In a two-dimensional bidomain model of ventricular tissue (phase-I Luo-Rudy ionic model with acute-ischaemia modifications), Boccia, Luther and Parlitz (2017) showed that sequences of far-field electric pulses terminate spiral waves pinned to an ischaemic heterogeneity over a substantially wider range of pacing parameters in anisotropic tissue than in isotropic tissue, with success extending to overdrive pacing frequencies at which the isotropic medium fails. The mechanistic origin of this anisotropic advantage was left unexplained, and the study was restricted to an electric field parallel to the myocardial fibres, a single heterogeneity radius, one conductivity dataset, and one ionic model. The open problem is to give a validated mechanistic account of far-field-pacing termination of pinned spirals in anisotropic cardiac tissue - in particular of why overdrive pacing succeeds on anisotropic substrates, and how unpinning success and the vulnerable window depend on the relative angle between the applied field and the fibre axis - together with an assessment of robustness across conductivity values and of the effect of residual post-termination wave activity.

Background

Re-entrant cardiac arrhythmias such as ventricular tachycardia and fibrillation are driven by rotating spiral waves of electrical excitation in the myocardium. When such a spiral anchors ('pins') to a region of reduced excitability and conductivity - for example an acutely ischaemic zone - it becomes remarkably stable and hard to remove. Conventional defibrillation delivers one high-energy shock, which is painful and can damage tissue; this motivates low-energy alternatives. One of them is far-field pacing (FFP): a sequence of weak electric-field pulses applied through remote electrodes. Cardiac tissue is described by the bidomain model, a continuum description that treats the intracellular and extracellular spaces as two interpenetrating domains with generally different conductivity ratios along and across the myocardial fibres ('unequal anisotropy ratios'). Because of this, an external field polarizes tissue near conductivity heterogeneities, turning an ischaemic region into a 'virtual electrode' that emits excitation waves; this polarization response does not arise in the single-domain approximation with equal anisotropy ratios, which is why far-field stimulation studies require the bidomain formulation.

Building on the demonstration by Pumir and Krinsky (1999) that an electric field can unpin a rotating wave from an obstacle, Bittihn et al. (2008) showed in a generic model of excitable media that far-field pacing terminates pinned spirals more reliably than locally applied anti-tachycardia pacing, and Bittihn et al. (2010) showed that unpinning succeeds only when a pulse arrives within a specific phase interval of the pinned spiral's rotation - the 'vulnerable window'. Shajahan et al. (2016) analysed how periodic far-field pulses scan and reset the phase of a pinned spiral. Experiments by Woods et al. (2006) confirmed virtual-electrode polarization around artificial heterogeneities, and Luther et al. (2011) demonstrated low-energy anti-fibrillation pacing (LEAP), terminating fibrillation in dogs with roughly an order of magnitude less energy than a single shock.

The membrane kinetics in such bidomain simulations are governed by an ionic cell model; Boccia, Luther and Parlitz (2017) used the phase-I Luo-Rudy ventricular model (Luo and Rudy 1991), modified for acute ischaemia by raising extracellular potassium from 5.4 to 14 mmol/l, reducing the sodium and L-type calcium conductances by 25%, and lowering tissue conductivities. They simulated a 4×44\times 4 cm sheet containing a circular ischaemic heterogeneity of radius 0.6 cm, used the conductivity measurements of Clerc (1976), and compared the outcomes of sequences of up to five far-field pulses between an isotropic medium and an anisotropic medium with the field parallel to the fibres, for pinned spirals both in their transient regime and in steady rotation. Their central findings were: (i) recruiting the heterogeneity as a virtual electrode required an electric field about five times stronger in the isotropic case; (ii) the set of pacing parameters (pacing period, initial phase, number of pulses) leading to successful termination was substantially larger in anisotropic tissue; (iii) while in isotropic media success was restricted to underdrive pacing, meaning a pulse period longer than the spiral rotation period TspT_{\mathrm{sp}} of about 380 ms, in anisotropic media it extended to overdrive pacing, meaning a period shorter than TspT_{\mathrm{sp}} (about 340-350 ms there); and (iv) unpinning with subsequent repinning of the spiral occurred only in the isotropic simulations, so the isotropically characterized unpinning/repinning mechanisms and vulnerable windows had no anisotropic counterpart yet.

The mechanism behind findings (ii)-(iv) was left unexplained. The authors offered only a preliminary conjecture - that the larger longitudinal conductivity, combined with the greater 'vulnerability' of spirals still in their transient regime, induces a drift of the spirals towards the tissue boundary where they annihilate - explicitly requiring additional simulations to test it. They further noted that only a field parallel to the fibres was modelled and that they expect the relative angle between field and fibres to be a 'critical parameter'; that measured bidomain conductivity values are inconsistent across the experimental literature with no consensus values; and that delivering pulses into quiescent tissue versus tissue with residual propagating waves could change the success rate. What is missing is a completed, mechanistically understood account of far-field pacing termination in anisotropic tissue.

Problem Statement

Using anisotropic bidomain simulations of cardiac tissue with a spiral wave pinned to an ischaemic heterogeneity - in the configuration of Boccia, Luther and Parlitz (2017) or a defensible generalization of it - determine, with validated numerical evidence and a mechanistic explanation:

  1. Why far-field pulse sequences terminate pinned spirals at overdrive pacing periods in anisotropic tissue while the corresponding isotropic medium fails, testing in particular the conjectured mechanism that transient-regime vulnerability combined with the larger longitudinal conductivity causes boundary-directed spiral drift and annihilation.
  2. How the unpinning and termination success, and the vulnerable window, depend on the relative angle between the applied electric field and the myocardial fibre orientation, of which only the parallel case has been studied.
  3. How the unpinning and repinning phenomena and the vulnerable-window concept established in isotropic media translate to anisotropic bidomain computations.
  4. How robust these conclusions are to the choice of bidomain conductivity dataset - measured values being inconsistent across the experimental literature with no consensus values - and how the success rate changes when pulses are delivered into quiescent tissue versus tissue with residual wave activity.

The four numbered items above are jointly posed aspects of a single research target: how the termination outcome (maintenance, unpinning or termination per delivered pulse) depends on pacing period, initial phase, pulse number and field-fibre angle, for both isotropic and anisotropic media and across the bidomain conductivity datasets and the quiescent-versus-residual-wave pulse-delivery conditions of item 4, together with a causal account of the isotropic-anisotropic differences that is validated by the simulations.

References

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