Open problems in adjoint-optimized ultra-low-energy defibrillation: singularity-targeting costs, waveform realizability, optimization stability, and synthesis without full state knowledge

Adjoint optimization of the applied electric field was shown by Garzón and Grigoriev (Chaos 34, 113110, 2024) to reduce the energy required to terminate fibrillation in a two-dimensional cardiac tissue model by three orders of magnitude relative to low-energy antitachycardia pacing, with the mechanism identified as electric-field-driven annihilation of pairs of phase singularities inside vulnerable windows. That demonstration, however, is offline and idealized: it assumes a known tissue model and complete knowledge of the initial state, uses a cost functional that penalizes terminal spatial gradients rather than the singularities themselves, relies on a Nesterov-accelerated gradient scheme whose hyperparameters must be carefully tuned to remain stable, and produces waveforms whose sub-millisecond temporal structure is destroyed by even mild smoothing. The open problem is to extend this adjoint-optimization approach along the directions its authors explicitly pose — implementing and evaluating cost functionals that target phase-singularity elimination directly, and finding accelerated optimization schemes that are both faster and stable — and along further open directions motivated by the limitations and results the authors report: how much smoothing and physical hardware limitation the optimal waveforms tolerate while retaining efficacy, whether the need for complete state knowledge can be removed by state-estimation, model-reduction, or rapid approximate synthesis, and how much further the defibrillation energy can be reduced among protocols meeting a fixed defibrillation success criterion.

Background

Cardiac fibrillation is a state of irregular, uncoordinated electrical activation of the heart muscle in which rotating waves of excitation break up into spatiotemporally chaotic patterns, and terminating it quickly is clinically essential: ventricular fibrillation is lethal within minutes, while atrial fibrillation is a major cause of stroke and heart failure. The standard therapy is defibrillation — delivery of a strong electric shock that depolarizes a large fraction of the tissue and resets the rhythm. Such shocks require substantial energy and cause intense pain and potential tissue damage, which is particularly problematic for implantable cardioverter defibrillators (ICDs), the implanted devices that must deliver these shocks repeatedly on limited battery power. This motivates the search for low-energy alternatives.

One such alternative is low-energy antitachycardia pacing (LEAP), which terminates fibrillation with a sequence of comparatively weak electric pulses delivered through the tissue (far-field stimulation); Fenton and co-workers showed both computationally and experimentally that such pulse trains can terminate atrial fibrillation at energies well below a single high-energy shock. The tissue substrate for this work is usually described by reaction–diffusion models of cardiac electrophysiology; the three-variable Fenton–Karma model (Fenton and Karma, 1998) is a simplified ionic model in which a voltage-like fast variable and two gating variables reproduce realistic action-potential propagation, restitution, and spiral-wave dynamics at low computational cost. In two dimensions, reentrant rotating waves are organized around phase singularities — points where the phase of the excitation cycle is undefined, around which the wave tip revolves — and fibrillating (chaotic) states correspond to multiple singularities of alternating topological charge being continuously created and annihilated. Anatomical heterogeneities (regions of impaired conduction) combined with an applied electric field produce virtual electrodes — localized polarization patterns near the heterogeneity boundaries — which perturb the wave dynamics. Certain phases of the local excitation cycle, known as vulnerable windows, are marked by exceptionally high sensitivity to small perturbations.

Garzón and Grigoriev (2024) posed defibrillation as an optimal-control problem constrained by the partial differential equations (PDEs) of the tissue model. Writing the tissue state as a vector field ${\bf u}(t,{\bf r})$ obeying a reaction–diffusion equation with the applied electric field $E(t)$ (a spatially uniform field entering through boundary conditions) as the control, they minimized a cost functional $\mathcal{L} = \frac{\alpha}{2} \mathcal{N} + \frac{1}{2}\mathcal{M}$, where $\mathcal{N} = \int_0^T E^2(t)\,dt$ measures the delivered energy, and $\mathcal{M} = \sum_i \gamma_i \int_\Omega |\nabla u_i({\bf r}, T)|^2\,d\Omega$ penalizes spatial gradients of the state, integrated over the tissue domain $\Omega$, at the terminal time $T$ (small terminal gradients correspond to a quiescent, defibrillated tissue). The minimization used the adjoint method, in which a backward-in-time adjoint field $\boldsymbol{\lambda}$, propagated from the terminal condition $\boldsymbol{\lambda}(T) = R{\bf w}(T)$, yields the functional derivative $\mathcal{G}(t) = \alpha E(t) + {\bf b}^{\mathsf{T}}\boldsymbol{\lambda}(t)$ of the cost with respect to the control at every time using a single forward and a single backward integration, instead of one full simulation per perturbed time point. Gradient descent on $E(t)$ was accelerated with the Nesterov accelerated gradient (NAG) method, a momentum scheme in which the update combines the current gradient with an extrapolated previous step weighted by a momentum parameter $\beta$, with step size $\Delta s$.

Their results, for the two-dimensional Fenton–Karma model with circular nonconducting heterogeneities: (i) a single properly timed biphasic pulse can defibrillate where a LEAP pulse train fails; (ii) adjoint optimization reduces the required energy $\mathcal{N}$ by roughly three orders of magnitude relative to LEAP, by exploiting vulnerable-window sensitivity to drive nearby oppositely charged phase singularities into mutual annihilation (weak fields cannot create new singularities, only steer existing ones); and (iii) all optimal waveforms found feature high-frequency oscillatory structure (overall period of order 10 ms with sub-millisecond detail) that is essential — applying a Gaussian smoothing with width $\sigma \approx 0.1$ ms to the optimized field destroys its defibrillating efficacy. Defibrillation success was declared using a fixed criterion: the tissue reaching a quiescent state (transmembrane voltage $u_1 < 0.1$ everywhere) within a bounded time ($5P = 1050$ ms, with $P \approx 210$ ms the characteristic period of the fibrillatory activity) after the pulse.

The authors state explicitly what remains unresolved. The approach 'is impractical': it requires an accurate model of the tissue and complete knowledge of the fibrillating state at the initial time, and the optimization cannot run in real time. The NAG acceleration 'requires a careful choice of hyperparameters $\beta$ and $\Delta s$ to ensure stability', and whether adaptive implementations (variable $\beta$, $\Delta s$) can be both faster than plain gradient descent and stable 'remains to be explored'. The cost functional penalizes terminal spatial gradients, whereas defibrillation could equivalently be achieved by eliminating phase singularities, which could be targeted directly — for instance by replacing the integrand of the terminal penalty with a positive-definite density $\mu = |\nabla u_i \times \nabla u_j|^2$ (with $i \neq j$) that is strongly peaked at the spiral-wave cores — a modification they expect to yield 'even more efficient defibrillation protocols' but which 'is worth being explored further'. And they express 'little doubt' the energy could be reduced further by better initial guesses for the descent, improved optimizers, and hyperparameter tuning, without characterizing how far. They also observe that, in their model, fibrillation 'appears to be a long chaotic transient rather than a sustained state', so that in the absence of any bounded termination requirement 'the lowest defibrillation energy is zero' — which is why a fixed success criterion is part of any meaningful energy-floor question.

Problem Statement

For two-dimensional reaction–diffusion models of cardiac tissue (such as the Fenton–Karma model with conduction heterogeneities, where a spatially uniform applied electric field $E(t)$ acts as the control), extend the adjoint-optimization approach to ultra-low-energy defibrillation beyond the offline, idealized demonstration of Garzón and Grigoriev (2024) by addressing the following open questions. Each item is a separate open question rather than a component of a single joint deliverable:

1. Singularity-targeting costs. Defibrillation corresponds to the elimination of phase singularities, yet the demonstrated cost functional $\mathcal{L} = \frac{\alpha}{2} \mathcal{N} + \frac{1}{2}\mathcal{M}$ penalizes terminal spatial gradients of the state through $\mathcal{M} = \sum_i \gamma_i \int_\Omega |\nabla u_i({\bf r}, T)|^2\,d\Omega$. Implement and evaluate cost functionals that target phase singularities directly, and determine whether they yield defibrillation protocols with lower energy $\mathcal{N} = \int_0^T E^2(t)\,dt$ than the gradient-penalty baseline for the same tissue model, initial fibrillating state, and defibrillation success criterion.

2. Physical realizability. All optimized waveforms found contain fine temporal structure whose removal by Gaussian smoothing of width $\sigma \approx 0.1$ ms abolishes defibrillating efficacy. Characterize the trade-off between waveform smoothness (as constrained by realizable hardware bandwidth) and defibrillating efficacy, i.e., determine how much temporal smoothing the optimal protocols tolerate and whether physically realizable approximations of the optimized fields remain effective.

3. Optimizer stability and speed. The Nesterov accelerated gradient scheme used for the outer minimization becomes unstable for poorly chosen hyperparameters (e.g., $\beta = 0.9$ for certain iterate ranges). Establish whether adaptive accelerated-gradient schemes (with variable $\beta$ and step size $\Delta s$) exist for this optimization that are simultaneously faster than plain gradient descent and stable.

4. Synthesis without full state knowledge. The optimization requires an accurate tissue model and complete knowledge of the fibrillating initial state, and cannot be run in real time. Develop and validate state-estimation, model-reduction, or rapid approximate synthesis procedures that produce effective defibrillating fields when the tissue state and/or model parameters are unknown, or show under what conditions approximate states and models suffice.

5. Energy floor. The authors express 'little doubt' that, using better initial guesses, improved optimizers, and hyperparameter tuning, the required defibrillation energy can be reduced further, without characterizing how far. Determine how much further the energy $\mathcal{N}$ can be reduced and characterize the achievable minimum among protocols that defibrillate under a fixed success criterion — in the source, the tissue reaching a quiescent state ($u_1 < 0.1$ everywhere) within a bounded time ($5P = 1050$ ms) after the pulse. Such a criterion is essential to the question's meaning: the source observes that fibrillation in its model is a long chaotic transient that eventually disappears on its own, so without a bounded termination requirement the lowest defibrillation energy is trivially zero.

References

1. A. Garzón and R. O. Grigoriev, "Ultra-low-energy defibrillation through adjoint optimization", Chaos 34 (11), 113110 (2024), DOI 10.1063/5.0222247, https://doi.org/10.1063/5.0222247 (preprint arXiv:2407.05115, https://arxiv.org/abs/2407.05115)
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