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Arno L, Kabus D, Dierckx H. Analysis of complex excitation patterns using Feynman-like diagrams. Sci Rep 2024; 14:28962. [PMID: 39578507 PMCID: PMC11584638 DOI: 10.1038/s41598-024-73544-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2023] [Accepted: 09/18/2024] [Indexed: 11/24/2024] Open
Abstract
Many extended chemical and biological systems self-organise into complex patterns that drive the medium behaviour in a non-linear fashion. An important class of such systems are excitable media, including neural and cardiac tissues. In extended excitable media, wave breaks can form rotating patterns and turbulence. However, the onset, sustaining and elimination of such complex patterns is currently incompletely understood. The classical theory of phase singularities in excitable media was recently challenged, as extended lines of conduction block were identified as phase discontinuities. Here, we provide a theoretical framework that captures the rich dynamics in excitable systems in terms of three quasiparticles: heads, tails, and pivots. We propose to call these quasiparticles 'cardions'. In simulations and experiments, we show that these basic building blocks combine into at least four different bound states. By representing their interactions similarly to Feynman diagrams in physics, the creation and annihilation of vortex pairs are shown to be sequences of dynamical creation, annihilation, and recombination of the identified quasiparticles. We draw such diagrams for numerical simulations, as well as optical voltage mapping experiments performed on cultured human atrial myocytes (hiAMs). Our results provide a new, unified language for a more detailed theory, analysis, and mechanistic insights of dynamical transitions in excitation patterns.
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Affiliation(s)
- Louise Arno
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500, Kortrijk, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Oude Markt 13, 3000, Leuven, Belgium
| | - Desmond Kabus
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500, Kortrijk, Belgium
- Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Albinusdreef 2, 2333 ZA, Leiden, The Netherlands
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Oude Markt 13, 3000, Leuven, Belgium
| | - Hans Dierckx
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Etienne Sabbelaan 53, 8500, Kortrijk, Belgium.
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Oude Markt 13, 3000, Leuven, Belgium.
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Kabus D, Cloet M, Zemlin C, Bernus O, Dierckx H. The Ithildin library for efficient numerical solution of anisotropic reaction-diffusion problems in excitable media. PLoS One 2024; 19:e0303674. [PMID: 39298417 DOI: 10.1371/journal.pone.0303674] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2024] [Accepted: 09/03/2024] [Indexed: 09/21/2024] Open
Abstract
Ithildin is an open-source library and framework for efficient parallelized simulations of excitable media, written in the C++ programming language. It uses parallelization on multiple CPU processors via the message passing interface (MPI). We demonstrate the library's versatility through a series of simulations in the context of the monodomain description of cardiac electrophysiology, including the S1S2 protocol, spiral break-up, and spiral waves in ventricular geometry. Our work demonstrates the power of Ithildin as a tool for studying complex wave patterns in cardiac tissue and its potential to inform future experimental and theoretical studies. We publish our full code with this paper in the name of open science.
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Affiliation(s)
- Desmond Kabus
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
- Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Leiden, The Netherlands
| | - Marie Cloet
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
| | - Christian Zemlin
- Division of Cardiothoracic Surgery, Department of Surgery, University of Washington School of Medicine, St Louis, MO, United States of America
| | - Olivier Bernus
- Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique de Bordeaux U1045, IHU Liryc, Hôpital Xavier Arnozan, Pessac, France
| | - Hans Dierckx
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
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Cloet M, Arno L, Kabus D, Van der Veken J, Panfilov AV, Dierckx H. Scroll Waves and Filaments in Excitable Media of Higher Spatial Dimension. PHYSICAL REVIEW LETTERS 2023; 131:208401. [PMID: 38039450 DOI: 10.1103/physrevlett.131.208401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 08/30/2023] [Indexed: 12/03/2023]
Abstract
Excitable media are ubiquitous in nature, and in such systems the local excitation tends to self-organize in traveling waves, or in rotating spiral-shaped patterns in two or three spatial dimensions. Examples include waves during a pandemic or electrical scroll waves in the heart. Here we show that such phenomena can be extended to a space of four or more dimensions and propose that connections of excitable elements in a network setting can be regarded as additional spatial dimensions. Numerical simulations are performed in four dimensions using the FitzHugh-Nagumo model, showing that the vortices rotate around a two-dimensional surface which we define as the superfilament. Evolution equations are derived for general superfilaments of codimension two in an N-dimensional space, and their equilibrium configurations are proven to be minimal surfaces. We suggest that biological excitable systems, such as the heart or brain which have nonlocal connections can be regarded, at least partially, as multidimensional excitable media and discuss further possible studies in this direction.
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Affiliation(s)
- Marie Cloet
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk 8500, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven 3000, Belgium
| | - Louise Arno
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk 8500, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven 3000, Belgium
| | - Desmond Kabus
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk 8500, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven 3000, Belgium
- Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Leiden 2333 ZA, Netherlands
| | | | - Alexander V Panfilov
- Department of Physics and Astronomy, Ghent University, Ghent 9000, Belgium
- Laboratory of Computational Biology and Medicine, Ural Federal University, Ekaterinburg 620002, Russia
- World-Class Research Center "Digital biodesign and personalized healthcare," Sechenov University, Moscow 119991, Russia
| | - Hans Dierckx
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk 8500, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven 3000, Belgium
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Leenknegt L, Panfilov AV, Dierckx H. Impact of electrode orientation, myocardial wall thickness, and myofiber direction on intracardiac electrograms: numerical modeling and analytical solutions. Front Physiol 2023; 14:1213218. [PMID: 37492643 PMCID: PMC10364610 DOI: 10.3389/fphys.2023.1213218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2023] [Accepted: 06/14/2023] [Indexed: 07/27/2023] Open
Abstract
Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R 2 (bipolar and parallel), and 1/R 3 (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters.
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Affiliation(s)
- Lore Leenknegt
- Department of Mathematics, KU Leuven Campus KULAK, KU Leuven, Kortrijk, Belgium
- iSi Health–KU Leuven Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
| | | | - Hans Dierckx
- Department of Mathematics, KU Leuven Campus KULAK, KU Leuven, Kortrijk, Belgium
- iSi Health–KU Leuven Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
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Ramlugun GS, Kulkarni K, Pallares-Lupon N, Boukens BJ, Efimov IR, Vigmond EJ, Bernus O, Walton RD. A comprehensive framework for evaluation of high pacing frequency and arrhythmic optical mapping signals. Front Physiol 2023; 14:734356. [PMID: 36755791 PMCID: PMC9901579 DOI: 10.3389/fphys.2023.734356] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2021] [Accepted: 01/09/2023] [Indexed: 01/24/2023] Open
Abstract
Introduction: High pacing frequency or irregular activity due to arrhythmia produces complex optical mapping signals and challenges for processing. The objective is to establish an automated activation time-based analytical framework applicable to optical mapping images of complex electrical behavior. Methods: Optical mapping signals with varying complexity from sheep (N = 7) ventricular preparations were examined. Windows of activation centered on each action potential upstroke were derived using Hilbert transform phase. Upstroke morphology was evaluated for potential multiple activation components and peaks of upstroke signal derivatives defined activation time. Spatially and temporally clustered activation time points were grouped in to wave fronts for individual processing. Each activation time point was evaluated for corresponding repolarization times. Each wave front was subsequently classified based on repetitive or non-repetitive events. Wave fronts were evaluated for activation time minima defining sites of wave front origin. A visualization tool was further developed to probe dynamically the ensemble activation sequence. Results: Our framework facilitated activation time mapping during complex dynamic events including transitions to rotor-like reentry and ventricular fibrillation. We showed that using fixed AT windows to extract AT maps can impair interpretation of the activation sequence. However, the phase windowing of action potential upstrokes enabled accurate recapitulation of repetitive behavior, providing spatially coherent activation patterns. We further demonstrate that grouping the spatio-temporal distribution of AT points in to coherent wave fronts, facilitated interpretation of isolated conduction events, such as conduction slowing, and to derive dynamic changes in repolarization properties. Focal origins precisely detected sites of stimulation origin and breakthrough for individual wave fronts. Furthermore, a visualization tool to dynamically probe activation time windows during reentry revealed a critical single static line of conduction slowing associated with the rotation core. Conclusion: This comprehensive analytical framework enables detailed quantitative assessment and visualization of complex electrical behavior.
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Affiliation(s)
- Girish S. Ramlugun
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique, Bordeaux, France
| | - Kanchan Kulkarni
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique, Bordeaux, France
| | - Nestor Pallares-Lupon
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique, Bordeaux, France
| | - Bastiaan J. Boukens
- Department of Physiology, Cardiovascular Research Institute Maastricht, University Maastricht, Maastricht, Netherlands,Department of Medical Biology, Amsterdam Cardiovascular Sciences, Amsterdam University Medical Center, University of Amsterdam, Amsterdam, Netherlands
| | - Igor R. Efimov
- Department of Biomedical Engineering, The George Washington University, Washington, DC, United States,Department of Biomedical Engineering, Northwestern University, Chicago, IL, United States,Department of Medicine, Northwestern University, Chicago, IL, United States
| | - Edward J. Vigmond
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux, UMR5251, Bordeaux, France
| | - Olivier Bernus
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique, Bordeaux, France
| | - Richard D. Walton
- IHU-Liryc, Fondation Bordeaux Université, Pessac-Bordeaux, France,Univ. Bordeaux, Inserm, Centre de Recherche Cardio-Thoracique, Bordeaux, France,*Correspondence: Richard D. Walton,
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Kabus D, Arno L, Leenknegt L, Panfilov AV, Dierckx H. Numerical methods for the detection of phase defect structures in excitable media. PLoS One 2022; 17:e0271351. [PMID: 35819963 PMCID: PMC9275727 DOI: 10.1371/journal.pone.0271351] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2022] [Accepted: 06/12/2022] [Indexed: 11/19/2022] Open
Abstract
Electrical waves that rotate in the heart organize dangerous cardiac arrhythmias. Finding the region around which such rotation occurs is one of the most important practical questions for arrhythmia management. For many years, the main method for finding such regions was so-called phase mapping, in which a continuous phase was assigned to points in the heart based on their excitation status and defining the rotation region as a point of phase singularity. Recent analysis, however, showed that in many rotation regimes there exist phase discontinuities and the region of rotation must be defined not as a point of phase singularity, but as a phase defect line. In this paper, we use this novel methodology and perform a comparative study of three different phase definitions applied to in silico data and to experimental data obtained from optical voltage mapping experiments on monolayers of human atrial myocytes. We introduce new phase defect detection algorithms and compare them with those that appeared in literature already. We find that the phase definition is more important than the algorithm to identify sudden spatial phase variations. Sharp phase defect lines can be obtained from a phase derived from local activation times observed during one cycle of arrhythmia. Alternatively, similar quality can be obtained from a reparameterization of the classical phase obtained from observation of a single timeframe of transmembrane potential. We found that the phase defect line length was (35.9 ± 6.2)mm in the Fenton-Karma model and (4.01 ± 0.55)mm in cardiac human atrial myocyte monolayers. As local activation times are obtained during standard clinical cardiac mapping, the methods are also suitable to be applied to clinical datasets. All studied methods are publicly available and can be downloaded from an institutional web-server.
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Affiliation(s)
- Desmond Kabus
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
- Laboratory of Experimental Cardiology, Leiden University Medical Center (LUMC), Leiden, The Netherlands
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
| | - Louise Arno
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
| | - Lore Leenknegt
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
| | - Alexander V. Panfilov
- Department of Physics and Astronomy, Ghent University, Ghent, Belgium
- Laboratory of Computational Biology and Medicine, Ural Federal University, Ekaterinburg, Russia
- World-Class Research Center “Digital biodesign and personalized healthcare”, Sechenov University, Moscow, Russia
| | - Hans Dierckx
- Department of Mathematics, KU Leuven Campus Kortrijk (KULAK), Kortrijk, Belgium
- iSi Health, Institute of Physics-based Modeling for In Silico Health, KU Leuven, Leuven, Belgium
- * E-mail:
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