Scroll waves are three-dimensional vortices which occur in excitable media. Their formation in the heart results in the onset of cardiac arrhythmias, and the dynamics of their filaments determine the arrhythmia type. Most studies of filament dynamics were performed in domains with simple geometries and generic description of the anisotropy of cardiac tissue. Recently, we developed an analytical model of fibre structure and anatomy of the left ventricle (LV) of the human heart. Here, we perform a systematic study of the dynamics of scroll wave filaments for the cases of positive and negative tension in this anatomical model. We study the various possible shapes of LV and different degree of anisotropy of cardiac tissue. We show that, for positive filament tension, the final position of scroll wave filament is mainly determined by the thickness of the myocardial wall but, however, anisotropy attracts the filament to the LV apex. For negative filament tension, the filament buckles, and for most cases, tends to the apex of the heart with no or slight dependency on the thickness of the LV. We discuss the mechanisms of the observed phenomena and their implications for cardiac arrhythmias.
Spiral and scroll waves are rotating patterns of activity in excitable media [
Several factors can induce drift of scroll waves in the heart. Among them are the anisotropy of cardiac tissue and the shape of the cardiac wall. It was shown for two-dimensional spiral waves on curved anisotropic surfaces [
For scroll waves, there are additional purely three-dimensional effects which are also likely to contribute to their dynamics in the heart. In particular, it has been shown that the scroll waves drift if their filaments are curved in space and, moreover, the filament length changes monotonically [
Recently, we have developed a model of the human heart LV. This model correctly describes the shape and myofibre rotation of the LV [
In this paper, we apply our anatomical LV for the study of scroll wave dynamics. We investigate how scroll wave filament dynamics is affected by the anisotropy ratio, thickness of myocardial wall, LV shape, and filament tension. We identify the attractors of filament motion and discuss the possible mechanisms which can account for the observed phenomena.
We used the AP model [
To investigate the effect of filament tension
The time course of a variable
Our anatomical LV model exhibits axisymmetry and uses a variant of spherical coordinates, where
A meridional section of the geometrical LV model. The case of the normal LV shape. Parameter values are
Every time unit in our model corresponds to 20 ms, and diffusion coefficients are chosen such that one space unit in our model corresponds to 1 mm. Throughout the simulations, the following geometry parameters were kept constant: longitudinal diffusion
Meridional sections of the geometrical LV model for various values of the parameters. The cases of the normal ((a), (c)) and the spherical LV shape ((b), (d)). ((a), (b)) shows the geometry for the cases with the minimal apical thickness (
In different simulations, we varied apical thickness
To initiate a spiral wave, we set the potential
The mesh used was a rectangular lattice in the coordinates
The program was written in C language, with OpenMP parallelization, compiled with GCC. Simulations were performed on two supercomputers under Scientific Linux 6.
During the simulations, the position of the scroll wave tip was recorded by finding the intersections of the iso-surfaces
Visualization of the results was done in Paraview, SharpEye, and Matlab.
We generated heart geometries of two shapes: elliptical (
Figure
Drift of a scroll wave in a model of human ventricles represented by trajectories of filament on a midmyocardial surface. (a) Initial position of a scroll wave at the midmyocardial surface. Different colors represent different values of the transmembrane voltage (variable
In the next section, we discuss in detail the type of this motion and its dependency on the model geometry and tissue anisotropy.
We will characterize the position of the filament by a thickness-averaged (i.e., mean) position and represent it as a point in
Figure
Final position
The vertical axis of Figure
General theoretical considerations predict that, in the case of positive filament tension, which we have for
From Figure
Figure
Now, let us try to separate the effects of different components of filament dynamics. First, we characterize the effect of the shape of the ventricle on the final position of the filament. If we compare the final position of the filament for both geometries, we find that, in a spherical shape, the filament is closer to the region of smaller thickness; for example, for
Secondly, let us consider the effect of anisotropy. We see that, for both shapes and all anisotropy ratios, an increase in the anisotropy ratio results in shifts of the filament towards the apex. Once again, the effect is more substantial for a normal shape, especially for 8 mm
Next, we characterize the trajectory of a scroll wave after approaching the attractor. In all cases, the scroll wave stabilizes at some latitude
Residual circumferential speed of the drifting filament after stabilization at the attractors
Figure
Final position
From Figure
Secondly, for the normal shape, we see almost no dependency on the LV thickness. For most parameter values, the scroll wave approaches the apex. However, for
We, however, observe a clear effect of anisotropy. In all cases, an increase in the anisotropy ratio resulted in a shift of the attractor to the apex. Thus, as for the positive filament tension, increased anisotropy tends to push the filament towards the apex.
The velocities of a scroll wave after approaching the attractor for negative filament tension shown in Figure
Residual circumferential speed of the drifting filament after stabilization at the attractors
We have also observed filament break-up for large anisotropy ratio and large apical thickness; that is,
Break-up of a scroll wave due to negative filament tension. Simulations are for
To understand the mechanisms of the observed phenomena, we performed a series of two-dimensional simulations in which we studied the drift of a spiral wave on a two-dimensional surface for the following cases: (a) a paraboloidal surface
Spiral wave drift on two-dimensional surfaces of different shape and anisotropy. Simulations in the Aliev-Panfilov model with parameter
For
The drift of the filaments studied in the previous section is a combination of three factors which can potentially contribute to the filament dynamics: the thickness of the medium, the anisotropy and the shape of the LV.
First, we consider the effect of wall thickness. From [
Stable filament positions
For the spherical LV model (
Next, we turn to the effect of LV shape and anisotropy. It was previously shown [
Now let us consider the case of the negative filament tension. The absence of a break-up for the negative filament tension can be explained by the dependency of this effect on the thickness of the tissue. In [
When the break-up is absent, we observe a drift of transmural filaments. However, in most of the cases, its final position is at the cardiac apex, and for stronger anisotropy, this tendency to go to the apex becomes stronger. Those results are opposite to the results of our two-dimensional simulations, which indicate that, in this case, both geometry and anisotropy repel two-dimensional spiral waves from the apex. This discrepancy can be understood by the observation that, for moderate wall thickness and negative filament tension, filaments will “buckle” and deform into an S-shape, after which they undergo precession [
Ratio of filament length
Buckled filament state after 60 s for
In this chapter, we have presented results on the drift of scroll wave filaments in an anatomical model of human ventricles and have studied the effect of shape, thickness and anisotropy of the ventricle on the drift pattern. We found that the results are substantially affected by the filament tension of the scroll wave.
In the case of the positive filament tension, one of the main determinants of the drift was the thickness of the myocardial wall and the filament tended to drift to the region of minimal thickness. However, in all cases, it never arrived to the point of minimal thickness and rotated at some small distance from it.
Another important determinant of filament drift was the anisotropy of the tissue. Its main effect in our simulations was the attraction of the scroll wave to the apex. The LV shape had a small effect on the results in terms of the direction of the drift. However, it affected the location of the attractor, especially when the gradient in the thickness was not large.
As cardiac tissue has a high excitability in normal conditions, one would expect that in normal conditions, the filament would be located close to the region of minimal thickness with a slight preference towards the apex, due to anisotropy effects. This information might be important for identifying sources of arrhythmias in the heart, with applications in the planning of successive clinical intervention.
We have also studied the case of negative filament tension. In that case, filaments generally behave chaotically, and this normally results in the break-up of scroll waves [
The mechanisms underlying the observed phenomena in the regime of positive filament tension can be partially explained by the existing theories of filament dynamics. As such filaments strive to minimize their length, they move to regions of minimal wall thickness. However, we found in our simulations that even in the isotropic case, the filaments did not exactly reach that minimum. Possible disturbing factors are filament twist [
In the regime of negative tension, the wall thickness proved in most cases to be insufficient for the development of a full three-dimensional break-up. Instead, we identified buckled filaments which also equilibrate at a given latitude, due to the axial symmetry of our LV model. A further theoretical consideration of the effects of shape and anisotropy on scroll wave dynamics would be nontrivial. Possible ways to approach this problem are to consider shapes with a small thickness and to use averaging methods as in [
We performed our simulations using the AP model, which provides a simplified description of cardiac tissue. Two-variable models allow researchers to easily obtain various regimes of filament tension, and they are much more efficient for large-scale numerical simulations. Therefore, two-variable models of cardiac tissue are widely used in studies of two-dimensional and three-dimensional dynamics of spiral waves in the heart (see, e.g., [
We have studied only filaments extending from the epicardial to the endocardial surface. It would also be interesting to study the dynamics of the intramural filaments. Such filaments can occur during the normal excitation of cardiac tissue and may have a complex shape and therefore complex dynamics [
In this chapter, we have studied the motion of scroll waves in a homogeneous model of cardiac tissue. It was shown that the heterogeneity of cardiac tissue substantially affects the motion of vortices and their dynamics [
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the paper.
The authors declare that there is no conflict of interests regarding the publication of this paper.
Hans Dierckx was funded by the F.W.O.-Flanders; his project includes the theoretical research of scroll waves. Sergei Pravdin, Vladimir S. Markhasin, and Alexander V. Panfilov were supported by The Russian Science Foundation (Project 14-35-00005 for Ural Federal University includes the numerical research of scroll waves). Sergei Pravdin was also supported by Ghent University (Grant 01SF1511 includes the development of algorithms and software for modelling electrophysiological activity of the heart using the analytical LV model). Simulations were performed at the HPC infrastructure of Ghent University (Belgium) and at the supercomputer Uran of Institute of Mathematics and Mechanics (Ekaterinburg, Russia).