Toward Prediction of the Local Onset of Alternans in the Heart

Abstract

A beat-to-beat variation in the cardiac action potential duration is a phenomenon known as alternans. Alternans has been linked to ventricular fibrillation, and thus the ability to predict the onset of alternans could be clinically beneficial. Theoretically, it has been proposed that the slope of a restitution curve, which relates the duration of the action potential to the preceding diastolic interval, can predict the onset of alternans. Experimentally, however, this hypothesis has not been consistently proven, mainly because of the intrinsic complexity of the dynamics of cardiac tissue. It was recently shown that the restitution portrait, which combines several restitution curves simultaneously, is associated with the onset of alternans in isolated myocytes. Our main purpose in this study was to determine whether the restitution portrait is correlated with the onset of alternans in the heart, where the dynamics include a spatial complexity. We performed optical mapping experiments in isolated Langendorff-perfused rabbit hearts in which alternans was induced by periodic pacing at different frequencies, and identified the local onset of alternans, B^onset. We identified two regions of the heart: the area that exhibited alternans at B^onset (1:1_alt) and the area that did not (1:1). We constructed two-dimensional restitution portraits for the epicardial surface of the heart and measured the spatial distribution of three different slopes (the dynamic restitution slope, $S_{dyn}^{RP}$, and two local S1-S2 slopes, S₁₂ and $S_{12}^{\max }$) separately for these two regions. We found that the S₁₂ and $S_{12}^{\max }$ slopes differed significantly between the 1:1_alt and 1:1 regions just before the onset of alternans, and $S_{dyn}^{RP}$ slopes were statistically similar. In addition, we found that the slopes of the dynamic restitution curve S_dyn 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 results demonstrate 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 duration (APD) is a phenomenon known as electrical alternans (Fig. 1, A and B). Alternans desynchronizes depolarization, increases dispersion of refractoriness, and creates a substrate for ventricular fibrillation (1–4). It has been shown that electrical alternans in single cells is the primary cause of T-wave alternans in the heart (1). Moreover, T-wave alternans is recognized as a precursor of ventricular arrhythmias, because it has been observed in a wide variety of clinical and experimental conditions associated with such arrhythmias (1,5–9).

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.

It has been suggested that one can determine the onset of alternans in cardiac myocytes by analyzing their responses to periodic stimulation and constructing a restitution curve (10–13) representing the nonlinear relationship between the APD and the preceding diastolic interval (DI). Furthermore, it has been proposed theoretically that a slope of the restitution curve equal to one predicts the onset of alternans in cardiac myocytes (13).

However, the actual dynamics of periodically paced cardiac myocytes are more complex, and usually the APD will depend on the entire pacing history (14–20), not just the preceding DI. This phenomenon, known as short-term memory, is an intrinsic property of cardiac myocytes. One of the main consequences of short-term memory is a dependence of the restitution curve on the pacing protocol used to obtain it. Investigators have used various pacing protocols experimentally to construct different restitution curves, with dynamic and S1-S2 restitution curves being the most common. In a dynamic protocol, steady-state (SS) APD and 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 of paced (S1) beats. Thus, the dynamic and S1-S2 restitution curves describe different aspects of cardiac dynamics (SS responses and responses to perturbations, respectively). In the presence of short-term memory, these different restitution curves have different slopes, and none of these have been clearly linked to the onset of alternans (15–18). Indeed, several experimental studies demonstrated the existence of alternans for a shallow restitution, and no alternans was observed for a steep restitution (11,12,19). Hence, it is widely accepted that individual restitution curves fail to predict the onset of alternans correctly.

To enable more accurate determination of the onset of alternans in periodically paced myocytes, a downsweep pacing protocol was developed theoretically (20) and implemented experimentally in small bullfrog cardiac tissue (21) and isolated guinea pig and rabbit myocytes (22). This protocol allows one to measure the restitution portrait of cardiac myocytes, which consists of several restitution curves measured simultaneously at various BCLs. Thus, the restitution portrait captures several aspects of cardiac dynamics simultaneously, in contrast to individual restitution curves. It was experimentally confirmed that one of the slopes measured in the restitution portrait, $S_{12}^{\max }$, is correlated with the onset of alternans in isolated rabbit and guinea pig myocytes (22).

However, the dynamics of periodically paced whole hearts are more complex, mainly due to the presence of a spatial component. Therefore, a single restitution curve measured in the heart does not reflect its spatiotemporal dynamics, and thus cannot correctly determine the onset of alternans. Various investigators have attempted to demonstrate the presence of spatial heterogeneity in the restitution properties of the heart using both individual restitution curves and restitution portraits (23–25), but a direct link between the onset of alternans in the heart and local restitution properties is still lacking.

Our main purpose in this study was to investigate the spatiotemporal formation of alternans in whole hearts and to determine whether the slopes measured in the restitution portrait are correlated with its onset. We induced alternans in isolated rabbit hearts by periodic pacing at different frequencies and identified its temporal and spatial local onset, B^onset. We constructed two-dimensional (2D) restitution portraits and dynamic restitution curves for the epicardial surface of the heart and measured the spatial distribution of several slopes separately for two different regions of the heart: the area that exhibited alternans at B^onset (1:1_alt) and the area that did not (1:1). We demonstrated that several slopes measured in the restitution portrait (S₁₂ and $S_{12}^{\max }$) were significantly different between the 1:1_alt 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 measured in the restitution portrait ($S_{dyn}^{RP}$) or restitution curve (S_dyn) were statistically similar between the 1:1 and 1:1_alt regions. Moreover, the quantitative values of all slopes were significantly different from the theoretically predicted value of one. These results demonstrate that the slopes measured in the restitution portrait correlate with the onset of alternans in 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 = 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 retrogradely perfused with warm (36 ± 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 μmol/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 μmol/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 = 5), 100 stimuli were applied at different BCLs B to achieve SS, starting from B = 300 ms in 20 ms decrements until B = 100 ms or until ventricular fibrillation occurred. In the perturbed downsweep pacing protocol (n = 5), the following steps were applied at each BCL B, starting with B = 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 B^LP = B + 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 B^SP = 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 B = 100 ms or until ventricular fibrillation occurred.

At each BCL, the APDs were measured at 80% repolarization during SS (A^SS) in both protocols, and after SP (A^SP) and LP (A^LP) in the perturbed downsweep pacing protocol. The preceding DI D^P was calculated at each pixel according to the following formula:

$$D^p=B^p-A^{SS}$$ (1)

where p denotes either SS, LP, or SP. Then, 2D APD and DI maps were created from the (D^P, A^P) 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 (A^SS, D^SS) responses measured at different BCLs (Table 1). We calculated the slope of the dynamic restitution curve (S_dyn) 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).

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)	$S_{12}^{\max }$	Measured at SP at each B
	SP: (ASP, DSP)
	SS: (ASS, DSS)	$S_{dyn}^{RP}$	Measured at SS from all B
Dynamic restitution curve	SS: (ASS, DSS)	Sdyn	Measured at SS from all B

At each pixel, the restitution portrait consists of a combination of the following responses: (A^SS,D^SS), (A^LP,D^LP), (A^SP,D^SP) measured at different BCLs (Table 1). Thus, as described previously (21,22), the following restitution curves can be visualized in the restitution portrait: 1), a unique dynamic restitution curve that consists of (A^SS,D^SS) responses measured at different B-values; and 2), several local S1-S2 restitution curves that consist of (A^SS,D^SS), (A^LP,D^LP), and (A^SP,D^SP) responses measured separately for each B-value. We calculated the slope of the dynamic restitution curve measured in the restitution portrait ($S_{dyn}^{RP}$) at each pixel by fitting (A^SS,D^SS) 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 ($S_{12}^{\max }$)) were measured.

Data analysis

Alternans was calculated at each pixel as a difference in APDs between odd and even beats, as described previously (25): ΔAPD = |APD_even − APD_odd| ≥ 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 ΔAPD maps were constructed to reveal the spatial distribution and amplitude 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 discordant alternans, the white color also indicates the presence of the nodal lines (26,27).

The local spatial onset of alternans, B^onset, 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 B^onset: the 1:1_alt 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 B_N was introduced as B_N = B−B^onset so that the onset of alternans would always occur at $B_N^{onset}$= 0 ms.

Group data are presented as the mean ± 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.

Results

Spatiotemporal evolution of alternans in the heart

We observed that alternans first appeared locally at B=B^onset in one of the ventricles and evolved spatially throughout the heart as the BCL decreased. Fig. 2 shows a representative example of the spatiotemporal evolution of alternans in the left ventricle of a rabbit heart at different values of B (top) or the corresponding normalized BCL B_N (bottom). The color bar represents the amplitude of alternans ΔAPD. Note the presence of 1:1 behavior (white) at large B, and the appearance of alternans (red) as B decreases. For instance, in this particular example, the onset of alternans occurred at B^onset = 170 ms ($B_N^{onset}$= 0 ms). At this specific BCL, the two regions, 1:1_alt and 1:1, were identified at the surface of the heart and projected back into all prior BCLs (see Fig. 2). Note that alternans spread throughout the heart as the BCLs became smaller than B^onset (negative B_N), and spatially concordant alternans became spatially discordant, as previously described (26,27). Spatially concordant and discordant alternans can be distinguished by their phases; however, a corresponding analysis is beyond the scope of this study and thus is not shown here. However, note that the presence of white color at B = 150 ms in Fig. 2 indicates the presence of the nodal line during spatially discordant alternans. The total area occupied by alternans (both spatially concordant and discordant) from all experiments is shown in Fig. 3 as a function of B_N. The dashed horizontal line represents the spatial threshold for alternans. Note that as B_N decreases, the total area occupied by alternans increases until the entire heart surface exhibits alternans.

Figure 2.

Representative example of the 2D ΔAPD maps in rabbit left ventricle at different B (top panel) and normalized B_N (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 B^onset (or $B_N^{onset}$= 0 ms). At B^onset, the two regions (1:1_alt and 1:1) are introduced and projected to all preceding B-values (black outlines).

Figure 3.

Mean percentage of total ventricular area occupied by alternans as a function of B_N for all rabbit ventricles (n = 5). The dashed horizontal line represents the spatial threshold for alternans.

2D restitution properties of the heart

We sought to determine whether the restitution properties of the two regions, 1:1_alt and 1:1, were different during periodic pacing, before the onset of alternans. To that end, we applied a perturbed downsweep pacing protocol and constructed 2D APD maps for SS (A^SS), LP (A^LP), and SP (A^SP) at various BCLs. Representative examples of these APD maps are shown in Fig. 4 A for large (B_n = 100 ms) and small (B_n = 10 ms) BCLs. The color bar represents the APDs, and the projected boundaries of the 1:1_alt and 1:1 regions are shown as black lines on each APD map. Note that for large B_N (top panel in Fig. 4 A), the spatial distributions of A^LP and A^SP were similar to A^SS for both the 1:1_alt and 1:1 regions. However, for small B_N (bottom panel in Fig. 4 A), only the spatial distribution of A^LP was similar to A^SS, whereas the spatial distribution of A^SP was visually different from A^SS for both the 1:1_alt and 1:1 regions.

Figure 4.

(A) Representative examples of 2D APD maps for A^LP, A^SS, and A^SP at B_n = 100 ms (top panel) and B_n = 10 ms (bottom panel). The color bar represents the duration of the action potential (in ms). The 1:1 and 1:1_alt regions are outlined by black curves. (B and C) The mean deviation of the (B) SP and (C) LP from SS as a function of B_N. The data were calculated separately for the 1:1_alt and 1:1 regions of the heart from all experiments (n = 5); ^∗ denotes statistically significant data (p < 0.05).

To quantify these results, we calculated the deviations of the APDs measured after SP and LP from the SS APD (ΔA^SP = A^SS − A^SP and ΔA^LP = A^LP − A^SS, respectively) at each pixel. Fig. 4, B and C, show the mean values of ΔA^SP and ΔA^LP, calculated separately for the 1:1_alt and 1:1 regions in all experiments, as a function of B_N. Note that for both regions the values of ΔA^SP and ΔA^LP increased as B_N decreased, but the magnitude of deviation was higher for ΔA^SP. Specifically, the maximum deviations occurred for the 1:1_alt region just before the onset of alternans, at B_n = 10 ms, were ΔA^SP = 8.6 ± 0.15 ms and ΔA^LP = 2.8 ± 0.14 ms (p < 0.05). Fig. 4 also demonstrates that the values of ΔA^SP and ΔA^LP were qualitatively similar for both 1:1_alt and 1:1 regions at all B_N-values except B_n = 10 ms. At B_n = 10 ms, both ΔA^SP and ΔA^LP were significantly higher for the 1:1_alt region than for the 1:1 region. In addition, the difference in ΔA^SP between the 1:1_alt and 1:1 regions (2.71 ± 0.52 ms) was significantly larger than the difference in ΔA^LP between these regions (0.78 ± 0.19 ms, p < 0.05). Therefore, immediately before the onset of alternans, ΔA^SP not only showed the highest maximal value, it also demonstrated a significant increase in regional heterogeneity. Hence, although both SP and LP lead to different APDs between the 1:1_alt and 1:1 regions, the SP could be considered 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 fixed value (10 ms), and thus the difference in ΔA^SP between the two 1:1_alt and 1:1 regions should lead to the different slope values calculated in the restitution portrait. Representative examples of the restitution portraits for two pixels taken from the 1:1_alt and 1:1 regions are compared in Fig. 5. Here, a unique SS restitution curve (red) and several local S1-S2 restitution curves for different B_N-values (blue) are shown along with the corresponding $S_{dyn}^{RP}$, S₁₂, and $S_{12}^{\max }$ slopes. The gray dotted lines are the equal BCL lines with slope = −1, and the solid black line shows slope = 1. Note that the dynamic restitution curves are almost parallel in both the 1:1_alt and 1:1 regions, indicating the absence of differences in $S_{dyn}^{RP}$ at all B_N-values. Similarly, local S1-S2 restitution curves are nearly parallel at large B_N (e.g., B_n = 100 ms), indicating that S₁₂ and $S_{12}^{\max }$ were similar between 1:1_alt and 1:1 regions as well. However, at small B_N (e.g., B_n = 10 ms), the difference between local S1-S2 restitution curves is pronounced, and both the S₁₂ and $S_{12}^{\max }$ slopes are visually steeper in the 1:1_alt region than in the 1:1 region. Therefore, Fig. 5 indicates that the S₁₂ and $S_{12}^{\max }$ slopes correlate with the local onset of alternans in the heart.

Figure 5.

Representative examples of the restitution portraits constructed for two pixels taken from the 1:1_alt (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 = −1 represent the equal-BCL lines. The solid line shows slope = 1. Three different slopes are present in the restitution portrait for each B: $S_{dyn}^{RP}$ and S₁₂ (measured at SS) and $S_{12}^{\max }$ (measured at the SP).

To quantify these results, we calculated the mean values of three slopes measured in the restitution portrait ($S_{dyn}^{RP}$, S₁₂, and $S_{12}^{\max }$) over the epicardial surface of the hearts in all experiments separately for the 1:1_alt and 1:1 regions. The data are shown in Fig. 6, A–C, as a function of B_N. Note that there is no significant difference between values of $S_{dyn}^{RP}$ calculated for both the 1:1_alt and 1:1 regions at any B_N (Fig. 6 A). However, the behavior of the S₁₂ and $S_{12}^{\max }$ slopes is different (Fig. 6, B and C). At large B_N, both S₁₂ and $S_{12}^{\max }$ are similar between the 1:1_alt and 1:1 regions, but as B_N decreases, the values of both slopes become significantly larger in the 1:1_alt region than in the 1:1 region (p < 0.05). Note that this significant difference appeared first for the S₁₂ slope (at B_n = 20 ms; Fig. 6 B) and later for the $S_{12}^{\max }$ slope (at B_n = 10 ms; Fig. 6 C). However, the difference in $S_{12}^{\max }$ values was significantly larger (0.4 ± 0.15) than the difference in S₁₂ values (0.19 ± 0.06, p < 0.05) between the 1:1_alt and 1:1 regions immediately before the onset of alternans (B_n = 10 ms). Therefore, both S₁₂ and $S_{12}^{\max }$ can serve as indicators of the local onset of alternans in the heart, although $S_{12}^{\max }$ can be considered as a more definite marker.

Figure 6.

Mean values of (A) $S_{dyn}^{RP}$, (B) S₁₂, (C) $S_{12}^{\max }$, and (D) S_dyn as a function of B_N. All slopes were calculated separately for the 1:1_alt (solid circles) and 1:1 (open circles) regions from all experiments (n = 5 for the restitution portrait, and n = 5 for the dynamic restitution curve); ^∗ denotes statistically significant data (p < 0.05).

To demonstrate that alternans can be induced in the heart by dynamic pacing alone (i.e., without LPs and SPs), we applied the dynamic pacing protocol in several rabbit hearts. Our results indicate that the formation of the alternans in the heart was similar to that described for the perturbed downsweep pacing protocols and shown in Figs. 1 and 2. The slopes of the dynamic restitution curve, S_dyn, calculated separately for the 1:1_alt and 1:1 regions are shown in Fig. 6 D, and indicate the absence of statistical significance between these regions at all B_N-values.

To determine whether any of the slopes measured in our experiments were comparable with the theoretically predicted value of one, we calculated the mean values of $S_{dyn}^{RP}$, S₁₂, and $S_{12}^{\max }$ measured in the restitution portrait, and S_dyn measured from the dynamic restitution curve, immediately before the onset of alternans, at B_n = 10 ms. Fig. 7 compares the values of all slopes for both the 1:1_alt and 1:1 regions with the value of one. Note that $S_{dyn}^{RP}$, S₁₂, and S_dyn were <1 in both the 1:1_alt and 1:1 regions, whereas $S_{12}^{\max }$ was >1 in 1:1_alt and <1 in the 1:1 region. Nevertheless, the values of all slopes are significantly different from the theoretical predicted value of one (p < 0.05).

Figure 7.

Mean values of $S_{dyn}^{RP}$, S_dyn, S₁₂, and $S_{12}^{\max }$ measured just before the onset of alternans at B_n = 10 ms in comparison with the theoretical predicted value of one at the onset of alternans. The data are taken from all experiments (n = 5 for both the restitution portrait and the dynamic restitution curve); ^∗ denotes statistically significant data (p < 0.05).

Discussion and Conclusions

In this study, we investigated the spatiotemporal formation of alternans in isolated rabbit hearts and sought to determine whether the slopes of the restitution curves measured in the restitution portrait are correlated with its onset. The major findings of our study are as follows: First, we observed that alternans is a complex spatiotemporal phenomenon in the heart. It initially appeared locally in some areas of the ventricles and evolved throughout the heart as BCL decreased. Second, we demonstrated that the slopes S₁₂ and $S_{12}^{\max }$ measured in the restitution portrait correlate with the local onset of alternans. Third, our results indicate that although S₁₂ and $S_{12}^{\max }$ correlate with the onset of alternans in the heart, their quantitative values are significantly different from the theoretically predicted value of one.

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

First, the presence of short-term memory in cardiac tissue makes the restitution properties dependent on the pacing protocols applied (28–32). As a consequence, the restitution curves measured by different pacing protocols have different slopes and fail to predict the onset of irregular cardiac rhythms correctly. For instance, stable 1:1 behavior is observed when the restitution curve is very steep (14,19), whereas the transition to alternans is observed in the presence of a shallow restitution curve (12,17). Investigators have made several attempts to reconsider the restitution hypothesis in the presence of short-term memory (20,29,30,33–35). In particular, the development of a new protocol for simultaneously measuring multiple aspects of cardiac dynamics, rather than individual restitution properties, led to the concept of a restitution portrait of cardiac tissue (20,21,35). The restitution portrait was incorporated experimentally in small bullfrog cardiac tissue and isolated rabbit and guinea pig myocytes, where spatial complexity was minimal (21) or absent (22), and the $S_{12}^{\max }$ slope was shown to successfully correlate with the onset of alternans.

The second major reason for the failure of the restitution hypothesis is the spatial complexity of the heart, the consequences of which are usually underestimated in experimental and clinical studies. Indeed, attempts to measure one or a few restitution curves for a given heart revealed that such restitution curves often represent average cardiac responses. To date, little attention has been given to the spatial distribution of the restitution properties of the heart and its correlation with the onset of alternans (18,25).

To our knowledge, this study represents the first attempt to correlate several slopes, measured both in the simple dynamic protocol and the more-sophisticated restitution portrait, with the local onset of alternans in the heart. It should be noted, however, that the restitution portrait does not represent the full spectrum of all possible aspects of cardiac 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 attempt so far to simultaneously measure several aspects of cardiac dynamics, as it combines both SS responses and responses from perturbations (35). Conventional measurements of these responses are obtained through the individual dynamic and S1-S2 restitution curves, respectively. In this study, we compared SS responses measured from the dynamic restitution curve (S_dyn) and the restitution portrait ($S_{dyn}^{RP}$), and demonstrated their similarity. It is not as easy to compare the standard S1-S2 restitution curve and local S1-S2 restitution curves from the restitution portrait, mainly because of the presence of an infinite number of S1-S2 restitution curves (one for each value of S1). Nevertheless, it has been shown theoretically that only points in the vicinity of the intersection of the S1-S2 restitution curve with the SS responses, and not the entire S1-S2 restitution curve, are important for determining the onset of alternans (20).

Our results indicate that although slopes S₁₂ and $S_{12}^{\max }$ predict the local onset of alternans in the heart, their values are significantly different from the theoretically predicted value of one. Specifically, S₁₂ < 1 in both the 1:1_alt and 1:1 regions, whereas $S_{12}^{\max }$ > 1 in the 1:1_alt region and $S_{12}^{\max }$< 1 in the 1:1 region. Many different studies have attempted to explain this discrepancy. The major factors that need to be taken into account are the presence of short-term memory and the concurrent existence of alternans in intracellular calcium cycling (15,26).

Finally, we note that, historically, the term “prediction of alternans” has been used with respect to the slopes of the restitution curve (13,34,36,38). However, we believe it would be more accurate to use the term “correlation with alternans”, since we have demonstrated that a statistically significant difference between restitution properties in two regions of the heart exists and is associated with the local onset of alternans. The first step toward actual prediction of irregular cardiac responses was made by Gelzer et al. (39), who applied theoretically derived pacing sequences in the heart to induce alternans and ventricular fibrillation.