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Circ Arrhythm Electrophysiol. 2019 Dec;12(12):e007569. doi: 10.1161/CIRCEP.119.007569. Epub 2019 Dec 9.

Renewal Theory as a Universal Quantitative Framework to Characterize Phase Singularity Regeneration in Mammalian Cardiac Fibrillation.

Author information

College of Medicine and Public Health (D.D., D.C., A.D.M., A.N.G.), Flinders University of South Australia, Adelaide, SA, Australia.
College of Science and Engineering (D.D., M.S., K.P.), Flinders University of South Australia, Adelaide, SA, Australia.
Department of Cardiology, Asklepios Clinic St Georg (P.K., C.M.).
Department of Cardiovascular Medicine, Flinders Medical Centre, Adelaide, SA, Australia (A.L., L.D., A.D.M., A.N.G.).
Department of Medicine, Montréal Heart Institute and Université de Montréal, Canada (F.X., M.A., S.N.).
Icahn School of Medicine, Mount Sinai, NY (B.S., F.G.A.).
School of Mathematical Sciences, University of Adelaide, SA, Australia (L.M.).
Department of Cardiology, University Medical Centre, Hamburg, Germany (S.W.).



Despite a century of research, no clear quantitative framework exists to model the fundamental processes responsible for the continuous formation and destruction of phase singularities (PS) in cardiac fibrillation. We hypothesized PS formation/destruction in fibrillation could be modeled as self-regenerating Poisson renewal processes, producing exponential distributions of interevent times governed by constant rate parameters defined by the prevailing properties of each system.


PS formation/destruction were studied in 5 systems: (1) human persistent atrial fibrillation (n=20), (2) tachypaced sheep atrial fibrillation (n=5), (3) rat atrial fibrillation (n=4), (5) rat ventricular fibrillation (n=11), and (5) computer-simulated fibrillation. PS time-to-event data were fitted by exponential probability distribution functions computed using maximum entropy theory, and rates of PS formation and destruction (λfd) determined. A systematic review was conducted to cross-validate with source data from literature.


In all systems, PS lifetime and interformation times were consistent with underlying Poisson renewal processes (human: λf, 4.2%/ms±1.1 [95% CI, 4.0-5.0], λd, 4.6%/ms±1.5 [95% CI, 4.3-4.9]; sheep: λf, 4.4%/ms [95% CI, 4.1-4.7], λd, 4.6%/ms±1.4 [95% CI, 4.3-4.8]; rat atrial fibrillation: λf, 33%/ms±8.8 [95% CI, 11-55], λd, 38%/ms [95% CI, 22-55]; rat ventricular fibrillation: λf, 38%/ms±24 [95% CI, 22-55], λf, 46%/ms±21 [95% CI, 31-60]; simulated fibrillation λd, 6.6-8.97%/ms [95% CI, 4.1-6.7]; R2≥0.90 in all cases). All PS distributions identified through systematic review were also consistent with an underlying Poisson renewal process.


Poisson renewal theory provides an evolutionarily preserved universal framework to quantify formation and destruction of rotational events in cardiac fibrillation.


atrial fibrillation; phase singularities; renewal theory; stochastic processes; ventricular fibrillation


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