Initiation and dynamics of a spiral wave around an ionic heterogeneity in a model for human cardiac tissue

Meander pattern of spiral wave and the spatial distribution of its cycle length

One of the most interesting dynamics of rotating spiral waves in an excitable medium is meandering. The tip of a meandering spiral wave moves along a complex trajectory, which often takes the shape of an epitrochoid or hypotrochoid with inward or outward petals. The cycle lengths (CLs) of a meandering spiral wave are not constant; rather, they depend on the meandering dynamics. In this paper, we show that the CLs take two mean values, outside T^{out} and inside T^{in} the meandering trajectory with a ratio of T^{in}/T^{out}=(n+1)/n for the inward and (n-1)/n for the outward petals, where n is the number of petals in the tip trajectory. We illustrate this using four models of excitable media and prove this result. These formulas are shown to be suitable for a meandering spiral wave in an anatomical model of the heart. We also show that the effective periods of overdrive pacing of meandering spiral waves depend on the electrode location relative to the tip trajectory. Overall, our approach can be used to study the meandering pattern from the CL data; it should work for any type of dynamics that produces complex tip trajectories of the spiral wave, for example, for a drift due to heterogeneity.

The physics of heart rhythm disorders

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.

Stochastic termination of spiral wave dynamics in cardiac tissue

Rotating spiral waves are self-organized features in spatially extended excitable media and may play an important role in cardiac arrhythmias including atrial fibrillation (AF). In homogeneous media, spiral wave dynamics are perpetuated through spiral wave breakup, leading to the continuous birth and death of spiral waves, but have a finite probability of termination. In non-homogeneous media, however, heterogeneities can act as anchoring sources that result in sustained spiral wave activity. It is thus unclear how and if AF may terminate following the removal of putative spiral wave sources in patients. Here, we address this question using computer simulations in which a stable spiral wave is trapped by an heterogeneity and is surrounded by spiral wave breakup. We show that, following ablation of spatial heterogeneity to render that region of the medium unexcitable, termination of spiral wave dynamics is stochastic and Poisson-distributed. Furthermore, we show that the dynamics can be accurately described by a master equation using birth and death rates. To validate these predictions in vivo, we mapped spiral wave activity in patients with AF and targeted the locations of spiral wave sources using radiofrequency ablation. Targeted ablation was indeed able to terminate AF, but only after a variable delay of up to several minutes. Furthermore, and consistent with numerical simulations, termination was not accompanied by gradual temporal or spatial organization. Our results suggest that spiral wave sources and tissue heterogeneities play a critical role in the maintenance of AF and that the removal of sources results in spiral wave dynamics with a finite termination time, which could have important clinical implications.

Intermittent trapping of spiral waves in a cardiac model

Spiral waves are found in many excitable systems and are thought to play a role in the incoherent electrical activation that underlies cardiac arrhythmias. It is well-known that spiral waves can be permanently trapped by local heterogeneities. In this paper, we demonstrate that spiral waves can also be intermittently trapped by such heterogeneities. Using simulations of a cardiac model in two dimensions, we show that a tissue heterogeneity of sufficient strength or size can result in a spiral wave that is trapped for a few rotations, after which it dislodges and meanders away from the heterogeneity. We also show that these results can be captured by a particle model in which the particle represents the spiral wave tip. For both models, we construct a phase diagram which quantifies which parameter combinations of heterogeneity size and strength result in permanent, intermittent, or no trapping. Our results are consistent with clinical observations in patients with atrial fibrillation that showed that spiral wave reentry can be intermittent.

Anisotropic shortening in the wavelength of electrical waves promotes onset of electrical turbulence in cardiac tissue: An in silico study

Several pathological conditions introduce spatial variations in the electrical properties of cardiac tissue. These variations occur as localized or distributed gradients in ion-channel functionality over extended tissue media. Electrical waves, propagating through such affected tissue, demonstrate distortions, depending on the nature of the ionic gradient in the diseased substrate. If the degree of distortion is large, reentrant activity may develop, in the form of rotating spiral (2d) and scroll (3d) waves of electrical activity. These reentrant waves are associated with the occurrence of lethal cardiac rhythm disorders, known as arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), which are believed to be common precursors of sudden cardiac arrest. By using state-of-the-art mathematical models for generic, and ionically-realistic (human) cardiac tissue, we study the detrimental effects of these ionic gradients on electrical wave propagation. We propose a possible mechanism for the development of instabilities in reentrant wave patterns, in the presence of ionic gradients in cardiac tissue, which may explain how one type of arrhythmia (VT) can degenerate into another (VF). Our proposed mechanism entails anisotropic reduction in the wavelength of the excitation waves because of anisotropic variation in its electrical properties, in particular the action potential duration (APD). We find that the variation in the APD, which we induce by varying ion-channel conductances, imposes a spatial variation in the spiral- or scroll-wave frequency ω. Such gradients in ω induce anisotropic shortening of wavelength of the spiral or scroll arms and eventually leads to instabilitites.

Chaotic tip trajectories of a single spiral wave in the presence of heterogeneities

Spiral waves have been observed in a variety of physical, chemical, and biological systems. They play a major role in cardiac arrhythmias, including fibrillation, where the observed irregular activation patterns are generally thought to arise from the continuous breakup of multiple unstable spiral waves. Using spatially extended simulations of different electrophysiological models of cardiac tissue, we show that a single spiral wave in the presence of heterogeneities can display chaotic tip trajectories, consistent with fibrillation. We also show that the simulated spiral tip dynamics, including chaotic trajectories, can be captured by a simple particle model which only describes the dynamics of the spiral tip. This shows that spiral wave breakup, or interactions with other waves, are not necessary to initiate chaos in spiral waves.

Reentry via high-frequency pacing in a mathematical model for human-ventricular cardiac tissue with a localized fibrotic region

Localized heterogeneities, caused by the regional proliferation of fibroblasts, occur in mammalian hearts because of diseases like myocardial infarction. Such fibroblast clumps can become sources of pathological reentrant activities, e.g., spiral or scroll waves of electrical activation in cardiac tissue. The occurrence of reentry in cardiac tissue with heterogeneities, such as fibroblast clumps, can depend on the frequency at which the medium is paced. Therefore, it is important to study the reentry-initiating potential of such fibroblast clumps at different frequencies of pacing. We investigate the arrhythmogenic effects of fibroblast clumps at high- and low-frequency pacing. We find that reentrant waves are induced in the medium more prominently at high-frequency pacing than with low-frequency pacing. We also study the other factors that affect the potential of fibroblast clumps to induce reentry in cardiac tissue. In particular, we show that the ability of a fibroblast clump to induce reentry depends on the size of the clump, the distribution and percentage of fibroblasts in the clump, and the excitability of the medium. We study the process of reentry in two-dimensional and a three-dimensional mathematical models for cardiac tissue.

Perpetuation of torsade de pointes in heterogeneous hearts: competing foci or re-entry?

KEY POINTS

The underlying mechanism of torsade de pointes (TdP) remains of debate: perpetuation may be due to (1) focal activity or (2) re-entrant activity. The onset of TdP correlates with action potential heterogeneities in different regions of the heart. We studied the mechanism of perpetuation of TdP in silico using a 2D model of human cardiac tissue and an anatomically accurate model of the ventricles of the human heart. We found that the mechanism of perpetuation TdP depends on the degree of heterogeneity. If the degree of heterogeneity is large, focal activity alone can sustain a TdP, otherwise re-entrant activity emerges. This result can help to understand the relationship between the mechanisms of TdP and tissue properties and may help in developing new drugs against it.

ABSTRACT

Torsade de pointes (TdP) can be the consequence of cardiac remodelling, drug effects or a combination of both. The mechanism underlying TdP is unclear, and may involve triggered focal activity or re-entry. Recent work by our group has indicated that both cases may exist, i.e. TdPs induced in the chronic atrioventricular block (CAVB) dog model may have a focal origin or are due to re-entry. Also it was found that heterogeneities might play an important role. In the current study we have used computational modelling to further investigate the mechanisms involved in TdP initiation and perpetuation, especially in the CAVB dog model, by the addition of heterogeneities with reduced repolarization reserve in comparison with the surrounding tissue. For this, the TNNP computer model was used for computations. We demonstrated in 2D and 3D simulations that ECGs with the typical TdP morphology can be caused by both multiple competing foci and re-entry circuits as a result of introduction of heterogeneities, depending on whether the heterogeneities have a large or a smaller reduced repolarization reserve in comparison with the surrounding tissue. Large heterogeneities can produce ectopic TdP, while smaller heterogeneities will produce re-entry-type TdP.

Global alternans instability and its effect on non-linear wave propagation: dynamical Wenckebach block and self terminating spiral waves

The main mechanism of formation of reentrant cardiac arrhythmias is via formation of waveblocks at heterogeneities of cardiac tissue. We report that heterogeneity and the area of waveblock can extend itself in space and can result formation of new additional sources, or termination of existing sources of arrhythmias. This effect is based on a new form of instability, which we coin as global alternans instability (GAI). GAI is closely related to the so-called (discordant) alternans instability, however its onset is determined by the global properties of the APD-restitution curve and not by its slope. The APD-restitution curve relates the duration of the cardiac pulse (APD) to the time interval between the pulses, and can easily be measured in an experimental or even clinical setting. We formulate the conditions for the onset of GAI, study its manifestation in various 1D and 2D situations and discuss its importance for the onset of cardiac arrhythmias.

Patient-derived models link re-entrant driver localization in atrial fibrillation to fibrosis spatial pattern

AIMS

The mechanisms underlying persistent atrial fibrillation (AF) in patients with atrial fibrosis are poorly understood. The goal of this study was to use patient-derived atrial models to test the hypothesis that AF re-entrant drivers (RDs) persist only in regions with specific fibrosis patterns.

METHODS AND RESULTS

Twenty patients with persistent AF (PsAF) underwent late gadolinium-enhanced MRI to detect the presence of atrial fibrosis. Segmented images were used to construct personalized 3D models of the fibrotic atria with biophysically realistic atrial electrophysiology. In each model, rapid pacing was applied to induce AF. AF dynamics were analysed and RDs were identified using phase mapping. Fibrosis patterns in RD regions were characterized by computing maps of fibrosis density (FD) and entropy (FE). AF was inducible in 13/20 models and perpetuated by few RDs (2.7 ± 1.5) that were spatially confined (trajectory of phase singularities: 7.6 ± 2.3 mm). Compared with the remaining atrial tissue, regions where RDs persisted had higher FE (IQR: 0.42-0.60 vs. 0.00-0.40, P < 0.05) and FD (IQR: 0.59-0.77 vs. 0.00-0.33, P < 0.05). Machine learning classified RD and non-RD regions based on FD and FE and identified a subset of fibrotic boundary zones present in 13.8 ± 4.9% of atrial tissue where 83.5 ± 2.4% of all RD phase singularities were located.

CONCLUSION

Patient-derived models demonstrate that AF in fibrotic substrates is perpetuated by RDs persisting in fibrosis boundary zones characterized by specific regional fibrosis metrics (high FE and FD). These results provide new insights into the mechanisms that sustain PsAF and could pave the way for personalized, MRI-based management of PsAF.

Post-repolarization refractoriness increases vulnerability to block and initiation of reentrant impulses in heterogeneous infarcted myocardium

Myocardial infarction causes remodeling of the tissue structure and the density and kinetics of several ion channels in the cell membrane. Heterogeneities in refractory period (ERP) have been shown to occur in the infarct border zone and have been proposed to lead to initiation of arrhythmias. The purpose of this study is to quantify the window of vulnerability (WV) to block and initiation of reentrant impulses in myocardium with ERP heterogeneities using computer simulations. We found that ERP transitions at the border between normal ventricular cells (NZ) with different ERPs are smooth, whereas ERP transitions between NZ and infarct border zone cells (IZ) are abrupt. The profile of the ERP transitions is a combination of electrotonic interaction between NZ and IZ cells and the characteristic post-repolarization refractoriness (PRR) of IZ cells. ERP heterogeneities between NZ and IZ cells are more vulnerable to block and initiation of reentrant impulses than ERP heterogeneities between NZ cells. The relationship between coupling intervals of premature impulses (V1V2) and coupling intervals between premature and first reentrant impulses (V2T1) at NZ/NZ and NZ/IZ borders is inverse (i.e. the longer the coupling intervals of premature impulses the shorter the coupling interval between the premature and first reentrant impulses); this is in contrast with the reported V1V2/V2T1 relationship measured during initiation of reentrant impulses in canine infarcted hearts which is direct.

IN CONCLUSION

(1) ERP transitions at the NZ-IZ border are abrupt as a consequence of PRR; (2) PRR increases the vulnerability to block and initiation of reentrant impulses in heterogeneous myocardium; (3) V1V2/V2T1 relationships measured at ERP heterogeneities in the computer model and in experimental canine infarcts are not consistent. Therefore, it is likely that other mechanisms like micro and/or macro structural heterogeneities also contribute to initiation of reentrant impulses in infarcted hearts.

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.

Small size ionic heterogeneities in the human heart can attract rotors

Rotors occurring in the heart underlie the mechanisms of cardiac arrhythmias. Answering the question whether or not the location of rotors is related to local properties of cardiac tissue has important practical applications. This is because ablation of rotors has been shown to be an effective way to fight cardiac arrhythmias. In this study, we investigate, in silico, the dynamics of rotors in two-dimensional and in an anatomical model of human ventricles using a Ten Tusscher-Noble-Noble-Panfilov (TNNP) model for ventricular cells. We study the effect of small size ionic heterogeneities, similar to those measured experimentally. It is shown that such heterogeneities cannot only anchor, but can also attract, rotors rotating at a substantial distance from the heterogeneity. This attraction distance depends on the extent of the heterogeneities and can be as large as 5-6 cm in realistic conditions. We conclude that small size ionic heterogeneities can be preferred localization points for rotors and discuss their possible mechanism and value for applications.