Spatiotemporal Organization of Electromechanical Phase Singularities during High-Frequency Cardiac Arrhythmias

Ventricular fibrillation (VF) is a life-threatening electromechanical dysfunction of the heart associated with complex spatiotemporal dynamics of electrical excitation and mechanical contraction of the heart muscle. It has been hypothesized that VF is driven by three-dimensional rotating electrical scroll waves, which can be characterized by filamentlike electrical phase singularities or vortex filaments, but visualizing their dynamics has been a long-standing challenge. Recently, it was shown that rotating excitation waves during VF are associated with rotating waves of mechanical deformation. Three-dimensional mechanical scroll waves and mechanical filaments describing their rotational core regions were observed in the ventricles by using high-resolution ultrasound. The findings suggest that the spatiotemporal organization of cardiac fibrillation may be assessed from waves of mechanical deformation. However, the complex relationship between excitation and mechanical waves during VF is currently not understood. Here, we study the fundamental nature of mechanical phase singularities, their spatiotemporal organization, and their relation with electrical phase singularities. We demonstrate the existence of two fundamental types of mechanical phase singularities: “paired singularities,” which are colocalized with electrical phase singularities, and “unpaired singularities,” which can form independently. We show that the unpaired singularities emerge due to the anisotropy of the active force field, generated by fiber anisotropy in cardiac tissue, and the nonlocality of elastic interactions, which jointly induce strong spatiotemporal inhomogeneities in the strain fields. The inhomogeneities lead to the breakup of deformation waves and create mechanical phase singularities, even in the absence of electrical singularities, which are typically associated with excitation wave break. We exploit these insights to develop an approach to discriminate paired and unpaired mechanical phase singularities, which could potentially be used to locate electrical rotor cores from a mechanical measurement. Our findings provide a fundamental understanding of the complex spatiotemporal organization of electromechanical waves in the heart and a theoretical basis for the analysis of high-resolution ultrasound data for the three-dimensional mapping of heart rhythm disorders.

Popular Summary

A normal heartbeat is orchestrated by the propagation of an electrical wave that triggers a synchronized mechanical contraction of the heart muscle. During life-threatening cardiac arrhythmias, in which something alters this steady rhythm, the electrical waves can become disorganized and stop the heart from pumping. Here, we demonstrate computationally that, in this setting, waves of mechanical contraction do not simply follow excitation waves but rather exhibit a higher degree of spatiotemporal disorganization.

Our work relies on phase mapping, a powerful tool for visualizing the progression of excitations and tissue deformation across the heart muscle. During a normal heartbeat, waves propagate across the heart in straight lines; at each point of the phase map, that propagation resembles a hand going around a clock, and the phase is continuous in space. During arrhythmias, however, excitation waves rapidly rotate around a singular point where the phase cannot be defined—as at the center of a clock.

Our results demonstrate that, in addition to these singular points associated with rotating excitation waves, phase maps of mechanical waves contain other singular points as well. This is because phase maps of electrical excitation and tissue deformation do not necessarily coincide. Electrical excitation signals a cell to contract, but the resulting force can deform tissue far from the point of excitation.

These results provide an improved basis for using high-resolution ultrasound images of heart-contraction waves to infer excitation-wave patterns underlying clinically important cardiac arrhythmias.

Figure 1

A cubical element of the MSM. (a) The mass of the cube $m$ is distributed equally among the eight cubes that meet at one corner. The edge and diagonal springs have a stiffness of $k/4$ and $k/2$, respectively. (b) A Kelvin-Voigt-type damping system is used where the edge and diagonal dampers have a damping constant of $c/4$ and $c/2$, respectively. (c) To add volume conservation to the model, a penalty pressure is applied to the faces of the cube. This pressure results in forces at each corner. (d) The compressive active contraction is applied along the fiber direction in each cube, and, based on the distances $a_1$ and $a_2$ with bilinear interpolation, it is redistributed on the masses.

Figure 2

A clockwise rotating spiral of active contraction and strain in a 2D isotropic elastic tissue. (a) Analytically derived active contraction field $T_a(x,y)$. (b) The corresponding strain field $\mathrm{Tr}(\epsilon )(x,y)$ is isotropic and closely follows the contraction field. (c) The mechanical phase map $\varphi (\epsilon )(x,y)$ reveals a mechanical phase singularity at the center of the rotor (red circle). (d) The complex amplitude $A(x,y)$ vanishes at the position of the mechanical phase singularity.

Figure 3

Focal active contraction and strain pattern in a 2D isotropic elastic tissue. (a) Circular ring-shaped focal pattern of active contraction $T_a(x,y)$. (b) The corresponding strain field $\mathrm{Tr}(\epsilon )(x,y)$ is isotropic and closely follows the circular active contraction field. (c) The mechanical phase map $\varphi (\epsilon )(x,y)$ exhibits a focal pattern and does not exhibit a phase singularity at any point. (d) The complex amplitude $A(x,y)$ does not vanish at any point.

Figure 4

Strain spiral wave pattern in an anisotropic 2D medium with fibers aligned along the $x$ axis. (a) Clockwise rotating spiral wave pattern of active contraction $T_a(x,y)$. (b) Strain along the fibers ${\epsilon }_{xx}$ vanishes in the horizontally aligned arms. Red circles and black stars with a white edge indicate the location of mechanical phase singularities calculated via the Hilbert transform and as the intersection of the ${\epsilon }_{xx}={\epsilon }_c$ contour (thin red dashed lines) and ${\partial }_t{\epsilon }_{xx}=0$ contours (thin blue lines), respectively, where ${\epsilon }_c$ is the average of the minimum and maximum values of ${\epsilon }_{xx}$. (c) Phase map of the strain $\varphi [{\epsilon }_{xx}(x,y)]$ shows one colocalized mechanical phase singularity at the center of the rotor pattern in (a) and two additional pairs in the top and bottom horizontally aligned spiral arms. (d) The amplitude $A$ of the complex transformed signal vanishes in the vicinity to all mechanical singularities (black regions). (e) Time series of ${\epsilon }_{xx}$ for points 1 and 2 in (b). (f) Phase diagram at points 1 and 2 from (e) derived via the Hilbert transform, where the horizontal axis is the original signal and the vertical axis is the signal shifted by $\pi /2$.

Figure 5

Focal wave in anisotropic medium with fibers aligned along the $x$ axis. (a) Active contraction field $T_a(x,y)$ of an elliptical ring-shaped focal pattern. (b) Strain ${\epsilon }_{xx}$ is stronger in the vertical than the horizontal arms due to fiber anisotropy. Red circles and blue stars indicate the location of phase singularities calculated by the Hilbert and contour-intersection method as in Fig. 4, respectively. (c) The phase map shows mechanical singularities emerge in the absence of electrical singularities. (d) The signal amplitude $A$ vanishes close to and around these singular points (black region).

Figure 6

Mechanism of formation of unpaired mechanical singularities in anisotropic tissue with fibers aligned along the $x$ axis. Active tension (a) and strain ${\epsilon }_{xx}$ (b) for a plane wave traveling in the $+y$ direction in a tissue with stress-free boundary conditions. Active tension (c) and strain ${\epsilon }_{xx}$ (d) for a target wave expanding in a tissue with stress-free boundary conditions. Active tension (e) and strain ${\epsilon }_{xx}$ (f) for a target wave expanding in a tissue with fixed boundary conditions ($u_x=u_y=0$ on all boundaries). Red circles and blue stars indicate the location of phase singularities calculated by the Hilbert and contour-intersection method as in Figs. 4 and 5, respectively. (g) and (h) show the same strain patterns as (d) and (f), respectively, in deformed coordinates.

Figure 7

Number of mechanical phase singularities for a clockwise rotating spiral in a heart muscle with anisotropic fiber architecture (fibers are along the $x$ axis) as a function of tissue size and tissue size to spiral wavelength ratio $L/{\lambda }_s$. Different vertical circles show the different numbers of phase singularities observed in one period. The intensity of the color in each circle exhibits the frequency of occurrence of that number of phase singularities. For instance, for ($L/{\lambda }_s=15$) during one period, 15, 17, and 19 phase singularities are observed. The color density of the circles shows that most of the time there are 17 phase singularities in the domain. The red squares show the average number of singularities during one period.

Figure 8

Electromechanical phase singularities in the presence of a single scroll wave in a 3D tissue ($1.2\times 1.2\times 1.2\mathrm{cm}$). (a) All fibers are uniformly aligned along the $x$ axis. (b) Fibers orthotropically stacked and rotating by 90°, parallel to the $x$ axis at the bottom and parallel to the $y$ axis at the top surface, respectively. Red lines represent lines of mechanical phase singularity or mechanical filaments, and the black line at the center represents the electrical filament or scroll wave core. The isosurface is presenting ${\overline{E}}_f={\overline{E}}_{xx}=0$, where ${\overline{E}}_f$ is the normalized strain along the fiber that is used to calculate the phase.

Figure 9

Electromechanical phase singularities in the presence of a single scroll wave with short wavelength in a 3D tissue ($4.5\times 4.5\times 1.2\mathrm{cm}$; inset figure is the top view) with all fibers being aligned along the $x$ axis. Red lines represent lines of mechanical phase singularity or mechanical filaments, and the black line at the center represents the electrical vortex filament or scroll wave core. The isosurface is presenting ${\overline{E}}_f={\overline{E}}_{xx}=0$, where ${\overline{E}}_f$ is the normalized strain along the fiber that is used to calculate the phase.

Figure 10

Electromechanical organization of filamentlike phase singularities on the surface of a 3D slab with a single scroll wave with small wavelength and fibers in the $x$ direction. (a) Transmembrane voltage in dimensionless units. (b) Contraction field enslaved to voltage. (c) Strain depicted as largest compressive eigenstrain $E_{xx}$. (d) Phase map $\varphi (E)$ from the Hilbert transform of the normalized strain signal. (e) Amplitude map $A(E)$ from Hilbert transform of the normalized strain signal. Amplitude vanishes in the neighborhood of phase singular points.

Figure 11

Organization of electromechanical filaments in the presence of fiber rotation: (a),(b) Single electrical scroll wave. (c),(d) Composition of multiple scroll waves due to electrical wave break. (e),(f) Focal wave pattern after application of a pulse at the center of the top surface. The isosurface is presenting ${\overline{E}}_f={\overline{E}}_{xx}=0$, where ${\overline{E}}_f$ is the normalized strain along the fiber. Insets show colocalized electromechanical filaments. In the case that electrical waves remain stable, one can see a clear colocalization between mechanical and electrical filaments. However, as scroll waves break, a more complex mechanism can be observed, but many of electrical filaments are still paired with a mechanical one.

Figure 12

Automated separation of paired (blue dots) from unpaired mechanical singularities (red crosses). Paired mechanical singularities colocalize with electrical singularities (black dots). The separation is performed by analyzing the temporal evolution of the strain surrounding a mechanical singularity. Phase singularities sampled over a short time interval (approximately 0.4 s) on the surface of 3D tissue during dynamics as shown in Figs. 9, 10, 11: (a) single scroll wave with fibers aligned along the $x$ axis; (b) single scroll wave with fibers rotated by 90°; (c) scroll wave chaos with wave breakup.

Figure 13

Characteristic electrical impulse and contraction shapes exhibited by the electrophysiology model described in Sec. 2b1 and Appendix pp2. In this graph, a 1D representation of the equations is solved using the forward Euler method. In this simulation, $D=1.1{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, $\mathrm{Re}=0.8$, $dx=3\times {10}^{-2}\mathrm{cm}$, and $dt=2.0\times {10}^{-4}\mathrm{s}$. Other parameters can be found in Table 1. The left vertical axis corresponds to the variables $U$ or $v$, and the right vertical axis represents the dimensionless magnitude of the active contracting strain $T_a$. The horizontal axis is time in the units of milliseconds.

Figure 14

A long bar with aspect ratio $59/5$ is used to study the mechanical properties of the MSM. (a) Two equal and opposite displacements along the bar’s long axis are applied to the ends of the bar. This loading is used to find the relation of the Young’s modulus and Poisson ratio with the penalty volumetric pressure and to find the stress-strain relation of the MSM. In this loading case, no boundary condition is introduced, because at all times the bar is in force and moment equilibrium. (b) A periodic displacement is applied to one end of the bar while the other end is kept fixed. This loading is used to investigate the storage and loss modulus of the MSM.

Figure 15

(Top) Young’s modulus and (bottom) Poisson’s ratio of the MSM at small strain $\epsilon =0.1\%$ and large strain $\epsilon =15\%$ as a function of dimensionless volumetric pressure $\overline{p}=p/E$. ${\overline{E}}_v$ is the dimensionless Young’s modulus which is equal to ${\overline{E}}_v=E_v/E$. $E_v$ is from Eq. (11), and ${\nu }_v$ is from Eq. (12).

Figure 16

Stress-strain curve of the MSM model. $\overline{p}$ represents the dimensionless volumetric pressure. Including the volumetric pressure increases the hyperelasticity.

Figure 17

The ratio between loss and storage modulus as a function of dimension less elastic diffusion. $D_e$ is $\eta /\rho$, where $\eta$ is the damping of the system and $\rho$ is the density of the material. $C=\sqrt{E/\rho }$ is the characteristic elastic wave speed, and $L$ is the length of the system (not the lattice dimension). The top $x$ axis shows the ratio between $D_e$ and the diffusion coefficient $D=1.1{\mathrm{cm}}^2/\mathrm{s}$ in the electrophysiology model. The relation between $E_l/E_s$ and damping of the system is linear in 0.1% strain and at zero damping; loss modulus is zero. The penalty volumetric pressure $p$ slightly decreases the $E_l/E_s$ ratio in high damping constants.