# User:Flavio H Fenton/Proposed/Restitution

## Overview

Restitution essentially describes a functional relationship between the duration of a cardiac action potential and the length of the quiescent interval (diastolic interval, or DI) preceding it. In many cases, this relationship reflects limited change in action potential duration (APD) over a broad range of long DIs and shortening of the APD at shorter DIs. APD restitution is therefore a form of adaptation to changes in rate. The APD of cardiac cells when the heart is at rest can be longer than the cycle length corresponding to the fastest heart rate. If the heart did not adapt to a significant rate increase by shortening the APD, and instead produced APDs of identical duration regardless of rate, this would result in block of the heart beat and skipped contractions when the heart beat extremely rapidly, such as during intense physical activity or fright. Instead, the heart’s ability to adapt to a faster rate by shortening both systole and diastole allows adequate contraction and pumping of blood under a wide range of physiological conditions. Mechanistically, restitution occurs because at fast rates not all ion channels are able to recover from inactivation fully before the next action potential occurs, resulting in decreased current.

An APD restitution curve is constructed by varying the DI and plotting the resulting APD. To first order, it reflects the underlying dynamics of the system during steady-state conditions. Nolasco and Dahlen (1968) were the first to describe APD restitution using this type of map model and showed that when the restitution curve slope was greater than one, oscillations in APD called alternans would result. During alternans, although the pacing cycle length remains constant, action potentials alternate in duration between long and short. Using the fact that the cycle length is equal to the sum of APD and DI, it is easy to see that for a constant cycle length during alternans, each long APD is followed by a short DI, which in turn is followed by a short APD and a long DI. Thus, the presence of a bifurcation in a plot of APD as a function of cycle length is directly correlated with a steep slope (greater than or equal to one) in the corresponding area of the restitution curve, where APD is plotted as a function of DI. In other words, for steep restitution slopes, small changes in DI are amplified into larger changes in APD, whereas for restitution slope values below one, small changes in DI are damped out over subsequent beats (Guevara et al., 1984).

Steep APD restitution curve slopes not only can give rise to alternans, but also can lead to breakup of spiral waves in tissue (Karma 1993; Karma 1994; Fenton et al., 2002). Indeed, a number of studies have shown that some pharmacological agents whose effects include a flattening of the APD restitution curve are associated with prevention or termination of fibrillation (Riccio et al., 1999; Garfinkel et al., 2000). However, a number of other studies (Echebarria and Karma, 2002; Tolkacheva et al., 2003; Cherry and Fenton, 2004; Cherry and Fenton, 2007) have shown that APD restitution curve slopes are only one factor contributing to the development of alternans and fibrillation, and that other properties, including cardiac memory and electrotonic effects, can suppress or enhance the induction of alternans.

Along with APD, other quantities adapt to changes in rate. For instance, action potential amplitude often diminishes, along with APD, at rapid heart rates. The resting membrane potential can increase as well during rapid pacing. In tissue, the conduction velocity of propagating waves also depends on rate and, in most cases, slows as the rate increases.

## Measuring restitution

In practice, measuring APD restitution is not completely straightforward. First, it requires a determination of the action potential duration. APD often is computed by recording the difference between the time of the upstroke and the time when repolarization to a certain percentage has occurred. Thus a voltage threshold corresponding to 10 percent of the action potential amplitude would correspond to a repolarization threshold of 90 percent and would give the value $$\rm APD_{90}\ .$$ It is possible to compute the voltage threshold corresponding to $$\rm APD_{90}\ ,$$ for instance, and to use that threshold throughout the series of action potentials being analyzed. However, increases in resting membrane potential and decreases in action potential amplitude at fast rates may make this method ineffective. In particular, it is possible for the resting membrane potential to increase above the threshold being used, even when action potential can still be produced. This difficulty can be avoided to some degree by using a threshold that varies with the amplitude and resting membrane potential of each action potential. However, even this technique is difficult to use during arrhythmias like fibrillation, in which the baseline drifts substantially and differentiating between action potentials and subthreshold responses may not always be feasible.

Second, different restitution protocols can be used, and different protocols in general produce different curves. This protocol dependence arises because in fact APD is not solely a function of the preceding DI, but instead is a weighted function of all preceding APDs and DIs. Two different protocols are commonly used. In the dynamic (or steady-state) restitution protocol (Koller et al., 1998), the preparation is paced at a given cycle length until steady-state is reached, and the APD and preceding DI (or, during alternans, two APDs with their preceding DIs) are recorded. Then the process is repeated with other cycle lengths. In the S1-S2 restitution protocol, the preparation is paced at a fixed cycle length S1 until steady-state is reached and is then perturbed by a stimulus (S2) after waiting a variable-length interval. The preparation is then paced at S1 until steady-state has been reached again and is then perturbed by a different S2. In this way, the DIs and APDs determined by the application of S2 are recorded as the restitution curve. The curve obtained using the S1-S2 protocol depends on the choice of the S1 cycle length, and therefore a family of curves over a range of S1 values can be obtained and can be used to quantify short-term memory (Cherry and Fenton, 2007).

An additional restitution protocol that has been used to obtain restitution information is the constant cycle length (constant-CL) curve. In this case, APDs and DIs are obtained and plotted over the time between a cycle length change and the achievement of steady state. While the dynamic and S1-S2 restitution curves usually have positive slopes in normal cardiac tissue, constant-CL restitution curves generally have negative slopes when measuring the trend toward steady-state following a change in cycle length. The restitution portrait (Kalb et al., 2004) provides a protocol that can be used to determine the dynamic restitution curve along with portions of S1-S2 and constant-CL restitution curves. In particular, constant-CL curves are measured both during the accommodation to new cycle lengths during the dynamic pacedown protocol and during the return to steady state following positive and negative perturbations to the cycle length that allow measurement of a portion of a local S1-S2 restitution curve.

The existence of many different protocols that can be used to obtain restitution curves leads to the question of which is the most relevant for predicting alternans and fibrillation. Indeed, a number of studies have shown cases where either a restitution curve slope is greater than one but alternans or fibrillation do not occur or a restitution curve slope is less than one but alternans or fibrillation nevertheless occur. Use of the restitution portrait also shows that none of the four different protocols tested provided a useful predictor of alternans in bullfrog myocardium (Kalb et al., 2004).

Some models of cardiac cells do not include this complicated dependence or have removed it explicitly so that APD is solely a function of the preceding DI, and in this case the restitution protocol used does not matter.

Measuring CV restitution has similar difficulties because DI still must be measured, and because dependencies on protocol have been observed for several models of cardiac tissue. Furthermore, measuring CV has additional difficulties. In anisotropic tissue, the longitudinal and transverse propagation directions may not be easy to determine, especially since fibers can curve and propagation may not occur along straight lines. Furthermore, it is necessary to compute velocity based on the difference in arrival time of a stimulus at sites a certain distance apart, while in reality, especially during discordant alternans, CV varies spatially. Hence, the measured velocity is actually the average of CV over the distance between the two sites, and variations in CV within that distance may be lost. It is also undesirable experimentally to measure CV between two very close sites because the short distance results in a small time difference that often has a larger experimental uncertainty, leading to a greater uncertainty in the CV value computed.

## Concordant and discordant alternans

When alternans occurs in tissue, different spatial patterns can develop. In the simplest case, the entire tissue experiences a long or short action potential together. However, a number of factors can promote the formation of more complicated spatial dynamics, when different regions of the tissue respond differently. The APD may vary spatially, but as long as the entire tissue still alternates long-short together, the tissue is said to exhibit spatially concordant alternans. When the tissue begins to alternate long-short in some areas but short-long in other areas, spatially discordant alternans is present. It is possible to determine whether alternans is concordant or discordant by plotting two successive APDs at all points in the tissue. In a one-dimensional cable, for instance, such a plot will show two non-intersecting lines of APD if the alternans is concordant, but will have one or more intersections during discordant alternans. The points of intersection are called nodes (or nodal lines or surfaces in two or three dimensions) and represent locations where the APD is constant from beat to beat. Nodes may be stationary or may travel, generally toward the site of stimulation. The number of nodes present can be used to further characterize and distinguish different states of discordant alternans.

Discordant alternans requires the presence of heterogeneity to occur. A static heterogeneity, such as a suitable voltage gradient, may be sufficient to induce discordance (Watanabe et al., 2001). However, in many cases a dynamic heterogeneity arises as a result of spatial variations in conduction velocity that occur because of conduction velocity restitution (Watanabe et al., 2001). Without such heterogeneity, alternans would always be spatially concordant. In many cases both concordant and discordant alternans can arise in the same type of preparation. Several factors facilitate the induction of discordant alternans, including increased tissue size, decreased pacing period, and an extremely steep conduction velocity restitution curve.

## Spiral wave breakup from steep APD restitution

Steep APD restitution curves can cause breakup of induced spiral waves. This breakup can occur either close to the tip and generally within the first few rotations of the spiral wave, or farther from the tip and often requiring a longer time to develop (Fenton et al., 2002). It is necessary in either case for the period of the spiral wave to fall within the region of steep APD restitution. In addition, as discussed later, steep APD restitution is neither necessary nor sufficient to produce spiral wave breakup.

### Breakup close to the tip

Spiral wave breakup close to the tip occurs primarily because of variations in velocity between the front and back of a wave (Courtemanche, 1996): the steepness of the restitution curve may reduce the propagation speed of the wave back. Essentially, a wavefront following closely behind an earlier wave may be forced to slow down. If the medium ahead of the front begins to recover more quickly, the wave may continue to propagate. If not, the wave front will collide with the back of the previous wave an will be annihilated. When this phenomenon occurs in the context of a spiral wave, localized block can produce wave break. Because this is a property inherent to the tissue dynamics, subsequent blocks and wave breaks are likely to occur as long as wave fronts are present, unless the spiral wave period is outside of the area of steep APD restitution.

### Breakup far from the tip

When the restitution curve is not so steep or the minimum DI before block occurs is very small, immediate breakup will not occur and instead discordant alternans can begin to develop. Because of spatial variations in APD and period, block may occur (Fox et al., 2002) in some region and produce wave breaks (Karma, 1993). Note that because increased size facilitates the induction of discordant alternans, a spiral wave initiated in a smaller domain may remain stable (Karma, 1994).

## Factors affecting the relation between restitution and alternans

The original analysis of Nolasco and Dahlen (1968) was performed using a one-dimensional map, and therefore is valid only under the assumption that each APD is determined solely by the previous DI. In practice, this assumption does not hold for cardiac cells and tissue. As a result, factors including electrotonic effects, cardiac memory, and conduction velocity restitution can facilitate or suppress alternans development. In addition, other mechanisms besides steep APD restitution may give rise to alternans.

### Electrotonic effects

In tissue, electrotonic effects can influence the magnitude of or even suppress alternans (as a function of conduction velocity restitution). In this case, the shape of the action potential plays an important role, as the diffusive electrotonic current depends on the voltage difference between adjacent sites. When the wave back is steep, it produces larger electrotonic currents during repolarization than a triangular action potential of similar duration (Cytrynbaum and Keener, 2002). These currents can have a stabilizing effect and can reduce the magnitude of alternans or suppress it entirely in tissue, even when isolated cells (and any sites in tissue that are stimulated directly) exhibit alternans (Cherry and Fenton, 2004). Action potential shape and the resultant electrotonic currents may provide an explanation for the results of Hall et al. (1999) and Banville and Gray (2002), in which alternans did not occur even when restitution curves with slopes greater than one were measured. In these preparations, action potentials repolarized quickly and had shapes that were more square than triangular.

Because it is possible for electrotonic effects in tissue to suppress alternans, alternans in a single cell is not a good predictor of alternans in tissue.

### Memory effects

Cardiac tissue does not depend exclusively on the previous DI but instead incorporates much of its short-term pacing history in determining the next APD, a property called cardiac memory. As a result it is possible that the steady-state or dynamic restitution curve, which has been suggested as the most relevant curve in determining the onset of alternans (Koller et al., 1998), will have a slope greater than one while other S1-S2 restitution curves may have shallower slopes. In this case, it is possible for alternans not to occur even in an isolated cell despite a steep APD restitution curve (Cherry and Fenton, 2004). A combination of restitution curve slopes can be used instead of the simple slope-greater-than-one condition to determine a history-dependent criterion for the onset of alternans (Tolkacheva et al., 2003), but even this improved criterion is not always accurate (Kalb et al., 2004).

Depending on the conduction velocity restitution curve, when the effects of memory are linked with electrotonic effects in tissue, it is possible for alternans to occur again (Cherry and Fenton, 2004). Because it is possible for alternans to occur in tissue even when it does not occur in a single cell, the absence of alternans in a single cell is not a good predictor of the absence of alternans in tissue.

### Conduction velocity restitution effects

While conduction velocity restitution is necessary for the induction of discordant alternans, the shape of the conduction velocity restitution curve is also important in determining how easily alternans can arise. Specifically, for a given size tissue, the likelihood of discordant alternans formation is inversely proportional to the quantity $$c^\prime/c^2\ ,$$ where $$c$$ is the conduction velocity and $$c^\prime$$ is the slope of the conduction velocity restitution curve (Echebarria and Karma, 2002). In other words, discordant alternans forms more easily when the conduction velocity restitution curve is extremely steeply sloped (and therefore flatter over a large range of long DIs).

Conversely, the alternans-suppressing electrotonic effects have the opposite dependence on $$c^\prime/c^2,$$ and are more likely to occur when the conduction velocity restitution curve has a more gradual slope over a large range of DIs (Cherry and Fenton, 2004). Therefore, interventions designed to prevent alternans by modifying conduction velocity restitution must consider that increasing the potential for alternans suppression also may decrease the minimum tissue size required for discordant alternans.

### Other causes of alternans

While the intricacies of alternans are discussed elsewhere, it is important to note that steep APD restitution is not the only mechanism that can give rise to APD alternans. Alternans in intracellular calcium have been shown to arise independently of voltage alternans but can in turn cause voltage alternans to occur (Chudin et al., 1999). In the context of regional ischemia, alternans may arise by purely electrotonic effects along the ischemic border zone (Bernus et al., 2005). Finally, splitting of the APD restitution curve into two branches can produce alternans even when the slopes of the two branches are both less than one (Cherry and Fenton, 2007).

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