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

Abstract

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 ($L/\xi )$.

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 $L/\xi$, computed by taking the ratio between system size ($L$, 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 $L/\xi$ 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). $L/\xi$ 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.

WHAT IS KNOWN?

• Atrial fibrillation (AF) and ventricular fibrillation (VF) are characterized by spatiotemporally disordered electromechanical states observed in the atria or ventricles.

• AF and VF episodes can exhibit varying durations, with some spontaneously ending quickly while others persist.

• To date, there is no overarching quantitative framework to help explain observable episode durations in self-terminating AF and VF.

WHAT THE STUDY ADDS

• We hypothesized that a unifying quantitative framework explaining observable episode durations in self-terminating AF and VF could be derived by considering systems exhibiting comparable transient disorder.

• Observable self-terminating AF and VF episode were found to be conformant with a power law based on the system size (derived from atrial or ventricular size) and spatial synchronization (the distance over which electrical activity remains correlated).

• This conformity to a power law suggests that the duration of self-terminating episodes can potentially be forecasted based on system size and spatial synchronization, providing a new methodological approach for understanding AF and VF dynamics.

Atrial fibrillation (AF) and ventricular fibrillation (VF) are characterized by spatiotemporally disordered electromechanical states observed in the atria or ventricles respectively.¹ A defining trait of these arrhythmias is variability of episode duration; some episodes of AF and VF spontaneously end, while others persist for extended periods. For instance, AF may be transient in some but sustain indefinitely in others.¹ Conversely, while some VF episodes can self-terminate under certain conditions, sustained VF beyond 10 minutes causes irreversible neurological damage and death.^2,3

To date, there is no overarching framework to explain observable episode durations in self-terminating AF and VF. If this gap could be addressed, it could offer answers to pressing clinical challenges, such as forecasting the likely duration of an AF or VF episode, understanding how to improve treatment success for persistent AF^4–6 or identifying defibrillation-resistant VF episodes.⁷

We reasoned that such an overarching framework for observable self-terminating episode durations in AF and VF could be derived by considering systems exhibiting comparable transient disorders.^8,9 In such systems (ie, coupled map lattices), episode durations have been proposed to conform with a power law,^8,9 where $\tau \left(L\right)$ represents the average duration, $P^{-L/\xi }$ the instantaneous probability of basin synchronization, which can be conceptualized as the probability that the electrical activity for a region in the ventricle or atria will organize, and $L/\xi$ the number of distinct subunits or independent regions in a system based on the system size (derived from atrial or ventricular size) and correlation length (the distance over which electrical activity remains synchronized):

$${\displaystyle \begin{array}{l}{\displaystyle \tau \left(L\right)\sim P^{-L/\xi }}\end{array}}$$ (1)

Such power law relationships have thus provided insights to not only characterize but also quantify the likely persistence of other complex systems.^8,9 We sought to test the hypothesis that the observable lifetimes of self-terminating AF and VF episodes conform to this power law.

METHODS

Datasets and Data Acquisition

The data that support the findings of this study are available from the corresponding author upon reasonable request. For example, human AF data used in this study are provided in the Supplemental Material.

Computer-Simulated Models of AF and VF

We simulated AF and VF using 3 models: (1) Aliev-Panfilov atrial model in 2-dimensional sheet, (2) Courtemanche atrial model in 3-dimensional (3D) left-atrial geometry, (3) and Luo-Rudy ventricular model in 2-dimensional sheet (details in the Supplemental Material). N=50 Aliev-Panfilov (APV) simulations of AF were run on a 100×100 grid in MATLAB with diffusion coefficient, 0.2 mm²/s; space step, 0.6 mm; and time step, 0.02 ms; using parameters adapted from 10 Tusscher and Panfilov.^10,11 The μ2 parameter, which controls action potential duration restitution/duration, was randomly varied across the simulations, introducing a random distribution to create areas of dynamical instability for spiral wave breakup, as well as episodes with eventual self-termination. For 3D left-atrial AF simulations using the Courtemanche model,^12,13 n=15 simulations were run using a left-atrial mesh constructed from segmentation of cardiac magnetic resonance images. Using this mesh, AF simulations were run on openCARP¹⁴ with specific conductivities outlined in the Supplemental Material. AF initiation involved 4 Archimedean spirals based on Roney et al,¹⁵ applied using the universal atrial coordinate system, leading to paired opposite chirality spirals on each of the posterior and anterior walls.¹⁵ Simulations of varying epoch lengths were achieved by perturbing potassium conductivities (I_kr, varied by a multiplier of ±50%) to achieve self-termination as described in the Supplemental Material. VF simulations using the Luo-Rudy model were also run in OpenCARP¹⁴ (n=50), with ionic parameters based on the 1991 article. Self-termination was also achieved by perturbing potassium conductivities (I_kr, varied by a multiplier of ±50%).¹⁶

Epicardial Sock Recordings of Human VF

The human VF study uses prospectively collected data as described by Nash et al.¹⁷ The study recruited n=12 patients (n=8 sustained VF, n=4 self-terminating VF) undergoing routine coronary bypass graft procedures or valve replacement surgery for ischemic heart disease with cross-clamp fibrillation. These studies were approved by the University College Hospital London Ethics Committee (Research Ethics Committee 01/0130) with written informed consent. Individual patient details are given in Supplemental Methods; Table S1. Recordings were obtained using a 256-electrode epicardial sock ($\mathrm{\Delta }x\approx 10$ mm interelectrode spacing) and unipolar electrograms (1 kHz sampling) were preprocessed as previously described. In a subset of patients, VF spontaneously terminated before the full 210 seconds without requiring defibrillation, defined as self-terminating episodes. This was compared with sustained VF episodes lasting the full duration. Additional study details are provided in Supplemental Methods.

Basket and High Definition Grid Recordings of Human AF

The human AF study extends our previously published studies.^18,19 The study was a multicenter observational design analyzing prospectively collected electrophysiological data acquired before ablation at 4 Australian hospitals (ACTRN12619001172190p), approved by the Southern Adelaide Local Health Network Human Research Ethics Committee (HREC110634). The inclusion criterion was AF undergoing ablation. Patient participation was by informed consent, with recruitment from Flinders University (baseline characteristics in Table S2). Basket catheter (19 patients) and high definition (HD)-grid recordings (42 patients) were performed as previously.¹⁸ Unipolar electrograms in both the LA and RA (1–500 Hz, 2000 Hz sampling) and surface electrocardiograms during spontaneous or induced AF lasting >1 minute were obtained using a 64-electrode basket catheters (Constellation, Boston Scientific, 48 mm [$\mathrm{\Delta }x=4$ mm], 60 mm [$\mathrm{\Delta }x=5$ mm]) and HD-grid a 3-3-3 mm catheter with 16 electrodes to record human AF ($\mathrm{\Delta }x=3$ mm; Supplemental Methods; Table S3). Spontaneously terminating AF episodes were defined as those that terminated before the full 1-minute without requiring defibrillation, while sustained recordings were defined as those lasting the full window of observation. Signal filtering, cleaning, and preprocessing are detailed further in Supplemental Methods.

Data Analysis and Computational Methods

Estimating Power Law Relationships in AF and VF Persistence

Power law scaling relations have been associated with the lifetimes of transiently occurring disordered states, expressed by^8,9:

$${\displaystyle \begin{array}{l}{\displaystyle \tau \left(L\right)\sim P^{-L/\xi }}\end{array}}$$ (1a)

This relationship can be explained upon considering the inherent properties of spatially extensive systems.^8,9 Within these systems, a characteristic correlation length, $\xi$, emerges in which the system’s constituent elements tend to act in a correlated or synchronized manner. However, as the distance defined by $\xi$ increases, the components begin to demonstrate independent behavior. Such a phenomenon implies the system can be conceptually divided into distinct subunits or independent regions, with their number represented by $L/\xi$. Consequently, the lifetime of the disordered state $\tau \left(L\right)$ tends to be governed by the probability of all independent regions synchronizing, given by $P^{-L/\xi },$ upon which the system converges to stable behavior.^8,9 Thus, systems with more independent regions are more likely to exhibit prolonged periods of disordered behavior (see Figure 1 for a numerical example). By examining the relationship between fibrillation episode durations and $L/\xi$, we thus aimed to ascertain if the dynamics of AF and VF mirrors the power law scaling commonly found in other complex nonlinear dynamical systems.

Figure 1.

Power law scaling in atrial fibrillation (AF) and ventricular fibrillation (VF) termination. A, Depicts a power law relationship associated with the average lifetime of transient disorder (chaos). The power law exhibits scale-invariance, revealing that alterations in 1 quantity produces proportional changes in another across multiple scales. This suggests that the same simple principles govern behavior across a wide range of scales and is what makes power laws applicable to diverse phenomena (a property known as universality). B, Outlines the hypothesis motivating this study, which reasons that there should be a power law scaling relationship between the number of independent regions (given by $\mathrm{L}/\mathrm{\xi }$) and time until termination ($\mathrm{\tau }\left(\mathrm{L}\right)$) in episodes of spontaneously terminating AF and VF. C, Provides a numerical example demonstrating why a greater number of independent regions ($\mathrm{L}/\mathrm{\xi }$) associates with longer lasting transient disorder, where P represents the probability of an independent region collapsing (here equal to 0.1). In the context of AF and VF, this can be related to electrical dynamics in the heart. More organized electrical activity (left) is associated with larger islands of coordinated activity, leading to a fewer number of independent regions. This is thus reasoned to associate with shorter durations of spatiotemporal disorder, hence shorter AF/VF episodes. The converse is true for less-organized electrical activity (right).

Mathematically, $L/\xi$ can be estimated using $D_i$ from equation (2) below. $D_i$ approximates spatial synchronization or correlation length $\xi$ using the 2-point spatial correlation length,${\xi }_2$²⁰:

$${\displaystyle \begin{array}{l}{\displaystyle D_i={\left(L/{\xi }_2\right)}^d}\end{array}}$$ (2)

$${\displaystyle \begin{array}{l}{\displaystyle \approx 4\pi r^2/\pi {{\xi }_2}^2}\end{array}}$$ (3)

Specifically, ${\xi }_2$ quantifies the distance over which 2 signals recorded at spatially separated electrodes or nodes, distanced $r$ units apart, maintain a strong cross-correlation. Spatial synchronization across these nodes can often be modeled by an exponential decay function²¹ (details in Supplemental Methods):

$${\displaystyle \begin{array}{l}{\displaystyle C\left(r\right)=e^{-r/{\xi }_2}}\end{array}}$$ (4)

The cross-correlation function between 2 sequences of normalized activations was calculated over windows shifted throughout the duration of the recording, with a range of lag values encompassing slightly >1 cycle length (≈170 ms for VF and ≈150 ms for AF). The maximum cross-correlation was given by the correlation coefficient for the sequence of activations between the 2 corresponding locations (S4). Relative system size, $L\approx (4\pi r^2/\pi )$, was estimated using atrial and ventricular volumes calculated using the radius $r$ from the convex hull of electrode co-ordinates from 1 of the following: (1) mapping catheter (for AF), (2) epicardial sock (VF), or (3) grid nodes (computer simulations; Supplemental Methods).

Statistical Fitting and Analysis

We assessed whether the association between $L/\xi$, which relates to the number of independent regions approximated by equations (1) and (2), and the observed AF and VF episode durations (ie, time until termination) exhibited a power law relationship through goodness of fit analysis among 3 potential scaling relationships: (1) a first-degree polynomial; (2) exponential; and (3) power law. This was assessed using Akaike information criterion, Bayesian information criterion, and coefficient of determination (R², significance=P<0.05; Supplemental Methods). The slope of the power law relationship in each model system ($m$), which reflects the likelihood of self-termination, and probability $P_{sync}$ of basin synchronization (ie, the probability that an independent subunit will collapse or synchronize, with larger $P_{sync}$ indicative of greater likelihood of synchronization). The slope of the power law, $m=\log \left(\frac{1}{P_{sync}}\right)$, which is positive for $P_{sync}<1$, was estimated using maximum likelihood. In comparison, the probability $P$ given in equation (1) refers to the aggregate probability of synchronization across the entire system.

To further establish potential relationships between $L/\xi$ and the persistence of AF and VF, differences in $L/\xi$ were also assessed in recordings of paroxysmal AF compared with persistent AF, as well as throughout 3 distinct phases of ischemic VF: (1) the perfusion stage (initial 30 seconds, with most organized electrical activity); (2) ischemia (following 150 seconds); and (3) reflow (last 30 seconds, most turbulent electrical activity) using independent samples t test (significance=P<0.05).

Comparison With Other Metrics Characterizing Arrhythmia Dynamics

To evaluate whether the relationships between $L/\xi$ and the persistence of AF and VF could be captured by other electrogram-based metrics established in literature, we evaluated 4 measures used to characterize arrhythmia dynamics. This was assessed by calculating the following on HD-grid human AF data: (1) dominant frequency,²² Shannon entropy,²³ mean bipolar voltage,²⁴ and peak-peak bipolar voltage.²⁵ Details on the calculation of these metrics are provided in Supplemental Methods.

RESULTS

Observable Computer Simulations of AF and VF Exhibit Episode Durations That Are Conformant With Power Law Scaling

The relationship between $L/\xi$ and simulated AF and VF episode durations (epoch lengths) conform to a power-law in all simulations (APV: R²=0.901; P<0.001), Luo-Rudy (R²=0.611; P<0.001), and 3D Courtemanche AF (R²=0.911; P<0.001) when compared with a first-degree polynomial (APV: R²=0.890; Luo-Rudy: R²=0.578; 3D Courtemanche AF: R²=0.794; P<0.001 all systems) and exponential relationship (APV: R²=0.864; Luo-Rudy: R²=0.572; 3D Courtemanche AF: R²=0.705; P<0.001 all systems; Figure 2A through 2C). This was also reflected by the Akaike information criterion and Bayesian information criterion in all models, as shown in the summary of the fitting statistics provided in Table S4.

Figure 2.

Computer simulated atrial fibrillation (AF) and ventricular fibrillation (VF) epoch lengths exhibit power law scaling. A through C, Observed simulated epoch lengths increased with larger $L/\xi$, following a scaling relationship that best fit to a power-law characterized by a slope, $m=\log \left(\frac{1}{P_{sync}}\right),$ and basin synchronization probability, $P_{sync}$. D through F, Terminating epochs of simulated AF and VF are associated with lower $L/\xi$ (i-iii).

Sustained 2-dimensional AF simulations run using the APV model were associated with higher $L/\xi$ (mean sustained, 23.78 [95% CI, 22.76–24.81]; mean terminating, 11.33 [95% CI, 10.23–12.43]; P<0.001). This was also seen in the Luo-Rudy simulations of VF (mean sustained, 36.47 [95% CI, 33.91–39.03]; mean terminating, 27.15 [95% CI, 25.59–28.72]; P<0.001) and 3D LA simulations of AF using the Courtemanche model (mean sustained, 1.1×10³ [95% CI, 424–1.89×10³]; mean terminating, 329.03 [95% CI, 216.05–442.01]; P=0.006; Figure 2D and 2F).

Observable Human AF and VF Episode Durations That Are Conformant With Power Law Scaling

Observable AF and VF self-terminating durations (epoch lengths) are conformant with a power-law relationship in basket-mapped human AF (R²=0.910; P<0.001), HD-grid mapped human AF (R²=0.924; P<0.001) and human VF (R²=0.840; P<0.001) when compared with a first-degree polynomial (basket: R²=0.898; HD: R²=0.899; VF: R²=0.731; P<0.001) and exponential relationship (basket: R²=0.897; HD: R²=0.920; VF: R²=0.681; P<0.001; Figure 3A through 3C). These trends were supported by the Akaike information criterion and Bayesian information criterion in all human model systems, as shown in Table S4.

Figure 3.

Human and ventricular fibrillation (VF) epoch lengths exhibit power law scaling. A through C, Observed epoch lengths increased with larger $L/\xi$, following a scaling relationship that best fit to a power-law characterized by a slope, $m=\log \left(\frac{1}{P_{sync}}\right),$ and basin synchronization probability, $P_{sync}$. D through F, Terminating epochs of atrial fibrillation (AF) and VF are associated with lower $L/\xi$ (i-iii).

Sustained episodes of basket-catheter mapped human AF were associated with higher lower $L/\xi$ (mean sustained, 34.38 [95% CI, 16.87–51.88]; mean terminating, 7.10 [95% CI, 4.77–9.44]; P<0.001). This was also seen in HD-grid mapped human AF (mean sustained, 78.78 [95% CI, 63.35–94.21]; mean terminating, 9.41 [95% CI, 4.49–14.33]; P=0.001) and human VF (mean sustained, 52.80 [95% CI, 27.13–78.48]; mean terminating, 10.30 [95% CI, 1.90–22.50]; P<0.001; Figure 3D and 3F). The effect of mapping density and field of view on $L/\xi$ estimation is detailed in Supplemental Methods.

Analysis of Power Law Slope and Basin Synchronization Probability

The power law slope and $P_{sync}$ in equation (1) indicate the likelihoods of self-termination and synchronization in a system’s regions, respectively. Computer simulations showed Luo-Rudy VF models with m=5.29 and $P_{sync}=0.005$, APV models with m=4.71 and $P_{sync}=0.009$, and 3D left-atrial Courtemanche AF models with m=2.98 and $P_{sync}=0.051$, reflecting varied synchronization probabilities and self-termination tendencies across models.

Interestingly, human clinical data exhibited similar trends in slopes and basin synchronization probabilities. VF recordings showed a slope m=5.16 and $P_{sync}$ =0.006, HD-grid mapped AF had m=2.31 and $P_{sync}$ =0.099, and basket-mapped AF m=3.99 and $P_{sync}=$ 0.019. These results align with the varying field of view of each modality. Notably, the low $P_{sync}$ in VF recordings using the electrode sock, which has the largest field of view, indicate a reduced likelihood of synchronization across all independent regions observed over the larger area. Conversely, smaller fields of view, as in HD-grid mapped AF, demonstrate higher synchronization probabilities across the observed independent regions. This was further validated through exploration of the effect of field of view in computer simulations (Supplemental Methods).

Relationship of $L/\xi$ With the Progression of Human Atrial and VF Persistence

$L/\xi$ was also further assessed in 380 epochs arising from n=24 patients with paroxysmal AF, and 282 epochs arising from n=18 persistent human patients with AF. Epochs from patients with paroxysmal AF were associated with consistently lower $L/\xi$ in paroxysmal AF (10.91 [95% CI, 4.25–17.56]) when compared with persistent AF (50.37 [95% CI, 43.73–57.01]; P<0.001; Figure 4A).

Figure 4.

L/xi throughout the evolution of atrial fibrillation (AF) and ventricular fibrillation (VF) and comparison to electrogram based metrics. A, Paroxysmal AF was associated with lower $L/\xi$ compared with persistent AF. B, As ischemic VF evolves and becomes more turbulent and disordered, $L/\xi$ increases. C through F, Electrogram-based metrics such as dominant frequency (DF), Shannon entropy, mean bipolar voltage, and peak-peak bipolar voltage showed a trend toward lower $L/\xi$ values in terminating AF/VF, but no statistically significant differences when compared in sustained vs spontaneously terminating AF and VF epochs.

Human VF episodes from n=12 patients undergoing cardiac surgery were also studied. Three distinct stages of VF were studied: (1) the perfusion stage (initial 30 seconds); (2) ischemia (following 150 seconds); and (3) reflow (last 30 seconds). Supplemental Methods and Figure S5 depict the propagation of electrical activity as VF evolves throughout the 3 stages, which can be seen to depict more turbulent behavior as VF progresses. This evolution was associated with larger $L/\xi$ (mean perfusion, 14.78 [95% CI, 5.33–24.67]); mean ischemia, 45.48 [95% CI, 33.36–57.12]); mean reflow, 82.06 [95% CI, 41.78–122.22]). Specifically, $L/\xi$ between the perfusion and ischemia (P=0.012), and perfusion and reflow (P<0.001), and ischemia and reflow stages were significantly different (P=0.011; Figure 4B).

Comparison to Other Electrogram-Based Metrics and Their Association to Self-Termination

When applied to human AF data, it was found that dominant frequency showed a trend toward lower values in spontaneously terminating epochs; however, no statistically significant differences were found when compared with sustained AF epochs (mean sustained, 7.11 Hz [95% CI, 6.46–7.76]; mean terminating, 6.522 Hz [95% CI, 5.74–7.31]; P=0.26; Figure 4C). Similar observations were made for other electrogram-based metrics, with a downward trend in values associated with terminating episodes, but no statistically significant differences observed for Shannon entropy (mean sustained, 4.59 bits [95% CI, 4.02–5.16]; mean terminating, 4.31 bits [95% CI, 3.85–4.77]; P=0.49), mean bipolar voltage (mean sustained, 4.5×10⁻⁴ mV [95% CI, 0.26×10⁻³–0.63×10⁻³]; mean terminating, 4.20×10⁻⁴ mV [95% CI, 0.14×10⁻³–0.70×10⁻³]; P=0.85), and peak-peak bipolar voltage (mean sustained, 0.57 mV [95% CI, 0.41–0.73]; mean terminating, 0.28 mV [95% CI, 0.22–0.35]; P=0.08; Figure 4D through 4F).

DISCUSSION

This study provides new insights into the dynamics of AF and VF, demonstrating that the time to termination of these arrhythmias conforms to a power law scaling relationship based on system size and spatial synchronization, characterized by $L/\xi$. This draws from observations in spatially extended systems where the durations of transient, spatiotemporally disordered behaviors also exhibit a power law relationship with system size and spatial synchronization.^8,9 By conceptualizing the heart as such a system, we propose a framework where the observable duration until the termination of fibrillatory episodes can be quantitatively assessed through estimation of $L/\xi$. This approach, previously unexplored in cardiac electrophysiology, was robustly supported across multiple computer-simulated models and clinical human AF and VF.

Theoretical and Clinical Implications

Power law scaling relationships could provide a valuable additional quantitative conceptualization to those currently available to measure fibrillatory dynamics. First, a potential benefit of this approach is that it could contribute to a holistic understanding of the variability in AF and VF durations. Although existing clinical theories of fibrillation offer an important role in understanding underlying AF and VF mechanisms, they do not specifically quantitatively address why any individual episode should have a propensity for a particular episode length. For instance, the multiple wavelet reentry theory posits that fibrillation is sustained by multiple wavelets of activation circulating throughout the heart,²⁶ while the rotor theory suggests that fibrillation may be driven by 1 or more rapidly rotating waves or rotors.²⁷ While these descriptions may provide a mechanism to explain how the arrhythmia sustains, they do not explain or quantify the variability in fibrillatory duration. Similarly, the focal source theory suggests that fibrillation may be triggered by rapid firing regions or foci,²⁸ which again is insightful mechanistically, but lacks an explanation about likely episode durations. The power law framework presented here bridges this gap and potentially helps explain the transition between paroxysmal and persistent AF.

In addition, this framework can be further applied to supplement the exploration of how substrate and ionic changes, such as fibrosis and ion channel expression, may translate to likely AF and VF durations. Notably in comparison, existing quantitative metrics of fibrillatory dynamics such as dominant frequency, entropy, fractionation, or voltage-based substrate characterization do not explain individual episode durations.^29,30 Here, these observations are linked to $L/\xi$, which is a determinant of the average lifetime of transient disorder in other systems.^8,9

Potential Clinical Applications

Predicting the duration of fibrillation holds significant clinical utility. The power law framework potentially addresses this gap by providing an overarching structure to explain observable episode durations in self-terminating fibrillation. By understanding likely episode durations, clinicians could make more informed decisions about the timing and nature of interventions. This is particularly relevant for distinguishing between paroxysmal and persistent AF, differentiated by $L/\xi ,$ as the former may benefit from less invasive management while the latter might require more aggressive ablation approaches or advanced rhythm control strategies. Enhanced prediction models could facilitate a more personalized approach to patient management, such as stratifying patients by their likelihood of spontaneous termination or aligning interventions with predicted risk profiles. This minimizes the potential for unnecessary interventions, while ensuring timely and appropriate treatment for those in need.

The clinical utility of the power law approach is further underscored by its compatibility with routine clinical data and the potential to leverage recent advances in catheter mapping. Quantification of other similar metrics such as wavelength has been a challenge for in vivo clinical implementation due to the need for the precise calculation of conduction velocity, effective refractory period, or action potential duration, which are difficult to obtain in human patients.³¹ In contrast, $L/\xi$ can be rapidly calculated from commonly recorded electrical signals such as the intracardiac electrogram and ECG.

Recent advances in electroanatomic mapping have also expanded the potential applications of spatial synchronization metrics. Novel array catheters^32–34 now allow simultaneous measurement of electrical synchronization information from a variety of potential configurations, historically limited in the era of linear mapping catheters.²¹ Additionally, advances in 3D electroanatomic mapping allow rapid and accurate volumetric estimation of cardiac chamber dimensions to determine system size.³⁵ With new systems integrating real-time export, measurement of $L/\xi$ also has the potential to be clinically available on existing commercial mapping platforms.

Further clinical studies are also underway to assess the ability of $L/\xi$ to forecast drug mechanisms, heart failure, and ablation outcome, as well as estimation from noninvasive surface ECG.

Comparison to Existing Electrogram-Based AF Metrics

In the context of fibrillation, $L/\xi$ can be thought of as an estimator of local electrophysical spatial synchronization, related to overall system size. In this study, we demonstrate that this metric appears to be a potential forecaster of the potential for the temporal persistence of fibrillatory episodes. This may be because the information captured by $L/\xi$ leverages the inherent spatial properties of AF and VF. While traditional metrics of fibrillatory dynamics, such as dominant frequency,²² entropy,²³ voltage,²⁴ and fractionation³⁶ are primarily reliant on local timing and electrogram properties, $L/\xi$ is an integrated measure over spatially distributed data points, taking into account the overall system size. This may potentially provide important nuanced insights into the system’s underlying dynamics, potentially allowing for more accurate consideration of the multi-scale and nonlinear behavior, and providing insight into the structural and functional organization of AF and VF as biophysical processes.

Origins of $L/\xi$ and the Power Law Scaling Relationship of Self-Terminating Disorder

A central aspect of our findings is the application of the parameter $L/\xi$ to understand the time to AF and VF termination. This approach draws on principles observed in spatially extended systems exhibiting transient, spatiotemporally disordered behavior.^8,9,37 The origins of $L/\xi$ in such systems underscore the duration of self-terminating disorder and its link to this ratio in a power-law fashion.⁸ In spatially extended systems with spatiotemporal disorder, neighboring sites within the characteristic correlation length $\xi$ exhibit coherent behavior, whereas sites beyond this length tend to act independently. This dichotomy allows for the division of the system into $L/\xi$ subunits, each behaving as an independent entity where $L$ represents the relative system size.⁸

The significance of this division lies in the probabilistic behavior of these subunits. When considering random initial conditions, the likelihood that all local basins of activity simultaneously synchronize to a globally stable state—a phenomenon captured by $P^{-L/\xi }$—follows a power-law relationship. This relationship intricately ties the number of synchronized independent units to the average duration of an episode. This power-law scaling has been instrumental in understanding the lifetime of episodes of supertransient turbulence in nonlinearly coupled systems.^8,9,38 Applying this framework to cardiac tissue, we propose that fibrillation episodes in the heart can be similarly analyzed. Here, $L/\xi$ offers a novel perspective in quantifying the average duration until termination of fibrillatory episodes. This perspective is potentially significant as, to our knowledge, it has not been previously applied in the context of cardiac tissue, where spatiotemporal dynamics play a crucial role.

By considering the heart as a spatially extended system with its own characteristic correlation length $\xi$, we can conceptualize the fibrillation dynamics in terms of coherent and independent behaviors across the tissue. The ratio $L/\xi$ effectively represents the scale at which these dynamics are observed, influencing the probability of the system transitioning from a disordered (fibrillating) state to an ordered (organized rhythm) state. The power law relationship $P^{-L/\xi }$, therefore, becomes a potentially useful framework in understanding the likelihood of fibrillation termination. This approach aligns cardiac electrophysiology with the broader framework of complex systems where similar scaling laws govern the transition between states.

Implications for Episode Length Prediction and the Influence of System Size

This study’s findings offer potentially useful insights for predicting AF and VF episode durations, with the power law relationship providing a framework to quantify the likely observable duration of self-terminating AF and VF based on estimates of $L/\xi$. The variation in $L/\xi$ ratios between patients with clinically persistent AF and those with paroxysmal AF, as well as in different forms of ischemic VF, also support the potential of the power law framework.

The power law slope and basin synchronization probabilities in computer-simulated and clinical AF and VF are also potentially revealing, as they provide insights into the dynamics of fibrillation at a subunit level, indicating how individual regions within the heart are likely to behave during an episode. These insights could further shed light on how likely fibrillatory episodes are to persist. The similarities in the slopes and basin synchronization probabilities between computer-simulated and clinical data also help validate the computational models used, but also potentially open avenues for further refining these models to replicate human cardiac behavior more accurately.

The findings presented are also notably supported by long-standing empirically derived principles in cardiac fibrillation. The critical mass hypothesis has long been a principle underlying the persistence of fibrillation, and the clinical association of increasing chamber size with the probability of AF maintenance is a generally accepted clinical principle.^39,40 The power law framework here supports the importance of system size, $L$, but also importantly quantifies how this influences AF and VF persistence, which is not offered by existing literature. Our findings also align AF and VF with transient disorder in other systems, whereby the duration of the disordered state tends to lengthen as the system’s size or spatial synchronization increases. In certain extensive systems, this disorganized period can even extend so long that it becomes challenging, if not impossible, to witness its cessation within practical observation times,⁹ which may also shed light on the nature of long-standing persistent AF.

Variability in Number of Independent Regions and Termination Dynamics

A notable aspect of our study is the absence of a fixed parameter, akin to Reynold’s number in fluid dynamics,⁴¹ which could universally determine the time to termination of spatiotemporal disorder in AF and VF. In fluid dynamics, the Reynolds number provides a reliable predictor for the onset of turbulence, derived from the intrinsic properties and flow conditions of the fluid.⁴¹ In contrast, here, we observe that the persistence of disorder in AF and VF is characterized by the number of independent regions within the cardiac tissue, determined by the ratio of system size ($L$) to correlation length ($\xi$). This ratio, $L/\xi$, varies considerably among systems, and even individuals. This variation and the lack of such comparable control parameter potentially align AF and VF instead with the principles of self-organized criticality, whereby a system naturally evolves to a critical state without the need for such an external tuning parameter.⁴²

Potential Links to Phase Transitions in Physics

The power law scaling relationship in observable AF and VF episode durations, characterized by $L/\xi$, draws intriguing parallels to concepts from the Berezinskii-Kosterlitz-Thouless (BKT) transition in physics, described by the XY model. The XY model is driven by the binding and unbinding of topological defects (vortex-antivortex pairs).⁴³ In cardiac tissue, these defects can be interpreted as regions of phase singularity or spiral waves, shown to be pivotal in the initiation and termination of fibrillation.⁴⁴

In the XY model, the concept of entropy relates to the interaction of these defects and the number of potential spots (independent regions) for defects to occupy.⁴⁵ The $L/\xi$ ratio becomes crucial in this context, as it effectively determines the landscape on which these singularities can form and interact. This approach thus allows us to consider the energy implications of various configurations of defects, offering a deeper understanding of the dynamics leading to fibrillation and its termination.

The observation that the number of independent regions, determined by $L/\xi$, influences the duration of fibrillation also aligns with Peierls’ argument, suggesting that the stability of a phase (ordered or disordered) in a system is critically dependent on the number of sites or regions.⁴⁶ In the context of fibrillation, this translates to the idea that the larger the number of independent regions, the more complex the interactions of defects, thereby influencing the likelihood and timing of the transition from a fibrillatory to a nonfibrillatory state.

Limitations

While our study provides potentially valuable insights for observable AF and VF durations, our findings rely partly on computer simulations and models, which, despite their sophistication, may not fully capture the complex physiological processes in human cardiac systems. The human data utilized, though robust, may also not fully represent the broader AF and VF population. Variations in patient demographics and cardiac conditions could influence the generalizability of our findings. Last, our study focuses on specific aspects of fibrillation dynamics. Other relevant factors, such as genetic predispositions and lifestyle influences, were not considered but are crucial in understanding AF and VF fully.

Conclusions

This study provides a new quantitative framework for understanding the observable durations of self-terminating AF and VF, which are shown to be conformant with a power law based on system size and correlation length. This could provide useful insights for the prediction and termination of AF and VF.

ARTICLE INFORMATION

Acknowledgments

We appreciate the comments from Prof. Emeritus Henry Greenside of Duke University, North Carolina, that helped to improve our article.

Sources of Funding

This work is supported by the National Health and Medical Research Council of Australia Project Grant (1063754). The National Heart Foundation of Australia (101188), and The Hospital Research Foundation (CP-Heart-006).

Disclosures

None.

Supplemental Material

Supplemental Methods

Tables S1–S4

Figures S1–S5