Wave emission on interacting heterogeneities in cardiac tissue

Cardiac arrhythmias, a precursor of fibrillationlike states in the beating heart, are associated with spiral waves, which are likely to become pinned to heterogeneities. Far-field pacing (FFP) is a promising method for terminating such waves by using heterogeneities in the tissue as internal pacing sites. In this study we investigated the role of multiple obstacles and their interaction during FFP. We show that a secondary nearby obstacle can significantly modulate the minimum electrical field in FFP. Further, we show that essentially the same effect can be observed in cardiac tissue culture, which is a powerful experimental model to simulate heart activity. Here, an isotropic cell distribution leads to domain formation of locally distributed depolarization sites. Both secondary obstacles and domain formation of local depolarization sites can modulate energy requirements to originate wave propagation on obstacles. Our theoretical result was confirmed by experiments with cardiomyocyte monolayers. This result may be useful for the future application of FFP to a real beating heart.

Figure 1

(Color online) Schematic representation of wave emission on heterogeneities. The electrical far-field, $E_{elec}$, is applied as indicated by the upper arrow. $\Phi$ is the membrane potential, and depolarized and hyperpolarized regions are marked as yellow (white) and blue (gray), respectively. (a) The change in the membrane potential for cells nearby an intracellular cleft is illustrated. Diagonal arrows illustrate outward and inward transmembrane current, respectively. (b) shows induced patterns of excitation on a round obstacle (cellular cleft) by an electrical field in a bidomain model. The direction of cell alignment is indicated by the white double-headed arrow. Dipole and hexapole components are shown on and near the obstacle located at the center of the figure, respectively. The obstacle radius is 2.7 mm. The profile of the white horizontal dashed line corresponds qualitatively to the membrane potential profile in (a).

Figure 2

(Color online) Neighboring obstacles under an applied electrical field. The radius of both obstacles is 1.5 mm, and the respective critical electrical field is $E_{min}\left(R=1.5\text{mm}\right)=445\text{mV}/\text{cm}$. The direction of the applied electrical field is indicated by the arrow, and the direction of cell alignment is indicated by the double-headed arrow. (a) shows influenced obstacles. The applied electrical field is $E=430\text{mV}/\text{cm}$. Only the lower obstacle shows wave propagation. (b) shows obstacles that are not influenced. The applied electrical field is $E=450\text{mV}/\text{cm}$.

Figure 3

(Color online) Effective electrical field of neighboring obstacles. (a) and (b) show illustrations of obstacle situations, with decreases and increases in $E_{min}$, respectively. Yellow (white) and blue (gray) indicate depolarized and hyperpolarized regions around the obstacles, respectively. (c) shows the change in the electrical field $\Delta E_{min}$ for a 2.25 mm-sized obstacle influenced by a 1.5 mm-sized obstacle depending on their relative positions. The color indicates the change in $E_{min}^1$ induced by the secondary obstacle: red shows an increase and blue shows a decrease in $E_{min}^1\left(R=2.25\text{mm}\right)=421\text{mV}/\text{cm}$. Contours for decrease and increase in $E_{min}$ are drawn for $-5$ and $-15\text{mV}/\text{cm}$, and for $+7$ and $+20\text{mV}/\text{cm}$ by dotted and solid white lines, respectively. The white double-headed arrow indicates the direction of cell alignment.

Figure 4

(Color online) Effect of WEH in isotropically distributed cell tissue. Cells are randomly orientated [see Eq. (5)]. The obstacle radius is 1.5 mm, and $E_{min}$ is 550 mV/cm.

Figure 5

Comparison of minimum electrical fields. (a) Probability of wave origination depending on obstacle size and electrical field. The distribution is plotted for obstacles of size 4.0, 3.75, 3.0, 2.25, 1.5, and 1.2 mm, corresponding to the distribution plotted from left to right, respectively. Solid lines are sigmoidal functions fitted to the respective obstacle size. Each data point represents at least 200 simulations. The upper and lower dashed lines in (b) show $E_{min}^{\perp }$ and $E_{min}^{\parallel }$ for aligned-cell tissue, respectively. For isotropic distributed cell tissue, 50%-wave probabilities are plotted as data points and the range of 10%-wave probability is plotted as error bars.

Figure 6

(Color online) Experiments on the minimum electrical far-field of WEH in cardiac tissue culture. (a) Snapshots of wave origination on the right side of the obstacle. Electrical-field stimulation at 1.2 V/cm was applied to a 1.85 mm-diameter obstacle. The anode and cathode are placed 4 cm from each other on the left and right sides of the observation chamber, respectively. The field of view is 11.6 mm. (b) The intensity profile recorded at the white mark emphasized by the white arrow in the first frame in (a). The intensity is scaled to the maximum intensity, $I_{max}$. (c) $E_{min}$ for a stimulation duration of 100 ms depending on the obstacle size. Equation (6) is plotted as a solid line.

Figure 7

(Color online) Comparison of the stimulation duration for the minimum electrical far-field in experiments with cardiac tissue. (a) Relationship between obstacle radius $R$ and the fraction of $E_{min}$ for stimulation durations of 10 ms and 100 ms. (b) Relationship between $E_{min}\left(S^{10}\right)$ and $E_{min}\left(S^{100}\right)$ corresponding to stimulation durations of 10 ms and 100 ms, respectively. The dotted lines correspond to the slopes of the obtained approximation in (a) for constant radii of 0.5, 0.9, and 1.85 mm, respectively. The upper dashed line is the reference for $E_{min}\left(S^{10}\right)=E_{min}\left(S^{100}\right)$, and the lower dashed line corresponds to a slope of 2.