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Jæger KH, Trotter JD, Cai X, Arevalo H, Tveito A. Evaluating computational efforts and physiological resolution of mathematical models of cardiac tissue. Sci Rep 2024; 14:16954. [PMID: 39043725 PMCID: PMC11266357 DOI: 10.1038/s41598-024-67431-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2024] [Accepted: 07/11/2024] [Indexed: 07/25/2024] Open
Abstract
Computational techniques have significantly advanced our understanding of cardiac electrophysiology, yet they have predominantly concentrated on averaged models that do not represent the intricate dynamics near individual cardiomyocytes. Recently, accurate models representing individual cells have gained popularity, enabling analysis of the electrophysiology at the micrometer level. Here, we evaluate five mathematical models to determine their computational efficiency and physiological fidelity. Our findings reveal that cell-based models introduced in recent literature offer both efficiency and precision for simulating small tissue samples (comprising thousands of cardiomyocytes). Conversely, the traditional bidomain model and its simplified counterpart, the monodomain model, are more appropriate for larger tissue masses (encompassing millions to billions of cardiomyocytes). For simulations requiring detailed parameter variations along individual cell membranes, the EMI model emerges as the only viable choice. This model distinctively accounts for the extracellular (E), membrane (M), and intracellular (I) spaces, providing a comprehensive framework for detailed studies. Nonetheless, the EMI model's applicability to large-scale tissues is limited by its substantial computational demands for subcellular resolution.
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Affiliation(s)
| | | | - Xing Cai
- Simula Research Laboratory, Oslo, Norway
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Jæger KH, Tveito A. A possible path to persistent re-entry waves at the outlet of the left pulmonary vein. NPJ Syst Biol Appl 2024; 10:79. [PMID: 39043674 PMCID: PMC11266599 DOI: 10.1038/s41540-024-00406-9] [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: 03/18/2024] [Accepted: 07/12/2024] [Indexed: 07/25/2024] Open
Abstract
Atrial fibrillation (AF) is the most common form of cardiac arrhythmia, often evolving from paroxysmal episodes to persistent stages over an extended timeframe. While various factors contribute to this progression, the precise biophysical mechanisms driving it remain unclear. Here we explore how rapid firing of cardiomyocytes at the outlet of the pulmonary vein of the left atria can create a substrate for a persistent re-entry wave. This is grounded in a recently formulated mathematical model of the regulation of calcium ion channel density by intracellular calcium concentration. According to the model, the number of calcium channels is controlled by the intracellular calcium concentration. In particular, if the concentration increases above a certain target level, the calcium current is weakened to restore the target level of calcium. During rapid pacing, the intracellular calcium concentration of the cardiomyocytes increases leading to a substantial reduction of the calcium current across the membrane of the myocytes, which again reduces the action potential duration. In a spatially resolved cell-based model of the outlet of the pulmonary vein of the left atria, we show that the reduced action potential duration can lead to re-entry. Initiated by rapid pacing, often stemming from paroxysmal AF episodes lasting several days, the reduction in calcium current is a critical factor. Our findings illustrate how such episodes can foster a conducive environment for persistent AF through electrical remodeling, characterized by diminished calcium currents. This underscores the importance of promptly addressing early AF episodes to prevent their progression to chronic stages.
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Affiliation(s)
| | - Aslak Tveito
- Department of Computational Physiology, Simula Research Laboratory, Oslo, Norway
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Jæger KH, Tveito A. The simplified Kirchhoff network model (SKNM): a cell-based reaction-diffusion model of excitable tissue. Sci Rep 2023; 13:16434. [PMID: 37777588 PMCID: PMC10542379 DOI: 10.1038/s41598-023-43444-9] [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: 06/02/2023] [Accepted: 09/24/2023] [Indexed: 10/02/2023] Open
Abstract
Cell-based models of excitable tissues offer the advantage of cell-level precision, which cannot be achieved using traditional homogenized electrophysiological models. However, this enhanced accuracy comes at the cost of increased computational demands, necessitating the development of efficient cell-based models. The widely-accepted bidomain model serves as the standard in computational cardiac electrophysiology, and under certain anisotropy ratio conditions, it is well known that it can be reduced to the simpler monodomain model. Recently, the Kirchhoff Network Model (KNM) was developed as a cell-based counterpart to the bidomain model. In this paper, we aim to demonstrate that KNM can be simplified using the same steps employed to derive the monodomain model from the bidomain model. We present the cell-based Simplified Kirchhoff Network Model (SKNM), which produces results closely aligned with those of KNM while requiring significantly less computational resources.
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Colman MA, Benson AP. A simple approach for image-based modelling of the heart that enables robust simulation of highly heterogeneous electrical excitation. Sci Rep 2023; 13:15119. [PMID: 37704647 PMCID: PMC10499818 DOI: 10.1038/s41598-023-39244-w] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2023] [Accepted: 07/21/2023] [Indexed: 09/15/2023] Open
Abstract
Remodelling of cardiac tissue structure, including intercellular electrical coupling, is a major determinant of the complex and heterogeneous excitation patterns associated with cardiac arrhythmias. Evaluation of the precise mechanisms by which local tissue structure determines global arrhythmic excitation patterns is a major challenge that may be critically important for the development of effective treatment strategies. Computational modelling is a key tool in the study of cardiac arrhythmias, yet the established approaches for organ-scale modelling are unsuitable to capture the impact of local conduction heterogeneities; a novel approach is required to provide this multi-scale mechanistic insight. We present a fundamentally simple yet powerful approach to simulate electrical excitation in highly heterogeneous whole-heart models that exploits the underlying discreteness of the myocardium. Preliminary simulations demonstrate that this approach can capture lower conduction velocities and reproduce wave breakdown and the development of re-entry in a range of conditions.
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Affiliation(s)
- Michael A Colman
- Faculty of Biological Sciences, School of Biomedical Sciences, University of Leeds, Leeds, UK.
| | - Alan P Benson
- Faculty of Biological Sciences, School of Biomedical Sciences, University of Leeds, Leeds, UK
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Jæger KH, Tveito A. Efficient, cell-based simulations of cardiac electrophysiology; The Kirchhoff Network Model (KNM). NPJ Syst Biol Appl 2023; 9:25. [PMID: 37316522 DOI: 10.1038/s41540-023-00288-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2023] [Accepted: 06/05/2023] [Indexed: 06/16/2023] Open
Abstract
Mathematical models based on homogenized representation of cardiac tissue have greatly improved our understanding of cardiac electrophysiology. However, these models are too coarse to investigate the dynamics at the level of the myocytes since the cells are not present in homogenized models. Recently, fine scale models have been proposed to allow for cell-level resolution of the dynamics, but these models are too computationally expensive to be used in applications like whole heart simulations of large animals. To address this issue, we propose a model that balances computational demands and physiological accuracy. The model is founded on Kirchhoff's current law, and represents every myocyte in the tissue. This allows specific properties to be assigned to individual cardiomyocytes, and other cell types like fibroblasts can be added to the model in an accurate manner while keeping the computing efforts reasonable.
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Nano-scale solution of the Poisson-Nernst-Planck (PNP) equations in a fraction of two neighboring cells reveals the magnitude of intercellular electrochemical waves. PLoS Comput Biol 2023; 19:e1010895. [PMID: 36791152 PMCID: PMC9974139 DOI: 10.1371/journal.pcbi.1010895] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Revised: 02/28/2023] [Accepted: 01/23/2023] [Indexed: 02/16/2023] Open
Abstract
The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters. Here we report solutions of the PNP equations in a fraction of two abuttal cells separated by a tiny extracellular space. We show that when only the potassium channels of the two cells are open, a stationary solution is reached with the well-known Debye layer close to the membranes. When the sodium channels of one of the cells are opened, a very strong and brief electrochemical wave emanates from the channels. If the extracellular space is sufficiently small and the number of sodium channels is sufficiently high, the wave extends all the way over to the neighboring cell and may therefore explain cardiac conduction even at very low levels of gap junctional coupling.
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Jæger KH, Edwards AG, Giles WR, Tveito A. Arrhythmogenic influence of mutations in a myocyte-based computational model of the pulmonary vein sleeve. Sci Rep 2022; 12:7040. [PMID: 35487957 PMCID: PMC9054808 DOI: 10.1038/s41598-022-11110-1] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2022] [Accepted: 04/12/2022] [Indexed: 11/09/2022] Open
Abstract
In the heart, electrophysiological dysregulation arises from defects at many biological levels (from point mutations in ion channel proteins to gross structural abnormalities). These defects disrupt the normal pattern of electrical activation, producing ectopic activity and reentrant arrhythmia. To interrogate mechanisms that link these primary biological defects to macroscopic electrophysiologic dysregulation most prior computational studies have utilized either (i) detailed models of myocyte ion channel dynamics at limited spatial scales, or (ii) homogenized models of action potential conduction that reproduce arrhythmic activity at tissue and organ levels. Here we apply our recent model (EMI), which integrates electrical activation and propagation across these scales, to study human atrial arrhythmias originating in the pulmonary vein (PV) sleeves. These small structures initiate most supraventricular arrhythmias and include pronounced myocyte-to-myocyte heterogeneities in ion channel expression and intercellular coupling. To test EMI's cell-based architecture in this physiological context we asked whether ion channel mutations known to underlie atrial fibrillation are capable of initiating arrhythmogenic behavior via increased excitability or reentry in a schematic PV sleeve geometry. Our results illustrate that EMI's improved spatial resolution can directly interrogate how electrophysiological changes at the individual myocyte level manifest in tissue and as arrhythmia in the PV sleeve.
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Affiliation(s)
| | | | - Wayne R Giles
- Simula Research Laboratory, Oslo, Norway.,Department of Physiology and Pharmacology, Cumming School of Medicine, University of Calgary, Calgary, Canada
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Louch WE, Perdreau-Dahl H, Edwards AG. Image-Driven Modeling of Nanoscopic Cardiac Function: Where Have We Come From, and Where Are We Going? Front Physiol 2022; 13:834211. [PMID: 35356084 PMCID: PMC8959215 DOI: 10.3389/fphys.2022.834211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2021] [Accepted: 01/31/2022] [Indexed: 11/24/2022] Open
Abstract
Complementary developments in microscopy and mathematical modeling have been critical to our understanding of cardiac excitation-contraction coupling. Historically, limitations imposed by the spatial or temporal resolution of imaging methods have been addressed through careful mathematical interrogation. Similarly, limitations imposed by computational power have been addressed by imaging macroscopic function in large subcellular domains or in whole myocytes. As both imaging resolution and computational tractability have improved, the two approaches have nearly merged in terms of the scales that they can each be used to interrogate. With this review we will provide an overview of these advances and their contribution to understanding ventricular myocyte function, including exciting developments over the last decade. We specifically focus on experimental methods that have pushed back limits of either spatial or temporal resolution of nanoscale imaging (e.g., DNA-PAINT), or have permitted high resolution imaging on large cellular volumes (e.g., serial scanning electron microscopy). We also review the progression of computational approaches used to integrate and interrogate these new experimental data sources, and comment on near-term advances that may unify understanding of the underlying biology. Finally, we comment on several outstanding questions in cardiac physiology that stand to benefit from a concerted and complementary application of these new experimental and computational methods.
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Affiliation(s)
- William E. Louch
- Institute for Experimental Medical Research, Oslo University Hospital and University of Oslo, Oslo, Norway
- K.G. Jebsen Centre for Cardiac Research, University of Oslo, Oslo, Norway
| | - Harmonie Perdreau-Dahl
- Institute for Experimental Medical Research, Oslo University Hospital and University of Oslo, Oslo, Norway
- K.G. Jebsen Centre for Cardiac Research, University of Oslo, Oslo, Norway
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Jæger KH, Tveito A. Deriving the Bidomain Model of Cardiac Electrophysiology From a Cell-Based Model; Properties and Comparisons. Front Physiol 2022; 12:811029. [PMID: 35069265 PMCID: PMC8782150 DOI: 10.3389/fphys.2021.811029] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2021] [Accepted: 12/13/2021] [Indexed: 11/23/2022] Open
Abstract
The bidomain model is considered to be the gold standard for numerical simulation of the electrophysiology of cardiac tissue. The model provides important insights into the conduction properties of the electrochemical wave traversing the cardiac muscle in every heartbeat. However, in normal resolution, the model represents the average over a large number of cardiomyocytes, and more accurate models based on representations of all individual cells have therefore been introduced in order to gain insight into the conduction properties close to the myocytes. The more accurate model considered here is referred to as the EMI model since both the extracellular space (E), the cell membrane (M) and the intracellular space (I) are explicitly represented in the model. Here, we show that the bidomain model can be derived from the cell-based EMI model and we thus reveal the close relation between the two models, and obtain an indication of the error introduced in the approximation. Also, we present numerical simulations comparing the results of the two models and thereby highlight both similarities and differences between the models. We observe that the deviations between the solutions of the models become larger for larger cell sizes. Furthermore, we observe that the bidomain model provides solutions that are very similar to the EMI model when conductive properties of the tissue are in the normal range, but large deviations are present when the resistance between cardiomyocytes is increased.
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Affiliation(s)
| | - Aslak Tveito
- Simula Research Laboratory, Oslo, Norway.,Department of Informatics, University of Oslo, Oslo, Norway
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