Observable Atrial and Ventricular Fibrillation Episode Durations Are Conformant With a Power Law Based on System Size and Spatial Synchronization.

BACKGROUND: Atrial fibrillation (AF) and ventricular fibrillation (VF) episodes exhibit varying durations, with some spontaneously ending quickly while others persist. A quantitative framework to explain episode durations remains elusive. We hypothesized that observable self-terminating AF and VF episode lengths, whereby durations are known, would conform with a power law based on the ratio of system size and correlation length ([Formula: see text]. METHODS: Using data from computer simulations (2-dimensional sheet and 3-dimensional left-atrial), human ischemic VF recordings (256-electrode sock, n=12 patients), and human AF recordings (64-electrode basket-catheter, n=9 patients; 16-electrode high definition-grid catheter, n=42 patients), conformance with a power law was assessed using the Akaike information criterion, Bayesian information criterion, coefficient of determination (R2, significance=P<0.05) and maximum likelihood estimation. We analyzed fibrillatory episode durations and [Formula: see text], computed by taking the ratio between system size ([Formula: see text], chamber/simulation size) and correlation length (xi, estimated from pairwise correlation coefficients over electrode/node distance). RESULTS: In all computer models, the relationship between episode durations and [Formula: see text] was conformant with a power law (Aliev-Panfilov R2: 0.90, P<0.001; Courtemanche R2: 0.91, P<0.001; Luo-Rudy R2: 0.61, P<0.001). Observable clinical AF/VF durations were also conformant with a power law relationship (VF R2: 0.86, P<0.001; AF basket R2: 0.91, P<0.001; AF grid R2: 0.92, P<0.001). [Formula: see text] also differentiated between self-terminating and sustained episodes of AF and VF (P<0.001; all systems), as well as paroxysmal versus persistent AF (P<0.001). In comparison, other electrogram metrics showed no statistically significant differences (dominant frequency, Shannon Entropy, mean voltage, peak-peak voltage; P>0.05). CONCLUSIONS: Observable fibrillation episode durations are conformant with a power law based on system size and correlation length.

Percolation theory as a conceptual framework to explain spontaneous atrial fibrillation termination: a pilot study.

Atrial fibrillation (AF) strikingly possesses the ability to abruptly transition into more organized electrical activity and spontaneously terminate, even after persisting for long periods. Despite being central to the clinical behavior and treatment of AF, these phenomena remain incompletely understood. In this paper, we hypothesized that the spontaneous termination of AF may represent a type of percolation phase transition, which is more likely to occur when a domain spanning cluster of refractory sites in the atrium are connected (called a 'percolation cluster'). This was assessed in n=50 computational simulations of AF developed using the Aliev-Panfilov (APV) 2-dimensional cell model. In self-terminating simulations of AF, it was found that the average refractory cluster size, χ(p) (median: 647.7), and the largest refractory cluster size, M1 (median: 1702.3), abruptly increased just prior to AF spontaneously terminating, indicating the onset of the formation of a domain spanning percolation cluster. In contrast, simulations of sustained AF did not demonstrate an increase in χ(p) (median: 463.0) and M1 (median: 1448.2). Self-terminating AF simulations also demonstrated hallmark properties characteristic of a percolation phase transition, such as an abrupt increase in χ(p) at the critical occupation probability pc. The cluster size distribution was also shown to obey a power law for various occupation probabilities p, also indicative of a percolation phase transition. Collectively, these properties suggests that the spontaneous termination of AF could be a form of percolation phase transition, which could provide new insights for AF treatment.

Markov modeling of phase singularity interaction effects in human atrial and ventricular fibrillation.

Atrial and ventricular fibrillation (AF/VF) are characterized by the repetitive regeneration of topological defects known as phase singularities (PSs). The effect of PS interactions has not been previously studied in human AF and VF. We hypothesized that PS population size would influence the rate of PS formation and destruction in human AF and VF, due to increased inter-defect interaction. PS population statistics were studied in computational simulations (Aliev-Panfilov), human AF and human VF. The influence of inter-PS interactions was evaluated by comparison between directly modeled discrete-time Markov chain (DTMC) transition matrices of the PS population changes, and M/M/∞ birth-death transition matrices of PS dynamics, which assumes that PS formations and destructions are effectively statistically independent events. Across all systems examined, PS population changes differed from those expected with M/M/∞. In human AF and VF, the formation rates decreased slightly with PS population when modeled with the DTMC, compared with the static formation rate expected through M/M/∞, suggesting new formations were being inhibited. In human AF and VF, the destruction rates increased with PS population for both models, with the DTMC rate increase exceeding the M/M/∞ estimates, indicating that PS were being destroyed faster as the PS population grew. In human AF and VF, the change in PS formation and destruction rates as the population increased differed between the two models. This indicates that the presence of additional PS influenced the likelihood of new PS formation and destruction, consistent with the notion of self-inhibitory inter-PS interactions.

The inspection paradox: An important consideration in the evaluation of rotor lifetimes in cardiac fibrillation.

Background and Objective: Renewal theory is a statistical approach to model the formation and destruction of phase singularities (PS), which occur at the pivots of spiral waves. A common issue arising during observation of renewal processes is an inspection paradox, due to oversampling of longer events. The objective of this study was to characterise the effect of a potential inspection paradox on the perception of PS lifetimes in cardiac fibrillation. Methods: A multisystem, multi-modality study was performed, examining computational simulations (Aliev-Panfilov (APV) model, Courtmanche-Nattel model), experimentally acquired optical mapping Atrial and Ventricular Fibrillation (AF/VF) data, and clinically acquired human AF and VF. Distributions of all PS lifetimes across full epochs of AF, VF, or computational simulations, were compared with distributions formed from lifetimes of PS existing at 10,000 simulated commencement timepoints. Results: In all systems, an inspection paradox led towards oversampling of PS with longer lifetimes. In APV computational simulations there was a mean PS lifetime shift of +84.9% (95% CI, ± 0.3%) (p < 0.001 for observed vs overall), in Courtmanche-Nattel simulations of AF +692.9% (95% CI, ±57.7%) (p < 0.001), in optically mapped rat AF +374.6% (95% CI, ± 88.5%) (p = 0.052), in human AF mapped with basket catheters +129.2% (95% CI, ±4.1%) (p < 0.05), human AF-HD grid catheters 150.8% (95% CI, ± 9.0%) (p < 0.001), in optically mapped rat VF +171.3% (95% CI, ±15.6%) (p < 0.001), in human epicardial VF 153.5% (95% CI, ±15.7%) (p < 0.001). Conclusion: Visual inspection of phase movies has the potential to systematically oversample longer lasting PS, due to an inspection paradox. An inspection paradox is minimised by consideration of the overall distribution of PS lifetimes.

Understanding the origins of the basic equations of statistical fibrillatory dynamics.

The mechanisms governing cardiac fibrillation remain unclear; however, it most likely represents a form of spatiotemporal chaos with conservative system dynamics. Renewal theory has recently been suggested as a statistical formulation with governing equations to quantify the formation and destruction of wavelets and rotors in fibrillatory dynamics. In this perspective Review, we aim to explain the origin of the renewal theory paradigm in spatiotemporal chaos. The ergodic nature of pattern formation in spatiotemporal chaos is demonstrated through the use of three chaotic systems: two classical systems and a simulation of cardiac fibrillation. The logistic map and the baker's transformation are used to demonstrate how the apparently random appearance of patterns in classical chaotic systems has macroscopic parameters that are predictable in a statistical sense. We demonstrate that the renewal theory approach developed for cardiac fibrillation statistically predicts pattern formation in these classical chaotic systems. Renewal theory provides governing equations to describe the apparently random formation and destruction of wavelets and rotors in atrial fibrillation (AF) and ventricular fibrillation (VF). This statistical framework for fibrillatory dynamics provides a holistic understanding of observed rotor and wavelet dynamics and is of conceptual significance in informing the clinical and mechanistic research of the rotor and multiple-wavelet mechanisms of AF and VF.

A governing equation for rotor and wavelet number in human clinical ventricular fibrillation: Implications for sudden cardiac death.

BACKGROUND: Ventricular fibrillation (VF) is characterized by multiple wavelets and rotors. No equation to predict the number of rotors and wavelets observed during fibrillation has been validated in human VF. OBJECTIVE: The purpose of this study was to test the hypothesis that a single equation derived from a Markov M/M/∞ birth-death process could predict the number of rotors and wavelets occurring in human clinical VF. METHODS: Epicardial induced VF (256-electrode) recordings obtained from patients undergoing cardiac surgery were studied (12 patients; 62 epochs). Rate constants for phase singularity (PS) (which occur at the pivot points of rotors) and wavefront (WF) formation and destruction were derived by fitting distributions to PS and WF interformation and lifetimes. These rate constants were combined in an M/M/∞ governing equation to predict the number of PS and WF in VF episodes. Observed distributions were compared to those predicted by the M/M/∞ equation. RESULTS: The M/M/∞ equation accurately predicted average PS and WF number and population distribution, demonstrated in all epochs. Self-terminating episodes of VF were distinguished from VF episodes requiring termination by a trend toward slower PS destruction, slower rates of PS formation, and a slower mixing rate of the VF process, indicated by larger values of the second largest eigenvalue modulus of the M/M/∞ birth-death matrix. The longest-lasting PS (associated with rotors) had shorter interactivation time intervals compared to shorter-lasting PS lasting <150 ms (∼1 PS rotation in human VF). CONCLUSION: The M/M/∞ equation explains the number of wavelets and rotors observed, supporting a paradigm of VF based on statistical fibrillatory dynamics.