# Incidence, Evolution, and Spatial Distribution of Functional Reentry During Ventricular Fibrillation in Pigs

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## Abstract

*Abstract*—Functional reentry has been hypothesized to be an underlying mechanism of ventricular fibrillation (VF); however, its contribution to activation patterns during fully developed VF is unclear. We applied new quantitative pattern analysis techniques to mapping data acquired from a 21×24 unipolar electrode array (2-mm spacing) located on the ventricular epicardium of 7 open-chest, unsupported pigs. Data epochs 4 seconds long beginning 1, 10, 20, 30, and 40 seconds after electrical induction were analyzed. Reentrant circuits were automatically identified and quantified. We found that 2.3% of activation pathways could unambiguously be classified as reentrant. From scaling analysis, an additional 28% of the pathways may also have been reentrant. Reentry was short-lived with 1.5±1.5 (mean±SD) complete cycles per circuit. The fraction of reentrant pathways, number of cycles per circuit, cycle duration, and area and perimeter of the cores all increased significantly as VF progressed. Core drift speed decreased significantly. Neither the orientation of the cores nor the direction of drift was well predicted by the epicardial fiber orientation (*r*^{2}=0.108 and 0.138, respectively, by linear regression). Reentrant circuits were clustered in regions of the epicardium. We conclude the following: (1) Epicardial reentry is relatively uncommon and short-lived during VF, suggesting either that sustained reentry is transmural or that mechanisms governing sustained reentry are relatively unimportant to the dynamics of VF. (2) Reentrant circuits become more common, larger, and longer-lived as VF progresses, which may explain a recently observed increase in VF organization during the first minute of VF. (3) The conditions necessary to induce and sustain reentry are distributed nonuniformly.

Functional reentry occurs when some event breaks an activation wave, placing the excitation wavefront in contact with its own refractory tail.^{1} In such a situation, the wavefront pivots around the wavebreak, circling about its tail.^{1} If uninterrupted, this pattern becomes a sustained, high-frequency source of wavefronts. Functional reentry has often been hypothesized as an underlying mechanism of ventricular fibrillation (VF).^{2} ^{3} ^{4} ^{5} ^{6} Although it has been clearly demonstrated at the onset of electrically induced VF,^{7} ^{8} ^{9} the contribution of functional reentry to activation patterns as VF progresses is less clear.

Our laboratory has recently developed a new series of computational methods for quantifying activation patterns observed in high-resolution epicardial mapping data.^{10} These methods are based on wavefront isolation, in which the individual activation wavefronts making up a VF episode are isolated from one another, allowing the episode to be quantified in terms of the statistical properties of the wavefronts and their interactions. The methods are automatic, allowing large VF mapping datasets to be objectively analyzed. In the present study, we extended these methods and applied them to mapping data acquired from an electrode array covering ≈20% of the epicardium. VF was studied in normal, healthy, in situ porcine hearts.

These new methods allow us to address, in a quantitative manner, several specific issues relating to the role of reentry in the mechanisms of VF. (1) Recently, many investigators have reported the presence of reentry during developed VF in normal, healthy hearts.^{2} ^{11} ^{12} ^{13} However, the incidence of reentry, ie, the number wavefronts that are reentrant compared with the number that are not, has never been reported. We compute this parameter, which is critical in determining the importance of reentry during VF. (2) Several studies have shown that after losing organization in the transition from Wiggers’ stage I to stage II, VF recovers organization in the remainder of the first minute.^{14} ^{15} ^{16} We recently reported that this reorganization is characterized by a gradual growth of spatial patterns and proposed that this growth is due to an increase in the size of the area circumscribed by reentrant wavefronts.^{15} In the present study, we test this hypothesis. (3) The orientation of cardiac muscle fibers is an important determinant of propagation patterns. We therefore test the hypothesis that the orientation and drift direction of reentrant wavefronts correlates with epicardial fiber orientation. (4) VF is widely believed to be perpetuated by a nonuniform dispersion of refractoriness.^{17} If this is true, we would expect reentry to occur in preferential locations. We therefore test whether reentry is uniformly distributed during VF.

## Materials and Methods

The use of experimental animals in the present study was approved by the Institutional Animal Care and Use Committee at the University of Alabama at Birmingham. All studies were performed in accordance with the guidelines established in the Position Statement of the American Heart Association on Research Animal Use adopted by the American Heart Association on November 11, 1984.

### Animal Preparation

Seven 32- to 43-kg pigs (Snyder Farms, Cullman, Ala) were studied; the hearts weighed 159±38 g. Detailed methods of animal preparation and instrumentation were described in our previous study.^{15} The data analyzed in the present study include data from the previous work. Briefly, the animals were anesthetized with sodium pentobarbital, intubated, and ventilated with a mixture of room air and oxygen. The heart was exposed through a median sternotomy and instrumented with a 504-electrode (21×24) plaque sutured to the anterior lateral right ventricular and medial left ventricular epicardium. The electrodes were 1-mm-diameter silver spheres with 2-mm spacing (on centers) in each direction. This spatial resolution has been shown to be appropriate for mapping VF.^{18} The plaque covered 20.16 cm^{2}, which is ≈20% of the epicardial surface. The ground reference for the unipolar recordings was attached to the right leg. Two stainless steel wires were sutured to the lateral left ventricle at least 2 cm from the plaque for use as bipolar pacing electrodes. A mesh electrode was sutured to the left ventricular apex, and a catheter electrode was inserted into the superior vena cava for use as rescue defibrillation electrodes.

### Data Acquisition

The 504 plaque electrodes were connected to a 528-channel mapping system.^{19} The unipolar electrograms were bandpass-filtered from 0.5 to 500 Hz, sampled at 2 kHz, and recorded to videotape with 14-bit resolution. Data were recorded continuously during S1–S2 stimulation and VF. From this datastream, permanent records 4 to 5 seconds in duration beginning 0, 10, 20, 30, and 40 seconds after induction were saved.

VF was induced using the programmed pacing protocol described previously.^{15} Cardiac perfusion was not maintained during VF. Six VF episodes were induced in each animal. A rescue biphasic shock at the minimum reliable defibrillation strength (typically 400 to 500 V) was delivered 45 seconds after each induction. A minimum of 15 minutes was allowed to elapse before VF induction was again attempted. On completion of the protocol, the pig was killed by electrically induced VF. The location of the recording plaque was marked, and the heart was excised, weighed, and fixed in 10% formalin.

### Determination of Epicardial Fiber Orientation

In 6 hearts, the heart wall under the plaque was cut out and the epicardium was dissected away using forceps. Under a dissecting microscope, suture material was inserted in the myocardium in alignment with the fibers at 16 sites (4×4 grid). The tissue was then placed under a video camera connected to an image processing system (Universal Imaging Corp). The fiber orientation at the 16 gridpoints, the coordinates of the gridpoints, and the coordinates of the 4 corners of the array were acquired. The fiber orientation at any point was determined from linear interpolation of these data.

### Quantitative Analysis of VF Activation Patterns

In each episode, 4-second epochs of VF mapping data beginning 1, 10, 20, 30, and 40 seconds after induction were extracted from the permanent recordings and transferred to a high-performance workstation (Silicon Graphics Inc) for analysis. Four to six episodes in each animal were successfully recorded and transferred. In total, 179 VF epochs (716 seconds) of VF were analyzed. To verify the integrity of the data before quantitative analysis, the activation patterns were visualized using animated maps of the first temporal derivatives of the 504 unipolar electrograms.

#### Wavefront Isolation

The basis for the analysis is the decomposition of the overall activation pattern into its constitutive units. This decomposition is described in detail elsewhere.^{10} Briefly, all 504 electrograms were differentiated using a 5-point digital filter. The discrete samples of the differentiated signals (one for each electrode every 0.5 ms) were stored in a 3-dimensional array, 2 of the dimensions corresponding to the dimensions of the plaque and the third to time. As in our previous studies,^{10} ^{15} Active samples were defined as those for which *dV/dt*<−0.5 V/s.

Individual wavefronts in the pattern were identified and isolated by grouping together active samples that were adjacent in space and time. A specially developed spatiotemporal filter that allows small discontinuities in wavefronts was then applied.^{10} ^{15} In this decomposition of the overall activation pattern, wavefronts, by definition, end after interaction with other wavefronts. For example, when 2 or more wavefronts collide and coalesce, the original wavefronts end at the time of the collision, and a new wavefront results. Conversely, when a wavefront fractionates into 2 or more parts, the original wavefront terminates, and 2 or more new wavefronts result. In this context, the timing of the wavefronts and their interactions can be summarized as a wavefront graph. Here, the word “graph” describes a mathematical construct built of line segments, or edges, connected at nodes. In a wavefront graph, each edge represents a wavefront. The horizontal coordinates of the 2 endpoints locate the wavefront in time. The nodes of the graph represent fractionations and collisions, which we collectively call contacts, because in the wavefront graph representation, wavefronts touch at these times. An example of a wavefront graph derived from 0.5 second of VF is shown in Figure 1⇓.

Because of the frequent contact events during VF, wavefronts typically have short durations, on the order of tens of milliseconds. However, if a wavefront terminates with a contact, the propagating activity associated with the wavefront does not end; it continues with the contact’s resultant wavefront(s). Tracing such a stream of propagating activity from its appearance on the array through a series of wavefronts and contacts to its disappearance might seem a more intuitive definition of a “wavefront.” However, because of the branching nature of the wavefront graphs (eg, Figure 1⇑), any particular wavefront might be part of many such streams that connect the appearance of activity with its disappearance. To contend with this ambiguity, we introduce a higher level of structure apparent in the wavefront graphs: families of wavefronts interrelated by contact events form subgraphs that are disconnected from the remainder of the graph. In graph theoretical terminology, these subgraphs are called “components,” and algorithms exist for automatically identifying them.^{20} Each component contains one or more routes, each of which is a set of one or more wavefronts that together connect the appearance of a stream of activity with its disappearance. In Figure 1⇑, there are 6 components, one of which is outlined by a dashed box. A single route through this component is shown by the large arrows. In the present study, we will use the number of components in a VF episode as a measure of the number of propagating “units.”

#### Initial Reentry Detection

After constructing the wavefront graph and identifying its components for a VF episode, we identified which components were reentrant with a 2-step method. In the first step, all routes through a component were processed in turn. For each route, we counted the number of electrodes that were activated by the wavefronts in the route. Next, we counted the number of activations associated with the route. We defined an activation as *dV/dt* at an electrode crossing the activation threshold at least 40 ms after any previous activation at that electrode. The 40-ms cutoff was chosen to be slightly shorter than the shortest refractory period observed during VF by Cha et al.^{7} For nonreentrant routes, the ratio of the number of activations to the number of electrodes activated was 1.0, indicating that no electrodes were activated more than once by the same route. Following Bollacker et al,^{21} we defined potentially reentrant routes as those for which this ratio was at least 1.1. Wavefront graph components were considered potentially reentrant if they contained at least one reentrant route. If a component had multiple reentrant routes, only the route with the largest ratio was used in the analyses that follow.

#### Reentry Confirmation and the Wavetip Path

Confirmation of the presence of reentry and our subsequent analyses use the concept of the wavetip, ie, the broken end of a reentrant wave. The path traced out by the wavetip defines many of the wave’s interesting dynamic properties. In the event that the wavetip path follows a fixed course from cycle to cycle, the region enclosed by the path, which is excitable but not excited by the wave, is known as the “core.”^{1} In computer modeling studies of functionally reentrant waves, the wavetip is typically defined as the crossing point of the contours of 2 variables, one determining the state of excitation and the other the state of recovery.^{22} In experimental preparations of cardiac tissue, the latter variable is generally not available, and so the wavetip path has been estimated by inspection of isochronal contour maps,^{23} analyzing frame stack plots of optically recorded transmembrane potential data,^{6} ^{24} manually locating the wavetip in animated displays of activation times picked from electrical mapping data,^{2} or by identifying phase singularities.^{12}

We devised a new method for accurately and automatically finding the wavetip path associated with a reentrant wavefront graph component. We define the wavetip path as the shortest possible path connecting active samples in each timestep of a reentrant component. Only active samples on the outer layer of the wavefronts in the component (ie, active samples that have at least one nonactive neighbor) may lie on the wavetip path. This idea is illustrated in Figure 2A⇓. Each block represents an active sample in a counterclockwise reentrant component. Each gray level represents a different timestep (note that only 4 timesteps of the entire reentrant component are shown). The shortest path running through a sample in each timestep is indicated by the circles and clearly tracks the pivoting end of the wave. Note that the hatched sample in the second timestep of Figure 2A⇓ is not on the outer layer of the component and hence cannot be on the wavetip path.

A network optimization algorithm was used to find wavetip paths (Figure 2B⇑ and 2C⇑). First, a directed graph was constructed in which a node represented every eligible active sample in the most-reentrant route (ie, the route with the largest ratio of activations to activated electrodes) through the wavefront graph component. The edges of this graph connected each active sample with all of the active samples in the succeeding timestep. Each edge was then weighted with the distance between the electrodes associated with the active samples connected by the edge. The graph therefore defined all possible paths that contain one active sample in each timestep. The length of any path was simply the sum of the weights of the edges along the path. A hypothetical wavefront containing 6 active samples in 3 timesteps is illustrated in Figure 2B⇑. The graph and associated edge weights constructed from this wavefront are shown in Figure 2C⇑.

The shortest path through such a graph is our desired wavetip path. To find it, we used a well-known algorithm from graph theory, Dijkstra’s shortest path algorithm.^{25} Given a starting node, Dijkstra’s algorithm finds the shortest path to the remaining nodes in the graph. We therefore ran the algorithm for each active sample in the first timestep, each time setting the active sample as the start node. The shortest path from a start node to an active sample in the last timestep, which also crossed itself at least once, was taken as the wavetip path. If none of the paths crossed themselves, then the component was deemed nonreentrant.

If there are multiple paths with the same length, then Dijkstra’s algorithm selects as the shortest the one that contains the fewest active samples. If there is still a tie, then the algorithm selects the first such it encounters. This order is dictated by the internal numbering of the active samples and so is effectively arbitrary.

Figure 2D⇑ illustrates an example of a wavetip path computed by this method. Each snapshot shows the active samples associated with a single reentrant component during VF (all other active samples were removed from the figure for clarity). The black sample in each frame is the current location of the wavetip, and the fine black line traces its path. This particular component completed one full cycle.

It is well-known that double potentials are often found adjacent to lines of block, the first deflection registering local activation and the second an electrotonic response to a wavefront on the opposite side of the line of block.^{26} If a wavetip were to follow a U-shaped course around a line of block, yet fail to reenter, the second deflection of a double potential could be misinterpreted as a second local activation, thus causing a complete reentrant circuit to be incorrectly registered. Our algorithms minimize this possibility in 2 ways. First, we require at least 10% of electrodes to be reactivated before reentry is recognized, and second, reentry is not confirmed unless the wavetip path crosses itself. These conditions help prevent sporadic activations incorrectly registered on the “wrong” side of a line of block from being interpreted as the completion of reentry.

#### Wavetip Path Loops

We next analyzed each wavetip path to identify closed loops. Figure 3A⇓ shows an example. We will refer to these loops as cycles, because for a stationary reentrant circuit, each such loop corresponds to a single traversal of the circuit. The cycle identification process began by searching pairs of segments in the wavetip path for intersections. When intersections were found, the involved segments were split, if necessary, so that the intersections only occurred at segment endpoints. This operation was implemented with an efficient line-sweep algorithm.^{27} Because of finite spatial and temporal resolution and the relatively slow speed of the wavetip, the wavetip was often located at the same electrode site for several timesteps. Thus, the time required to traverse a segment of the wavetip path was often greater than the timestep (0.5 ms). If a segment was split at an intersection, the durations of the 2 new segments were fractions of the original duration in proportion to their spatial length.

Next, the wavetip path was traversed, one segment at a time, keeping track of which segment endpoints had been visited. When an endpoint was visited for the second time, the segments intervening the first and second visits were identified as members of a cycle (dashed line in Figure 3A⇑). An ambiguity arises for reentrant circuits with more than one cycle. Consider the wavetip path in Figure 3B⇑. There are clearly 2 cycles, yet because the wavetip does not follow a repeating course, it is not immediately obvious where each cycle begins and ends. To contend with this ambiguity, we allow cycles to nest, that is, an inner cycle (eg, the bold line in Figure 3B⇑) can begin and end after an outer cycle (eg, the dashed line in Figure 3B⇑) has begun, but before it has ended. Unlimited levels of nesting are allowed. This definition always yields measurable cycles, even for complex wavetip paths with multiple overlapping cycles of differing sizes.

#### Core Excitability

To determine whether reentry was functional or anatomical, we identified the electrodes that were encircled by each cycle and checked to see if they were ever activated either in the same VF dataset or in other datasets from the same animal. Unactivated electrodes would suggest an anatomical obstacle in the interior of the circuit.

#### Cycle Quantification

Once the segments of the cycles were identified, the perimeters and durations of the cycles were calculated by summing the lengths and durations of their segments. Segments belonging to nested cycles were not included in the sums for the corresponding outer cycles. The areas and centroids of the cycles were computed as well.

We computed cycle orientation and aspect ratio by forming an inertia tensor: where *x*_{i} and *y*_{i} are the coordinates of the *i*^{th} wavetip location relative to the centroid of the cycle. The eigenvectors of this tensor define the principal axes of the loop, which are the orthogonal axes about which the cycle, if cut out of a rigid material, would spin without wobbling.^{28} For example, if a cycle were elliptical, the principal axes would correspond to the major and minor axes. We computed the principal lengths and widths of the cycles by finding the intersections between the principal axes and the wavetip path. Aspect ratios (length:width) were derived from these principal lengths. The orientation of the principal axes of the cycle in Figure 3A⇑ is indicated by the L-shaped symbol. The centroid is at the angle of the “L.” The arms of the “L” are 33% of the respective principal lengths. The aspect ratio of this cycle is 1.51.

#### Drift Velocity

If a wavetip path completed 2 or more cycles, we estimated the average drift velocity of the core by plotting the *x*- and *y*-coordinates of the cycle centroids against the average of the cycle’s starting and ending times. Straight lines were fitted to these data by least-squares. The slopes of these 2 lines gave the average *x*- and *y*- drift velocity components from which average drift speed and direction were computed.

## Results

In the 179 VF epochs analyzed, there were a total of 19 441 components. Of these, 448 (2.3%) were reentrant. In the 448 reentrant circuits, there were a total of 681 complete cycles (1.5±1.5 cycles per circuit). Very few circuits had more than 1 or 2 cycles, but one had 18 (Figure 4⇓). We measured the duration, area, perimeter, and aspect ratio of each cycle, and for components with multiple cycles, the average core drift speed, and the fraction of time the wavetip spent between closed cycles relative to time spent in cycles. Values are presented as mean±SD for these parameters in the Table⇓.

In 651 of the cycles, all of the electrodes in the core region were activated at least once by other wavefronts in the same dataset. In the remaining 30 cycles, the 1 or 2 core electrodes that were unactivated in the same dataset were activated in other datasets recorded from the same animal. This indicates that the core regions were excitable and that all reentry was functional, although some circuits may have been anchored to heterogeneities too small to be detected with our spatial resolution (2 mm).

We tested the dependence on time of 7 parameters: the fraction of components that were reentrant, the number of cycles per reentrant circuit, the duration of the cycles, the area of the cycles, the perimeter of the cycles, the aspect ratio of the cycles, and the drift speed of the core. Because the distribution of some of these parameters was markedly nonnormal, we used the nonparametric Kruskal-Wallis *H* statistic to test for a time effect.^{29} The fraction of reentrant components, number of cycles, and area, perimeter, and duration of cycles all increased significantly between 1 and 40 seconds after induction (Figure 5A⇓ through 5E). Aspect ratio did not change (Figure 5F⇓), and drift speed decreased (Figure 5G⇓).

To determine if the reentrant cycles were aligned with the epicardial fibers, in 6 of the hearts, we estimated the fiber orientation at the centroid of each cycle. The orientation of the principal axis of the cycles was poorly predicted by the corresponding epicardial fiber orientation (Figure 6⇓). The coefficient of determination (*r*^{2} value) for a simple linear regression was 0.108, indicating that only ≈10% of the variance in cycle orientation was accounted for by a linear fit to fiber orientation. Because the orientation of the cycles becomes arbitrary as the aspect ratio approaches 1, we repeated the linear regression using (aspect ratio−1) to weight each observation. The strength of the linear association in this analysis did not change.

We also sought a linear relationship between the direction of core drift and fiber orientation. In 6 hearts, we found a reentry center for each reentrant circuit by averaging the cycle centroids. We then determined the fiber orientation at these sites. If necessary, the direction of drift was reversed so that the difference in orientation was always less than 90°. The relationship between the 2 variables was weak, with *r*^{2}=0.138 by simple linear regression (Figure 7⇓).

To determine if reentry occurs in preferred locations, we plotted the spatial locations of the reentry centers found above for each heart individually and for all 7 hearts together (Figure 8⇓). In each heart, the reentry centers tended to cluster in specific regions (Figure 8A⇓ through 8G), although these regions did not appear to be consistent from heart to heart (Figure 8H⇓). To rigorously test if the reentry centers differed from a uniform random distribution, we implemented a 2-dimensional Kolmogorov-Smirnov (KS) goodness-of-fit test.^{30} This test determines, within specified confidence bounds, whether a sample of points was drawn from a uniform probability density function. Given that reentry centers were unlikely to fall near the boundary of the array, because of the finite radius of the cycles, to avoid biasing the KS test, we computed the mean of all principal axis lengths (both major and minor) and offset the KS analysis window by half of this length (5.2 mm) from all boundaries. The resulting window is shown by the dashed box in Figure 8H⇓. In all 8 cases, the distribution of reentry centers differed significantly (*P*<0.05) from a uniform random distribution. The correlation coefficient of the distribution of all reentry centers (Figure 8H⇓) is −0.21 (*P*<0.0001).

## Discussion

### Major Findings

The major findings of the present study are as follows. (1) Complete reentrant circuits are present on the epicardium during VF but are relatively uncommon and most persist for only 1 or 2 cycles. (2) The characteristics of the reentrant circuits change with time, with the circuits becoming more common, larger, longer-lived, and more slowly drifting as VF progresses. (3) The epicardial fiber orientation is only weakly correlated with the principal axes of the reentrant pathways and the direction in which the centers of the reentrant pathways drift. (4) The reentrant pathways are not uniformly distributed spatially; rather, they tend to cluster in specific regions that vary from heart to heart.

These findings were established using a new series of quantitative methods for analyzing cardiac mapping databased on graph theory and computational geometry. Because these methods are automatic and objective, the results are unaffected by observer bias or interobserver variability. The results can be affected by user-specified parameters; however, the sensitivity of the methods to these parameters has been characterized,^{10} and appropriate values were chosen and consistently applied.

### Limitations

#### Duration and Spacing of Data Epochs

In our previous study,^{15} we analyzed data epochs of 0.5 second in duration. This choice was dictated by our use of the “multiplicity” parameter for quantifying organization.^{16} The analyses used in the present study have no such upper limit on epoch duration; thus, dividing each VF episode into contiguous epochs would have been appropriate. However, because of technical limitations of the data archival system, the data epochs we could transfer to a computer for analysis were limited to 4 to 5 seconds. We therefore chose 4 seconds as the length of our analysis window. This temporal sampling—4 seconds of every 10—should not have affected our results because (1) 4 seconds is much longer than the durations of the reentrant circuits we found, almost all of which were well under 1 second (the longest-lived outlier persisted for about 2 seconds); and (2) our previous studies indicate that no additional trends in VF patterns are revealed by temporal resolution finer than 10 seconds.^{10} ^{15} ^{16}

#### Mapping Limitations

Many of the limitations of the present study are common to extracellular electrical mapping studies from epicardial arrays. (1) The array covered only 20% of the epicardium, and so complete activation pathways could not be determined. As we discuss in the following section, this must be considered in our estimate for the incidence of reentry. It may also have biased our estimate of the number of cycles per reentrant circuit, because the cycle count of a nonstationary circuit near the edge of the array will be erroneously low if the circuit drifts out of the mapped region. However, this is unlikely to have significantly affected our results, because so few circuits had multiple cycles (Table⇑) and the drift rate for circuits that did was relatively slow (≈4 mm per cycle). In addition, because data were recorded only from the anterior right ventricle and a small portion of the left ventricle, activation patterns in other regions of the heart could have been markedly different. (2) Because mapping was confined to the surface of the heart, intramural reentry could not be detected. The statistical properties of reentry could have been different if, for example, the mapping plane cut through the heart wall. (3) Our spatial resolution was 2 mm, which is in the upper range of that recommended by Bayly et al.^{18} Finer details of the reentrant cycles may have emerged with greater electrode density. (4) We only analyzed data up to 44 seconds after induction. Subsequent changes in activation patterns were not studied.

#### Figure⇑-of-Eight Reentry

Components arising from figure-of-eight reentry have 2 counter rotating wavetip paths. However, because our wavetip tracking method finds only the single shortest wavetip path for each reentrant component, these components could not be identified as such. Thus, if a component had 2 wavetips, only the parameters of the shortest wavetip path were included in our results. Assuming that the 2 wavetips in a figure-of-eight pattern behave similarly (eg, have similar cycle area and duration), this should not have affected our results. In particular, our results relating to the incidence of reentry are not affected, because they are based on counting reentrant components (ie, the complex of wavefronts associated with the paired wavetips) not the wavetips themselves.

#### Activation Rate and Cycle Duration

In the present study, overall activation rates, as estimated from the reciprocal of the cycle durations shown in Figure 5C⇑, are faster than the average activation rates estimated in our previous publication.^{15} We attribute this difference primarily to 2 factors. (1) When reentry is nonstationary, as it was in the present study, the duration of closed cycles underestimates the period of the repeating pattern, which should include the duration of a closed cycle plus the interval until the beginning of the next closed cycle. We found that in wavetip paths with multiple cycles, time between cycles was 17% of cycle duration (Table⇑). Wavetip paths with a single cycle typically include a significant interval exclusive of the closed cycle as well (eg, Figure 3A⇑). Therefore, as a rough estimate, activation rate at a reentrant site should be corrected upward by 17%. (2) The activation rates in the earlier study were computed using data from the entire mapped region, including regions where block was occurring, and the local rate was therefore slower. However, estimating overall activation rate from sites of reentry biases the computation to specific regions and times where activation is probably at its fastest and therefore overestimates the overall activation rate.

In Figure 5C⇑, it is apparent that a few of the cycle durations are quite short—shorter, in fact, than the 40-ms refractory period we used to detect potentially reentrant components. We attribute these short cycles (16 of 681) to failure of our algorithms to screen out false reentry due to double potentials (see Materials and Methods). Although the effect is minor, these short cycles also bias the overall activation rate upward.

### Contribution of Reentry to VF Activation Patterns

In the present study, the contribution of reentry to the overall activation pattern was quantified by the percentage of wavefront graph components that completed at least one reentrant cycle (2.3%; of all datasets, 448 of 19 441 components). This is a lower bound; many of the nonreentrant components may in fact have been due to reentrant circuits whose wavetip path was either partially or totally out of the mapped region. Using a scaling argument, we can estimate an upper bound for the percentage of epicardial wavefronts that were due to reentry. Because of the finite size of the reentrant cycles, the area sampled for reentrant circuits is not the total mapped region but is rather the area obtained by offsetting all 4 boundaries by half of the mean principal length of the cycles. This region (shown by the dotted line in the lower right panel of Figure 8⇑) has an area of 1188 mm^{2} and covers ≈12% of the epicardium (assuming a total epicardial area of 10 000 mm^{2}). Assuming a spatially uniform distribution of reentrant pathways, we therefore estimate that there were a total of 448/0.1188=3771 reentrant pathways, of which 3771−448=3323 were not located in our mapped region. Assuming further that these 3323 circuits each persisted for 1.5 cycles, and that each cycle emitted a wave that propagated into the mapped region, we estimate that 4985 of the observed components were due to reentrant circuits not fully contained within the mapped region. Thus, as many as (4985+448)/19 441=27.9% of the observed components may have been due to epicardial reentry. This estimate is an upper bound, because it is unlikely that all wavefronts generated anywhere on the heart could propagate into the mapped region. Thus, the true percentage of reentrant wavefront graph components is likely to lie between 2.3% and 27.9%.

There are 3 possible classifications for the nonreentrant components. (1) They may have been due to reentry that did not complete a full cycle. (2) They may have been due to intramural reentry. Reentry of this type cannot be detected from epicardial recordings: when a wavefront emanating from an intramural reentrant source reaches the epicardium, it spreads away in all directions and cannot be distinguished from a wavefront generated by a nonreentrant source. (3) They may have been truly nonreentrant. A wave of the third type must either extend all the way across the recording array or close on itself in an expanding ring. If it does not, then a wavebreak exists, and the wave is therefore functionally reentrant (ie, type 1 above).^{1} ^{24} ^{31} Gray et al^{12} recently reported that for every wavebreak (phase singularity in their terminology) that completed at least one full circuit, there were four that did not. This estimate implies that a large percentage—perhaps all—of wavefront graph components not identified as reentrant were in fact due to reentry of type 1 above.

### Temporal Changes in Reentry Characteristics

In our previous study,^{15} we found that the size of spatial patterns increased between 10 and 40 seconds after induction. Based on the theory of excitable media,^{31} we proposed that this change was due to growth in the size of reentrant circuits secondary to diminishing excitability and lengthening refractory periods due to metabolic changes caused by early ischemia in the unsupported fibrillating hearts. Our present data strongly support this hypothesis: between 1 and 40 seconds after induction, cycle duration, area, and perimeter increased (Figure 5C⇑ through 5E). During this same period, the stability of reentry appeared to increase, with reentrant wavefront graph components becoming more common and persisting for more cycles (Figure 5A⇑ and 5B⇑). The apparent increase in reentry incidence may have been secondary to the increase in persistence; ie, as the duration of the reentrant components increased, more components completed their first cycle and could therefore be identified as reentrant. We did not observe any changes consistent with the loss of organization usually reported for the first few seconds of VF.^{15} This was probably because our first data epoch began too late (1 second after induction) and was too long (4 seconds) to resolve these early changes.

### Cycle Aspect Ratio and Orientation

The aspect ratio of the reentrant cycles did not change with time (Figure 5F⇑), suggesting that aspect ratio was determined by some structural characteristic of the heart, such as fiber orientation, that does not vary with time. However, a correlation analysis to determine how well the orientation of the cycles was predicted by the epicardial fiber orientation found a weak relationship (Figure 6⇑) with only ≈10% of the variance in cycle orientation accounted for by the epicardial fiber orientation. This finding was surprising, because previous studies have reported that reentrant cores align well with the fiber orientation.^{2} ^{24} One explanation for this is that cycle orientation is determined by fiber orientation, but that the fiber orientations at all depths through the ventricular wall contribute, not just the epicardial orientation. Thus, reentrant circuits in the 2-dimensional preparation used by Pertsov et al^{24} would be expected to align more closely with the epicardial fibers. In the study by Lee et al,^{2} only two thirds of reentrant circuits were reported to align with fibers, and in these, the 2 angles were not quantified. Thus, more detailed analysis may have yielded results similar to ours.

### Core Drift Speed

Movement of the reentrant core has been observed both in experimental preparations and in models^{32} and has been attributed to an interplay between the excitability and refractory period of homogeneous tissue (ie, meander),^{32} gradients in electrophysiological properties,^{32} or gradients in fiber orientation.^{33} In the present study, we found that the drift speed decreased as VF progressed (Figure 5G⇑), and furthermore, that the direction of drift correlated only weakly with epicardial fiber orientation (*r*^{2}=0.138), which is consistent with previous reports.^{24} ^{34} ^{35} These data suggest that core movement was due to either meander or functional gradients that diminished as VF progressed.

### Spatial Distribution of Reentry

The centers of the reentrant circuits were not uniformly distributed over the mapped region; rather, in each heart, there were specific regions in which reentry tended to cluster (Figure 8A⇑ through 8G). This departure from a uniform random distribution was statistically significant. When the reentry centers from all hearts were plotted together, the distribution was still significantly nonuniform, although the clustering was less striking (Figure 8H⇑). Some of the variation from heart to heart may have resulted from small changes in the location of the electrode array. The correlation coefficient of the overall distribution was −0.21, indicating that reentry was most likely to be observed along a line roughly perpendicular to the left anterior descending coronary artery. These data indicate that there were specific structural or functional heterogeneities that either made reentrant wavebreaks more likely to form in particular locations, or stabilized existing wavebreaks, making them more likely to complete a full cycle. A structural feature that may be involved is the attachment of papillary muscles to the bulk myocardium. Such sites have been implicated in the formation and stabilization of reentry.^{36}

### Implications for Mechanisms of VF

The results from our present and previous studies^{15} indicate that although wavebreaks may be very common during reentry, the associated wavefronts complete a full cycle relatively infrequently, and circuits completing more than 2 cycles are even less common (Figure 4⇑). There are 2 possible interpretations of this finding. (1) If persistent reentrant circuits drive VF, then they are either transmural or located in another region of the heart. (2) Phenomena governing the initiation and termination of reentry play a larger role in the dynamics of VF in in situ hearts than phenomena, such as rapid drift,^{6} that involve the behavior of persistent reentrant circuits.

There are 2 broad classes of theory that address the dynamics of wavefronts during VF. The classical model relies on an underlying patchy heterogeneity of recovery properties.^{17} When a wavefront propagates over a region with delayed recovery, part of the wavefront may block, creating one or more wavebreaks. Uniformly decreasing excitability and lengthening mean refractory periods in such a medium make block more likely to occur and should therefore lead to a more complex and highly fractionated activation pattern.^{37} More recent proposals hold that the proliferation of wavefronts during VF occurs when reentrant wavefronts break up into multiple child reentrant wavefronts. One proposed mechanism for this phenomenon is spiral breakup, which is caused by oscillatory instabilities that occur when the slope of the tissue’s action potential duration restitution curve exceeds 1.0.^{38} ^{39} ^{40} ^{41} Other proposed mechanisms involve the development of bends in the central filament of 3-dimensional reentrant vortices. When these bends contact the tissue’s boundaries, they are annihilated, effectively cutting the filament in two.^{42} Most of these newer mechanisms share the property that decreasing the excitability of the medium stabilizes reentrant circuits and retards their proliferation,^{40} ^{42} although one proposed mechanism makes the opposite prediction.^{43} Thus, our finding that reentry stabilizes as fibrillation proceeds and the tissue becomes depressed through ischemia provides experimental support for the notion that VF is maintained by the dynamic proliferation of reentrant wavefronts.

On the other hand, the classic nonuniform dispersion of refractoriness hypothesis predicts that reentry should occur at preferred sites, whereas the newer mechanisms do not depend on patchy failure and therefore predict a uniform distribution of reentry. Thus, our finding that reentry is nonuniformly distributed would seem to contradict our conclusion of the previous paragraph. The two findings may be reconciled by considering that spiral breakup has been reported to occur only in specific tissue property ranges.^{5} Thus, spiral breakup may be responsible for wavebreak formation, but the conditions in which it occurs may be nonuniformly distributed. Likewise, the filament instability model relies on filament bends coming into contact with the heart’s boundaries. The complex shape of the heart is likely to predispose this event to occur at specific locations.

## Acknowledgments

This work was supported in part by grants from the National Institutes of Health (HL-28429 and HL-33637), the Whitaker Foundation, and the American Heart Association (9820030SE).

- Received August 31, 1998.
- Accepted February 12, 1999.

- © 1999 American Heart Association, Inc.

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- Incidence, Evolution, and Spatial Distribution of Functional Reentry During Ventricular Fibrillation in PigsJack M. Rogers, Jian Huang, William M. Smith and Raymond E. IdekerCirculation Research. 1999;84:945-954, originally published April 30, 1999https://doi.org/10.1161/01.RES.84.8.945
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- Incidence, Evolution, and Spatial Distribution of Functional Reentry During Ventricular Fibrillation in PigsJack M. Rogers, Jian Huang, William M. Smith and Raymond E. IdekerCirculation Research. 1999;84:945-954, originally published April 30, 1999https://doi.org/10.1161/01.RES.84.8.945Permalink: