Spatial Autocorrelation Dimension (Di ) for the Characterisation of Atrial Fibrillatory Dynamics

AF is a heart rhythm disorder characterised by spatiotemporally turbulent electrical activation of the cardiac atria. Multiple theories of how AF sustains and terminates have been proposed, but there is no single integrated and universally accepted mechanistic understanding.¹

The proposed theories broadly involve the following:

• Focal triggers, including repetitive ectopic discharges, specifically from the pulmonary veins. This theory has been widely accepted and is the basis for ablation for AF, which involves electrically isolating the pulmonary veins from the rest of the cardiac atria; however, this treatment often fails.^2,3
• Re-entry, including multiple wavelets (multiple circular waves forming, propagating and being destroyed throughout the cardiac atria) and rotors (one or more rotational centres, or topological defects, from which spiral waves rotate outwards).^4–7 The behaviour of these wavelets and rotors can be predicted using principles of topological defect chaos using renewal theory, but their ablation has not improved outcomes.^8–10
• Transmural conduction, which involves electrical dissociation across transverse layers of the atrial wall and active involvement of the epicardium in the maintenance of AF.¹

We propose the notion that AF is a disordered process, and therefore abides by the same principles displayed by other disordered or turbulent systems, rather than being confined to any one of the above proposed theories, which, taken individually, are each scientifically plausible and supported by evidence.

Chaos theory is the study of apparently random behaviour in systems governed by deterministic laws.¹¹ That is, irregular processes that appear to be random, but that are, in fact, governed by an underlying hidden set of rules, are said to be in a state of chaos.¹² Spiral defect chaos has been observed in various pattern-forming systems in nature, including physical, chemical and biological systems.^13,14 Spiral waves occur in spatially sufficiently large excitable systems. These waves can become unstable and break up to create multiple, drifting, spiral waves that constantly form and disappear in a state termed spiral defect chaos.

Throughout the history of AF enquiry over the past century, a common theme of unstable re-entry has been observed by many investigators.^4,15–17 As such, we consider the principles that underlie spiral defect chaos may be usefully applied to the understanding of atrial fibrillatory dynamics.^14,18

Scientific Origin of Spatial Autocorrelation Dimension, D_i: Incorporation of System Size and Correlation Length

In the case of AF, we designate the entity of spatial autocorrelation dimension, D_i. The D_i of a system of sustained disorder (AF) can be considered as a measure of the number of independent regions (or overall complexity) and depends on the ratio of system size to correlation length, as displayed in Equation 1:

where L is atrial size in the context of AF, ξ₂ is a two-point correlation length (a constant relating the cross-correlation of electrograms [EGMs] as a function of distance) and d is spatial dimensionality, which is considered to be 2 for the cardiac atria (Figures 1, 2).¹⁹

We hypothesised that D_i may be a means of characterising AF and therefore may account for the relative persistence or collapse of spatiotemporal turbulence in the heart; that is, whether AF has the propensity to sustain or terminate.

Atrial size is a critical determinant of AF clinical behaviour, as discussed in subsequent sections. From a theoretical viewpoint, numerical simulations suggest that sufficiently large homogeneous systems will be extensively chaotic.¹⁹ System complexity has been hypothesised to increase with the system size, which is consistent with Equation 1.¹⁹

The loss of spatial synchronisation of electrical activity that occurs in AF is a standard qualitative clinical observation. Spatial synchronisation may be quantified using the spatial correlation length, ξ, a measure of the correlation between different regions of the system. Parts of a large system will be uncorrelated, and hence independent, when separated by distances larger than ξ. It has been hypothesised that small spatial correlation lengths relative to the size of the system would indicate multiple independent regions.¹⁹ In practical terms, the two-point correlation length (ξ₂) is the entity that can be measured clinically and is used in Equation 1.

In 1994, Egolf and Greenside proposed in Nature that the number of independent regions could be determined by using the equation D = L/ξ_c, where L is size, ξ_c is the chaos correlation length and d is spatial dimensionality.¹⁹ In that paper, the authors proposed that given the fibrillating pig heart has a radius (r) of 25 mm and a ξ_c of 6 mm, there are approximately 70 (see Equation 2) independent regions in pig cardiac VF.^19,20 We postulate that a similar notion may be used to characterise human AF.

It seems plausible that a greater number of regions behaving independently in a turbulent system (either due to a large absolute system size or small independent regions) would lead to more extensive disorder, which is potentially more likely to sustain. Supporting this notion, in other spatially extended disordered systems, the probability of collapse of spatiotemporal disorder is given by P^L/ξ, where P is the instantaneous probability of each independent region synchronising (with values <1 in AF simulations), indicating that the likelihood of collapse of disorder (analogous to termination in the case of AF) depends on the ratio of system size to correlation length (i.e. the number of independent regions), with the power law relationship indicating that a small change in the number of independent regions has a relatively large effect on the probability of disorder collapsing.^21–23 In terms of episode duration, this is given in Equation 3, which we have recently validated in AF.^21–23

In further detail, we have recently reported that differences in D_i may account for the relative persistence or collapse of spatiotemporal turbulence in the heart, with high D_i observed in sustained episodes of AF.²¹

A major advantage of D_i is that it relates AF, a system that may be described according to the principles of spiral defect chaos, to multiple other comparable systems in nature. It incorporates both system size and wavelet domain size, fundamental biophysical properties of the cardiac fibrillatory process, to give an overall representation of system complexity. D_i has been shown, across multiple models (computational models and human data) and multiple pathologies (AF and VF), to predict termination of fibrillation.²¹ It is not restricted to a certain physical process (rotors, wavelets, focal discharges), nor does it provide explicit evidence for a particular mechanism of AF.²⁰ It also has the potential to be measured in real time in the electrophysiology laboratory, giving it a practical application during catheter ablation to guide therapy.²⁴

Practical Aspects of D_i Measurement

The computation of D_i requires the elements of Equation 1: a measure of atrial size and the two-point correlation length (ξ₂), estimating spatial synchronisation. Both values can be determined from intracardiac mapping using standard equipment, with a unique value for each measured location within the atria. An overview of the computational methods is shown in Figure 2.

System size, L, is estimated from the 3D geometry created during electroanatomical mapping; the convex hull is applied to electrode coordinates from the mapping catheter to determine the chamber volume and the radius, r, is then calculated from that volume measurement.²¹

Based on methods described previously, the two-point correlation length ξ₂ can be calculated as follows:²⁵

1. For each chosen anatomical location of a multipolar mapping catheter across the right atrium and left atrium (LA), individual unipolar EGMs are recorded from the electrodes on the mapping catheter (16 EGMs in the case of the HD Grid (Abbott) multipolar mapping catheter). The morphology of each of these EGMs is compared, pairwise, to the 15 neighbouring EGMs to determine the cross-correlation (CC; i.e. morphological similarity between the pairs of EGMs).
2. The distance between each pair of electrodes, l, is determined from the x, y, z coordinates of the electrodes, using the Euclidean distance formula.
3. The CC is then plotted against the distance, l, and fitted to an exponential. This relationship is described in Equation 4:

4. The equation is solved for x₂ to attain the two-point correlation length, ξ₂, the constant relating CC to l.

Once r and ξ₂ are computed, these values are used to calculate D_i according to Equations 1 and 2.

Importance of Atrial Size in AF

As shown in Equations 1 and 2, there are two variables that determine D_i in the setting of AF: atrial size, L, and the two-point correlation length, ξ₂. This section focuses on atrial size as a historic indicator of AF behaviour and outcomes.

Introduction and Overview

The critical mass hypothesis is the concept that a critical tissue size is needed to support cardiac fibrillation and was postulated initially in 1914.¹⁵ In 1959, Moe et al. wrote ‘obviously, a large mass of tissue can support a larger total number of independent wavelets’ and the larger the number of wavelets, the greater the likelihood of sustained AF.⁴

There is extensive evidence that supports the notion of atrial size being important in the pathogenesis, behaviour and management of AF.²⁶ This holds true across early experimental work, computational modelling and animal models, in which larger animals have more easily inducible AF, and the higher prevalence of AF in nature.²⁷ In humans, many predisposing factors to AF have a common downstream effect of resulting in enlarged atrial chambers; these include greater weight and height, as well as cardiac disease (cardiomyopathy, hypertension, valve disease), which are all associated with an increased risk of AF.²⁶ Increased atrial size reduces the likelihood of long-term sinus rhythm with any treatment, including antiarrhythmic drugs, cardioversion and ablation.^28–32 Atrial size is also important in the transition of paroxysmal to persistent AF.³³

The most effective ablation strategies in persistent AF are those that reduce contiguous electrical mass, when added to standard pulmonary vein isolation (PVI).^26,34 As such, the efficacy of PVI may be due, in part, to a reduction in the contiguous atrial size available for AF to sustain.

Atrial Size in Determining AF Behaviour: Evidence From Simulations

Simulations of AF have supported the notion that effective atrial size is important in determining AF behaviour; however, they have limitations that affect their direct application to a human AF model. The simulations may not use anatomically correct atrial geometry and may not account for tissue heterogeneity or anatomical barriers.

Findings of computer simulations support the notion that adequate electrically contiguous space is required to support AF. In 2006, Qu et al. performed simulations of AF and found that in 2D tissue the duration of fibrillation increases exponentially as tissue area increases.³⁵ This is consistent with the notions posed in Equations 1 and 2. For a given 2D area, Qu et al. found that square shapes behaved differently to rectangular shapes, indicating shape, not just size, may be important when applying this notion to the complex structure of a human atrium.³⁵ Subsequent simulations in 2012 demonstrated that the likelihood of terminating multiple wavelet re-entry is proportional to the ratio of the boundary length to tissue area, supporting the notion of contiguous area being relevant to AF persistence.³⁶ It was also reported that AF termination requires linear lesions from the tissue edge to the spiral wave core and that meandering spiral waves terminate upon collision with a boundary, supporting the idea of the importance of the size of the electrically active domain in determining propensity to termination.³⁶ The efficacy of linear lesions was also reported to vary directly with the regional density of spiral waves, indicating that the size and number of independently functioning units may be important, as in Equation 1.³⁶

In 2014, the notion of ‘fibrillation number’ was first reported in the cardiac literature in the context of VF.³⁷ In 2015, this idea was presented in the AF literature as the LA diameter divided by wavelength (where the wavelength is calculated as the atrial effective refractory period multiplied by conduction velocity).³⁸ A higher fibrillation number implies a greater vulnerability to fibrillation. Simulations found that the fibrillation number in AF was linearly correlated with fibrillation maintenance time.³⁸ To date, fibrillation number is not a concept that has been widely reported in the literature, and it relies on assumptions regarding AF mechanism. Fibrillation number does have the advantage of incorporating atrial size, as well as a measure of wavelength in the assessment of tendency to AF. In practice, however, wavelength is not measurable with standard clinical tools and therefore has limited clinical utility.

Further simulation experiments in 2015 found that in multiple wavelet re-entry, tissue size and shape determine the duration of re-entrant episodes.³⁹ It was reported that the number of waves coexisting on a given tissue follows a normal distribution; therefore, the mean and variance could be used to calculate the probability of spontaneous termination. Carrick et al. developed the ‘fibrillogenicity index’, derived from tissue size and shape, among other features (action potential duration, resistance and capacitance), to predict multiple wavelet re-entry half-life.³⁹ Although these findings are mechanistically informative and support the importance of tissue size and shape in the behaviour of simulated AF, many variables are not readily measurable in clinical human AF and clinical utility is therefore limited.

Simulations of spatially extended excitable systems have used techniques from statistical physics to determine the mean episode duration of spiral defect chaos, using generic models of cardiac fibrillation.¹⁴ In that report, it was demonstrated that the duration depends exponentially on domain size.¹⁴

Importance of Atrial Size: Evidence from Animal Models

Evidence from animal models supports the link between increasing atrial size and increased AF predisposition. In 2005, a study in a canine model reported that the probability of sustained AF was significantly associated with increasing tissue area, width and weight, among other factors (including decreasing effective refractory period and wavelength).⁴⁰ Another study in a pig model (2013) also found that the probability of sustained AF was increased with increased tissue area (and decreasing effective refractory period).⁴¹ In 2021, fibrillation number, which incorporates LA size, was reported to predict sustained AF in a sheep model.⁴²

Importance of Atrial Size: Evidence From Human Data

Importance of Left Atrium Size in AF Prognosis

Chamber size has been shown to be an important independent predictor of AF clinical phenotype, independent of traditional electrophysiological properties. A 2006 study evaluated patients undergoing LA ablation for accessory pathways, paroxysmal AF and persistent AF.³³ Chamber size was larger in the group with persistent AF than in the group with paroxysmal AF, but other parameters thought to be associated with AF likelihood (the shortening of refractoriness and slowing of conduction velocity) did not differ between patients with paroxysmal and persistent AF.³³

A larger LA predicts AF at the end of AF ablation. The fibrillation number (incorporating LA size) was significantly higher among patients in whom AF was inducible after AF ablation than among those in whom AF was not inducible.³⁸ In addition, there was a linear relationship between fibrillation number and AF inducibility (the latter being quantified by the pacing cycle length required to induce AF).³⁸ A 2019 meta-analysis of studies evaluating ablation for persistent AF showed that LA size is the most important predictor of freedom from AF; in fact, the authors of that analysis reported a 4% reduction of freedom from AF with every 1-mm increase in LA diameter.³⁴ Contrary to clinical intuition, the duration of AF, as measured by the distinction between persistent and longstanding persistent AF, did not predict freedom from AF.³⁴ In a study of patients undergoing a redo ablation for AF and presenting with durable PVI, LA size was the only significant predictor of ablation outcome.⁴³

Atrial defibrillation threshold is theoretically related to critical atrial mass. A higher atrial defibrillation threshold, indicating a larger critical mass for sustaining AF, was associated with increased AF recurrence in a 2022 case series of AF ablation patients with non-paroxysmal AF.⁴⁴ A high threshold was a risk factor for post-catheter ablation recurrence in the group of patients with longstanding persistent AF.⁴⁴

More Ablation (Less Electrically Contiguous Atrial Area) is Associated With Less AF

AF can occur in electrically contiguous areas of cardiac atrial tissue. It seems logical that more extensive atrial ablation, leaving smaller electrically contiguous areas available for the fibrillatory process to sustain, would be favourable. Indeed, this is supported by the highly efficacious surgical maze ablation procedures, which represent the most extensive means of volume reduction of electrically active atrial tissue.^45–47 This demonstrates that the reduction of effective atrial area available for the fibrillatory process to sustain is of benefit, supporting contiguous atrial size as a factor that determines the sustainability of the fibrillatory process.

In 2014, more ablation lines were linked to greater reductions in AF inducibility and the atrial defibrillation threshold.⁴⁸ A randomised trial of persistent AF patients found that extended PVI (with greater atrial volume electrically isolated) was more effective in reducing symptomatic AF recurrence than PVI and ganglionic plexus ablation.⁴⁹ A 2019 meta-analysis suggested that LA appendage and posterior wall isolation are associated with superior long-term freedom from AF compared with other ablation strategies (including lines, complex fractionated atrial EMGs and ganglionic plexus ablation).³⁴

Although posterior LA wall isolation reduces the critical mass for the maintenance of AF, this has not translated to improvements in clinical outcomes.⁵⁰ The addition of posterior wall isolation to PVI is not beneficial in an unselected population with persistent AF, but may be beneficial in certain patients, such as those with rapid atrial activity on the posterior wall.^51,52 Posterior wall isolation may have distant effects, including decreasing the rotors and multiple wavelets in the anterior wall, inferior wall and LA appendage.⁵⁰ Together, these findings suggest that there is an interaction between individual patient-level electroanatomical phenotype, the specific anatomical location of the ablation and AF behaviour, indicating that a general approach of prespecified volume reduction in all-comer patients is an inadequate approach for AF management.

In paroxysmal AF patients undergoing PVI, those without recurrence had a significantly larger postablation low-voltage encircled area (32% of the total LA surface area) than those with recurrence (21%).⁵³ A randomised trial of complete PVI versus deliberately incomplete PVI demonstrated lower recurrence in the complete PVI group (62% versus 79%).⁵⁴ Interestingly, when this group of patients was invasively restudied, the rates of electrical connection in both groups were higher than the recurrence of AF in both groups.⁵⁴ This suggests that complete electrical disconnection is perhaps not the only mechanism of PVI, and that pleiotropic effects may also be at play.

Combined endocardial and epicardial ablation has been reported to have favourable effects on AF burden.⁵⁵ This was concluded from a randomised controlled trial in which the group randomised to epicardial ablation received substantial posterior LA epicardial ablation and subsequent transcatheter endocardial ablation, whereas the control group underwent the standard transcatheter approach only (PVI, LA roofline, CTI line).⁵⁵ This finding may be due, at least in part, to a reduction in the atrial tissue mass available to sustain AF.

Heterogeneous Processes Result in Downstream Atrial Enlargement, but the Principles of D_i Can be Applied Across all Aetiologies

Many underlying physiological and pathological processes can result in atrial dilation. Regardless of the underlying process, the resultant state of AF can be understood by the described principles of D_i. A vast array of pathologies has the final common physiological endpoint of dilated atria. These pathologies can be acute or chronic, secondary to pressure or volume overload, due to direct infiltration or connective tissue abnormality and have different histopathologic changes, such as fibrosis, hypertrophy or inflammation. These changes can be reversible or irreversible.

There is growing evidence that genetic and epigenetic factors contribute to AF risk.⁵⁶ Genes regulating repolarisation, excitability, automaticity, fibrosis and calcium handling are implicated in AF, although there is no clear direct understanding of the link between genotype, epigenetics and the AF phenotype.⁵⁶ Various genetic and epigenetic signals combine for clinical AF to occur, and the resulting arrhythmia can be described according to physical principles, regardless of the inciting upstream contributors. As described earlier, a major advantage of an approach based on the principles of D_i is that it does not assume a certain gene, clinical risk factor or mechanistic process, but instead characterises an individual’s AF as a biophysical process.

Electrical remodelling can also occur independent of atrial dilation, indicating that atrial size is not the only important predictor of AF behaviour.²⁶ Depending on the cause, atrial structural changes can manifest varying degrees of electrical heterogeneity and therefore varying electrical ‘substrate’ for AF. Atrial fibrosis may contribute to both atrial structural and electrical remodelling in AF.⁵⁷ Fibroblasts comprise 75% of cardiac cells by number but only 10–15% by volume, so although fibroblast proliferation of increases atrial size, the effect of fibrosis is much more profound.⁵⁸ Fibroblasts produce excess extracellular matrix proteins, which can interrupt cardiomyocyte–bundle continuity, leading to local conduction disturbances and re-entrant arrhythmia. Fibroblasts can also electrically couple to myocytes and increase the heterogeneity of repolarisation, increasing the chances of re-entry.⁵⁸

Why Personalised Atrial Volume Reduction May Improve AF Outcomes

A logical conclusion of the notion of atrial size being of key importance in the behaviour of AF is to simply reduce atrial size as much as possible. Problematically, this discounts the important mechanical function of the atria. In sinus rhythm, atrial contraction provides an ‘atrial kick’ that leads to approximately 10% of cardiac output. The loss of this mechanical function in AF can be a reason for symptoms. Preservation of LA mechanical function needs to be considered if substantial atrial debulking ablation procedures are considered for AF. In one study of extensive ablation, abnormalities in LA mechanical function occurred if the lesion area exceeded 25% of the total LA surface area.⁴⁹ In addition, extensive atrial ablation, such as with surgical maze or endocardial–epicardial combined procedures, is more invasive with higher risks of complications and so is not suitable for all patients with AF. Finally, extensive linear ablation increases the risk of organised macro-re-entrant atrial arrhythmias.

It would be a fundamental advance to determine, in advance of ablation, the exact degree of atrial size reduction required to change the behaviour and outcomes of AF. In some patients, this information may render intervention futile, whereas in others it may guide intervention to minimise collateral harm. D_i may assist in this personalised approach to AF ablation.

Role of Spatial Synchronisation in AF Maintenance

Although the assessment of spatial synchronisation using the notion of the breakdown of similarity of EGMs with distance has been reported, ‘correlation length’ has not previously been systematically applied in the AF literature. The length constant,⁵⁹ activation space constant and wavelength have been reported, with marked differences in estimates depending on the method used.^25,60–63 These parameters do not conceptualise AF according to the principles of spatiotemporal turbulence, and the marked variability is potentially due to the limitation of modelling, relying on the notion of re-entrant wavelets as the entity that sustains AF. In the multiple wavelet theory of AF, tissue wavelength determines the minimum size of a re-entrant wavelet.¹⁷ Therefore, tissue wavelength would be expected to determine the minimum distance over which sequences of activations remain similar during AF. Contrastingly, in the mother rotor or stable rotor theory of AF, AF is sustained if the tissue is larger than the core of the stable rotor. Instead, AF may be analysed as a system of spatiotemporal turbulence, sustained due to spiral wave breakup caused by dynamic instabilities. Using this framework, AF may sustain for a longer period if there is a greater number of independent areas in the system. The correlation length, or the length over which the signals are correlated, determines the quantity of independent regions for a system of a given size. The two-point correlation length (ξ₂) describes the extent of cross-correlation or similarity between two neighbouring sites; it has recently been measured in AF and reported to be significantly higher in epochs of spontaneously terminating simulations versus sustained simulations of AF and VF.²¹

Wavelength: Inconsistent Relationship with Clinical Behaviour and Challenges of Measurement

Tissue wavelength has been related to the inducibility of AF and the effect of antiarrhythmic medications in animal studies. In 1988, Rensma et al. determined in dogs that the wavelength of the atrial impulse is equal to the refractory period multiplied by the conduction velocity.⁶¹ There was a gradient of wavelengths required for induction of increasingly disorganised arrhythmias; greatest was normal rhythm (14–18 cm), followed by premature beats (graded according to the degree of prematurity), then premature beats that evoked rapid repetitive responses (critical shortening of the wavelength below 12.3 cm), then atrial flutter (induced below 9.7 cm) and then AF (induced below 7.8 cm).⁶¹ The tissue wavelength measurement would plausibly relate to the correlation length measurement, because electrical signals measured from tissue activated by the same wave would be expected to be correlated, although there are substantial differences in the methods of measurement and the underlying concept. The effects of acetylcholine, propafenone, quinidine and d-sotalol on wavelength, refractory period and conduction velocity have also been studied; the combination of refractory period and the conduction velocity (as expressed in the wavelength) was found to be more reliable than the individual components in predicting the induction of the different arrhythmias and the effect of antiarrhythmic drugs.⁶¹

One problem with the use of wavelength to predict AF behaviour is that it cannot be measured directly with standard electrophysiological catheters or tools. Wavelength can only be measured directly using high-resolution electrical mapping for simultaneous conduction velocity and effective refractory period estimation.⁶⁴ This can only be performed with experiments involving a large number of electrodes, or from computer simulations to track local action potentials (depolarisation and repolarisation) over the entire atria.⁶³ This is the gold standard of wavelength measurement. Other analytical methods have been used to calculate wavelength, with conflicting results.⁶³ One such method for calculating wavelength is placing two electrodes in the free wall of the right atrium and pacing from these electrodes in sinus rhythm to measure the conduction velocity between them, as well as the effective refractory period.⁶² However, a major flaw of this method is the inability to measure wavelength in AF. There have been contradictory reports, with some studies reporting that wavelength is not an independent predictor of AF inducibility and others reporting that longer wavelength is associated with fewer spontaneous or induced atrial arrhythmias.^62,64

Activation Space Constant

The notion of an activation space constant, the area over which signals in AF are well correlated, was developed from human right atrial studies as a measure of the wavelet’s domain.^25,60 With increased distance between recording sites, the sequences of activation would be expected to become less similar as it becomes less likely that the two sites are excited by the same wavelet. In AF, the correlation between electrodes was found to decrease with interelectrode spacing, which did not hold true for sinus rhythm and atrial flutter (conditions with a single wave front in which the wave front should be associated).^25,60 The authors of those studies reported that correlation as a function of distance during AF fit a decaying exponential and reported the activation space constant as this measure.^25,60 The activation space constant was found to relate to the clinical course of AF, being shortest in chronic AF, longer in paroxysmal AF (PAF) and longest in newly induced AF (no clinical history of AF).²⁵ Both procainamide and adenosine during AF altered the space constant.⁶⁰ These findings support the notion that the propensity for AF to spontaneously terminate reduces with smaller wavelet domains.

Activation space constant values differ from tissue wavelength, but are consistent with historic estimates of wavelength. The estimated minimal critical wavelength necessary to support AF in humans has historically been estimated at 12 cm.¹⁷ The activation space constant was derived from a linear array of electrodes in the right atrium and therefore reflects the mean cross-sectional diameter of a re-entrant wavelet, not the circumference of a re-entrant loop.^25,60 If a single wavelet is assumed to be a circle, the minimum circumference is determined by tissue wavelength and the average cross-section is approximated by its diameter. With this assumption, the activation space constant may then be viewed as proportional to tissue wavelength, with wavelength equal to 2π (activation space constant), giving a mean (±SD) estimate of tissue wavelength of 13 ± 4 cm.^25,60 This is consistent with historical estimations.¹⁷

Although the papers cited above were published in the mid-1990s and showed promise, there were multiple competing paradigms in the AF literature at the time.^25,60 In 1998, the notion of pulmonary vein drivers was reported.² This motivated the development of PVI as an ablation technique. Subsequently, in 2002, the concept of rotors was developing, which led to the development and study of focal ablation targets.^5,9,65 Ultimately, neither of these theories has resulted in ablative therapies with adequate long-term outcomes, especially in patients with non-paroxysmal AF.⁶⁶

Correlation Dimension

The notion of EGMs being analysed as part of non-linear dynamic systems has been investigated in several studies using a chaos theory-based technique known as correlation dimension. Correlation dimension quantifies the level of randomness of a strange attractor and has been used to assess the organisation and complexity of EGMs. Correlation dimension and correlation entropy were used to classify AF based on unipolar epicardial EGMs recorded at the free wall of the right atrium.⁶⁷ Correlation dimension was used by Censi et al., who found that measurement of organisation during AF should be based on estimation of the non-linear coupling between two sites.⁶⁸ The estimated correlation dimension was used to determine that a higher right atrial organisation before catheter ablation was associated with AF termination within the LA, and that during ablation the organisation level increased prior to termination.⁶⁹ Correlation dimension was also found to discriminate between different levels of atrial activity organisation during AF pacing therapies.⁷⁰ Correlation ‘dispersion’ has also been studied as an index to automatically estimate fractionation of bipolar AF EGMs.⁷¹

Areas of Repetitive Activity as a Measure of Synchronisation

It has been reported that synchronised regions (regions of repetitive activity [REACT]) were largest in atrial tachycardia, smaller in terminating AF and smallest in non-terminating AF, in keeping with the notion that smaller synchronised regions (i.e. poorer spatial synchronisation) determine clinical response.⁷² Fundamentally, that study involved investigation of the correlation of unipolar EGMs in shape over time, such that synchronous EGMs (with a higher REACT value) were defined as more self-similar repeating EGM patterns (i.e. similar EGMs with each consecutive AF cycle). The authors evaluated both the global REACT value and the percentage area with a high (³0.6) REACT value and found that both were higher in terminating than sustained AF during acute AF ablation.⁷² These findings demonstrate promising results using the notion of synchronisation to characterise AF. However, the methods used evaluate temporal synchronisation (i.e. self-similarity over time) and are fundamentally different to the notion of D_i, which incorporates spatial synchronisation (i.e. self-similarity with increasing distance).

Correlation Length

Correlation length was first cited in the cardiac fibrillation literature in 1993 and has been measured in pig VF (1993), guinea pig VF (2003), and human VF (2002).^20,59,73 A study of goats in 2000 reported that the area covered by a ‘coherent patch of fibrillating atrial tissue’ during AF may correspond to the extension of the spatial domain of a fibrillation wavelet.⁷⁴ Larger association lengths resulted from fewer and larger re-entrant circuits and, interestingly, there were important differences in the spatial organisation between the right and left atria.⁷⁴ In another study, cibenzoline (an antiarrhythmic medication) was found to enhance the spatial organisation of AF by increasing the correlation length in goats.⁷⁵ In 2017, a computer-simulated ablation in a 3D AF model noted that the correlation length scale increased when AF organised to atrial flutter.⁷⁶

To date, there have been no studies evaluating correlation length in the human LA during AF. A similar metric has been measured in humans during VF (reported as the length constant) and in human right atria (reported as the activation space constant).^25,59,60 A major advantage of correlation length is that it relates AF, a system that may be described using the principles of spiral defect chaos, to multiple other comparable systems in nature. It is not restricted to a certain physical process (rotors, wavelets, focal discharges), nor does it provide explicit evidence for a particular mechanism of AF.²⁰ It can be feasibly measured using standard clinical equipment, and it has the potential to be measured in real time, giving it a practical application during catheter ablation to guide therapy.

Alternative Approaches to the Characterisation of Atrial Fibrillatory Dynamics

Other means of characterising AF dynamics that have demonstrated promising results are Shannon entropy, dominant frequency (DF) and Granger causality analysis. Shannon entropy was proposed as a mechanistically based approach to assist in rotor mapping.⁷⁷ DF is determined by spectral analysis and frequency mapping to localise sites of high-frequency activity during AF, and sites of high DF have been thought to be important in the maintenance of AF.⁷⁸ These approaches assume that AF is sustained by rotors; however, rotor ablation has not improved clinical outcomes.^9,10 Further, mapping studies of persistent AF have reported that localised drivers or sustained rotors were not detected, questioning this mechanistic theory.⁷⁹ Granger causality, a tool for quantifying causal relationships between complex time series, has been validated in human AF to measure global fibrillation organisation and map rotational drivers.⁸⁰ Importantly, this work highlights that the interdependence of neighbouring regions is a marker of AF organisation, in keeping with the key principle that spatial synchronisation is important in AF dynamics. Further research is needed to link novel approaches to existing paradigms, and a variety of approaches may be beneficial to further our understanding of the atrial fibrillatory process, as well as to improve clinical outcomes for patients with AF.

Clinical Implications

D_i is a metric that combines the fundamental properties of the fibrillatory process, system size and spatial synchronisation, and is readily measurable in clinical practice. It has been associated with both acute termination of AF and outcomes after ablation, and can now be measured in real time in the electrophysiology laboratory.²¹

D_i as a Putative Treatment Target

Clinical outcomes of AF have stagnated despite advances in mapping and ablation technology, and personalising AF ablation therapy may be critical to incremental improvement in ablation outcomes. D_i represents a means of describing individual patient electroanatomical phenotype. An individualised ablation strategy based on this metric requires prospective evaluation.

Atrial size may be modulated via relief of volume and pressure overload with diuresis and medical therapy for heart failure, valve intervention where appropriate, treatment of hypertension, lifestyle interventions including weight loss and exercise, treatment of sleep apnoea, AF ablation and surgical atrial volume reduction.^81–85 Further to the simple anatomical size of the atria, the surface area of electrically available contiguous atrial tissue for AF to sustain is more specifically relevant as a determinant of fibrillatory behaviour. This is evidenced by the success of the surgical maze procedure (i.e. very extensive ablation) and supported by simulations of AF in which self-terminating behaviour depends on the size, shape and boundary conditions.^35,36,86 However, linear ablation in an all-comer population with persistent AF has not been shown to reduce recurrent AF and, therefore, a more nuanced use of ablation lesion sets requires further thought.¹⁸

Spatial synchronisation is hypothesised to reflect the local electrophysiological milieu, which encompasses the local cellular and membrane function, speed of conduction and local fibrosis. It is possible that antiarrhythmic agents may impact spatial synchronisation, although this has not been directly studied. The autonomic nervous system, inflammation and genetic factors may be involved in the development of spatial dyssynchrony and may represent treatment targets to improve synchronisation (i.e. propensity to self-terminate). As we further understand the distribution of spatial synchronisation in AF across the complex 3D electroanatomical environment of the cardiac atria, local anatomical areas, such as those traditionally thought to represent drivers or triggers of AF may be important. It remains to be studied how ablation may impact spatial synchronisation, if at all; PVI reduces atrial surface area for AF to sustain, which may be a key reason for procedural success. There may be additional impacts of ablation on spatial synchronisation in local and remote locations in the cardiac atria after ablation.

We would hypothesise that incremental system size reduction may be necessary in patients with high D_i to achieve rhythm control. Given the power law relationship in Equation 3, we would hypothesise that a small reduction in system size may lead to a substantial increase in the likelihood of AF termination (i.e. a decrease in episode length). Therefore, personalised ablation lesion sets with the aim of reducing electrically contiguous surface area warrant specific evaluation. This will be of particular importance in the new era of pulsed field ablation, in which large areas of atrial tissue can be ablated more rapidly and with lower risk of collateral tissue damage.

Importantly, the ratio of system size to spatial synchronisation may emerge as a treatment target to develop personalised treatment approaches for an individual’s AF pathobiology, to maximise efficacy and minimise futility and harm. Artificial intelligence is a rapidly expanding area of interest that can quickly evaluate large volumes of information and has the capability to advance the detection, diagnosis, risk stratification and treatment of AF.⁸⁷ Artificial intelligence is ideally positioned to further this work, and an artificial intelligence-guided treatment algorithm would require prospective clinical study.

Examples of D_i Measured in Clinical Practice

A clinical example is shown in Figure 3: two patients, both with persistent AF and mild left ventricular systolic dysfunction. The first patient has a low mean D_i of 4 and had no recurrence in the first year after AF ablation, despite many comorbidities, including severe LA enlargement. In contrast, the second patient, with a high mean D_i of 235 but only moderate LA enlargement and no other comorbidities, had 37 recurrences in the first year of follow-up, despite the use of amiodarone, with many episodes lasting multiple days. This highlights the potential incremental information that can be gained from detailed characterisation of the fibrillatory process at an individual level. Such information could be studied in future as a tool to tailor personalised ablation strategies to maximise success while limiting complications from unnecessary additional ablation.

Conclusion

System size and spatial synchronisation determine the behaviour of sufficiently large chaotic systems. There is empirical and clinical evidence that atrial size and spatial synchronisation of atrial electrical signals may determine the behaviour of AF, thereby indicating that the principles of spiral defect chaos may be applied to understand the cardiac fibrillatory process.