As already shortly reviewed in the introductory Section 1.2.1 cardiac arrhythmia is in most cases thought to be driven by rotors. During ventricular fibrillation, electrical excitation waves propagate continuously through the heart in a spiral like fashion. At the centre of each spiral wave exists a point around which the wave rotates. Fur-thermore, locally the heart goes through an oscillation like activation which can be associated with a phase. This phase will change continuously in time, however, in space it is continuous only in most places. The centre of each rotation contains a discontinuity much like a spiral staircase always has a pole at its centre. This point7 is the phase singularity (PS) and a hallmark of cardiac arrhythmia. In earlier cardiac literature other names, such as pivoting points [28] or wave breaks [88] have been used.
These names will be used synonymously here, because conceptually they both describe the spiral cores and differ only in how the phase is defined. These differences will be discussed below.
This chapter gives a short overview over different methods of defining PS and identifying them. It further describes the approach developed and used for the data analysis in this thesis.
7The identification as a single point is practical but not necessarily correct. Indeed, linear reentries are common in cardiac tissue and a PS is “smeared” out in this case with the phase map being noncontinuous on a line.
4.4 Phase Singularity Identification and Tracking 43
4.4.1 Overview
Phase singularity analysis can be separated into three parts, though, depending on the algorithm used, each may not be obviously separated:
1. defining the phase map
2. identifying the phase singularities 3. tracking phase singularities.
Although none of these steps may be expected to be particularly difficult, in practice each one poses its own challenges, as evidenced by continuing research in developing techniques to analyze PS and the multitude of methods employed in practice. Many of these methods, not including additional variations in tracking methods, are listed in the Appendix C.2 (p. 161). The main reasons for such difficulties are:
1. Video noise will cause variations in the phase.
2. Discrete pixels mean that phase gradients become phase differences. However, unlike phase gradients, phase differences are ill defined. For example, a phase difference of −𝜋 is identical to one of 𝜋. By definition, however, close to a PS the gradient in the phase map must become large. Such large gradients mean that large phase differences are likely to be found. These large differences can then become problematic since their correct sign is unclear.
3. Complex activation patterns additionally complicate tracking by causing fast moving PS.
In the following sections I will briefly review some of the methods used for finding the phase map and PS. Then some of the issues and limitations will be discussed in more detail. Finally, the method of phase singularity identification and tracking used in this thesis will be presented and the different analysis steps described.
4.4.2 Typical Definitions for the Phase
In the Section 1.2 the local oscillator behaviour was shown exemplarily for the Bär-Eiswirth model in Figure 1.3. In such a two variable model, a phase may be defined directly based on the two variables – 𝑢 and 𝑣 in the Bär-Eiswirt model. During the dynamics, the local oscillators rarely come close to the unstable point at𝑢 = 0.65 and 𝑣 = 0.45 (compare Fig. 1.3) and thus the phase was defined as the angle around it.
Defining the Phase using Embedding
In simulations heart cells are often described by more than two variables while in experiments commonly only one variable – the membrane potential – is recorded.
However, for defining a phase a second variable is necessary since an angle needs to be defined. Thus, the second variable needs to be reconstructed. This can be achieved by using the delay embedding technique from nonlinear time series analysis [89]. Delay
embedding defines a time delay 𝜏𝑑 and then uses not just the membrane potential as a function of time 𝑉m(𝑡) but also the value at a delay 𝑉m(𝑡 + 𝜏𝑑). The value for 𝜏𝑑 has to be chosen based on the typical oscillation period and APD. 𝜏𝑑 = 25ms has for example been used in [38]. After defining two variables, the phase can be found as before. For this the mean value ⟨𝑉m⟩ = 1/𝑇 ⋅ ∑ 𝑉m(𝑡) may be used to give the formula:
𝜑 =atan2(𝑉m(𝑡 + 𝜏𝑑) − ⟨𝑉m⟩, 𝑉m(𝑡) − ⟨𝑉m⟩). (4.1) Defining the Phase through Wavebreaks
Another method of defining phase singularities (or wavebreaks) is to find the points where the wavefronts and wavebacks connect [28]. Here the wavefront is defined as the area that was just activated while the waveback is defined as the area where the action potential just dropped again. Such points where wavefront and -back meet will always exist for a stable spiral reentry. Effectively, it can be thought of as defining four distinct phases based on the comparison of the active areas in two consecutive (or time delayed) frames:
1. Activation phase: If a region was not active in the previous frame but is active in the current frame.
2. Active phase: If a region is considered active both in the current and the previous frame.
3. Refractory phase: If a region was active in the previous frame but is not active in the current frame.
4. Excitable phase for regions not active in either frame.
If each of these phases is assigned a phase value differing by 𝜋/2 from the previous one this also defines a phase map similar to those of other approaches.
Defining the Phase via Hilbert Transform
Another method to overcome the difficulty when only a single variable𝑢– such as the membrane potential – is measured is to use the Hilbert transform of 𝑢:
𝐻𝑢(𝑡) = 1 𝜋 ∫
∞
−∞
𝑢(𝜏 )
𝑡 − 𝜏 d𝜏 (4.2)
In the Fourier domain the Hilbert transform causes a phase shift by 90°. It thus allows the definition of the analytic signal:
𝑢𝑎(𝑡) = 𝑢(𝑡) + 𝑖 ⋅ 𝐻𝑢(𝑡) (4.3)
= |𝑢(𝑡) + 𝑖 ⋅ 𝐻𝑢(𝑡)| ⋅ 𝑒𝑖⋅𝜑. (4.4) With the value for 𝜑defining the phase.
4.5 Problems and Limitations of Singularity Identification 45