Abstract
This work aimed at the detection of rotor centers within the atrial cavity during atrial fibrillation on the basis of phase singularities. A voxel based method was established which employs the Hilbert transform and the phase of unipolar electrograms. The method provides a 3D overview of phase singularities at the endocardial surface and within the blood volume. Mapping those phase singularities from the inside of the atria at the endocardium yielded rotor center trajectories. We discuss the results for an unstable and a more stable rotor. The side length of the areas covered by the trajectories varied from 1.5 mm to 10 mm. These results are important for cardiologists who target rotors with RF ablation in order to cure atrial fibrillation.
1 Introduction
Atrial fibrillation (AF) is the most common sustained cardiac arrhythmia [11]. During AF, the atria are typically excited more than 350 times per minute as the refractory periods of atrial cells are compressed [6]. Different excitation patterns occur. This work deals with the task of determining the centers of rotatory excitation patterns on the basis of simulated intraatrial electrograms in three-dimensional space.
The behavior of rotor filaments within ventricular and atrial tissue has already been investigated [1, 2, 5]. Especially in clinical practice, electrograms cannot be recorded within tissue. Instead, catheters are placed on the endocardium and within blood. The detection of rotor centers and their trajectory within blood generates new challenges. The signal amplitude for example decreases with increasing distance to the source on the endocardium. In this work, we introduce a method which is capable of detecting rotor centers within the whole atrial volume. Considering the rotor centers near the endocardium leads to the question, how the rotor center trajectories look like, which difficulties occur with respect to their detection in simulated signals and and in clinical data. Two rotors – an unstable and a more stable one – are discussed.
Different methods already exist for the detection of rotor centers [4]. This work focuses on a phase based approach. A phase singularity (PS) is a point in a phase map which is surrounded by the complete range of monotonically increasing phase values from −π toπ. At the PS itself, the phase is undefined [9]. In the following, rotor centers are assumed to be situated at PSs. The position of rotor centers is of interest in clinical practice [7] as creating myocardial lesions near to the rotor center is believed to alleviate or cure AF.
2 Material and methods
2.1 Simulation setup
A personalized anatomical model of the atria and simulation results were taken from earlier work [3]. The model included complete rule-based fiber orientation and was adjusted on the basis of the Courtemanche-Raimirez-Nattel model for cardiac cells [3]. The model consists of 263 x 328 x 283 voxels with a length of 0.33 mm. Extracellular potentials within blood resulted from transmembrane voltages by forward calculation. The extracellular potentials were interpreted as virtual intracardiac electrograms.
2.2 Hilbert transform
In this work, phase values of the signals were calculated with the help of the Hilbert transform which is defined as follows: Choose a filter G (f) which turns the phase of negative input frequencies forward by 90° respectively +π and the phase of positive input frequencies backward by 90° respectively −π. Constant components are suppressed by G(f). If a signal y(t) passes G(f), the Hilbert transform H{y(t)} results at the output of the filter. Calculationswere based on the MATLAB (MathWorks, Inc.) implementation of the Hilbert transform.
2.3 Phase calculation based on the Hilbert transform
For a signal with periodic properties, a measure of the signals progression [9] within a period at a certain point in time can be an aspired attribute. The instantaneous phase represents one possibility to map that position to the interval from −π to π. As the actual phase calculation with the inverse tangent is not independent of constant components, we subtracted the mean value of the signal before proceeding [8]. Let ỹ(t) be
with
The phase φ(t) of the complex-valued analytic signal z(t) was calculated with the help of the inverse tangent:
real(z(t)) and imag(z(t)) denote the real and imaginary part of z(t).
2.4 Detection of phase singularities in three-dimensional volumes
To begin with, a method was introduced which was capable of detecting PSs in 2D space. In a 2D pixel model, a square composed of four neighboring pixels had to meet two conditions if it enclosed a PS (see Figure 1):
A phase jump from about −π to about +π connects two pixels which share a common edge (path A).
The phase outlines a monotone course while tracking path B.
Two conditions define a PS in a 2D square of four pixels. Path A connects two pixels via a phase jump and the phase outlines a monotone course along path B. For reasons of graphical simplicity, each pixel is represented by a node in its center.
Nine planes can be extracted from each cube consisting of eight voxels: three planes parallel to the coordinate axes of the voxel lattice and six planes in diagonal directions. For reasons of graphical simplicity, each voxel is represented as a node in its center.
In a 3D voxel space, the task of detecting PSs became more difficult. The orientation of the rotor and thus the plane in which the PS might be located, was not predefined anymore. As the introduced model consisted of voxels, it provided a cubic structure. For each voxel, a cube was assembled with 7 neighboring voxels by walking along the coordinate axes (see Figure 2).
In order to detect PSs with unknown orientation, several planes had to be taken into account. Figure 2 shows three planes in parallel to the coordinate axes and six planes in diagonal directions. For each voxel, these nine planes were extracted and checked for the existence of a PS as described above. The resulting PSs characterize a 3D rotor, which is also called spiral wave.
2.5 From phase singularities to rotor center trajectories
The trajectory of the rotor center at the endocardium was read out after the detection of the PSs in the voxel model as described above. A triangular mesh represented the endocardium. PSs which belonged to the same rotor center filament were united into a group. The PS which was located closest to the endocardium was projected onto the endocardial surface for each group and all time steps.
3 Results
3.1 Intraatrial electrograms and corresponding phase signals
Figure 3 shows exemplarily two unipolar intraatrial electrograms. Both were recorded near a rotor center. Beneath each electrogram, the phase signal is shown which was calculated on the basis of the Hilbert transform.
The first and the third plot depict intraatrial electrograms (EGM). The second and the fourth plot show the corresponding phase signals.
The upper phase signal in Figure 3 shows the desired linear shape with phase jumps from −π to π. The lower phase signal deviates from that linear shape which is over-laid by ripples. In addition, the phase begins to oscillate near the phase jumps.
3.2 Phase singularities and rotor center trajectories
The phase values of one exemplary time step are mapped on the endocardial surface in Figure 4. The rotor center was detected with the help of the method described in section 2.4.
Rotor center trajectories of two rotors were investigated with a sample rate of 1 kHz. Rotor 1 was located in the right atrium near the tricuspid valve. Rotor 1 was observed in the time interval from 3.6 s to 5.0 s. Time is referenced to the start of the simulation at 0 s. During that time, rotor 1 rotated with a cycle length of about 145 ms and showed approximately nine and a half revolutions. The turns were regular and stable. The rotor center trajectory is depicted in Figure 5. The trajectory was restricted to a small range of approximately 1.5 mm in x-direction and approximately 3 mm in y-direction. In z-direction no significant movement occurred because the endocardium was almost orthogonal to the z-axis.
Excerpt of the endocardium with phase map. The colors visualize the phase at each point. The white arrow points at the rotor center.
Rotor center trajectory for stable rotor 1. The rotor centers for early points in time are depicted in blue. Late points in time are depicted in yellow. The last revolution is aditionally marked with a black line which connects the rotor centers of the single time steps. An excerpt of the curved endocardium is shown in gray.
Rotor 2 was located at the roof of the left atrium and was obverved from 0.6 s to 2.0 s. During that time, rotor 2 rotated with a cycle length of about 190 ms and showed approximately seven revolutions. The behavior of rotor 2 was more unstable than the behavior of rotor 1. The rotor center trajectory is depicted in Figure 6 for points in time between 1.27 s and 1.60 s. For early points in time the rotor center showed stationary characteristics. Later on it moved along an oval shaped curve across the endocardium, before it returned to its initial stationary state. For approximately 3 % of those time steps the algorithm did not yield a proper rotor center. The area covered by the overall rotor center trajectory of rotor 2 measured more than 10 mm in x-direction, approximately 4 mm in y-direction and approximately 3 mm in z-direction.
Rotor center trajectory for unstable rotor 2. The rotor centers for early points in time are depicted in blue. Late points in time are depicted in yellow. An excerpt of the curved endocardium is shown in gray.
4 Discussion
Methods for the detection of rotor center filaments in ventricular and atrial tissue had already been published [1, 2, 5]. Likewise, the rotor centers and trajectories in 2D simulations were discussed in other work [10]. This work describes and evaluates a method which is capable of detecting PSs in the atrial cavity of a realistic heart model. The point of view changes from cardiac tissue to blood, which is closer to clinical applications. Furthermore we describe the trajectories of rotor tips at the curved endocardial surface.
We showed that our phase based approach is capable of detecting rotor centers in the atrial cavity. Nevertheless, the phase signals did not always fulfill the desired linear shape. Oscillating phase values near the phase jump occurred. The algorithm which searches for PSs will fail if fluctuations become to large. The same holds for ripples which overlaid the linear part of the phase signal. Especially the criterion of a monotone course of phase values along path B (see Figure 1) was threatened by ripples. As the extent of ripples and oscillations was dependent on the shape of the electrogram, this could lead to problems in clinical applications. Further investigations need to show, whether other methods can overcome those disadvantages of the phase calculation with the inverse tangent and the Hilbert transform.
Rotor centers showed different temporal trajectories for different rotors. The algorithm did not yield rotor centers for approximately 3 % of the investigated time steps for rotor 2 which partially explains the gap in the trajectory shown in figure 6. Additionally neighboring activation patterns caused the unstable behavior of the trajectory of rotor 2. The fact that the rotor center was not fixed in space for rotor 2 needs to be paid special attention. Keeping in mind that rotor centers are targets for the creation of lesions in electrophysiological studies, the measurements should not be restricted to a single point in time. Instead the rotor center should be followed for a period of time to be sure about its trajectory.
Author’s Statement
Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.
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