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. 2022 Sep 19:978:1-45.
doi: 10.1016/j.physrep.2022.06.003. Epub 2022 Jul 6.

The physics of heart rhythm disorders

Wouter-Jan Rappel  1
Affiliations

The physics of heart rhythm disorders

Wouter-Jan Rappel. Phys Rep. .

Abstract

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Keywords: Arrhythmias; Fibrillation; Heart rhythm disorders; Modeling.

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Figures

Figure 1:
Figure 1:
Schematic representations of the heart. A. The relevant anatomical parts with arrows indicating direction of blood flow: deoxygenated blood enters the right atrium through the venae cavae and is pumped by the right ventricle to the lungs through the pulmonary artery. Oxygenated blood returns to the heart through the pulmonary veins and is circulated to the rest of the body through the aorta. B. Schematic representation of the electric system of the heart, shown here in blue. 1. Sinoatrial node 2. Atrioventricular node 3. Bundle of His 4. Left bundle branch 5. Left posterior fascicle 6. Left-anterior fascicle 7. Left ventricle 8. Ventricular septum 9. Right ventricle 10. Right bundle branch (By J. Heuser - self made, based upon Image:Heart anterior view coronal section.jpg by Patrick J. Lynch).
Figure 2:
Figure 2:
Electrophysiology of a myocyte. A. Fluorescent image of a human cardiac myocyte where α-actinin is labeled in green and the nucleus is stained in blue. Scale bar: 10 μm. (From [24]). B. Schematic representation of a ventricular cardiac myocyte and some of its ion channels, ion transporters, signaling cascades, and structure. RyR indicates ryanodine receptors; SR, sarcoplasmic reticulum; Mito, mitochondria; CaMK, calcium-calmodulindependent protein kinase; NCX, Na/Ca exchanger; PLM, phospholemman; PLB, phospholamban; ATP, ATPase; PKA, protein kinase A; β-AR, β-adrenergic receptor; IKr, IKs, IK1, potassium channels; and ICl(Ca), a calcium-activated Cl current. (From [25]) C. Typical action potential shape, along with the most important currents during each phase of the action potential.
Figure 3:
Figure 3:
Cell arrangement and fiber orientation in human hearts. A. A micrograph showing the orientation of cardiac myocytes in a slice of the ventricular right wall. The dark regions represent the nuclei of the cells and the cytoplasm is red (Regents of University of Michigan Medical School (§2012). B. Fiber orientation in the left ventricle of an ex vivo human heart obtained using DTMRI and a color coding representing the local fiber angle. Left-handed (right-handed) spiraling fibers are shown in blue (green) and darker values correspond to larger local angles. (From [43]). C. Fiber orientations in the LA in 2 different hearts, viewed posteriorly over the roof. The distance of the fiber relative to the endocardial shell is indicated using the color bar so that endocardial (epicardial) fibers are colored yellow (red). LIPV: left inferior pulmonary vein; LSPV: left superior pulmonary vein; RIPV: right inferior pulmonary vein; and RSPV: right superior pulmonary vein. (From [44]).
Figure 4:
Figure 4:
The FHN model. A: Nullclines (black) of the FHN model, along with the the streamlines of the phase portrait and the equilibrium point (yellow dot). A small stimulus initiates the red decay trajectory, marked by 1, whereas a large stimulus initiates the red excitation trajectory, marked by 2. Parameter values of the model (see text): ϵ = 0.08, a = 0.7, and b = 0.8, resulting in a fixed point located at v=−1.2 and w=−0.625. B: Time trace of the variable v corresponding to the trajectory #1 and #2 in A.
Figure 5:
Figure 5:
Model fitting. A: Clinical AP morphology from an AF patient compared to model AP morphologies obtained by fitting the model parameters for an AF patient. The average clinical AP morphology is shown as a dashed line while the FK model is shown in blue and the KKT model is in red (From [73]). B: Schematic representation of a cardiac cell activated by stimuli with period T. As a result, the membrane potential increases rapidly, followed by a gradual return to the resting potential. The APD (action potential duration) measures the time at which the potential is above a certain threshold while the DI (diastolic interval) measures the time interval between the end of one and the start of another action potential. C: Clinically determined APD and its polynomial fit for an AF patient (open and closed symbols, respectively) and the results from a fitting procedure for the FK (blue) and KKT (red) model. D: CV restitution curves of several ionic models and of measurements in a guinea pig (solid lines) together with those obtained from the fitted FK model (From [70]). (BR: Beeler Reuter [76], MBR: modified Beeler Reuter with speed-up calcium, MLR-I: Luo-Rudy with speed-up calcium [77], GP: Guinea pig [78]). E: As in C, but now for the clinically determined CV restitution curve.
Figure 6:
Figure 6:
Comparison of propagation speed for planar and curved fronts. A: Activation fronts plotted every 50ms following line stimulus in the middle of the rectangular domain using the FK model (parameter set #8) [87]. B: As in A, but now following a point stimulus. Due to the curvature of the wavefront, the propagation velocity is smaller than for the planar front in A.
Figure 7:
Figure 7:
Alternans in cardiac myocytes. A: Cobweb diagram illustrating a stable fixed point, shown in red, on a APD restitution curve with slope < 1. The dynamics of a small perturbation in DI, corresponding to the blue point, can be followed using the restitution curve APDn+1 = f(DIn) and the relationship APDn + DIn = T (dashed line) and is shown to approach the fixed point, indicating stability. B: As in (C) but now for a restitution curve with a slope larger than 1. In this case, a point on the curve is unstable, as can be seen by following the path of the perturbation. C: Example of alternans in a single cell. Shown here is the response of the FK model (parameter set #4 from Ref. [87]) to periodic pacing with a period of 220 ms. The resulting APD oscillates between long and short.
Figure 8:
Figure 8:
Calcium alternans in experiments and simulations. A: Linescans in a rat ventricular myocyte in response to 3 consecutive voltage pulses, visualizing Ca2+ using a fluorescent indicator. From [147]. B: Normalized Ca2+ release in a 2D slice of a simulated cell, with green/red indicating minimal/maximal Ca2+ release. The pattern shows alternans, with a small release in beat 445 and a large release in beat 446. From [154]. C: A line scan plot of cytoplasmic Ca2+ from a simulation of coupled CRU networks, illustrating miniwaves during the release beats. From [148].
Figure 9:
Figure 9:
Initiation of spiral waves using a premature stimulus. A: Schematic drawing of a stimulus (white circle) that follows a planar wave that propagated from left to right. Proper timing of this stimulus results in wave block to the right, indicated by the red symbol, but not to the left, as indicated by the green arrows. As the newly created wave propagates in the retrograde direction, tissue in the right-half of the domain continues to recover, eventually leading to reentry and spiral wave formation. B-E: Snapshots of a simulation showing the formation of a figure-of-eight reentry pattern as the result of a properly timed extra stimulus. Panel B shows the premature stimulus along with the planar wave propagating from left to right. Panel C demonstrates that only propagation in the retrograde direction is possible while panel D illustrates the ensuing reentry into the recovered tissue. Panel E shows the figure-of-eight reentry pattern. Snapshots were created using the FK model and numbers correspond to time in ms.
Figure 10:
Figure 10:
Alternans in spatially extended domains. A: Initiation of discordant alternans in an 8 cm long computational cable using the Beeler-Reuter model (vertical axis denotes space and horizontal axis denotes time). All stimuli were applied at the top end of the cable, using constant 310 ms intervals. After multiple stimuli, the top and bottom of the cable are alternating out of phase. From [175]. B: Spatially discordant APD alternans in simulated 2D cardiac tissue using the Luo-Rudy model and a pacing cycle of 220 ms. The top panel displays the action potential at sites a and b, showing a long-short pattern. The lower panel shows that the spatial APD distribution, color-coded with blue for short and red for long, for two consecutive beats. The white line represents the nodal line, which separates the out of phase domains and where alternans is absent. C: Sequence of events following a premature ectopic beat (asterisk) that can result in reentry. The beat is initially blocked when it attempts to propagate into the region with long APD. It can reenter this region, however, after it has propagated along the nodal line, resulting in a figure-eight re-entry. B&C from [173].
Figure 11:
Figure 11:
Examples of wave breakup in models with heterogeneities. A: Snapshots showing successive times in a simulation of the FHN in a domain that contains which a heterogeneous zone of refractoriness, indicated by the red lines. Due the heterogeneity, the wave is originally blocked but then reenters the heterogeneous zone, after which a spiral wave is formed. After [180]. B: Reentry by interaction of an activation wave with a circular and fibrotic region in a computational model using the modified Beeler-Reuter model. Total area 77 cm2. From [186]. C: Initiation of reentry in a patient-specific model shown in 4 snapshots, 60ms apart. Reentry is shown using white arrows and is due to fibrosis in that region. From [187].
Figure 12:
Figure 12:
Spiral wave breakup. A-D: Snapshot of a simulation of the FK model in which a spiral wave breaks up due to the Doppler shift that is induced by the meandering tip. This shift results in wave block (B) and spiral wave breakup (C-D). The voltage is displayed using a color map with orange/black corresponding to polarized/depolarized tissue, respectively. From [87].
Figure 13:
Figure 13:
Spiral tip dynamics A: Snapshot of a simulation of a single spiral with the voltage shown using a gray scale with white (black) corresponding to depolarized (repolarized) tissue. The intersection of the red and green lines is marked by a yellow dot and represents a phase singularity and can be identified as the spiral wave tip. B: Example of spiral wave breakup in a simulation, with voltage shown using a color scale, activation fronts shown as green line, recovery lines shown as black lines, tips of clockwise rotating spiral waves as black dots and tips of counter clockwise rotating spiral waves as white dots. C&D: Snapshots of counterclockwise rotating spiral waves for two different parameter sets of the FK model. In these snapshots, the voltage is color-coded with red corresponding to high and low corresponding to low values, and the tip trajectory is shown in white. E: Comparison of tip trajectories obtained using the full models (upper row) and the single particle model (lower row). The left and middle column correspond to the trajectories from the FK model shown in C&D while the right column corresponds to a tip trajectory from the KKT model. Scale bars are 0.5 cm. From [221].
Figure 14:
Figure 14:
Complex spiral tip dynamics. A: Phase diagram of the spiral wave dynamics in the FK model in the presence of two identical heterogeneities. Representative tip trajectories, corresponding to the three dots, are shown as black lines, with the location of the heterogeneities indicated as red crosses. The tip dynamics in the yellow region has a positive Lyapunov exponent (λ = 7.1) and is thus chaotic. B: Schematic representation of a particle moving in a potential landscape with two wells, representing tissue heterogeneities. C: Phase diagram of the single particle model in the presence of the potential landscape shown in panel B. The phase diagram is qualitatively similar to the one of the full model, presented in panel A. D-E: Snapshots of a simulation of the FK model in the presence of a circular heterogeneity with a radius of 0.275 cm, plotted by the dashed white line. Snapshot D represents the start of the simulation interval while snapshot E represents its end. The trajectory of the spiral tip (cyan dot) is plotted in red and shows that the spiral wave is temporarily trapped by the heterogeneity. F: Distance r from the tip to the center of the heterogeneity as a function of time. Trapped intervals are indicated by the red bars. G: Trajectory of an intermittently trapped particle, moving on a potential surface with a single well. H: Distance r of the particle to the minimum of the potential well as a function of time. Trapped intervals during are indicated by the red bars. Panel A-C from [221]; panel D-H from [228].
Figure 15:
Figure 15:
Filament stability and fiber twist. A: Simplified three-dimensional wedge geometry with rotating anisotropy. B: Break up of filaments for a slab of dimension 5cmxcm and with a thickness of 0.9 cm. The rotation rate in this simulation was taken to be 12°/mm. From [88].
Figure 16:
Figure 16:
Experimental and clinical mapping of spiral wave patterns. A: Voltage sensitive dye recordings of reentry in in a cultured monolayer of neonatal rat cardiac cells. The colors indicate the normalized voltage level, with blue being the resting state (0%) and red the peak action potential (100%). From [260]. One full rotation of the reentry is illustrated, with frames 20 ms apart. B: Snapshot of the phase map on a rabbit heart surface during sustained fibrillation, obtained using a voltage sensitive dye. The phase map reveals two counter-rotating spiral waves, indicated by the white and black dots. From [213]. C: Activation maps during sustained atrial fibrillation in the RA of a human explanted heart, recorded using a voltage sensitive dye. The activation is simultaneously measured on the endo- and epicardial surface and reveals a reentry pathway on both surfaces. From [261]. D: Isochronal maps of the LA, separated by 237 days and obtained using a basket electrode, demonstrate temporal conservation of a spiral wave in human AF. Time in ms. Targeted ablation at the spiral tip eliminated AF. From [262]. E: Mechanical filaments (red) inside pig ventricles (grey shaded) between endocardial (endo) and epicardial (epi) heart surface during ventricular fibrillation imaged using 4D ultrasound. From [263].
Figure 17:
Figure 17:
Low-energy defibrillation. A: Sawtooth potential in a simulation of a strand of 30 cells and an intracellular junction resistance of 50 MΩ (From [290]). B: Upper panel: Unipolar anodal stimulation at the center of a thin computational sheet with equal intra- and extracellular anisotropy ratios. In this case, the changes in Vm have identical sign, with a shape that is given by the the fiber orientation (taken here to be along the horizontal direction, as indicated by the arrow). Lower panel: Identical stimulus but now with unequal intra- and extracellular anisotropy ratio. Changes in Vm are both positive and negative, with tear-drop shaped regions of depolarization (called virtual cathodes). (From [122].) C: Example of termination of spiral waves by wave emission from heterogeneities in a simulation using the Luo-Rudy model. Electric field pulses initiated waves from the obstacles, resulting in the termination of the rotating waves. (From [303].) D: Snapshots of the electrical activation of a dog’s atrium, visualized using optical mapping, during low-energy defibrillation (From [304]). The upper row visualizes AF while the lower row corresponds to the activation maps after each low energy pulse (LEAP). The last panel shows the removal of fibrillation, resulting in tissue that is at rest.
Figure 18:
Figure 18:
Modeling ablation of fibrillation. A-F: Examples of ablation mechanisms that lead to acute termination of atrial fibrillation (Adapted from [316]). In (AD), a slow-conducting zone is sandwiched between nonconducting regions (hatches). The resulting figure-of-eight reentry can be abruptly terminated by ablating the slow-conducting isthmus. E: Simulation of termination in a high excitability region, indicated by the dashed circle, that is surrounded by low excitability tissue. The excitability is a linear function of the radial direction. Before ablation (left panel), a wave is attached to a circular obstacle. After ablation, the wavefront finds itself in a region of low excitability, detaches from the obstacle, and migrates towards the edge of the computational domain, resulting in the termination of reentry (right panel). F: Simulation of termination in a region of low excitability. Within the dashed circle, the tissue excitability is a linear function of the radius and is lowest at the obstacle. Before ablation, a counter clockwise rotating activation front is attached to a nonconducting zone (left panel). After ablation (middle and left panels), the wavefront encounters the wave back, resulting in block (double bars in right panel). The wavefront detaches from the ablation lesion and terminates. Scale bars, 1 cm. (G-H) Targeted ablation based on fibrotic tissue regions (From [211]). (G) Posterior (top) and anterior (bottom) views of a patient-specific atrial mode as reconstructed from segmented LGE-MRI scans, including the distribution of fibrotic tissue shown in green. (H) Sites of ablation delivery shown as red dots in the left atrium of the patient in (G). The dashed ellipse indicates the locations ablated based on the locations of persistent reentry circuits, identified by computational analysis.
Figure 19:
Figure 19:
Termination statistics in cardiac models. A: Termination times as a function of the system size for the FK model with periodic boundary conditions plotted as red symbols. Also shown are the results from the master equation approach (black symbols) and from the closed-form expression obtained using the WKB analysis (solid line). B&C: Transition rates for the FK model computed using direct simulations with periodic boundary conditions. Shown are the birth and death rates for n → n + 2 (B) and n → n − 2 (C) computed in a square geometry of various sizes. Error bars represent standard deviation of the mean value. D: The W±2 rates, normalized by the area of the domain, as a function of the density of tips. The rates collapse onto a single curve for all computational domain sizes. E: Simulation of spiral wave break up in a 7.5cmx7.5cm domain. The potential is shown using a gray scale and spiral wave tips are indicated by red symbols. The central portion, with radius 0.75 cm, inside the white circle has reduced excitability, which results in a stable spiral wave source. Scale bar: 1cm. F: Snapshot of the activation pattern after the ablation of the heterogeneous circular region in A. The virtual ablation lesion, shown in green, results in the removal of the spiral wave from the central region. All non-conducting boundaries are shown as orange lines. Panels A-D from [362], Panels E&F from [314].
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