Abstract
Catheter ablation strategies for treatment of cardiac arrhythmias are quite successful when targeting spatially constrained substrates. Complex, dynamic, and spatially varying substrates, however, pose a significant challenge for ablation, which delivers spatially fixed lesions. We describe tissue excitation using concepts of surface topology which provides a framework for addressing this challenge. The aim of this study was to test the efficacy of mechanism-based ablation strategies in the setting of complex dynamic substrates.
We used a computational model of propagation through electrically excitable tissue to test the effects of ablation on excitation patterns of progressively greater complexity, from fixed rotors to multi-wavelet re-entry. Our results indicate that (i) focal ablation at a spiral-wave core does not result in termination; (ii) termination requires linear lesions from the tissue edge to the spiral-wave core; (iii) meandering spiral-waves terminate upon collision with a boundary (linear lesion or tissue edge); (iv) the probability of terminating multi-wavelet re-entry is proportional to the ratio of total boundary length to tissue area; (v) the efficacy of linear lesions varies directly with the regional density of spiral-waves.
We establish a theoretical framework for re-entrant arrhythmias that explains the requirements for their successful treatment. We demonstrate the inadequacy of focal ablation for spatially fixed spiral-waves. Mechanistically guided principles for ablating multi-wavelet re-entry are provided. The potential to capitalize upon regional heterogeneity of spiral-wave density for improved ablation efficacy is described.
Introduction
The fundamental properties of propagation required to support re-entry were elucidated in 1913 by Mines1: re-entry comprises a closed circuit whose length is greater than the excitation wavelength. This has formed the basis for treatment strategies of re-entry ever since; anti-arrhythmics aim to increase wavelength, while ablation aims to physically interrupt the circuit. In 1914 Garrey established the mass hypothesis of fibrillation: ‘ … the duration of the fibrillatory conduction being in direct proportion to the mass of the tissue.’2 Even in this early work Garrey realized that, ‘ … circuits can exist in large masses but not in sufficiently small ones.’ In fact, 4 years earlier Erlanger noted that fibrillation was impossible in sufficiently narrow strips of tissue.3 Nevertheless, although the application of Mines' premise to fixed anatomic re-entrant circuits is straightforward, it is not immediately apparent how this paradigm can be extended to treat multi-wavelet re-entry with its complex, dynamic, and spatially varying circuits. Indeed, it would be another 70 years before Cox employed these fundamental principles to develop the maze procedure; ‘ … [we must] place incisions in the atria close enough together that macro-re-entrant circuits could not form between them.’4
The surgical maze procedure, and the subsequent catheter ablation techniques which it inspired, has proven effective in preventing atrial fibrillation (AF) in many patients. Unfortunately, AF persists despite these procedures in a significant number of patients with advanced electrical disease. Furthermore, there is not uniform agreement about the appropriate ablation strategy for patients who are refractory to pulmonary vein encircling combined with roof and mitral isthmus ablation lines. On the basis of present understanding of the mechanisms responsible for re-entrant arrhythmias, we have developed a conceptual framework that identifies the general requirements for a successful treatment strategy. This framework provides a topological description of re-entry that defines the lesion distribution required to terminate and prevent re-entry of progressively greater complexity. Here, we test these principles on re-entrant rhythms generated by a computational model of cardiac excitation.
Conceptual framework
Re-entrant rhythms persist by repeatedly looping back to re-excite tissue in a cycle of perpetual propagation rather than by periodic de novo impulse formation. Because of cardiac refractory properties, activation cannot simply reverse directions. Instead, re-entry requires separate paths for departure from and return to each site, analogous to an electronic circuit.
The components of re-entrant circuits can vary, the anatomic and physiologic constituents falling along a continuum from lower to higher spatio-temporal complexity. At the lower end of the spectrum, circuits are composed of permanent anatomically defined structural elements such as a region of scar tissue. Circuit components need not be permanent and structural, however. They can also be functional and therefore transient, such as occurs when electrical dissociation between adjacent fibres allows formation of separate conduction paths.5
Figure 1 illustrates these concepts using simulated patterns of atrial activity produced by a computational model we have developed6 (the model is described briefly below in Methods). Spiral-waves are an example of functional re-entrant substrate (Figure 1B) created when source–sink relationships at the spiral centre create a core of unexcited tissue around which rotation occurs.7 This can occur even in homogeneous and fully excitable tissue in which, based on the timing and distribution of excitation, groups of cells in separate phases of refractoriness create the separate paths which link to form a circuit. The simplest spiral-waves have spatially fixed cores, whereas more complex examples have cores that meander throughout the tissue.7 At the most complex end of the re-entry spectrum, spiral-waves encountering spatially varying refractoriness can divide to form distinct daughter-waves, resulting in multi-wavelet re-entry.8
Re-entrant circuits of increasing complexity. (A) Scar mediated re-entry (structural inner-boundary). (B) Spiral-wave re-entry (functional inner-boundary) (C) Multi-wavelet re-entry (diffuse, temporally varied functional inner-boundaries). Note in (A) the region between the leading edge of excitation and the trailing edge of refractoriness must be excitable; if ‘head-meets-tail’ conduction fails as refractory tissue connects the inner-boundary to the tissue edge.
Re-entrant circuits can be defined topologically. The left atrium, for example, can be seen as a bounded and interrupted plane. The annulus forms the edge or outer-boundary of the plane; the pulmonary veins form holes or discontinuities ‘interrupting’ the plane. An interrupted plane forms a circuit for re-entry. Anyplace within the tissue, across which current does not flow forms a discontinuity. The discontinuity can be structural (Figure1A) or functional (Figure 1B). Topologically, two surfaces are homomorphic (the same) if by stretching but not cutting or pasting one can be transformed into the other (Figure 2). All bounded surfaces with an inner-discontinuity are homomorphic and functionally identical. As a result of tissue refractory properties a structurally uninterrupted plane may nonetheless be capable of forming a circuit (around an inner-discontinuity 2° to physiological conduction block). One can describe a physical topology (defined by the tissue structural geometry) and a functional topology (defined by the physiologically possible paths of activation).
Surface topology and re-entry. (A) Uninterrupted plane (outer-boundary only). (B) Interrupted plane (disconnected inner-boundary). (C) Several time-steps after a lesion has connected wave-tip to tissue-edge (eliminating inner-boundary). Note in (A) with nothing to divide activation a single wave traverses the plane (and will extinguish at the right edge); in (B) functional block at the wave-tip (star) causes interruption of the plane, creating a circuit and allowing re-entry. In (C) a linear lesion, by connecting the edge to the inner-boundary, converts the surface topology into an uninterrupted (though geometrically more convoluted) plane and increases the total boundary length and length-to-area ratio. (A) (uninterrupted plane) and (C) (wave-tip connected to outer edge) are homomorphic, whereas (A) and (B), and (B) and (C), are heteromorphic.
Re-entry requires a complete circuit; disruption causes propagation to cease. As such, termination of re-entry requires ‘breaking’ its circuit. This provides a unifying formulation for perpetuation and termination of all re-entrant rhythms. If its circuit is disrupted, a rhythm is extinguished, whether by prolongation of wavelength beyond circuit-length (e.g. anti-arrhythmic medication) or by structural circuit disruption (e.g. ablation). Topologically this is equivalent to conversion from an interrupted plane (capable of re-entry) to an uninterrupted plane (which is not). From a topological perspective, despite their differences in shape, a surface with no inner-discontinuity and one with a discontinuity that is connected to the tissue edge are equivalent (Figure 2A and C are homomorphic).
Circuit-based ablation strategies
As with any re-entrant rhythm, termination of multi-wavelet re-entry requires circuit disruption. Unfortunately, the circuits involved in multi-wavelet re-entry are spatially complex and temporally varying, so actually achieving circuit disruption is a pragmatic challenge. Nevertheless, the topologic perspective on re-entry provides a conceptual framework for deciding where ablation lesions should be placed so as to yield the greatest likelihood of terminating multi-wavelet re-entry. The goal of ablation is to reconfigure the tissue's topology (structural and functional) into that of an uninterrupted plane.
The simplest situation is presented by a fixed spiral-wave. The leading edge of excitation in spiral-wave re-entry is curved, and curvature increases progressively until source–sink relationships at the spiral centre result in propagation failure. The point of propagation failure at the wave-tip forms the inner-edge of the spiral-wave around which rotation occurs (Figure 2B). Circuit disruption requires a lesion to span from the tissue edge to the inner-discontinuity at the wave-tip (Figure 2C).
The situation becomes more complicated with a meandering spiral-wave, as it is difficult to determine where a fixed lesion can be placed that is guaranteed to transect the re-entrant circuit. Movement of the spiral-core provides a means to overcome this pragmatic hurdle; spiral meander can result in collision of the wave-tip with a boundary. In fact, the probability of termination (tip/boundary collision) increases as the ratio of total boundary length to tissue area increases.
Linear ablation lesions that are contiguous with the tissue edge can increase the boundary length area ratio and thereby increase the probability of termination. Note that such lesions do not change the topology of the atrial tissue; it remains an uninterrupted plane, but with a boundary that becomes progressively long and tortuous as the number of lesions increases. As tissue regions with shorter wavelength are more likely to contain spiral-waves,9 linear lesions in these areas are more likely to cross a tip-trajectory and cause spiral-wave termination. The crucial point remains, however, that all lesions must be placed such that the topology of the atrial tissue remains that of an uninterrupted plane.
Methods
Computational modelling
In order to test our ablation strategies, we needed a computer model of propagation through excitable media whose emergent behaviour includes formation of stable and meandering spiral-waves as well as multi-wavelet re-entry. Furthermore, to test the impact of linear ablation (lines of electrically inert cells) on propagation during multi-wavelet re-entry, we needed a model with sufficiently small computational burden that multiple simulations of extended periods of excitation could be run in a manageable amount of time. To achieve these goals, we used our previously described hybrid between a physics-based and a cellular automaton model.6
In brief, cells are arranged in a two-dimensional grid each cell connected to its four neighbours (up, down, left, and right) via electrically resistive pathways. Each cell has an intrinsic current trajectory (Im—equivalent to net trans-membrane current) that follows a prescribed profile when the cell becomes excited. Excitation is elicited either when the current arriving from the four neighbouring cells accumulates sufficiently to raise the cell voltage (Vm—equivalent to trans-membrane voltage) above a specified threshold or when the cell receives sufficient external stimulation (pacing). Once excited, a cell remains refractory (i.e. cannot be re-excited) until Vm repolarizes to the excitation threshold. The duration of a cell's refractory period is thus determined by the duration of its action potential. Following the absolute refractory period, there is a period of relative refractoriness during which excitation can occur but with decreased upstroke velocity. Each cell's intrinsic action potential morphology (voltage vs. time) is modulated by its prior diastolic interval and lowest achieved voltage at the time of its depolarization. This modulation confers restitution upon upstroke velocity and action potential duration (APD). Tissue heterogeneity is represented by an APD that varies randomly about a set mean. This mean itself can also vary across the tissue.
This model does not include all the known biophysical details of individual cardiac cells that are included in some of the most advanced current computational models of cardiac electrophysiology. Nevertheless, it does incorporate the key behavioural features of individual cells that are required to reproduce realistic global conduction behaviour. Most important for the present application, this behaviour includes source–sink relationships with wave curvature-dependent conduction velocity and safety factor, and the potential for excitable but unexcited cells to exist at the core of a spiral-wave. Our model thus combines the computational expediency of cellular automata with the realism of much more complicated models that include processes at the level of the ion channel.
Phase maps
The voltage matrices from each time-step of the cardiac model simulations were saved. Each voltage map was converted into a phase map as previously described.10,11 In brief, we perform a Hilbert transform to generate an orthogonal phase-shifted signal from the original signal at each coordinate of the tissue space–time plot (x, y, t). From the original and phase-shifted signals, we calculate the phase at each coordinate at each time-step.
Phase singularities and spiral-wave-tip density
The location of the leading edge of each excitation wave was determined for each time-step of the simulation (based on the coordinates at which each cell first crossed the threshold for excitation).
Phase singularities were identified using the algorithm described by Iyer and Gray.12 Phase singularity sites were considered to represent a spiral-wave-tip if (i) all phases surrounded the singularity in sequence (from –Π to Π) and (ii) the phase singularity was located at the end of a leading edge of activation. Space–time plots of the phase singularities were created to delineate wave-tip trajectory. The total number of spiral-wave tips (measured during each time-step over the sampling interval) divided by the space–time volume was defined as the spiral-wave-tip density.
Spiral-waves were initiated by rapid pacing from two sites in close proximity with an offset in the timing of impulse delivery. Spiral-waves were spatially stable in the setting of homogeneous tissue (all cells identical) with a shorter wavelength than circuit length. As APD was increased (wavelength ≥ circuit length) the spiral-waves began to meander. Multi-wavelet re-entry resulted when APD was randomized across the tissue. A region with higher spiral-wave density resulted when a patch of tissue with shorter mean APD was created.
Statistical analysis
Descriptive data are reported as mean ± standard deviation. T-test was performed with the PRISM (version 5.03) statistical software (GraphPad Inc., La Jolla, CA, USA). P values of < 0.05 were considered to be statistically significant.
Results
We first studied the ablation of fixed spiral-waves. Consistent with our topological view of ablation strategies, and contrary to a recent report,13 we found that lesions placed at the centre of spiral-waves simply converted a functional discontinuity into a structural one, but did not result in termination of re-entry. Termination required placement of a lesion spanning from the tissue edge to the wave-tip. Even a single excitable cell between the lesion and wave-tip was sufficient for spiral-wave perpetuation. However, lesions needed only connect the tissue edge to the outermost extent of the tip-trajectory. Meandering spiral-waves terminated when their wave tips collided with a lesion.
We next studied the ablation of meandering spiral-waves and multi-wavelet re-entry. In a set of simulations, cellular APD was randomly varied about a mean value (75 ± 25 ms) to produce a random spatial distribution of refractoriness. Using this approach, we initiated varied patterns of multi-wavelet re-entry by varying APD distribution. To test the hypothesis that mobile spiral-waves and multi-wavelet re-entry are terminated through probabilistic collisions with the tissue boundary, we examined the duration of multi-wavelet re-entry as a function of the ratio of total boundary length to tissue area. In rectangular sections of tissue, we varied width and height (and thus boundary length) while keeping area fixed (Figure 3, video/online supplement). The average duration of multi-wavelet re-entry increased progressively as the ratio of boundary length to area was decreased (average duration 2.2 ± 1.7 × 103 time-steps at a ratio of 0.26; average duration 4.0 ± 2.1 × 106 time-steps at a ratio of 0.125. (Note that simulations at the lowest ratio were truncated at 5.0 × 106 time-steps; only 2 of 10 simulations terminated within this time frame.)
Impact of length to area ratio on duration of multi-wavelet re-entry. Keeping tissue area fixed (at 1,600 mm2) the length and height were varied such that length/area ratio decreased from top to bottom: 0.26, 0.2125, 0.145, and 0.125, respectively.
In a related set of experiments, we tested the effects of linear ablation on the average duration of multi-wavelet re-entry. Tissue area was 800 mm2 and baseline boundary length to area ratio (prior to ablation) was 0.15. As ablation lines were added to the tissue (Figure 4), the average time to termination of multi-wavelet re-entry was progressively decreased; zero lines produced termination in 2.2 ± 2.9 × 105 time-steps; one line in 4.2 ± 8.5 × 104 time-steps; two lines in 1.6 ± 2.5 × 103 time-steps; three lines in 575 ± 67 time-steps.
Ablation lines increase length/area ratio and decrease the duration of multi-wavelet re-entry. Clockwise from top left average duration decreased: zero lines 2.2 ± 2.9 × 105 time-steps; one line in 4.2 ± 8.5 × 104 time-steps; two lines in 1.6 ± 2.5 × 103 time-steps; three lines in 575 ± 67 time-steps.
Regional variation in ablation efficacy
Finally, we tested the effects of heterogeneous tissue properties on the effectiveness of ablation lesions. When the central area of the tissue had shorter average APD (90 ± 15 ms) than the surrounding regions (115 ± 10 ms), spiral-waves meandered and were preferentially located within the central short refractory period zone. The mean spiral-wave-tip density (measured as the number of cores per space-time volume) was significantly higher in the central shorter APD zone (4.73 ± 0.71 cores/mm2 ms) vs. the surrounding tissue (2.95 ± 0.18 cores/mm2 ms), P < 0.001. In tissue without a central short APD zone there was no significant difference in spiral-wave-tip density between the central zone and the periphery (1.58 ± 0.65 vs. 1.57 ± 0.52 cores/mm2 ms, P = 0.963).
The regional efficacy of ablation for termination of multi-wavelet re-entry was assessed by delivering ablation lesions either to the central zone (high spiral-wave density) or the surrounding tissue (low spiral-wave density). In both cases, the total boundary length to area ratio was kept fixed (0.13) by varying only the location but not length or number of ablation lines. Hundred percent of episodes terminated when lesions were delivered inside the central zone compared with 76% termination following ablation outside the central zone (n = 25). The average time to termination was shorter when lesions were placed within the central zone (1.0 ± 1.2 × 104 time-steps) compared with lesions placed only in the surrounding tissue (7.5 ± 8.2 × 104 time-steps), P = 0.0005. Conversely, when the tissue periphery had shorter average APD (90 ± 15 ms vs. 115 ± 10 ms), spiral-wave-tip density was higher within that area (4.05 ± 0.64 cores/mm2 ms) compared with the central longer APD zone (3.29 ± 0.52 cores/mm2ms), P < 0.001. Ablation resulted in termination of multi-wavelet re-entry in 80% of simulations (whether lesions were placed in the central zone (n = 25) or in the periphery (n = 25)). The average time to termination was shorter when lesions were placed in the peripheral high spiral-wave density zone (1.7 ± 2.8 × 104) compared with the central low spiral-wave density zone (4.9 ± 3.7 × 104 time-steps, P = 0.009).
Discussion
Studies in animals,14 simulations in computational models,14,15 and multi-electrode array mapping during AF in humans,8 strongly suggest that complex forms of re-entry are involved in the perpetuation of AF. The framework presented here offers a perspective for understanding treatment of multi-wavelet re-entry—when it succeeds and when it fails. Anti-arrhythmic medications alter atrial functional topology and ablation alters atrial structural topology. The requirements for rendering atrial functional topology incapable of supporting re-entry are prescribed by the framework of circuit interruption/inner-boundary elimination. The pragmatic realization of this goal in the complex functional and structural architecture of the atria is a tremendous challenge.
In multi-wavelet re-entry spirals terminate when all spiral-cores collide with an outer boundary. The probability of collision, and therefore termination, is increased as the ratio of total boundary length to area is increased. The addition of linear ablations (contiguous with the tissue edge) can substantially increase the probability of termination. Ablation lines are most likely to result in termination of multi-wavelet re-entry when delivered to areas of high spiral-wave density. Interestingly, the addition of linear scar resulted in increased dispersion of refractoriness (through reduced electrotonic interactions, data not shown). Although this alone enhances the propensity for re-entry, because scars were contiguous with an outer boundary they did not provide a circuit and hence were not arrhythmogenic.
The standard practice of pulmonary vein isolation16 combined with linear ablation that connects the isolated veins to each other and to the mitral annulus17 converts left atrial structural topology into that of an uninterrupted plane, and cures AF in ∼75% of cases.18 It is therefore reasonable to predict that the addition of only a few more appropriately placed ablation lines (to increase the probability of spiral-core vs. boundary collision) should further significantly reduce the ability of the atria to support multi-wavelet re-entry.
The key question, then, is where to place additional lesions in order to optimally treat the ∼25% of patients not responsive to current strategies. The topological framework provides a guide for answering this question, the goal being to distribute linear lesions that provide the greatest likelihood of producing wave extinction while at the same time minimizing total lesion length. Although it remains to be determined which sites provide the greatest lesion efficiency in human AF, our results suggest that targeting sites with high spiral-wave-tip density will result in improved outcome.
The topological framework also helps to shed light on some controversial aspects of AF ablation. For example, there are numerous reports of improved outcome with addition (or sole use) of focal ablation that targets complex fractionated atrial electrograms.19–22 On the basis of the findings in this study, one would predict that focal ablation would create new potential re-entrant circuits (unless lesions are continued to an atrial boundary). Interestingly, it has been suggested that fractionated electrograms identify sites of autonomic ganglia.23–25 Consequently, destruction of ganglia may be successful by producing autonomic withdrawal thereby increasing wavelength. If the wavelength increase is greater than the path length of the circuits created by focal ablation the net result will be anti-arrhythmic. Thus, ganglia ablation can be considered to be a modification to atrial functional topology acting like an interventional anti-arrhythmic ‘medication’.
Ultimately, one would like to be able to apply a topological analysis to individual patients to prospectively identify those requiring additional ablation lesions and to design optimal ablation strategies that are patient specific. Unfortunately, this is not yet possible with current mapping technology.26–28
Clinical implications
The evidence supporting a critical role of re-entry in AF perpetuation is compelling. Thus, a paradigm for understanding and eliminating re-entry, particularly its most complex forms, represents a step toward development of a more effective ablation strategy. On the basis of currently available information, we feel that an ablation strategy which includes pulmonary vein isolation along with linear lesions guided by the concepts of spiral-core trajectory and distribution has the greatest likelihood of AF cure.
Limitations
The framework developed here will require validation in biologic experiments. Computer simulations are incapable of validating hypotheses; rather, simulations can delineate the implications of the assumptions upon which they are based. However, these simulations have the advantage of allowing precise and independent manipulation of the parameters that effect re-entry.
Another limitation of our approach is the use of two-dimensional tissue. Future experiments will have to incorporate a third dimension in order to study the effects of tissue thickness on multi-wavelet re-entry and its treatment. Finally, in this study we examined only the role of continuous propagation; we have not addressed the role of impulse formation in AF. It has been widely appreciated that focal firing, mostly from the pulmonary veins, plays a critical role in initiation of AF. Thus isolation of the pulmonary veins is a component of the majority of ablation strategies.
Conclusions
Re-entrant arrhythmias require a complete circuit or wave propagation extinguishes. Thus, treatment of re-entry, regardless of its complexity, requires ‘breaking’ the arrhythmia circuit. In multi-wavelet re-entry spiral-waves annihilate upon collision with a tissue boundary. The likelihood of wave annihilation is therefore proportional to the ratio of total boundary length to tissue area. Linear ablation lesions can extend the tissue boundaries, increasing the boundary length to area ratio and the probability of termination. In heterogeneous tissue spiral-wave density is higher in some regions than others; as a result the efficacy with which lesions result in termination is greatest in regions of high spiral density.
Acknowledgement
The authors thank Burton Sobel, MD, for his thoughtful help in developing this work and preparing the manuscript.
Conflict of interest: Dr Spector receives research support from Medtronic, Biosense Webster and St Jude; consults for Medtronic and Biosense Webster. Dr Bates receives research support from Medtronic.
