GENESIS OF DISCORDANT ALTERNANS IN A DISCRETE CARDIAC FIBER
by Matthew Mian
Discordant alternans is an electrical feature of cardiac tissue that recently has been linked to the onset of ventricular arrhythmias. In the following analysis, we extend two existing mechanisms for the initiation of discordant alternans to the physiologically relevant case of a discretized cardiac fiber. Our simulations confirm these mechanisms of initiation, lending support to an existing theory for APD alternans genesis.
Introduction
Electrical alternans is a term that describes any alternate-beat variation of cardiac tissue that is reflected on the electrocardiogram. Although first characterized nearly a century ago, electrical alternans has only recently been linked to arrhythmogenesis. Notably, alternans has been shown to be a possible harbinger of ventricular arrhythmias and sudden cardiac death.1,2 For this reason, alternans has become a subject of particular interest in experimental and theoretical electrophysiology. Elucidation of the mechanisms by which alternans initiates could be important to the design of antiarrhythmic control schemes and pharmaceutical therapies.
One type of electrical alternans is that of action potential duration (APD). APD alternans typically develops when cardiac tissue is stimulated at a high rate. Under such conditions, cardiac tissue may develop a variation in APD that reflects the alternating length of the preceding recovery periods (diastolic intervals).
Figure 1: APD alternans in a ventricular cardiac fiber. The duration of action potentials
alternates from beat to beat when the tissue is paced at a high rate.
Discordant alternans, a special case of the alternans phenomenon, refers to the simultaneous existence of two spatially distinct regions exhibiting alternans of opposite phases. With regard to action potential duration, discordant alternans has been linked to both QRS and T-wave alternans of the ECG.2 Electrical alternans of the T-wave, in particular, is associated with ventricular fibrillation.3,4 While there has been considerable speculation on the origin of discordant alternans, the theory underlying its genesis is poorly understood.
Figure 2: Discordant alternans in a cardiac fiber. The vertical axis denotes the length of the cable (8cm).
Membrane potential (in mV) is visualized as color, with scale at right. APD at the ends of the fiber alternates out of phase.
Recently, Watanabe et. al. showed that discordant APD alternans can be generated in spatially homogenous tissue via two different mechanisms.5 The first scenario involves an ectopic premature stimulus while the second entails a complex interaction of APD and conduction velocity. While this discovery represents an important step in characterizing discordant alternans, the study has several limitations.
Watanabe et al. model cardiac tissue as a homogenous and continuous medium, although its structure is known to be heterogeneous. Cardiac tissue is discretized into individual cells. Gap junctions, the channels that connect neighboring cells, provide electrical pathways for conduction. However, gap junctions also offer a higher effective resistivity than the intracellular space (cytoplasm). As a result, conduction velocity is reduced when a propagating action potential encounters a cellular junction. This contrasts with propagation in continuous media, where conduction velocity is spatially uniform.
We first review the work of Watanabe et al. by reproducing their results in a continuous fiber. We then consider the assumption of spatial homogeneity. Is the behavior reported by Watanabe et al. preserved in a fiber formulated to reflect the inhomogeneity of real cardiac tissue? We seek to address this question using computational simulation and analysis.
Methods
Cardiac tissue is modeled as a one-dimensional monodomain, where the equation governing membrane potential is given by
( 1 )
where Cm = 1.0 μF/cm2 is the membrane capacitance, Iion is the sum of ionic currents (μA/cm2), Vm is the transmembrane voltage (mV), β is the surface to volume ratio (cm-1), and σi is the conductivity coefficient (mS/cm). We include an additional scale factor of 2 to account for the difference between geometrical and capacitive areas.
Following Watanabe et al., we used the Beeler-Reuter (BR) ion channel model.6 All simulations are performed using the CARDIOWAVE software package. For faster computation, we employ a semi-implicit method with a conjugate gradient solver to integrate Equation 1. We choose step-sizes of 10 μm and 0.02 msec. APD and DI were computed as time spent above and below the membrane potential of -80 mV, respectively.
For fiber discretization, we follow the approach of Shaw and Rudy.7 Fibers are 8 cm in length and are divided into cells of length 100 μm, with each cell containing 10 nodes.
Figure 3: Fiber discretization scheme; image courtesy of Shaw and Rudy7.
Gap junctions are inserted between cells. The specific gap junction resistance is chosen to be 1.5 Ωcm2 and intracellular resistivity (ρi) is 150 Ωcm. These choices give an effective cellular resistivity of 300 Ωcm. Continuous fiber cells contain no gap junctions, and have an effective resistivity of 300 Ωcm. All cells are assumed to be cylindrical, with a radius of 11 μm.
Results and Discussion
“Ectopic Focus” scenario
Watanabe et al. proposed two mechanisms for the initiation of discordant alternans in homogenous tissue. The first mechanism involves a single ectopic beat. An ectopic beat is any beat which originates in a location other than the normal stimulus site (i.e. the sinus node). In the heart, ectopic beats are generally benign, but they can be associated with ventricular tachycardia and other arrhythmias.
To simulate an ectopic beat, we apply a stimulus to the terminal end of the fiber. The stimulus is timed carefully so that it is applied just after the terminal end has repolarized from a normal beat. This is the approach taken by Watanabe et al., and we note that it immediately gives rise to discordant APD alternans (Figure 4).
Figure 4: Genesis of discordant APD alternans via an ectopic beat. Vertical axis denotes the fiber length (8cm), where the top is the normal pacing site and the bottom is the ectopic site. Membrane potential (mV) is visualized as color. An ectopic stimulus is applied at t=500 msec, resulting in the immediate initiation of discordant APD alternans. We may interpret the APD at any point along the fiber as the horizontal width of the colored depolarization band. The dark bands are the diastolic intervals (resting periods between stimuli).
Discordant alternans is generated in the above scenario because the ectopic depolarization wavefront encounters regions of increasing DI (dark blue area) as it travels back up the cable to the stimulus site. Reflecting the length of the preceding rest interval, APD also increases up the cable. This effect is maintained (albeit in reverse orientation) in subsequent beats originating from the normal pacing site.
Given this mechanism of genesis, is it possible for a discrete cardiac fiber to display APD alternans? In essence, this scenario uses a carefully timed ectopic beat to take advantage of an existing geometric distribution of diastolic intervals. Since there is nothing about this DI distribution that is unique to a continuous cable, we suspect that an ectopic focus should be sufficient to generate APD alternans in a discrete fiber.
Indeed, simulations confirm that it is possible to use an ectopic focus to sustain APD alternans in a discrete fiber. While a discrete fiber requires a slight modification of the stimulus strength and timing, the ectopic focus method is still successful (we omit a visual presentation of this result as it is qualitatively equivalent to that in Figure 4). Again, it makes sense that a discrete fiber should behave like a continuous fiber in this instance. Since the requirement for alternans in this scenario is a geometric distribution of diastolic intervals, fiber discretization is not a critical issue.
“Sinus Node” scenario
The second mechanism proposed by Watanabe et al. involves a complex interaction of action potential duration and conduction velocity. Although this second scenario initiates discordant alternans less rapidly than does the first mechanism, it is unique in that it occurs spontaneously. Furthermore, this “sinus node” mechanism involves a shift from concordant alternans (which are spatially in phase) to discordant alternans (out of phase). Such transitions reflect a state of increased dispersion and refractoriness, which predisposes the heart to reentry and fibrillation.
The sinus node scenario is illustrated for a continuous fiber in Figure 5. Stimuli are applied at one end of the cardiac fiber every 310 msec. Concordant APD alternans appears immediately, reflecting a regular variation in the diastolic interval. Reduction in DI not only affects APD, but also conduction velocity. As the DI preceding the second beat is reduced, conduction down the fiber is slowed. Slow conduction means that the depolarization wavefront encounters increasing DI as it travels down the cable. In other words, the delayed arrival of the wavefront gives the terminal end of the cable time to recover from the previous beat. Ultimately, this process is manifested as an increase in APD near the terminal end of the fiber (Figure 5, upper panel).
With time, the increased APD at the terminal end of the cable is magnified. Eventually, the conduction velocity effect gives rise to a reversal of APD spatial distribution. At this point, the two ends of the cable still alternate in APD, but they are out of phase (Figure 5, lower panel). This state of discordant alternans is henceforth maintained (barring an ectopic stimulus or a pacing irregularity).
Figure 5: Sinus node scenario illustrates the concordant to discordant alternans transition. Upper panel: regular, rapid pacing produces spatially concordant APD alternans. Reduced conduction velocity leads to variation in APD at the terminal fiber end, which ultimately results in a reversal of spatial distribution (lower panel). Both panels show membrane potential (mV) in color vs. position and time.
For discordant alternans, APD varies continuously down the fiber. Given that there is a reversal of spatial distribution between beats, the intermediate value theorem requires that, between any two beats, there exists a position at which APD does not change. We refer to this position as a “node.” From Figure 5, we see that a node forms near the terminal end of the fiber and then moves toward the stimulus site. The formation of a node marks the onset of discordant alternans.
Figure 6: Formation and movement of a node. During a period of concordant alternans (upper panel), there is no node. With time, a node appears at the terminal end of the fiber and moves toward the stimulus (middle and lower panels), eventually halting at a steady-state position. In an 8 cm fiber, the steady-state position is approximately 2 cm from the stimulus site.
Next, we characterize the behavior of a discrete fiber in response to the pacing protocol described above. Unlike the ectopic focus scenario, the sinus node mechanism is not dependent on a geometric distribution of diastolic intervals. Thus, it is not initially clear how the APD spatial distribution evolves in a fiber that is discretized.
When simulations are performed, however, we find that discordant alternans is induced in a long discrete fiber at high pacing rates. Thus, the alternans observed in the continuous cable is preserved in a medium designed to be more reflective of the actual structure of cardiac tissue.
Figure 7: Discordant alternans genesis in a discrete fiber. The leftmost pair of action potentials represent beats 9 and 10 after the initiation of rapid pacing. A node develops near the terminal end of the fiber (lower end) and travels up toward pacing site. This represents close qualitative agreement with the scenario seen in the continuous fiber.
Despite the qualitative agreement in the results for a continuous and discrete fiber, we note a difference between these two cases. In the discrete fiber, node formation takes longer than it does in a continuous fiber of equivalent resistivity.
We visualize this delay in Figure 8. The upper two panels represent a discrete fiber and the lower two represent an equivalent continuous fiber. Spatial APD distributions reveal that the APD node in the discrete fiber forms later and lags slightly behind the node of its continuous counterpart.
Figure 8: Comparison of node formation and movement in discrete and continuous fibers. Upper two panels represent the discrete fiber while the lower two represent the continuous fiber. Second and fourth plots show corresponding APD distributions for beats 29 and 30. The node in the discrete fiber is formed later and lags behind.
What could account for the observed delay? Watanabe et al. report that the velocity restitution function is critical to the sinus node mechanism. Velocity restitution refers to the dependence of conduction velocity (θ) on the length of the previous diastolic interval (DI). This relation is typically modeled with an equation of the form:
( 2 )
Yet, when restitution data are gathered for the continuous and discrete fibers, the fit parameters for Equation 2 (i.e. θmax, k, and τ) are nearly identical (<1% difference). Clearly, there must be another factor contributing to the observed delay. Perhaps the discrete fiber’s physiological role is that of a safety mechanism which acts to prevent alternans initiation. While this may seem an attractive hypothesis, such a suggestion is not warranted by the data here. Furthermore, the delays observed never exceed 2 beats. For a large simulation, numerical approximations could drive a node initiation from one set of beats to the next. Also, we observe that the discrete node travels along the fiber just as quickly as does the continuous node. Indeed, while the initial delay might not be significant, future work may help to explain why it is not captured by the restitution function.
The results of our analysis are significant because they establish that a phenomenon observed in a homogenous preparation is preserved in a realistic tissue formulation. Although action potential propagation is fundamentally different in continuous and discrete media, the mechanisms proposed by Watanabe et al. for discordant alternans genesis are confirmed in both cases. Furthermore, the requirements for discordant alternans initiation in the two mechanisms are not extensive. The ectopic focus scenario entails only a single ectopic stimulus, while the sinus node scenario requires a medium of sufficient size (this could perhaps be reduced by increasing fiber resistivity to slow conduction). Both mechanisms, of course, require a rapid pacing rate. While these mechanisms may be relatively straightforward, they represent an important step towards characterizing the role of restitution in discordant alternans genesis.
Conclusions
The confirmation of the ectopic focus and sinus node mechanisms in a substrate formulated to mimic the behavior of an actual cardiac cable lends support to the theory of Watanabe et al. Our simulations provide evidence that the genesis of discordant APD alternans is neither a consequence of the homogenous tissue assumption nor an artifact of particular numerical or mathematical approaches. Further work on this topic may seek to clarify and rationalize the delays observed in the discrete fiber and to extend our analysis to a two-dimensional sheet.
References
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