) from all experiments is shownin Fig. 3 as a function ofBN. The dashed horizontal line repre-sents the spatial threshold for alternans. Note that as BN

decreases, the total area occupied by alternans increases untilthe entire heart surface exhibits alternans.

2D restitution properties of the heart

We sought to determine whether the restitution properties ofthe two regions, 1:1alt and 1:1, were different during peri-odic pacing, before the onset of alternans. To that end, weapplied a perturbed downsweep pacing protocol and con-structed 2D APD maps for SS (ASS), LP (ALP), andSP (ASP) at various BCLs. Representative examples of theseAPDmaps are shown in Fig. 4 A for large (Bn¼ 100 ms) andsmall (Bn ¼ 10 ms) BCLs. The color bar represents the

FIGURE 2 Representative example of the 2D DAPD maps in rabbit left

ventricle at different B (top panel) and normalized BN (bottom panel)

values. The color bar represents the amplitude of alternans (red) and 1:1

responses (white). The local spatial onset of alternans occurs at Bonset (or

BonsetN ¼ 0 ms). At Bonset, the two regions (1:1alt and 1:1) are introduced

and projected to all preceding B-values (black outlines).

FIGURE 3 Mean percentage of total ventricular area occupied by alter-

nans as a function of BN for all rabbit ventricles (n ¼ 5). The dashed hori-

zontal line represents the spatial threshold for alternans.

FIGURE 4 (A) Representative examples of 2D APD maps for ALP, ASS,

and ASP at Bn ¼ 100 ms (top panel) and Bn ¼ 10 ms (bottom panel). The

color bar represents the duration of the action potential (in ms). The 1:1

and 1:1alt regions are outlined by black curves. (B and C) The mean devi-

ation of the (B) SP and (C) LP from SS as a function of BN. The data were

calculated separately for the 1:1alt and 1:1 regions of the heart from all

experiments (n ¼ 5); * denotes statistically significant data (p < 0.05).


APDs, and the projected boundaries of the 1:1alt and 1:1regions are shown as black lines on each APD map. Notethat for large BN (top panel in Fig. 4 A), the spatial distribu-tions of ALP and ASP were similar to ASS for both the 1:1altand 1:1 regions. However, for small BN (bottom panel inFig. 4 A), only the spatial distribution of ALP was similarto ASS, whereas the spatial distribution of ASP was visuallydifferent from ASS for both the 1:1alt and 1:1 regions.

To quantify these results, we calculated the deviations ofthe APDs measured after SP and LP from the SS APD(DASP ¼ ASS � ASP and DALP ¼ALP �ASS, respectively) ateach pixel. Fig. 4, B and C, show the mean values of DASP

andDALP, calculated separately for the 1:1alt and 1:1 regionsin all experiments, as a function of BN. Note that for bothregions the values of DASP and DALP increased as BN

decreased, but the magnitude of deviation was higher forDASP. Specifically, the maximum deviations occurred forthe 1:1alt region just before the onset of alternans, atBn ¼ 10 ms, were DASP ¼ 8.6 5 0.15 ms and DALP ¼2.8 5 0.14 ms (p < 0.05). Fig. 4 also demonstrates that thevalues of DASP and DALP were qualitatively similar forboth 1:1alt and 1:1 regions at all BN-values except Bn ¼10ms.AtBn¼ 10ms, bothDASP andDALPwere significantlyhigher for the 1:1alt region than for the 1:1 region. In addition,the difference in DASP between the 1:1alt and 1:1 regions(2.715 0.52 ms) was significantly larger than the differencein DALP between these regions (0.78 5 0.19 ms, p < 0.05).Therefore, immediately before the onset of alternans, DASP

not only showed the highest maximal value, it also demon-strated a significant increase in regional heterogeneity.Hence, although both SP and LP lead to different APDsbetween the 1:1alt and 1:1 regions, the SP could be consid-ered as a stronger indicator for the onset of alternans.

Toward prediction of the local onset of alternans

In our pacing protocol, both SP and LP were set to be a fixedvalue (10 ms), and thus the difference in DASP between the

two 1:1alt and 1:1 regions should lead to the different slopevalues calculated in the restitution portrait. Representativeexamples of the restitution portraits for two pixels takenfrom the 1:1alt and 1:1 regions are compared in Fig. 5.Here, a unique SS restitution curve (red) and several localS1-S2 restitution curves for different BN-values (blue) areshown along with the corresponding SRPdyn, S12, and Smax

12

slopes. The gray dotted lines are the equal BCL lines withslope ¼ �1, and the solid black line shows slope ¼ 1.Note that the dynamic restitution curves are almost parallelin both the 1:1alt and 1:1 regions, indicating the absence ofdifferences in SRPdyn at all BN-values. Similarly, local S1-S2restitution curves are nearly parallel at large BN (e.g.,Bn ¼ 100 ms), indicating that S12 and Smax

12 were similarbetween 1:1alt and 1:1 regions as well. However, at smallBN (e.g., Bn ¼ 10 ms), the difference between local S1-S2restitution curves is pronounced, and both the S12 and Smax

12


FIGURE 5 Representative examples of the restitution portraits con-

structed for two pixels taken from the 1:1alt (solid circles) and 1:1 (open

circles) regions. The SS restitution curve is shown in red, and the local

S1-S2 restitution curves for different B-values are shown in blue. The

gray dotted lines with slope ¼ �1 represent the equal-BCL lines. The solid

line shows slope ¼ 1. Three different slopes are present in the restitution

portrait for each B: SRPdyn and S12 (measured at SS) and Smax12 (measured at

the SP).

FIGURE 6 Mean values of (A) SRPdyn, (B) S12, (C) Smax12 , and (D) Sdyn as

a function of BN. All slopes were calculated separately for the 1:1alt (solid

circles) and 1:1 (open circles) regions from all experiments (n ¼ 5 for the

restitution portrait, and n ¼ 5 for the dynamic restitution curve); * denotes

statistically significant data (p < 0.05).

FIGURE 7 Mean values of SRPdyn, Sdyn, S12, and Smax12 measured just before

the onset of alternans at Bn ¼ 10 ms in comparison with the theoretical pre-

dicted value of one at the onset of alternans. The data are taken from all

experiments (n ¼ 5 for both the restitution portrait and the dynamic resti-

tution curve); * denotes statistically significant data (p < 0.05).

872 Cram et al.

slopes are visually steeper in the 1:1alt region than in the 1:1region. Therefore, Fig. 5 indicates that the S12 and Smax

12

slopes correlate with the local onset of alternans in the heart.To quantify these results, we calculated the mean values

of three slopes measured in the restitution portrait (SRPdyn,S12, and Smax

12 ) over the epicardial surface of the hearts inall experiments separately for the 1:1alt and 1:1 regions.The data are shown in Fig. 6, A–C, as a function of BN.Note that there is no significant difference between valuesof SRPdyn calculated for both the 1:1alt and 1:1 regions atany BN (Fig. 6 A). However, the behavior of the S12 andSmax12 slopes is different (Fig. 6, B and C). At large BN,

both S12 and Smax12 are similar between the 1:1alt and 1:1

regions, but as BN decreases, the values of both slopesbecome significantly larger in the 1:1alt region than in the1:1 region (p < 0.05). Note that this significant differenceappeared first for the S12 slope (at Bn ¼ 20 ms; Fig. 6 B)and later for the Smax

12 slope (at Bn ¼ 10 ms; Fig. 6 C).However, the difference in Smax

12 values was significantlylarger (0.4 5 0.15) than the difference in S12 values(0.19 5 0.06, p < 0.05) between the 1:1alt and 1:1 regionsimmediately before the onset of alternans (Bn ¼ 10 ms).Therefore, both S12 and Smax

12 can serve as indicators of thelocal onset of alternans in the heart, although Smax

12 can beconsidered as a more definite marker.

To demonstrate that alternans can be induced in the heartby dynamic pacing alone (i.e., without LPs and SPs), weapplied the dynamic pacing protocol in several rabbit hearts.Our results indicate that the formation of the alternans in theheart was similar to that described for the perturbed down-sweep pacing protocols and shown in Figs. 1 and 2. Theslopes of the dynamic restitution curve, Sdyn, calculatedseparately for the 1:1alt and 1:1 regions are shown in


Fig. 6 D, and indicate the absence of statistical significancebetween these regions at all BN-values.

To determine whether any of the slopes measured in ourexperiments were comparable with the theoretically pre-dicted value of one, we calculated the mean values of SRPdyn,S12, and Smax

12 measured in the restitution portrait, and Sdynmeasured from the dynamic restitution curve, immediatelybefore the onset of alternans, at Bn¼ 10 ms. Fig. 7 comparesthe values of all slopes for both the 1:1alt and 1:1 regionswith the value of one. Note that SRPdyn, S12, and Sdyn were<1 in both the 1:1alt and 1:1 regions, whereas Smax

12 was>1 in 1:1alt and <1 in the 1:1 region. Nevertheless, thevalues of all slopes are significantly different from the theo-retical predicted value of one (p < 0.05).


DISCUSSION AND CONCLUSIONS

In this study, we investigated the spatiotemporal formation ofalternans in isolated rabbit hearts and sought to determinewhether the slopes of the restitution curves measured in therestitution portrait are correlated with its onset. The majorfindings of our study are as follows: First, we observed thatalternans is a complex spatiotemporal phenomenon in theheart. It initially appeared locally in some areas of the ventri-cles and evolved throughout the heart as BCL decreased.Second, we demonstrated that the slopes S12 and Smax

12

measured in the restitution portrait correlate with the localonset of alternans. Third, our results indicate that althoughS12 and S

max12 correlate with the onset of alternans in the heart,

their quantitative values are significantly different from thetheoretically predicted value of one.

The restitution hypothesis (i.e., the idea that the slope ofthe restitution curve can be used to determine the onset ofalternans) was developed theoretically decades ago (13).From an experimental standpoint, this hypothesis was veryattractive because it allows complex cardiac rhythms to bepredicted from the dynamical behavior of periodicallypaced cardiac tissue. However, the restitution hypothesiswas compromised in several experimental studies (15–18)due to the intrinsic complexity of the dynamical behaviorof cardiac tissue. There are two major reasons for the failureof the restitution hypothesis, as discussed below.

First, the presence of short-term memory in cardiac tissuemakes the restitutionpropertiesdependent on thepacingproto-cols applied (28–32). As a consequence, the restitution curvesmeasured by different pacing protocols have different slopesand fail to predict the onset of irregular cardiac rhythmscorrectly. For instance, stable 1:1 behavior is observed whenthe restitution curve is very steep (14,19), whereas the transi-tion to alternans is observed in the presence of a shallow resti-tution curve (12,17). Investigators have made several attemptsto reconsider the restitution hypothesis in the presence ofshort-termmemory (20,29,30,33–35). In particular, the devel-opment of a new protocol for simultaneously measuringmultiple aspects of cardiac dynamics, rather than individualrestitution properties, led to the concept of a restitution portraitof cardiac tissue (20,21,35). The restitution portrait was incor-poratedexperimentally in small bullfrog cardiac tissue and iso-lated rabbit and guinea pig myocytes, where spatialcomplexitywasminimal (21) or absent (22), and theSmax

12 slopewas shown to successfully correlatewith theonset of alternans.

The second major reason for the failure of the restitutionhypothesis is the spatial complexity of the heart, the conse-quences of which are usually underestimated in experi-mental and clinical studies. Indeed, attempts to measureone or a few restitution curves for a given heart revealedthat such restitution curves often represent average cardiacresponses. To date, little attention has been given to thespatial distribution of the restitution properties of the heartand its correlation with the onset of alternans (18,25).

To our knowledge, this study represents the first attempt tocorrelate several slopes, measured both in the simpledynamic protocol and the more-sophisticated restitutionportrait, with the local onset of alternans in the heart. Itshould be noted, however, that the restitution portrait doesnot represent the full spectrum of all possible aspects ofcardiac dynamics. For instance, it does not incorporate S1-S2-S3 responses and constant-memory pacing (36,37).However, it can be considered as the most successful attemptso far to simultaneously measure several aspects of cardiacdynamics, as it combines both SS responses and responsesfrom perturbations (35). Conventional measurements ofthese responses are obtained through the individual dynamicand S1-S2 restitution curves, respectively. In this study, wecompared SS responses measured from the dynamic restitu-tion curve (Sdyn) and the restitution portrait (SRPdyn), anddemonstrated their similarity. It is not as easy to comparethe standard S1-S2 restitution curve and local S1-S2 restitu-tion curves from the restitution portrait, mainly because ofthe presence of an infinite number of S1-S2 restitution curves(one for each value of S1). Nevertheless, it has been showntheoretically that only points in the vicinity of the intersec-tion of the S1-S2 restitution curve with the SS responses,and not the entire S1-S2 restitution curve, are important fordetermining the onset of alternans (20).

Our results indicate that although slopes S12 and Smax12

predict the local onset of alternans in the heart, their valuesare significantly different from the theoretically predictedvalue of one. Specifically, S12 < 1 in both the 1:1alt and1:1 regions, whereas Smax

12 > 1 in the 1:1alt region andSmax12 < 1 in the 1:1 region. Many different studies have at-

tempted to explain this discrepancy. The major factors thatneed to be taken into account are the presence of short-term memory and the concurrent existence of alternans inintracellular calcium cycling (15,26).

Finally, we note that, historically, the term ‘‘prediction ofalternans’’ has been used with respect to the slopes of therestitution curve (13,34,36,38). However, we believe itwould be more accurate to use the term ‘‘correlation with al-ternans’’, since we have demonstrated that a statisticallysignificant difference between restitution properties in tworegions of the heart exists and is associated with the localonset of alternans. The first step toward actual predictionof irregular cardiac responses was made by Gelzer et al.(39), who applied theoretically derived pacing sequencesin the heart to induce alternans and ventricular fibrillation.

This work was supported by the American Heart Association (Scientist

Development Grant 0635061N to E.G.T.) and the National Science Founda-

tion (grant PHY0957468 to E.G.T.).

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Toward Prediction of the Local Onset of Alternans in the Heart