Mechanisms of Discordant Alternans and Induction of Reentry in Simulated Cardiac Tissue

Electrical Restitution

To study APD and CV restitution effects, we modified LR1 kinetics to produce 2 types of APD restitution and 3 different types of CV restitution, for a total of 6 combinations. One APD restitution type had slope >1 over a wide range of DIs (Figure 1⇑, A through C), and the other had slope <1 everywhere (Figure 1⇑, D through F). CV restitution types differed in the range of DIs over which CV varied: either the normal range for the LR1 model (Figure 1⇑, A and D), a shortened range (Figure 1⇑, B and E), or a broadened range (Figure 1⇑, C and F). Details of the parameter changes are stated in the Figure 1⇑ legend.

Modulation of Dispersion by Premature Stimulus

In intact guinea pig ventricle, Laurita et al¹⁸ found that dispersion of APD during a premature stimulus decreased to a minimum before progressively increasing as the coupling interval was shortened. To investigate the roles of electrical restitution and electrophysiological heterogeneity, we compared electrophysiologically homogeneous and heterogeneous tissue by use of various APD and CV restitution combinations shown in Figure 1⇑.

Homogeneous Tissue

For homogeneous tissue, the dispersion of APD (ς) was almost zero for long S₁S₂ coupling intervals, increasing monotonically as the coupling interval decreased (Figure 2A⇓). There was no dip in ς before the increase, because ς was already near zero. The S₁S₂ coupling interval at which ς increased was determined by CV restitution. Broadening the range of DIs over which CV changed caused ς to increase at longer S₁S₂ coupling intervals (▵), and narrowing the range had the opposite effect (○). Figure 2C⇓ shows the spatial distribution of APD in homogeneous tissue with steep APD restitution plus normal CV restitution (corresponding to Figure 1A⇑) during baseline pacing and during a premature stimulus at 3 different S₁S₂ coupling intervals.

• Download figure
• Open in new tab
• Download powerpoint

Figure 2.

APD dispersion (ς) versus S₁S₂ coupling interval. A, Homogeneous tissue with steep APD restitution and either normal (•), narrowed (○), or broadened CV restitution (▵). B, Heterogeneous tissue with steep APD restitution and either normal (•) or broadened (▵) CV restitution and with shallow APD restitution and normal CV restitution (○). C and D, Spatial APD distributions in homogeneous (C) and heterogeneous (D) tissue at different S₁S₂ coupling intervals for steep APD restitution plus normal CV restitution (as in Figure 1A⇑).

Transition From Concordant Alternans to Discordant Alternans

To study how concordant and discordant APD alternans develop, we rapidly paced homogeneous or heterogeneous 2D tissue with different APD and CV restitution characteristics.

Homogeneous Tissue

Figure 3⇓ shows the pseudo-ECG, APD alternans, and CL alternans at different pacing cycle lengths (PCLs) in homogeneous tissue. At PCL=300 ms, there was no alternans. At PCL=220 ms, concordant APD alternans developed, but no CL alternans. At PCL=180 ms, CL alternans began and APD alternans became discordant. During concordant alternans, the pseudo-ECG showed only T-wave alternans, without QRS alternans. With discordant alternans, both the T-wave and the QRS complex alternated. This result is very similar to the findings of Pastore et al⁹ (their Figures 4⇓ and 6⇓). Thus, preexisting electrophysiological heterogeneity is not required for concordant or discordant alternans or for QRS or T-wave alternans.

• Download figure
• Open in new tab
• Download powerpoint

Figure 3.

Transition from concordant (middle panels) to discordant (right panels) alternans in homogeneous tissue during constant pacing for steep APD restitution plus normal CV restitution (as in Figure 1A⇑). A, Pseudo-ECG illustrating no alternans (PCL=300 ms), T-wave alternans (PCL=220 ms), and QRS plus T-wave alternans (PCL=180 ms). B and C, Spatial APD distribution (B) and ΔCL (=CL−PCL) distribution (C) for 2 successive beats at different PCLs. D, Membrane voltage records from 2 different sites in tissue, showing discordant alternans. 1, 10, and 20 mark corresponding beats at each site.

• Download figure
• Open in new tab
• Download powerpoint

Figure 4.

A through C, Maximum differences in APD (•) and CL (○) in homogeneous tissue versus heart rate with steep APD restitution and either normal (A), narrowed (B), or broadened (C) CV restitution. Downward and upward arrows indicate onset of concordant and discordant alternans, respectively. D through F, APD dispersion (ς) of 2 successive beats (○ and •) vs heart rate. D and E, Homogeneous (D) and heterogeneous (E) tissue, with steep APD restitution plus normal CV restitution (as in Figure 1A⇑), respectively. F, Heterogeneous tissue, with shallow APD restitution plus normal CV restitution (as in Figure 1D⇑).

Figure 4⇑, A through C summarizes the maximum differences in APD (•) and CL (○) versus PCL in homogeneous tissue with steep APD restitution and different CV restitution properties (corresponding to Figure 1⇑, A through C, respectively). With normal CV restitution (Figure 4A⇑), CL alternans occurred after APD alternans. With CV restitution narrowed (Figure 4B⇑), APD alternans occurred, but CL alternans never developed. With CV restitution broadened (Figure 4C⇑), APD alternans and CL alternans occurred simultaneously. CL alternans was invariably associated with the onset of discordant APD alternans. For flat APD restitution (corresponding to Figure 1⇑, D through F), neither APD nor CL alternans was observed at any PCL, although a mild dispersion of APD was present, similar to Figure 3B⇑.

Discordant Alternans and the Induction of Reentry

To explore the relationship between electrical alternans and reentry, homogeneous and heterogeneous tissues were paced from the lower left corner for 15 beats at a fixed PCL (S₁), followed by a premature stimulus (S₂) to induce reentry.

Homogeneous Tissue

In homogeneous tissue, an S₂ stimulus delivered at the S₁ site did not induce reentry. To induce reentry, S₂ had to be delivered at a different site from S₁ to break symmetry. Accordingly, S₂ was applied along the diagonal from the lower left to the upper right corners. Figure 5⇓, A and B, summarizes the vulnerable window for induction of reentry versus distance of S₂ along the diagonal. At long PCLs without APD alternans (Figure 5A⇓), the S₂ either propagated or failed but never induced reentry. Only when the PCL was short enough to induce discordant alternans could reentry be induced by an S₂ in a proper position (Figure 5B⇓). Thus, induction of reentry by a premature stimulus during discordant alternans did not require preexisting electrophysiological heterogeneity if S₂ was delivered at a different site from S₁.

• Download figure
• Open in new tab
• Download powerpoint

Figure 5.

A and B, Phase diagram in space-time for vulnerability to reentry by a premature stimulus (S₂) in homogeneous tissue with steep APD restitution plus normal CV restitution (as in Figure 1A⇑) at 2 PCLs: 300 ms (A) and 180 ms (B). Fifteen S₁ beats were delivered to lower left corner of tissue. S₂ was delivered at time t after the last S₁ stimulus at distance r along diagonal line from lower left to upper right corners. C through F, Phase diagram for vulnerability to reentry by a premature stimulus (S₂) as a function of S₁ PCL in heterogeneous tissue under the following conditions: for C, steep APD restitution plus normal CV restitution (as in Figure 1A⇑); for D, shallow APD restitution plus normal CV restitution (as in Figure 1D⇑); for E, as in A, except with decreased heterogeneity (β=0.4 in Equation 2); for F, as in B, except with increased tissue heterogeneity (in Equation 2, β=0 for area of 1-cm radius in center, β=0.8 elsewhere). S₁ and S₂ were delivered at same site. P indicates that wave excited by S₂ propagates throughout tissue without reentry; F, that S₂ cannot stimulate a wave; F_R, that wave stimulated by S₂ can propagate but is blocked a distance away from pacing site; R, that S₂ induces reentry; and R_T, that S₂ induces transient reentry.

Heterogeneous Tissue

In heterogeneous tissue, an S₂ delivered at the S₁ site could induce reentry, because symmetry was broken by the preexisting tissue heterogeneity. Figure 6⇓ illustrates reentry induction by an S₂, which is similar to the example shown in guinea pig heart⁹ (see their Figure 8). Conduction block occurred at a location where APDs were in their long phase during discordant alternans (Figure 6B⇓, middle). This local conduction block resulted in figure-eight reentry (fourth panel) and subsequent breakup into a fibrillation-like state (fifth panel).

• Download figure
• Open in new tab
• Download powerpoint

Figure 6.

Induction of reentry in heterogeneous tissue with steep APD restitution plus normal CV restitution (as in Figure 1A⇑). S₁ (15 beats at PCL=170 ms) and S₂ (S₁S₂=120 ms) were delivered at same site. A, Membrane potential at different sites in tissue, with conduction block initiating reentry indicated by arrows. B, Snapshots of membrane voltage during last 2 S₁ beats (beats 14 and 15) and at various times after S₂ was delivered, showing initiation of figure-eight reentry leading to fibrillation. C, Spatial APD distribution for S₁ beats 14 and 15, showing discordant alternans.

Induction of reentry by this protocol depended on both electrical restitution properties and the degree of preexisting electrophysiological heterogeneity, in addition to the S₁ and S₁S₂ intervals. Figure 5⇑, C through F, summarizes the phase diagrams for several restitution combinations. For steep APD restitution plus normal CV restitution (as in Figure 1A⇑), there was a large vulnerable window for S₂ to induce reentry at short PCLs, which narrowed and disappeared as the PCL increased (Figure 5C⇑), similar to homogeneous tissue. If the degree of preexisting electrophysiological heterogeneity was reduced (by setting β=0.4 in Equation 2), there was still a large vulnerable window for reentry (Figure 5E⇑). In addition, a new phase was observed in which S₂ excited a propagating wave that was blocked a distance away from the pacing site.

If APD restitution was flattened (corresponding to Figure 1D⇑), reentry could not be induced with the same degree of preexisting electrophysiological heterogeneity (Figure 5D⇑). However, if the preexisting heterogeneity was increased to produce a steep local electrophysiological gradient, a small vulnerable window was observed (Figure 5F⇑). In addition, a new phase occurred in which reentry was only transient. Thus, with sufficiently large electrophysiological heterogeneity, reentry could be induced even when APD restitution was flat enough to prevent discordant (or even concordant) alternans.

Mechanisms of Initiation of Reentry

It is generally argued that discordant alternans facilitates induction of reentry by increasing dispersion of refractoriness. However, a detailed mechanistic explanation is lacking. To this end, Figure 7A⇓ illustrates schematically 2 successive waves in the tissue. During wave propagation, both wavefront conduction velocity (CVF) and the waveback conduction velocity (CVB) vary in time and space. Conduction fails when the wavefront velocity is below a critical velocity CVF_c.²⁰ For local conduction block, eg, at location a in Figure 7A⇓, CVB₁ must be slower than CVF_c at a, such that the wavefront of the second wave can approach closer and closer to the waveback of the first wave until CVF₂ becomes smaller than CVF_c. However, local wavebreak does not necessarily guarantee reentry. Wavebreak at point a may not lead to a reentry because the wavelength (and hence refractoriness) of the tissue in this region is long and the broken wave does not have sufficient surrounding excitable tissue nearby to execute a full turn. In contrast, wavebreak at point b is more likely to induce reentry, because the wavelength and refractory period are short. From this argument, it is easy to conceptualize the conditions favoring reentry: (1) a sufficiently slow propagating waveback to cause conduction failure; (2) inhomogeneities to cause conduction failure locally; and (3) alternation of wavelength, or refractory period, to facilitate reentry.

• Download figure
• Open in new tab
• Download powerpoint

Figure 7.

A, Schematic of successive waves in tissue, illustrating wavefront (thick lines) and waveback (thin lines) propagation (see text). B through D, Wavefront (solid lines) and waveback (dashed lines) velocities along tissue diagonal line (r) for 2 successive beats (thick and thin lines, respectively) in homogeneous tissue with steep APD restitution plus normal CV restitution (as in Figure 1A⇑). Dashed-dotted lines indicate wavefront velocity at which conduction fails (CVF_c). B, Concordant alternans at PCL=220 ms. C and D, Discordant alternans at PCL=180 ms. C shows only waveback velocities, illustrating that 1 beat develops negative waveback velocity. Inset shows voltage in space, and arrows indicate propagation directions. D shows wavefront and waveback velocities of 1 beat (thick lines) and wavefront velocity of next beat (thin line). E, Wavefront and waveback velocities in markedly heterogeneous tissue with shallow APD restitution plus normal CV restitution described in Figure 5F⇑. Arrow indicates that slow waveback occurs near boundary of heterogeneity.

With respect to slow propagation of a waveback, the relationship between CVB and CVF of a wave can be deduced as follows: the time for the waveback at position r to propagate a distance Δr is the time that the wavefront propagates this same distance [Δr/CVF(r)] plus the APD difference between position r+Δr and position r [ΔAPD(r)=APD(r+Δr)−APD(r)], ie, Δr/CVF(r)+ΔAPD(r). Therefore, the waveback velocity is given by the expression where APD′(r)≈ΔAPD(r)/Δr is the spatial derivative of APD. Equation 4 is similar to the equation derived by Courtemanche.²¹ From Equation 4, if the gradient APD′(r)>0, then CVB(r)<CVF(r). If APD′(r)<0 but APD′(r) · CVF(r)>−1, then CVB(r)>CVF(r). If APD′(r)<0 but APD′(r) · CVF(r)<−1, then CVB(r)<0<CVF(r). Therefore, the back of a wave can propagate either slower or faster than its front, depending on the spatial gradient of APD. Two ways to produce large spatial APD gradients are discordant alternans and tissue heterogeneity.

Figure 7⇑, B through D, shows CVF and CVB during pacing in homogeneous tissue with steep APD restitution plus normal CV restitution (corresponding to Figure 1A⇑). At PCL=220 ms (Figure 7B⇑), concordant alternans was present. Both CVF and CVB were almost the same except at the boundaries and were much larger than CVF_c. At PCL=180 ms (Figure 7⇑, C and D), discordant alternans was present, and CVB varied greatly over space. In 1 of the 2 alternating beats, CVB changed from large positive to large negative velocity (Figure 7C⇑). The negative velocity was due to the large decrease of APD along the direction of propagation (see Equation 4), which is illustrated in the inset at the bottom of Figure 7C⇑. In a later beat (Figure 7D⇑), the waveback propagated very slowly at first (CVB≪CVF and CVF_c) and then became faster than the wavefront. In contrast, Figure 7E⇑ shows CVF and CVB for heterogeneous tissue, with shallow APD restitution but steep gradient of electrophysiological heterogeneity in Figure 5F⇑. A narrow area showing slow waveback propagation (much slower than CVF_c) is present at one boundary of the sharp heterogeneity, and CVB varies from positive to negative at the other boundary. The cause of the negative velocity is similar to that in Figure 7C⇑.

Modulation of Dispersion by a Premature Stimulus

The mechanism by which a premature stimulus modulates dispersion agrees with the conjecture by Laurita et al¹⁸ : a short S₁S₂ coupling interval results in a short DI for the premature beat. The premature beat thus propagates more slowly than the normal beats, so that DIs along the propagation direction increase, which, in turn, result in longer APD (see Figure 2C⇑). In homogeneous tissue, this results in a monotonic increase in dispersion as S₁S₂ decreases. However, in electrophysiologically heterogeneous tissue, both APD restitution and preexisting heterogeneity are important for modulation of dispersion. If APD restitution is steep, the long APD is markedly shortened by a short DI, whereas the shorter APD is only modestly shortened by a longer DI. This results in less dispersion (Figure 2B⇑). CV restitution changes the coupling interval and thus changes the dispersion.

Mechanism of Concordant and Discordant APD Alternans

For APD alternans (concordant or discordant) to occur, an APD restitution slope >1 is required, even in electrophysiologically heterogeneous tissue. During pacing from the same site, to proceed from concordant to discordant alternans requires that the PCL be short enough to engage CV restitution. CV restitution explains why discordant alternans is associated with QRS as well as T-wave alternans. Neither concordant nor discordant alternans requires preexisting electrophysiological heterogeneity; they can arise solely from the dynamics of electrical restitution. (See the Appendix for formal analysis of the mechanism.) Although for large heterogeneous initial conditions, transient discordant alternans can be initiated without engaging CV restitution, alternans will finally become concordant.

Facilitation of Reentry

Discordant alternans markedly increases dispersion of refractoriness and increases the ability of a premature stimulus to cause localized wavebreak and induce reentry, even in completely homogeneous tissue. Preexisting electrophysiological heterogeneity is not required if the S₂ extrastimulus is delivered at a different site from the S₁ pacing site. In the absence of discordant alternans, preexisting electrophysiological heterogeneity can also cause localized wavebreak.

Although reentry can be induced with or without steep APD restitution, the vulnerability is different. When APD restitution is shallow, wavebreak occurs only in regions with very large preexisting heterogeneity. Because APD and wavelength are long, reentry cannot be easily induced (Figure 5⇑), eg, at location a in Figure 7A⇑. The vulnerable window of reentry is much larger when APD restitution is steep enough to produce discordant alternans (Figure 5⇑), because during discordant alternans, wavebreak always occurs in the waves with short APD and wavelength (eg, at location b in Figure 7A⇑).

In summary, heterogeneity breaks symmetry and also can cause wavebreak by waveback slowing; APD restitution causes the dynamical instability, resulting in wavelength oscillation; CV restitution converts concordant APD alternans into discordant alternans, which results in further waveback slowing and increased dispersion of refractoriness.

Clinical Implications

Combined with experimental evidence relating discordant APD alternans to T-wave alternans and increased vulnerability to ventricular arrhythmias,^{1 2 3 4 5 6 7 8 9 10} our findings further emphasize the importance of dispersion of refractoriness to induction of ventricular reentry. Most importantly, our analysis suggests that dispersion of refractoriness arising from purely dynamic factors, namely APD and CV restitution, is at least as important as, if not more important than, preexisting electrophysiological heterogeneity for enhancing susceptibility to ventricular arrhythmias. This is therapeutically encouraging, because restitution properties are potentially modifiable by drugs. Drugs altering electrical restitution to reduce dynamic dispersion represent a promising antiarrhythmic strategy.

Mechanism of Discordant Alternans

Assume APD and CV restitution are valid in a spatially uniform system: In Equation 5, CL_n(r) can be expressed as Starting from a uniform initial condition, we can derive from Equations 5 and 6 the following.

1. For long PCLs without APD alternans, the system remains uniform.

2. For PCLs at which APD alternates but CV restitution is not engaged, CV_n+1(r)=CV_n(r). In this case, the integral in Equation 6 is zero, ie, ΔCL_n(r)=0. Therefore, CL_n(r)=PCL everywhere in space, which indicates from Equation 5 that APD_n(r) and DI_n(r) must both be uniform in space, ie, uniform concordant alternans occurs.

3. For PCLs in a range in which both APD alternans occurs and CV restitution is engaged, assume that the system is in a stable alternating state such that DI_n+2(r)=DI_n(r)>DI_n−1(r)=DI_n+1(r) at position r. According to Equation 6, ΔCL_n(r)=−ΔCL_n+1(r) and DI_n(r)=CL_n(r)−APD_n(r); we then have

Rearranging and substituting Equation 7, Uniform concordant alternans, with APD_n+1(r)−APD_n(r) constant over space, is impossible, because the integral in Equation 8 changes with r. Nonuniform concordant alternans can exist when CV_n+1(r)−CV_n(r) or r is small, for which Equation 8 can be satisfied. However, when either CV_n+1(r)−CV_n(r) or r is large, then the integration in Equation 8 becomes larger and larger, and Equation 8 cannot hold for concordant alternans. Therefore, CV_n+1(r)−CV_n(r) must change its sign along r so as not to violate Equation 8, resulting in discordant alternans.