M/M/Infinity Birth-Death Processes – A Quantitative Representational Framework to Summarize and Explain Phase Singularity and Wavelet Dynamics in Atrial Fibrillation

Abstract

RationaleA quantitative framework to summarize and explain the quasi-stationary population dynamics of unstable phase singularities (PS) and wavelets in human atrial fibrillation (AF) is at present lacking. Building on recent evidence showing that the formation and destruction of PS and wavelets in AF can be represented as renewal processes, we sought to establish such a quantitative framework, which could also potentially provide insight into the mechanisms of spontaneous AF termination.ObjectivesHere, we hypothesized that the observed number of PS or wavelets in AF could be governed by a common set of renewal rate constants λf (for PS or wavelet formation) and λd (PS or wavelet destruction), with steady-state population dynamics modeled as an M/M/∞ birth–death process. We further hypothesized that changes to the M/M/∞ birth–death matrix would explain spontaneous AF termination.Methods and ResultsAF was studied in in a multimodality, multispecies study in humans, animal experimental models (rats and sheep) and Ramirez-Nattel-Courtemanche model computer simulations. We demonstrated: (i) that λf and λd can be combined in a Markov M/M/∞ process to accurately model the observed average number and population distribution of PS and wavelets in all systems at different scales of mapping; and (ii) that slowing of the rate constants λf and λd is associated with slower mixing rates of the M/M/∞ birth–death matrix, providing an explanation for spontaneous AF termination.ConclusionM/M/∞ birth–death processes provide an accurate quantitative representational architecture to characterize PS and wavelet population dynamics in AF, by providing governing equations to understand the regeneration of PS and wavelets during sustained AF, as well as providing insight into the mechanism of spontaneous AF termination.

Reconceptualising Atrial Fibrillation Using Renewal Theory: A Novel Approach to the Assessment of Atrial Fibrillation Dynamics

“…[ 52 ] In both examples, consistent with our data (Figure 2), an exponential distribution of PS lifetimes and rotors can be seen. [ 17 , 18 ] This is consistent with the notion of quasi-stationary decay dynamics of unstable re-entrant circuits. PersAF = persistent AF.…”

“…This has recently been examined by consideration of the autocorrelation of PS interval series, which converge to zero in the case of non-zero interval lags. [ 55 ]…”

“…Thus far, we have demonstrated it is possible to use the renewal approach in simulated, experimental and human AF. [ 17 , 18 ]…”

Atrial fibrosis and substrate based characterization in atrial fibrillation: Time to move forwards

“…In addition, our group has recently proposed a novel technique to quantify fibrillatory dynamics in patients with AF 152,153 . We demonstrated that the RA inter‐formation times, RA inter‐destruction times and lifetimes of RA in animal models and human models with persistent AF follow an exponential distribution, consistent with an underlying renewal process 152 .…”

Spatial concentration and distribution of phase singularities in human atrial fibrillation: Insights for the AF mechanism

“…An issue that is not fully resolved is whether basket‐type catheters may be effective in the mapping of PSs. While in general, it has been suggested that basket catheters may have the theoretical risk of false detection of PSs, we have recently shown that rate constants of PS formation and destruction detected by basket catheters can accurately predict the number and population distribution of observed PS, in line with a property of scale invariance under transformation 47 . This issue would therefore be considered as under debate in the literature at the current time.…”