Synchronization of Ventricular Fibrillation

The synchronization that is described in [55,44] and in the previous section is phase synchronization: Basically the whole tissue does locally the same oscillations in the phase space. However, in my numerical work of chapter2 for example far field pulses were used to control the system and the frequency of excitation was uniform in the

4.1 Control by Periodic Pulse Sequences

0.60 0.65 0.70 0.75 0.80 0.85 0.90 maximum activated area

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(a) (b)

0.55 0.60 0.65 0.70 0.75 0.80 0.85 maximum activated area

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(c) (d)

Figure 4.7: These diagrams present the dependence of the maximum activated area on the voltage of the pulses and on the ratio of the pacing frequencyf_p and the mean dominant frequencyfd

opt. The unstructured distribution of data points is depicted on the right in 2D-histograms with a logarithmic color map. On the left, the maximum activated area is depicted color coded with the voltage and the frequency ratio on the axes. The diagrams (a) and (b) display the rabbit, diagrams (c) and (d) the pig dataset. At each point in the displayed parameter space the average of all data points that lie in a surrounding disc is depicted. The disc is displayed in the lower left in black. If less than 25 data points contribute to a mean value, no value is displayed. Generally, the maximum activated area increase with the voltage. However, for large voltages a dependency on the frequency ratio emerges, that is more clearly visible in the pig data. Here, the maximum activated area is increased near the frequency ratio of 1.1.

4 Results

medium in the synchronized state, but the phases were not. Thus, I will investigate frequency synchronization caused by periodic pulses in the following. How periodic pulses can synchronize the dynamics of VF in the sense that the heart tissue is every where excited with the same frequency as the pacing frequency, but not necessarily at the same time.

In order to quantify the synchronization, I measure the fraction of the heart surface that has a dominant frequency matching the frequency of the pacing. This measure will be calledsynchronized area fraction in the following. Refer to the lower row of fig. 4.6 for an example where the pulse sequence leads to a spatially homogeneous distribution of the dominant frequency. Almost the whole tissue is excited with a frequency that equals the pacing frequency, i.e. the synchronized area fraction in the last frame is close to 1.

Please note, that this measure differs significantly from the maximum activated area used in the previous section as a full frequency synchronization can for example be achieved by pulses that cause planar waves that travel over the whole muscle. In this case, the whole tissue would not be excited at once, but with the same frequency. We will thus look at a different kind of control in this section compared to the previous one.

Figure 4.8 shows the mean synchronized area fraction in the last eighth of a pulse sequence depending on the frequency ratiofp/fd

optand the voltage of the pulses. The figure is structured as the corresponding ones before: On the right the unstructured distribution of data points is depicted with a logarithmic color scale. The top row and bottom row show data from experiments with rabbit and pig hearts, respectively.

Due to variations of the synchronized area fraction for very similar parameter values, the average value of the data points in a disc centered at the respective location is displayed. The disc is shown in the lower left of the diagrams.

The main qualitative features are similar to those found in the corresponding plots with the maximum activated area as a measure: The synchronized area fraction increases with the voltage and this increase occurs at smaller voltages the closer the frequency ratio is to 1.1. However, while there was no clear frequency dependency below ≈35 V in the pig data and below ≈ 20 V in the rabbit data, fig. 4.8 shows larger synchronized area fractions around 1.1 also in this range of low voltages.

Figure4.7 illustrates for which voltages and frequency ratiosf_p/f_d^opt a periodic pulse sequence has a high chance to change the dynamics drastically, such that large fractions of the heart surface are activated simultaneously. In contrast, fig.4.8 demonstrates, how the dynamics are changed by periodic pulse sequences in terms of the frequency with which the tissue is activated. In most areas of the parameter space that is shown in fig. 4.8, the structure is strikingly simple and very similar for both data from experiments with rabbit and pig hearts: Contour lines form wedges made of straight lines with the tip being between a frequency ratio of 1.0 and 1.2.

Figure 4.8 consists of data from many experiments. However the same qualitative features can be found in single experiments as figs.4.9and 4.10 illustrate. The data of single experiments is difficult to interpret as only very few data points exist. Still,

4.1 Control by Periodic Pulse Sequences

0.1 0.2 0.3 0.4 0.5 0.6 0.7 synchronized area fraction

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(a) (b)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 synchronized area fraction

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(c) (d)

Figure 4.8: These diagrams present the dependence of the synchronized area fraction on the voltage of the pulses and on the ratio of the pacing frequencyf_p and the mean dominant frequencyfd

opt. The unstructured distribution of data points is depicted on the right in 2D-histograms with a logarithmic color map. On the left, the synchronized area fraction averaged over the last eighth of the pacing is depicted color coded with the voltage and the frequency ratio on the axes. The diagrams (a) and (b) display the rabbit, diagrams (c) and (d) the pig dataset. In the left diagrams, the depicted value is the mean of data points in a surrounding disc with a size that is displayed in the lower left in black. If less than 25 contribute to a mean value, no value is displayed. Generally, the synchronized area fraction increase with the voltage. Additionally, there is a strong dependency on the frequency ratio. The synchronized area fraction is the largest near the frequency ratio≈1.1.

4 Results

most diagrams for single experiments seem to be in agreement with the structure shown in fig.4.8. These 30 diagrams are not included in this thesis but can be viewed online[76].

0.2 0.4 0.6 0.8 1.0

synchronizedareafraction

Figure 4.9: The synchronized area fraction that is caused by periodic pulse sequences is shown depending on the voltage of the pulses and on the frequency ratio fp/fd. The data points shown as circles are the actual recordings, while the remaining colored area is created from an interpolation with a 1/r radial function, with r being the distance of two points in the diagram.

The depicted data is from an experiment with a pig heart on 15 Feb 2018.

The animal model is a chronic heart failure model. The pulses were applied using two plate electrodes.

Due to the clear structure in fig.4.8, it might be tempting to expect a deterministic behavior for the synchronization, i.e. that a certain parameter combination always leads to a specific synchronized area fraction. However, due to the stochastic nature of VF, values can vary considerably for single recordings. This is illustraded by fig.A.5 which is analogous to fig. 4.8, except that nearest neighbor interpolation is used instead of averaging. FigureA.5 on page107 depicts the same data but with nearest neighbor interpolation instead of averaging.

Together, Figures 4.7 and 4.8 allow to make predictions, what kind of control can be expected for pulse sequences depending on the ratio f_p/f_d^opt and the voltage of the pulses. Especially, they make the prediction that a pulse sequence with a frequency ratiofp/f_d^opt ≈1.1 will have the largest effect, both in terms of frequency synchronization and activation.