Elimination of scroll waves in three-dimensional cardiac-tissue models by control currents confined to a two-dimensional surface

Mulimani, Alageshan, and Pandit (Phys. Rev. Research 2, 023155, 2020) showed that a convolutional neural network (CNN) can detect spiral waves in pseudocolor plots of the transmembrane potential of two-dimensional (2D) cardiac-tissue models, and that a heat map of the spiral-wave cores then guides low-amplitude Gaussian current pulses that eliminate both unbroken and broken spiral waves - a mathematical analog of low-energy defibrillation. They verified that their CNN detects scroll waves (the three-dimensional, 3D, generalization of spiral waves) in 3D simulation domains, but stated that the elimination of such 3D scroll waves by applying currents only on a 2D surface of a 3D domain remains a significant challenge. This challenge remains open: later work improved 2D elimination (U-Net-guided currents, reinforcement learning, genetic-algorithm-optimized pacing, adaptive optogenetic illumination), demonstrated 3D scroll-wave reconstruction from surface observations, achieved low-energy 3D termination by non-localized global field pulses, and demonstrated closed-loop in-vitro machine-learning-plus-optogenetics control in 2D monolayers - but no published study eliminates scroll waves in a 3D cardiac-tissue domain using localized, low-amplitude currents restricted to a 2D surface in walls thick relative to the electrotonic space constant, whether guided by learned detection or not (surface-confined elimination has so far been achieved only in thin slabs - Sridhar and Sinha, EPL 81, 50002 (2008); Shajahan, Nayak, and Pandit, PLoS ONE 4, e4738 (2009)). The problem posed here is to determine whether such surface-confined elimination is possible, to construct a working protocol, and to characterize the parameter and geometry regimes in which it succeeds or fails.

Background

Life-threatening ventricular arrhythmias - ventricular tachycardia (VT) and ventricular fibrillation (VF) - are associated with rotating electrical excitation waves in heart muscle. In mathematical models, cardiac tissue is described by reaction–diffusion partial differential equations (PDEs) of the monodomain type, in which a transmembrane potential $V(x,t)$ evolves under ionic current terms $f(V,g)$ and diffusion; the ionic kinetics range from simple two-variable caricatures such as the Aliev–Panfilov model to ionically realistic human-ventricular models such as the ten Tusscher–Panfilov (TP06) model. In two dimensions these PDEs support spiral waves, whose repeatedly broken and re-formed multiple-wave states (spiral turbulence) are the standard mathematical analogs of VF; in three dimensions the rotating waves become scroll waves, organized around one-dimensional curves of phase singularity called filaments, and their turbulent states are the analogs of VF in the thick ventricular wall. Conventional defibrillation delivers a strong electric shock that painfully and sometimes damagingly resets the whole tissue, motivating a research program on low-energy alternatives: Sinha, Pande, and Pandit (Phys. Rev. Lett. 86, 3678, 2001) suppressed spiral turbulence in a VF model by low-amplitude stimuli delivered on a coarse mesh of electrodes; Luther et al. (Nature 475, 235, 2011) showed that pulsed electric fields can terminate arrhythmias with far less energy than a single shock; and optogenetics - expressing the light-gated ion channel channelrhodopsin in cardiac myocytes so that illumination acts as a spatially patterned current source - enables light-induced termination of spiral-wave arrhythmias in vitro (Bingen et al., Cardiovasc. Res. 104, 194, 2014) and optogenetic defibrillation of 3D human whole-heart simulations (Bruegmann et al., J. Clin. Invest. 126, 2016). Within this program, Mulimani, Alageshan, and Pandit (Phys. Rev. Research 2, 023155, 2020) contributed a deep-learning-assisted control scheme for 2D domains: a CNN classifier distinguishes pseudocolor images of $V$ with spiral waves from those without; a heat map of the spiral-wave cores is built from the CNN outputs; and low-amplitude Gaussian current pulses, applied only where the heat map has high intensity, eliminate both unbroken and broken spiral waves in several cardiac models, with phase diagrams in the (pulse duration $t_{def}$, spatial width $\sigma$) plane charting where the Gaussian-control scheme succeeds for each current amplitude $I_{def}$. Their study also demonstrated CNN-based detection of scroll waves in 3D simulation domains and an optogenetic variant requiring a minimum of about 15% light-modified cells. The unresolved step, stated explicitly in that work's conclusions, is actuation in three dimensions: currents applied on a two-dimensional surface of a 3D domain - the geometric situation of an epicardial sheet or illuminated heart surface facing a thick wall - have not been shown to eliminate scroll waves, and surface-applied currents are expected to attenuate with depth, so it is unclear whether low-amplitude surface control can reach and annihilate filaments buried in the bulk.

Problem Statement

Consider a three-dimensional domain $\Omega$ (for example, a slab or a ventricular-geometry domain) governed by a monodomain reaction–diffusion model for cardiac tissue - reaction–diffusion equations for the transmembrane potential with ionic current source terms, ranging from simple two-variable excitable models such as the Aliev–Panfilov model to ionically realistic human-ventricular models such as the ten Tusscher–Panfilov (TP06) model - whose state supports scroll waves, i.e., rotating three-dimensional excitation waves organized around one-dimensional phase-singularity curves called filaments, or scroll-wave turbulence (their multiply broken and re-formed states). Control is restricted to a two-dimensional surface $S$ of the domain (for example, an epicardial boundary plane, i.e., a surface of the heart wall): low-amplitude, spatially localized time-dependent stimulus currents with prescribed spatial profiles may be applied only at points of $S$, in the spirit of the low-amplitude Gaussian control pulses of the source study's 2D scheme; global defibrillation-strength shocks or transmural depolarization of the entire wall do not qualify as admissible control. The problem is to determine whether, and under what conditions, such surface-confined currents can eliminate all scroll waves in $\Omega$ - that is, drive the tissue to quiescence (a spatially uniform resting state) or to a non-reentrant rhythm (a rhythm not sustained by rotating waves) that persists after the control is switched off. Such elimination may be impossible - established by arguments and, where feasible, by direct numerical simulations showing that no admissible surface-confined current protocol terminates the scroll waves - or it may be possible, in which case the problem is to exhibit a concrete control protocol (amplitudes, durations, spatial profiles, surface-coverage pattern, and timing relative to the ongoing wave dynamics), to demonstrate its success in direct numerical simulations on 3D domains, and to characterize systematically the regimes - in the control parameters (current amplitude $I_{def}$, pulse duration $t_{def}$, spatial extent, coverage fraction) and in the domain properties (wall thickness relative to the electrotonic space constant, i.e., the length scale over which sub-threshold voltage disturbances decay passively through the tissue; anisotropy, i.e., direction-dependent conductivity; heterogeneity; ionic model) - in which elimination succeeds or fails. The question concerns the general 3D surface-confined control problem rather than a single engineered case: the elimination must accordingly be established in more than one model, geometry, or heterogeneity pattern, including at least one wall thick relative to the electrotonic space constant.

References

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