Wave trains induced by circularly polarized electric fields in cardiac tissues

Clinically, cardiac fibrillation caused by spiral and turbulent waves can be terminated by globally resetting electric activity in cardiac tissues with a single high-voltage electric shock, but it is usually associated with severe side effects. Presently, a promising alternative uses wave emission from heterogeneities induced by a sequence of low-voltage uniform electric field pulses. Nevertheless, this method can only emit waves locally near obstacles in turbulent waves and thereby requires multiple obstacles to globally synchronize myocardium and thus to terminate fibrillation. Here we propose a new approach using wave emission from heterogeneities induced by a low-voltage circularly polarized electric field (i.e., a rotating uniform electric field). We find that, this approach can generate circular wave trains near obstacles and they propagate outwardly. We study the characteristics of such circular wave trains and further find that, the higher-frequency circular wave trains can effectively suppress spiral turbulence.

Scientific RepoRts | 5:13349 | DOi: 10.1038/srep13349 trains and CPEF. Furthermore, we present a successful application of using a higher-frequency circular wave train induced by a low-voltage CPEF to suppress spiral turbulence, and also discuss its suppression mechanism.

Results
To describe the electric activity of cardiac tissues, we consider the following Luo-Rudy model 42 :  I  I  I  I  I  I  2 ion N a s i where V is the membrane potential, C m is the membrane capacitance, D is the diffusion current coefficient, and I ion is the total ionic currents which consist of a fast sodium current I Na , a slow inward current I si , a time-dependent potassium current I K , a time-independent potassium current I K1 , a plateau potassium current I Kp , and a time-independent background current I b . In mono-domain models, the general effect of an external electric field on an obstacle can be expressed as a Neumann boundary condition 36,43 : n⋅ (∇ V − E) = 0, where n is the normal vector to the obstacle boundary, and E is the external electric field. Through this paper without loss of generality, we choose E = (E x , E y ) as a counter-clockwise rotating , and E 0 , ω CPEF are its strength and angular frequency, respectively.
In the following, we use a two-dimensional quiescent medium with a circular obstacle of radius R in its center to do the simulation. We find that, as shown in Fig. 1a,b, with CPEF at a weak strength E 0 and certain frequency ω CPEF , the de-polarization and hyper-polarization induced by CPEF near the obstacle rotate synchronously with the rotating CPEF, and the membrane potential pattern is distributed similarly as Chinese "ancient Taijitu" 41 . When E 0 increases above some threshold, the de-polarization begins to emit a wave as shown in Fig. 1c,d. Then the two ends of the wave propagate oppositely along the obstacle, and they quickly collide with each other and finally can form a circular wave propagating outwardly as shown in Fig. 1e. With the continued effect of CPEF, the second circular wave can be formed and it can also propagate outwardly, then the third one and so on can also emerge (see Fig. 1f). That is, a circular wave train can be generated and can continually propagate outwardly. Similarly, we also observe a circular wave train induced by CPEF near a circular obstacle in a modified FitzHugh-Nagumo model 44 , thus the above results may be model-independent. Therefore, we can recognize that an obstacle under CPEF can act as a pacing electrode and generate the circular wave train.
As for such a circular wave train induced by CPEF, for convenience, we focus on studying the formation time (T) of the first circular wave to reflect the capability of CPEF to induce the circular wave train. As illustrated in Fig. 2, with a given angular frequency of CPEF, we can find T is highly related to R and E 0 . In details, with a given E 0 , the reciprocal of T(1/T) will change when R increases. As shown in Fig. 2a, with E 0 = 1.0 V/cm, we can see R has a threshold at 0.04 cm, and 1/T has a sharp up jump. Below this threshold, 1/T is zero, which means no wave can be induced by CPEF near such small obstacles. Once R increases above this threshold, 1/T begins to decrease. And with the continual increasing of R, 1/T decreases more and more slowly. On the other hand, with a given R, 1/T will also change when E 0 increases. As shown in Fig. 2b, with R = 0.24 cm, E 0 also has a threshold at 0.45 V/cm, and 1/T also has a sharp up jump. Below this threshold, 1/T is zero and no wave can be induced by CPEF because the electric strength is not strong enough to exceed the de-polarization threshold. Once E 0 increases above this threshold, 1/T begins to increase slowly. And if E 0 becomes large enough (e.g. E 0 = 0.85 V/cm as illustrated in Fig. 2b), 1/T tends to be a constant.
With proper R and E 0 , the circular waves can be continuously induced by CPEF and then can form a circular wave train with angular frequency ω cir . Since the circular wave train is forcedly excited by CPEF with ω CPEF , ω cir should highly depend on ω CPEF . In Fig. 3a, to study the relation between ω cir and ω CPEF , we gradually increase ω CPEF from 0.065 rad/ms which can be seen as a minimum (Below this minimum, CPEF cannot generate a stable circular wave train). When ω CPEF is set as this minimum, a stable circular wave train can be generated and its angular frequency ω cir is synchronized with ω CPEF , and the ratio ω CPEF /ω cir is about 1:1 as shown in Fig. 3b. However, when ω CPEF increases above 0.07 rad/ms, ω cir has a down jump (see Fig. 3a) which leads the ratios ω CPEF /ω cir are no longer 1:1 but locked at nearly 2:1 (see Fig. 3b). The reason for this is that, CPEF rotates so fast that after forming a circular wave, the medium around the obstacle has not yet recovered to be excitable. Thus the rotating de-polarization of CPEF cannot excite another wave in its present round until the medium recover to be excitable again in its next round, and so forth. If ω CPEF continues to increase, ω cir will also increase but the ratios ω CPEF /ω cir are always locked at nearly 2:1 until ω CPEF reaches 0.17 rad/ms. When ω CPEF becomes larger than 0.17 rad/ms, ω cir faces another down jump and subsequently another increasing with ω CPEF , and the ratios ω CPEF /ω cir are locked at nearly 3:1. In a word, as illustrated in Fig. 3, CPEF always keeps nearly n:1 (n = 1,2,3) angular frequency relation with the induced circular wave train, which has been widely reported in the pattern formation domain 45 Besides, we also measure the dominant angular frequency of the spiral turbulence ω tur in the same medium and the same obstacle. We can see from Fig. 3a that, in some angular frequency ranges of ω CPEF , ω cir is higher than ω tur . As the high-frequency waves may invade low-frequency domain [49][50][51][52][53][54][55][56] , we believe the circular wave trains with such higher frequencies can suppress spiral turbulence [12][13][14][15][16][17][18] .
To test this idea we choose E 0 = 1.0 V/cm, ω CPEF = 0.14 rad/ms for CPEF which can generate the circular wave train with angular frequency ω cir higher than ω tur (see Fig. 3a). In Fig. 4, we numerically simulate spiral turbulence with a circular obstacle of R = 0.24 cm and use it as the initial state (t = 0). In the beginning of applying CPEF, due to the disturbance of the nearby turbulent waves, the emitted waves near the obstacle fail to form circular waves. Nevertheless, by continuously emitting waves under CPEF, circular waves begin to emerge. Then at t = 1000 ms, a full circular wave is formed and squeezes out the nearby turbulent waves, following which more and more circular waves can be gradually formed. Later at t = 1800 ms, there are only few turbulent waves left. Finally at t = 2800 ms, all the turbulent waves are driven away out of the boundary. In addition, the medium can recover to a quiescent state after stopping CPEF. Furthermore, we find the circular wave trains induced by CPEF can successfully suppress spiral turbulence as long as ω cir > ω tur . Similarly in the modified FitzHugh-Nagumo model, the higher-frequency circular wave train induced by CPEF near a circular obstacle is also obtained and it can also successfully suppress the spiral turbulence.
Furthermore, we also testify the ability of CPEF suppressing three-dimensional scroll turbulence in Luo-Rudy model. As illustrated in Fig. 5, we numerically simulate scroll turbulence with a spherical obstacle of R = 0.24 cm as the initial state (t = 0). Then under CPEF, emitting waves continuously emerge and collide with the turbulent waves (t = 650 ms). Later at t = 890 ms, a full spherical wave is formed and squeezes out the nearby turbulent waves. Finally at t = 1100 ms, all the turbulent waves are driven away out of the boundary. Similarly, we also observe that, in the modified FitzHugh-Nagumo model, a spherical wave train induced by CPEF near a spherical obstacle can successfully suppress the three-dimensional scroll turbulence.

Discussion
In this section, we discuss the mechanism about successfully suppressing the spiral turbulence by a high-frequency circular wave train induced by CPEF. And we can owe this success to the contributions of the rotating de-polarization and hyper-polarization induced by CPEF, thus the membrane potential at the boundary of the obstacle would successively and periodically go through both effects. Under the influence of CPEF (E 0 = 1.0 V/cm, ω CPEF = 0.14 rad/ms, the same as those in Fig. 4), we study the variation of the membrane potential at an arbitrary position on the obstacle boundary in a quiescent medium, e.g., the membrane potential V1 in Fig. 6a. We find V1 would be depolarized to the excited state, and then forced to recover to the excitable state quickly by the hyper-polarization as shown in Fig. 6b. Due to the diffusion of V1, the nearby membrane potential V2 would also be directly affected. Although the second stimulus from V1 fails to de-polarize V2, the effect of hyper-polarization from V1 still makes it recovered to the excitable state quickly. Thus V2 is able to be de-polarized by the third stimulus from V1. Further, the membrane potential V3 would be affected by the diffusion of V2 and thus forms periodic excitations (i.e., the circular wave train). Hence every two rounds of the rotating de-polarization and hyper-polarization induced by CPEF can stimulate a circular wave. The angular frequency of the circular wave train ω cir is 0.072 rad/ms, which is higher than ω tur . Therefore, the circular wave train induced by CPEF with such a high angular frequency can be used to suppress the spiral turbulence.
As shown in Fig. 1A of Ref. 35, using UEF can also generate circular waves near the obstacle in a quiescent medium. However, UEF can hardly utilize the same mechanism as CPEF to induce a higher-frequency circular wave train for suppressing the spiral turbulence. We employ a series of UEF pulses to the quiescent medium with the same electric strength and angular frequency as CPEF. In Fig. 6c, we find the membrane potential V1 excited by the de-polarization induced by UEF cannot be forced to recover to the excitable state quickly due to the lack of the hyper-polarization in the same position, thus the nearby membrane potential V2 will have to go through a relatively long excited time. Hence the membrane potential V2 and thereby V3 can only be stimulated for every three UEF pulses. As illustrated in Fig. 6d, the membrane potential V4 is only affected by the hyper-polarization induced by UEF which cannot induce stimuli and the existing stimuli actually come from V1. So the ratio ω UEF /ω cir is about 3:1 and ω cir is about 0.047 rad/ms, which is lower than ω tur . Therefore UEF at ω UEF = 0.14 rad/ms cannot induce a higher-frequency circular wave train to suppress the spiral turbulence.
In order to verify whether UEF with other ω UEF can induce higher-frequency circular wave trains, we measure ω cir in a large region of ω UEF in the same quiescent medium and the same obstacle as in Fig. 3a. As shown in Fig. 7a, most of the circular wave trains induced by UEF have ω cir < ω tur and thus cannot suppress the spiral turbulence. Comparing it to the case of CPEF in Fig. 3a and taking the angular frequency ranges of 0.13 rad/ms ≤ ω ≤ 0.17 rad/ms for instance, we find every two rounds of CPEF can   stimulate a circular wave, but UEF at the same angular frequency would need three pulses to stimulate a circular wave (may refer to Figs 3b and 7b), and thus ω cir (UEF) < ω tur < ω cir (CPEF).
In other words, the main difference of CPEF from UEF is the rotation. Because of the rotation of CPEF, the medium will be affected by both de-polarization and hyper-polarization. While using UEF, only the de-polarization can affect the medium. Therefore, with the same angular frequency of both external electric fields (ω = 0.14 rad/ms in Fig. 6), the ratio ω CPEF/ ω cir is about 2:1, as shown in Fig. 6b. However, the ratio ω UEF/ ω cir is about 3:1, as illustrated in Fig. 6c. Hence the circular wave train induced by the rotating CPEF has the angular frequency of ω cir (CPEF) = 0.072 rad/ms which is higher than ω cir (UEF) = 0.047 rad/ms. And this difference of ratios between CPEF and UEF exists in a wide region of the angular frequency ω of both external electric fields. Comparing with the Figs 3a and 7a, the rotating CPEF can maintain the phase-locking state at the ratio of 2:1 in a longer region than UEF. And in some part of the ratio of 2:1 (i.e., 0.13 rad/ms ≤ ω ≤ 0.17 rad/ms), the circular wave trains induced by CPEF have higher frequencies than the dominant frequency of turbulence and can be used to terminate fibrillation. But in the same region (i.e.,0.13 rad/ms ≤ ω ≤ 0.17 rad/ms), the circular wave trains induced by UEF are at the ratio of 3:1, and thus have lower frequencies than the dominant frequency of turbulence, and cannot be used to terminate fibrillation.
Moreover, further simulations indicate the waves induced by UEF cannot form the circular waves in the turbulent waves and thereby the induced circular wave trains in the quiescent medium with relatively high angular frequencies (ω cir = 0.07 rad/ms in Fig. 7) also cannot suppress the spiral turbulence as in Fig. 4. This may owe to the fact that the de-polarization and hyper-polarization induced by UEF cannot rotate and thus the waves can only be induced in a fixed position near the obstacle (e.g., only V1 in Fig. 6c affected by the de-polarization can emit waves while V4 in Fig. 6d cannot). Conversely, the de-polarization and hyper-polarization induced by CPEF can rotate and thereby the waves can be emitted in any position on the obstacle boundary (e.g., the membrane potential at the arbitrary position on the obstacle boundary has the same variation as V1 in Fig. 6b). Hence CPEF can effectively generate the circular wave trains and eventually suppress the turbulent waves. Therefore, although the circular wave trains can be induced by UEF in a quiescent medium, they cannot form the circular waves in the presence of turbulence waves as in Fig. 4 and thereby UEF can hardly utilize the same mechanism as CPEF to suppress spiral turbulence.
To conclude, CPEF can effectively generate the higher-frequency circular wave trains near obstacles. And this capability is closely related to the strength and the angular frequency of CPEF and the size of obstacles. Moreover, the circular wave trains induced by CPEF have a wide application prospect. An important application is that the higher-frequency circular wave trains induced by CPEF can be used to suppress spiral turbulence, which may provide a promising alternative to terminate fibrillation. Additionally, CPEF has been realized in Belousov-Zhabotinsky reaction by applying two ACs onto two pairs of field electrodes perpendicular to each other 40 . Similarly, it will also be easily realized in cardiac tissues by replacing DCs to ACs in the experimental preparation of Fig. 5D in Ref. 20. Hence we believe this approach will have strong practical value in heart clinical treatments, and its effectiveness and applicability in bi-domain model and in real cardiac tissues will need to be further studied.

Methods
In Luo-Rudy model, to add the introduced boundary condition into the circular boundary of the obstacle in Cartesian coordinates, we adopt the phase field method 36,43 . Considering the effect of an external electric field on the obstacle, equation (1)  where C m = 1 μ F/cm 2 , D = 0.001 cm 2 /ms, the total ionic currents I ion are determined by ionic gates, whose gating variables are obtained as solutions to a coupled system of nonlinear ordinary differential equations, and the parameters are modified as in Ref. 57. In Cartesian coordinates, equation (3) is integrated on the 10 cm × 10 cm two-dimensional medium and 5 cm × 5 cm × 2 cm three-dimensional medium which are large enough to sustain the turbulence [57][58][59] with no-flux boundary conditions via Euler method, and the central difference method is applied to compute the Laplacian term ∇ 2 V and the gradient terms φ ∇( ) ln , ∇ V. The space and the time step in two-dimensional domain are Δ x = 0.015 cm, Δ y = 0.015 cm and Δ t = 0.005 ms, respectively. And the space and the time step in three-dimensional domain are Δ x = 0.02 cm, Δ y = 0.02 cm, Δ z = 0.02 cm and Δ t = 0.01 ms, respectively.