Introduction

In hearts, spiral and turbulent waves may cause serious cardiac deceases, such as fibrillation1,2,3,4,5,6,7. At present, the clinically effective method for terminating fibrillation uses a single high-voltage electric shock to reset all electric activity in cardiac tissues8,9,10, but it is usually associated with severe side effects9,10,11. Besides this method, a theoretical effort uses local fast pacing delivered via injecting a signal on a chosen area of the heart12,13,14,15,16. Although this approach can numerically generate a higher-frequency wave train to suppress spiral turbulence, it is not easy to be realized in real cardiac tissues17,18.

Recently, a promising alternative called wave emission from heterogeneities (WEH) or far-field stimulation is proposed19,20,21. It exploits the fact that, applying an external electric field onto a whole piece of cardiac tissue can lead to de-polarizations and hyper-polarizations (so-called Weidmann zones22) near obstacles. These obstacles can be considered as conductivity heterogeneities inherently in cardiac tissues such as blood vessels, ischemic regions and smaller-scale discontinuities23. If the de-polarizations are supra-threshold, these obstacles can act as virtual electrodes or second sources24,25,26,27,28,29,30. Previous works focused on WEH in response to the uniform electric field (UEF)19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36, which applies a sequence of low-voltage UEF pulses onto field electrodes. Nevertheless, WEH induced by UEF can only emit waves locally near obstacles in turbulent waves. So it requires multiple obstacles to activate more areas and progressively synchronize the whole myocardium to terminate fibrillation19,20,21.

Compared to UEF, the circularly polarized electric field (CPEF) has shown its unique ability to control spirals and turbulence in chemical systems37,38,39, which has been verified in the Belousov-Zhabotinsky reaction40; with a different mechanism in cardiac tissues, CPEF can also unpin the anchored spirals41. In this paper, we study WEH in response to CPEF and find it can generate circular wave trains (target waves) near obstacles and they can propagate outwardly. We study the capability of CPEF to induce such circular wave trains in a quiescent medium and analyze the angular frequency relation between the circular wave 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 model42:

where V is the membrane potential, Cm is the membrane capacitance, D is the diffusion current coefficient and Iion is the total ionic currents which consist of a fast sodium current INa, a slow inward current Isi, a time-dependent potassium current IK, a time-independent potassium current IK1, a plateau potassium current IKp and a time-independent background current Ib. In mono-domain models, the general effect of an external electric field on an obstacle can be expressed as a Neumann boundary condition36,43: 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 = (Ex, Ey) as a counter-clockwise rotating CPEF, where , and E0, ωCPEF 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 E0 and certain frequency ωCPEF, 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 E0 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 model44, 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.

Figure 1
figure 1

Circular waves induced by CPEF in a two-dimensional quiescent medium.

The angular frequency of CPEF ωCPEF = 0.14 rad/ms and the obstacle radius R = 0.24 cm. (a,b), Distribution of the membrane potential induced by CPEF with strength E0 = 0.1 V/cm. (c,d), A wave induced by de-polarization under CPEF with E0 = 1.0 V/cm. (e), The first circular wave induced by CPEF. (f), The second and the third circular waves induced by CPEF. The patterns in (b,d) are the enlarged views of those in (a,c), respectively. Regions D and H represent de-polarization and hyper-polarization induced by CPEF, respectively. The black curved arrows mean that CPEF rotate counter-clockwise.

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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 E0. In details, with a given E0, the reciprocal of T(1/T) will change when R increases. As shown in Fig. 2a, with E0 = 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 E0 increases. As shown in Fig. 2b, with R = 0.24 cm, E0 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 E0 increases above this threshold, 1/T begins to increase slowly. And if E0 becomes large enough (e.g. E0 = 0.85 V/cm as illustrated in Fig. 2b), 1/T tends to be a constant.

Figure 2
figure 2

The formation time (T) of the first circular wave induced by CPEF in a two-dimensional quiescent medium.

(a), The relation between 1/T and the obstacle radius R, where the strength of CPEF E0 = 1.0 V/cm. (b), The relation between 1/T and E0, where R = 0.24 cm. The angular frequency of CPEF ωCPEF = 0.14 rad/ms in both (a,b).

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With proper R and E0, the circular waves can be continuously induced by CPEF and then can form a circular wave train with angular frequency ωcir. Since the circular wave train is forcedly excited by CPEF with ωCPEF, ωcir should highly depend on ωCPEF. In Fig. 3a, to study the relation between ωcir and ωCPEF, we gradually increase ωCPEF from 0.065 rad/ms which can be seen as a minimum (Below this minimum, CPEF cannot generate a stable circular wave train). When ωCPEF is set as this minimum, a stable circular wave train can be generated and its angular frequency ωcir is synchronized with ωCPEF and the ratio ωCPEF/ωcir is about 1:1 as shown in Fig. 3b. However, when ωCPEF increases above 0.07 rad/ms, ωcir has a down jump (see Fig. 3a) which leads the ratios ωCPEF/ωcir 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 ωCPEF continues to increase, ωcir will also increase but the ratios ωCPEF/ωcir are always locked at nearly 2:1 until ωCPEF reaches 0.17 rad/ms. When ωCPEF becomes larger than 0.17 rad/ms, ωcir faces another down jump and subsequently another increasing with ωCPEF and the ratios ωCPEF/ωcir 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 domain45,46,47,48.

Figure 3
figure 3

The angular frequency relations between the circular wave train and CPEF in a two-dimensional quiescent medium.

(a), The obstacle radius R = 0.24 cm, the strength of CPEF E0 = 1.0 V/cm and the angular frequency of CPEF 0.065 rad/ms ≤ ωCPEF ≤ 0.22 rad/ms. The dashed line with solid circles represents the angular frequency of the circular wave trains ωcir. The dash-dotted line represents the dominant angular frequency of the spiral turbulence ωtur in the same medium and the same obstacle. (b), The ratios of ωCPEF over ωcir correspond to the data in (a).

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Besides, we also measure the dominant angular frequency of the spiral turbulence ωtur in the same medium and the same obstacle. We can see from Fig. 3a that, in some angular frequency ranges of ωCPEF, ωcir is higher than ωtur. As the high-frequency waves may invade low-frequency domain49,50,51,52,53,54,55,56, we believe the circular wave trains with such higher frequencies can suppress spiral turbulence12,13,14,15,16,17,18.

To test this idea we choose E0 = 1.0 V/cm, ωCPEF = 0.14 rad/ms for CPEF which can generate the circular wave train with angular frequency ωcir higher than ωtur (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 ωcir > ωtur. 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.

Figure 4
figure 4

Suppression of two-dimensional spiral turbulence by CPEF.

The obstacle radius R = 0.24 cm and the dominant angular frequency of the spiral turbulence ωtur is the same as that in Fig. 3a. The strength of CPEF E0 = 1.0 V/cm and the angular frequency of CPEF ωCPEF = 0.14 rad/ms. (a), The CPEF is applied from t = 0. (b), t = 1000 ms. (c), t = 1800 ms. (d), t = 2800 ms.

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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.

Figure 5
figure 5

Suppression of three-dimensional scroll turbulence by CPEF.

The obstacle radius R = 0.24 cm. The strength of CPEF E0 = 1.8 V/cm and the angular frequency of CPEF ωCPEF = 0.14 rad/ms. (a), The CPEF is applied from t = 0. (b), t = 650 ms. (c), t = 890 ms. (d), t = 1100 ms.

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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 (E0 = 1.0 V/cm, ωCPEF = 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 ωcir is 0.072 rad/ms, which is higher than ωtur. Therefore, the circular wave train induced by CPEF with such a high angular frequency can be used to suppress the spiral turbulence.

Figure 6
figure 6

The variations of the membrane potentials V under CPEF or UEF in a two-dimension quiescent medium.

(a), The locations of the membrane potentials V1-V6. V1, V4 are the membrane potentials on the obstacle boundary. V2, V5 are the membrane potentials near the obstacle boundary. V3, V6 are the membrane potentials far away from the obstacle boundary. CPEF rotates counter-clockwise and UEF is horizontal. (b), Under CPEF, the strength E0 = 1.0 V/cm and the angular frequency ωCPEF = 0.14 rad/ms. (c,d), Under UEF, E0 = 1.0 V/cm, ωUEF = 0.14 rad/ms and the pluse duration is 10 ms. The red arrows indicate the effects of the de-polarizations and hyper-polarizations.

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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 ωUEF/ωcir is about 3:1 and ωcir is about 0.047 rad/ms, which is lower than ωtur. Therefore UEF at ωUEF = 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 ωUEF can induce higher-frequency circular wave trains, we measure ωcir in a large region of ωUEF 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 ωcir < ωtur 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 ωcir(UEF) < ωtur < ωcir(CPEF).

Figure 7
figure 7

The angular frequency relations between the circular wave trains and UEF in a two-dimension quiescent medium.

(a), The obstacle radius R = 0.24 cm, the strength of UEF E0 = 1.0 V/cm, the angular frequency of UEF 0.065 rad/ms ≤ ωUEF ≤ 0.22 rad/ms and the pluse duration is 10 ms. The dashed line with solid circles represents the angular frequency of the circular wave trains ωcir. The dash-dotted line represents the dominant angular frequency of the spiral turbulence ωtur in the same medium and the same obstacle. (b), The ratios of ωUEF over ωcir correspond to the data in (a).

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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 ωCPEF/ωcir is about 2:1, as shown in Fig. 6b. However, the ratio ωUEF/ωcir is about 3:1, as illustrated in Fig. 6c. Hence the circular wave train induced by the rotating CPEF has the angular frequency of ωcir(CPEF) = 0.072 rad/ms which is higher than ωcir(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 (ωcir = 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 other40. 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 method36,43. Considering the effect of an external electric field on the obstacle, equation (1) can be adapted as

where Cm = 1 μF/cm2, D = 0.001 cm2/ms, the total ionic currents Iion 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 turbulence57,58,59 with no-flux boundary conditions via Euler method and the central difference method is applied to compute the Laplacian term ∇2V and the gradient terms , ∇V. The space and the time step in two-dimensional domain are Δx = 0.015 cm, Δy = 0.015 cm and Δt = 0.005 ms, respectively. And the space and the time step in three-dimensional domain are Δx = 0.02 cm, Δy = 0.02 cm, Δz = 0.02 cm and Δt = 0.01 ms, respectively.

Additional Information

How to cite this article: Feng, X. et al. Wave trains induced by circularly polarized electric fields in cardiac tissues. Sci. Rep. 5, 13349; doi: 10.1038/srep13349 (2015).