Ventricular fibrillation, the disorganized electrical activity that causes sudden cardiac death, can be modeled as spiral- and scroll-wave turbulence in excitable media representing cardiac tissue. GPU-based direct numerical simulations of two-variable cardiac models (Panfilov and Aliev-Panfilov) showed that the spatial spectra EV(k)E_V(k) and Eg(k)E_g(k) of the transmembrane potential and recovery variable follow approximate power laws and that structure functions suggest multiscaling, but with exponents far less well determined than in fluid turbulence. Three linked questions remain open: (1) are the approximate spectral exponents and the extended-self-similarity multiscaling exponent ratios universal, i.e. model- and parameter-independent within quantified error bars, once simulations with larger domains, finer resolution, and additional models are performed, and is genuine multiscaling present at all; (2) can a phenomenological scaling theory analogous to Kolmogorov's 1941 (K41) theory of fluid turbulence be formulated for spiral- and scroll-wave turbulence, identifying the fundamental parameters that would govern it; and (3) can these computational statistical properties be connected to experimental or clinical data on cardiac fibrillation?

Background

Cardiac tissue is an excitable medium: each cell undergoes a nonlinear excursion of its transmembrane potential VV (depolarization, plateau, repolarization, rest) followed by a slow recovery governed by a gating-type variable gg, and the electrical state diffuses from cell to cell through gap junctions. Waves of excitation propagating in such a medium can organize into rotating spiral waves (in two dimensions) or scroll waves (their three-dimensional counterparts, organized around curved filaments). When a spiral or scroll wave becomes unstable, it breaks up into a disordered, statistically steady state of many interacting broken waves — spiral- or scroll-wave turbulence — which is the in-silico counterpart of ventricular and atrial fibrillation, the arrhythmia responsible for sudden cardiac death. Spiral-wave breakup in excitable media was established in the early 1990s (Bär and Eiswirth, 1993), experimental drifting spiral waves in isolated cardiac muscle date to Davidenko et al. (1992), and statistics of defects in a reaction-diffusion system were studied by Hildebrand, Bär and Eiswirth (1995).

Classical fluid turbulence provides the analytical template the source invokes. In statistically steady, homogeneous, isotropic fluid turbulence, the energy spectrum E(k)E(k) (the contribution of each Fourier wavenumber kk to the kinetic energy) follows a power law E(k)∼k−5/3E(k) \sim k^{-5/3} over the inertial range between the forcing scale and the viscous dissipation scale, a result derived from Kolmogorov's 1941 dimensional phenomenology (K41; Kolmogorov, 1941). Structure functions Sp(r)=⟨∣δV(r)∣p⟩S_p(r) = \langle |\delta V(r)|^p \rangle of field increments scale as rζpr^{\zeta_p}; a single scaling exponent for all pp indicates simple scaling, while systematic pp-dependence of ζp\zeta_p (multiscaling, caused by intermittency) is seen in real turbulence. The extended-self-similarity (ESS) method enlarges the measurable scaling range by plotting SpS_p against Sp′S_{p'} instead of against rr; the slope yields the exponent ratio ζp/ζp′\zeta_p/\zeta_{p'}.

Rajany, Gupta, Panfilov and Pandit (arXiv:1705.09935, 2017) applied this fluid-turbulence toolkit, for the first time, to spiral- and scroll-wave turbulence in two two-variable models of cardiac tissue: the Panfilov model and the Aliev-Panfilov model, reaction-diffusion equations for VV and a slow recovery variable gg. Using large-domain direct numerical simulations (DNSs) on graphics processing units (domains of 4096×40964096 \times 4096 in 2D and 6403640^3 in 3D), they found that once turbulence is initiated by breaking a large spiral or scroll wave, the resulting state is statistically steady and, far from boundaries, statistically homogeneous and isotropic, with no external forcing required; the dimensionless ratio L/λL/\lambda of domain size to plane-wave wavelength acts as a Reynolds-number-like control parameter. They defined the electrical analogs of the energy spectrum, EV(k)E_V(k) and Eg(k)E_g(k), for the potential and recovery fields, and found them spread over decades of kk with approximate power laws: in 2D, mV≃−2.7m_V \simeq -2.7 (Panfilov) and −3.2-3.2 (Aliev-Panfilov), mg≃−1.9m_g \simeq -1.9 and −3.1-3.1; in 3D, mV≃−3.8m_V \simeq -3.8 and −3.9-3.9, mg≃−1.8m_g \simeq -1.8 and −4.4-4.4. These values are model-dependent at the resolution achieved, and the authors state explicitly that whether they are universal "requires detailed, high-resolution studies in other models and conditions", that the limited power-law ranges leave genuine multiscaling of the structure functions unsettled ("can be settled only by DNSs that use much larger domains and longer runs"), and that if the exponents are not universal — plausibly because of the many parameters of excitable-media models — then "we cannot develop, in any obvious way, the spiral-turbulence analog of the K41 phenomenological theory". The closest precedent for such a program is the Kuramoto-Sivashinsky equation, a pattern-forming PDE whose two-dimensional spectral properties were shown to be universal by Jayaprakash, Hayot and Pandit (1993). The authors also emphasize that none of their questions can be answered from direct experimental or clinical data yet, and express the hope that the study stimulates such experimental work.

Problem Statement

For the statistically steady, statistically homogeneous and isotropic spiral-wave (2D) and scroll-wave (3D) turbulent states of reaction-diffusion models of cardiac tissue, determine:

  1. (Universality) Whether the approximate power-law exponents of the spatial spectra EV(k)E_V(k) and Eg(k)E_g(k) of the transmembrane potential and recovery variable, and the multiscaling exponent ratios ζp/ζ2\zeta_p/\zeta_2 of their structure functions (as obtained by the extended-self-similarity method), are universal — that is, independent of the model (beyond the Panfilov and Aliev-Panfilov models studied, e.g. detailed ionic models), of parameter regimes, and of resolution, within quantified error bars — and whether genuine multiscaling is present at all. This requires direct numerical simulations with substantially larger domains, higher resolution, and longer runs than those of the source study, together with error analysis adequate to distinguish multiscaling from simple scaling.

  2. (Scaling theory) Whether a phenomenological scaling theory for spiral- and scroll-wave turbulence, analogous to Kolmogorov's 1941 theory for fluid turbulence, can be formulated: specifically, whether the observed spectral and structure-function scaling can be derived from a small set of fundamental governing parameters (candidates include the ratio L/λL/\lambda of domain size to plane-wave wavelength, the action-potential duration, and diffusivity), or, conversely, a supported argument showing that the many-parameter structure of excitable-media models precludes such a theory.

  3. (Experimental bridge) Whether these computational statistical properties can be connected to experimental or clinical data on cardiac fibrillation.

These three questions jointly serve one objective: establishing whether cardiac electrical turbulence admits the same kind of compact, universal, turbulence-style statistical characterization that K41 provides for fluid turbulence. The universality determination (1) supplies the empirical constraint any theory (2) must respect, and both generate quantitative predictions that the experimental comparison (3) would test.

References

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