Mechanisms of vortices termination in the cardiac muscle

We propose a solution to a long-standing problem: how to terminate multiple vortices in the heart, when the locations of their cores and their critical time windows are unknown. We scan the phases of all pinned vortices in parallel with electric field pulses (E-pulses). We specify a condition on pacing parameters that guarantees termination of one vortex. For more than one vortex with significantly different frequencies, the success of scanning depends on chance, and all vortices are terminated with a success rate of less than one. We found that a similar mechanism terminates also a free (not pinned) vortex. A series of about 500 experiments with termination of ventricular fibrillation by E-pulses in pig isolated hearts is evidence that pinned vortices, hidden from direct observation, are significant in fibrillation. These results form a physical basis needed for the creation of new effective low energy defibrillation methods based on the termination of vortices underlying fibrillation.

VNB, 0000-0001-6617-4677 We propose a solution to a long-standing problem: how to terminate multiple vortices in the heart, when the locations of their cores and their critical time windows are unknown. We scan the phases of all pinned vortices in parallel with electric field pulses (E-pulses). We specify a condition on pacing parameters that guarantees termination of one vortex. For more than one vortex with significantly different frequencies, the success of scanning depends on chance, and all vortices are terminated with a success rate of less than one. We found that a similar mechanism terminates also a free (not pinned) vortex. A series of about 500 experiments with termination of ventricular fibrillation by E-pulses in pig isolated hearts is evidence that pinned vortices, hidden from direct observation, are significant in fibrillation. These results form a physical basis needed for the creation of new effective low energy defibrillation methods based on the termination of vortices underlying fibrillation.

Pinned vortices
In this paper, we investigate termination of multiple vortices in the heterogeneous cardiac muscle. The difficulties arise owing to the interaction of vortices. We investigate the excitation dynamics in the vicinity of the cores of pinned vortices. This allows us to draw conclusions about the overall dynamics. When the VW of a vortex is hit by the E-pulse, this vortex is displaced to a new position. If the vortex was situated close to the tissue boundary, it is terminated. Our aim is that VW of every vortex is hit by an E-pulse ('all vortices are terminated'). Wave patterns were calculated using the Barkley model: in a rectangular domain with circular holes, with no-flux boundary conditions at the outer boundaries. Pulses of electric field E are implemented as in [26] using the boundary conditions n · (∇u − E) = 0 at the boundaries of the holes. The numerical integration used an explicit Euler scheme with a time step of 1.6 × 10 −3 and central-difference approximation of Laplacian with a space step of 1/6. The Barkley model is formulated in non-dimensional units; for presentation purposes, we postulate that the time unit of the Barkley model is 20 ms and the space unit of the Barkley model is 0.5 mm; this gives physiologically reasonable time and space scales. Figure 1 shows termination of two pinned vortices by E-pacing (see also the movie in the electronic supplementary material). This can be achieved generically, for any parameter of the vortices, without knowing their geometric location and time positions of the VWs.
To hit the VW with an E-pulse, the phase scanning (figure 2a,b) should be performed with steps 0 < s < VW. Thus, the VW duration (at the chosen E, see figure 4h) determines suitable values of s. Then, the number of pulses N required to cover the whole period of a vortex is N ≥ T v /s, where T v is the period of the vortex, s = T − T v is the scanning step, and T is the period of E-pacing. This gives the E-pacing period T = s + T v . Thus, all parameters of E-pacing (E, N, T) can be set following equations  (c) s < 0 for faster pacing T < T v , the scanning moves in the opposite direction. E-pulse reaches the excitable gap 'e' , excites an AP thus resetting the rotation phase, and all subsequent pulses get into the same phase [25,27]. It does not reach the VW.
Minimum energy for termination of a pinned vortex is achieved when the electric field strength is chosen so that the normalized VW(E) = 1/N, where N is the number of pacing pulses. The maximal success rate is achieved when the pacing frequency f = f best , where f best is the frequency for which the normalized scanning stepŝ = 1/N. When f < f best , i.e. T > T best , the scanning stepŝ > VW, and the VW may be missed while scanning, thereby decreasing the success rate. When f > f best , so T < T best , the scanning stepŝ < VW = 1/N, and not all phases are scanned. This also decreases the success rate.
What does interaction of vortices change here? In cardiac muscle, the fastest vortex entrains slower vortices if there is a normal wave propagation between them. Then, only one frequency remains; this facilitates vortices termination. But entrainment ceases if the fastest vortex is terminated before the slower vortices, and then the frequency of the system changes (period increases). Here, two wave scenarios are possible, which we describe for the case of just two vortices with periods T v1 and T v2 , such that T v1 > T v2 .
1. If the periods of the two vortices are not much different, so that T v2 < T v1 < T, then the pacing is still under-driving, and the slower vortex (T v1 ) can be terminated by E-pacing with same period T (figures 1 and 2b), provided that termination conditions (2.1) are met for the slower vortex. 2. If however, the periods of the two vortices are so much different that then the pacing with the same period is no longer under-driving, but over-driving. And overdrive pacing will typically entrain the remaining vortex rather than eliminate it.
For successful termination of fibrillation, the E-pacing period should be increased to a higher value T 2 , such that T v1 < T 2 . Thus, vortices can be terminated in any case. Experiments [16] underestimated the potential of the method as this mechanism was not known yet.

Free vortices
Below, we describe a mechanism terminating a high frequency free vortex by electric field pacing. Numerical and theoretical publications state it is very easy: usual local pacing (anti-tachycardia pacing; ATP) with frequency higher than a free vortex frequency terminates a free vortex. It works in experiment and in clinics, but only for low frequency vortices. Classic ATP cannot terminate ventricular fibrillation (VF), cannot terminate high frequency rotating waves, including free rotating waves. Waves emitted from a pacing electrode propagate along the whole tissue only for low frequency. For higher frequencies, the Wenckebach rhythm transformation occurs in a heterogeneous cardiac tissue. By contrast, an electric field penetrates everywhere, without frequency limitations, and only requires local heterogeneities to act as virtual electrodes. This mechanism can be used for terminating a high frequency free vortex. A free (not pinned) vortex can be terminated when its moving core passes not very far (at distance L λ, where λ is the wavelength) from a defect in the medium, serving as a virtual electrode, figure 3 (this illustration uses the same mathematical model for the excitable medium and for the action of the electrical field as in the previous subsection). The success rate increases as distance L decreases. A mechanism reliably terminating a free rotating wave was found in 1983 [28]: waves with a frequency higher than the frequency of a rotating wave, induce its drift and termination on the boundary. Cardiologists used a high-frequency pacing (ATP) well before the mechanism was understood. But ATP can not terminate high frequency rotating waves. This fundamental limitation is overcome by the mechanism of a free vortex termination proposed here. This mechanism depends on wave emission induced from a defect induced by the electric field. The electric field penetrates everywhere, hence no restriction on its efficacy is imposed by the maximal frequency of propagating waves in any part of the cardiac tissue. An increased amplitude of electric field |E| results in defibrillation. There is a classical explanation: electric field should be increased to the value where it terminates all propagating waves. A physical explanation is: the wave emission is induced from a larger number of defects when the electric field is increased [14]. Figure 4g shows another mechanism: the size of the excited region increases with the electric field, and the duration of the VW increases with it.   excitation phase can be defined as the angle in the polar coordinates in the phase plane of the reaction kinetics, centred at a suitably chosen point in the 'No Mans Land', in terminology of [30]. Likewise, topological singularity in two dimensions is defined as a point such that any sufficiently small onedimensional contour surrounding it has a topological charge. We illustrate the relationship between these concepts and VW using time-separation analysis for the FitzHugh-Nagumo (FHN) equations: Here f (u) = Au(1 − u)(u − α), and 1 is a small parameter permitting the time-scales separation (for details of relevant formalisms, see review [31]). The wavefront propagation velocity θ can be estimated by assuming that the slow variable v is approximately constant across the wavefront. The propagation of the front is then described by equation (2.3) alone, where v is a constant parameter. Transforming the independent variables such that ξ = x − θ t makes equation (2.3) an ordinary differential equation: which together with boundary conditions u(∞) = u 1 , u(−∞) = u 3 , where u 1 = u 1 (v) and u 3 = u 3 (v) are, respectively, the lowest and highest roots of f (u) = v, define θ as a function of v (figure 4b). Here, velocity Vulnerability is a cardiological term coined for initiation of fibrillation by an electric pulse. In the physical language, vulnerability in one dimension can be related to a change of the topological charge, and in two dimensions and three dimensions to creation of new phase singularites. In one dimension, this phenomenon happens when the current injection nucleates a wave propagating in only one direction, figure 4d. This is in contrast with the generic case, where the topological charge is conserved, when the new wave propagates in two directions, figure 4c, or new wave is not nucleated at all (not shown).
where VW is the normalized duration of the VW, then the jth vortex is considered terminated; (iii) if neither vortex is terminated, then the slower vortice's phase is enslaved by the faster one's, φ 2 n+1 = (φ 1 n+1 − D) mod 1, where D is a fixed phase delay; and (iv) if both vortices are terminated, iterations stop and E-pacing is deemed successful. Figure 5 shows the success rate of termination of two vortices as a function of the normalized frequency of E-pacing. The graphs represent results of Monte Carlo simulations of the axiomatic model described above, with random initial phases of vortices and two variants for the choice of frequencies: (i) normal distributions of parametersT 1 = 1 ± 0.1 andT 2 = 1.6 ± 0.05 (mean ± s.d.), 'different frequencies'; and (ii) the same parameters forT 1 , andT 2 enforced very close toT 1 , namelyT 2 = (1 + 10 −6 )T 1 , 'close frequencies', with other parameters fixed at EG = 0.4, VW = 0.2, D = 0.25. The success rate of termination of two vortices with significant difference in frequencies as per equation (2.2) is seen in figure 5 to be threefold lower than that for vortices with insignificant difference in frequencies. This happens because when the leading (fastest) vortex is terminated first, the same E-pacing period T appears below the period T 1 of the resting slower vortex, see equation (2.2). Thus, the resting vortex cannot be terminated (figure 2c).  shows graphs forf < 1, (c) shows graphs forf ≥ 1. Graphs (b,c) and the experimental curve in (a) are calculated from data in [38]. Image (a) indicates that in about a half of fibrillation experiments, the frequencies of the vortices were not significantly different. The optimal pacing frequencyf = 0.77 is below the arrhythmia frequency (f < 1) as it should be for terminating pinned vortices. These experiments provide evidence that pinned vortices, hidden from direct observation, are significant in fibrillation.
In figure 6a, the experimental curve is fit by the blue theoretical curve much better than by either of the two theoretical curves in figure 5. It indicates that in about a half of fibrillation experiments, the frequencies of the vortices were not significantly different.  Figure 6a,b shows that the optimal pacing frequencyf = 0.77 is below the arrhythmia frequency (f < 1) as it should be for terminating pinned vortices. Note that elimination of a free, rather than pinned, vortex by inducing its drift via the mechanism described in [28], requires the pacing frequency to be above the arrhythmia frequency,f > 1. These experiments provide evidence that pinned vortices, hidden from direct observation, are significant in fibrillation. In particular, they show that the VW mechanism is an explanation for the high success rate of VF termination using electric field pacing.

Discussion
In this paper, after more than 25 years of research, we propose a solution to a problem; how to terminate multiple vortices in the cardiac tissue hidden from direct observation. In order to control vortices, two problems should be overcome: both, the geometric positions of their cores, and the positions of their critical time windows, are not known during fibrillation. The first problem we have solved previously using an electric field pulse to excite the cores of all pinned vortices simultaneously. Approaches to solve the second problem are being developed. One of them is based on the phase scanning of all pinned vortices in parallel to hit the critical time window of every pinned vortex. In this paper, we investigate the related physical mechanisms using simple two variable models as well as a detailed ionic model of the cardiac tissue. A similar mechanism terminates also a free (not pinned) vortex, when the vortex's core passes not very far from a defect.
Even though it is widely believed that the success of defibrillation has a probabilistic nature, we have shown that termination of one vortex can be achieved deterministically, in any case. This can be achieved generically, for any parameters of the vortex, without knowing its geometric location and timing of its VW. All that is needed is to set the parameters of E-pacing (E, N, T) according to equations (2.1). Termination of an arrhythmia becomes probabilistic when two or more vortices are involved. If there is normal wave propagation between the two vortices, and the slower vortex is enslaved by the faster one, then the E-pacing protocol described in [16] cannot control which of the vortices will be terminated first. If the slower vortex is terminated first, the frequency of the system does not change, and both vortices are terminated deterministically, in any case. If, by chance, the faster vortex is terminated first, the frequency of the system changes, and the remaining slower vortex may be not terminated if conditions (2.2) are satisfied.
Here, we investigated two extreme cases: permanently pinned vortices and permanently free vortices. There is no sharp transition between them in heterogeneous media with different size pinning centres. In cardiac muscle, there are heterogeneities of all sizes, including those to which vortices pin weakly. A weakly pinned vortex is pinned for some time only, then leaves the pinning centre and moves as a free vortex, again for some time. When moving and meeting a pinning centre, it may pin to it, or may reach the boundary of the tissue and disappear.
Three-dimensional models are widely used in investigation of wave patterns induced by rotating waves, e.g. [10]. A three-dimensional mechanism of defibrillation was described in [35][36][37]. Study of vortices termination in two-dimensional models is a necessary step for developing an understanding of mechanisms of three-dimensional vortices termination in the heart. Termination of vortices underlying fibrillation is only a small part of a problem preventing and curing the cardiac arrhythmias where a combination of molecular and dynamics approaches is prominent [39].
In conclusion, we have shown mechanisms of terminating pinned and free vortices by electric field pulses when the geometric positions of their cores, and the phases of rotation are not known. These results form the physical basis for creation of new effective methods for terminating vortices underlying fibrillation.
Ethics. The study was reviewed and approved by the ethics committee, permit no. 33.9-42052-04-11/0384, Lower Saxony State Office for Customer Protection and Food Safety.