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Mortensen P, Gao H, Smith G, Simitev RD. Addendum: Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2021; 38:292-298. [PMID: 33959762 DOI: 10.1093/imammb/dqab005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2021] [Revised: 03/26/2021] [Accepted: 04/14/2021] [Indexed: 11/13/2022]
Abstract
The analytical theory of our earlier study (Mortensen et al., 2021, Math. Med. Biol., 38, 106-131) is extended to address the outstanding cases of fibroblast barrier distribution and myocyte strait distribution. In particular, closed-form approximations to the resting membrane potential and to the critical parameter values for propagation are derived for these two non-uniform fibroblast distributions and are in good agreement with numerical estimates.
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Affiliation(s)
- Peter Mortensen
- School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, UK.,Institute of Cardiovascular & Medical Sciences, University of Glasgow, Glasgow G12 8TA, UK
| | - Hao Gao
- School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, UK
| | - Godfrey Smith
- Institute of Cardiovascular & Medical Sciences, University of Glasgow, Glasgow G12 8TA, UK
| | - Radostin D Simitev
- School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QQ, UK
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Mortensen P, Gao H, Smith G, Simitev RD. Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2021; 38:106-131. [PMID: 33412587 DOI: 10.1093/imammb/dqaa014] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Revised: 11/20/2020] [Accepted: 12/08/2020] [Indexed: 02/07/2023]
Abstract
The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait-all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte-fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.
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Affiliation(s)
- Peter Mortensen
- School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK, and Institute of Cardiovascular and Medical Sciences, University of Glasgow, Glasgow G12 8TA, UK
| | - Hao Gao
- School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK
| | - Godfrey Smith
- Institute of Cardiovascular and Medical Sciences, University of Glasgow, Glasgow G12 8TA, UK
| | - Radostin D Simitev
- School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK
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Marcotte CD, Biktashev VN. Predicting critical ignition in slow-fast excitable models. Phys Rev E 2020; 101:042201. [PMID: 32422766 DOI: 10.1103/physreve.101.042201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2019] [Accepted: 03/06/2020] [Indexed: 11/07/2022]
Abstract
Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.
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Affiliation(s)
- Christopher D Marcotte
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, EX4 4QJ Exeter, United Kingdom
| | - Vadim N Biktashev
- Department of Mathematics, University of Exeter, EX4 4QF Exeter, United Kingdom
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Starý T, Biktashev VN. Fast-slow asymptotics for a Markov chain model of fast sodium current. CHAOS (WOODBURY, N.Y.) 2017; 27:093937. [PMID: 28964114 DOI: 10.1063/1.5003013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We explore the feasibility of using fast-slow asymptotics to eliminate the computational stiffness of discrete-state, continuous-time deterministic Markov chain models of ionic channels underlying cardiac excitability. We focus on a Markov chain model of fast sodium current, and investigate its asymptotic behaviour with respect to small parameters identified in different ways.
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Affiliation(s)
- Tomáš Starý
- College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, United Kingdom
| | - Vadim N Biktashev
- College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, United Kingdom
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Gray RA, Pathmanathan P. A Parsimonious Model of the Rabbit Action Potential Elucidates the Minimal Physiological Requirements for Alternans and Spiral Wave Breakup. PLoS Comput Biol 2016; 12:e1005087. [PMID: 27749895 PMCID: PMC5066986 DOI: 10.1371/journal.pcbi.1005087] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2016] [Accepted: 07/21/2016] [Indexed: 11/19/2022] Open
Abstract
Elucidating the underlying mechanisms of fatal cardiac arrhythmias requires a tight integration of electrophysiological experiments, models, and theory. Existing models of transmembrane action potential (AP) are complex (resulting in over parameterization) and varied (leading to dissimilar predictions). Thus, simpler models are needed to elucidate the "minimal physiological requirements" to reproduce significant observable phenomena using as few parameters as possible. Moreover, models have been derived from experimental studies from a variety of species under a range of environmental conditions (for example, all existing rabbit AP models incorporate a formulation of the rapid sodium current, INa, based on 30 year old data from chick embryo cell aggregates). Here we develop a simple "parsimonious" rabbit AP model that is mathematically identifiable (i.e., not over parameterized) by combining a novel Hodgkin-Huxley formulation of INa with a phenomenological model of repolarization similar to the voltage dependent, time-independent rectifying outward potassium current (IK). The model was calibrated using the following experimental data sets measured from the same species (rabbit) under physiological conditions: dynamic current-voltage (I-V) relationships during the AP upstroke; rapid recovery of AP excitability during the relative refractory period; and steady-state INa inactivation via voltage clamp. Simulations reproduced several important "emergent" phenomena including cellular alternans at rates > 250 bpm as observed in rabbit myocytes, reentrant spiral waves as observed on the surface of the rabbit heart, and spiral wave breakup. Model variants were studied which elucidated the minimal requirements for alternans and spiral wave break up, namely the kinetics of INa inactivation and the non-linear rectification of IK.The simplicity of the model, and the fact that its parameters have physiological meaning, make it ideal for engendering generalizable mechanistic insight and should provide a solid "building-block" to generate more detailed ionic models to represent complex rabbit electrophysiology.
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Affiliation(s)
- Richard A. Gray
- Division of Biomedical Physics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, Maryland, United States of America
- * E-mail:
| | - Pras Pathmanathan
- Division of Biomedical Physics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, Maryland, United States of America
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Shimayoshi T, Cha CY, Amano A. Quantitative Decomposition of Dynamics of Mathematical Cell Models: Method and Application to Ventricular Myocyte Models. PLoS One 2015; 10:e0124970. [PMID: 26091413 PMCID: PMC4474442 DOI: 10.1371/journal.pone.0124970] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2014] [Accepted: 03/19/2015] [Indexed: 11/18/2022] Open
Abstract
Mathematical cell models are effective tools to understand cellular physiological functions precisely. For detailed analysis of model dynamics in order to investigate how much each component affects cellular behaviour, mathematical approaches are essential. This article presents a numerical analysis technique, which is applicable to any complicated cell model formulated as a system of ordinary differential equations, to quantitatively evaluate contributions of respective model components to the model dynamics in the intact situation. The present technique employs a novel mathematical index for decomposed dynamics with respect to each differential variable, along with a concept named instantaneous equilibrium point, which represents the trend of a model variable at some instant. This article also illustrates applications of the method to comprehensive myocardial cell models for analysing insights into the mechanisms of action potential generation and calcium transient. The analysis results exhibit quantitative contributions of individual channel gating mechanisms and ion exchanger activities to membrane repolarization and of calcium fluxes and buffers to raising and descending of the cytosolic calcium level. These analyses quantitatively explicate principle of the model, which leads to a better understanding of cellular dynamics.
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Affiliation(s)
- Takao Shimayoshi
- Graduate School of Informatics, Kyoto University, Kyoto, Japan
- * E-mail:
| | - Chae Young Cha
- Oxford Centre for Diabetes, Endocrinology and Metabolism, University of Oxford, Oxford, United Kingdom
| | - Akira Amano
- College of Life Sciences, Ritsumeikan University, Kusatsu, Shiga, Japan
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Abstract
The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.
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Affiliation(s)
- Trine Krogh-Madsen
- Greenberg Division of Cardiology, Department of Medicine, Weill Cornell Medical College, New York, New York 10065, USA.
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Fink M, Niederer SA, Cherry EM, Fenton FH, Koivumäki JT, Seemann G, Thul R, Zhang H, Sachse FB, Beard D, Crampin EJ, Smith NP. Cardiac cell modelling: observations from the heart of the cardiac physiome project. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2010; 104:2-21. [PMID: 20303361 DOI: 10.1016/j.pbiomolbio.2010.03.002] [Citation(s) in RCA: 108] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2009] [Revised: 12/06/2009] [Accepted: 03/04/2010] [Indexed: 10/19/2022]
Abstract
In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.
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Affiliation(s)
- Martin Fink
- Department of Physiology, Anatomy and Genetics, University of Oxford, OX1 3JP, United Kingdom
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Asymptotics of Conduction Velocity Restitution in Models of Electrical Excitation in the Heart. Bull Math Biol 2010; 73:72-115. [DOI: 10.1007/s11538-010-9523-6] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2009] [Accepted: 02/12/2010] [Indexed: 11/26/2022]
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