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Book Cover
E-book
Author Franzone, Piero Colli, author

Title Mathematical cardiac electrophysiology / Piero Colli Franzone, Luca Franco Pavarino, Simone Scacchi
Published Cham : Springer, [2014]
©2014

Copies

Description 1 online resource (xiv, 397 pages) : illustrations (some color)
Series MS & A, 2037-5255 ; volume 13
MS & A (Series) ; volume 13
Contents 880-01 1 Basic cardiac anatomy and electrocardiology -- 2 Mathematical models of cellular bioelectrical activity -- 3 Mathematical models of cardiac cells arrangements: the Bidomain model -- 4 Reduced macroscopic models: the Monodomain and Eikonal models -- 5 Anisotropic cardiac sources -- 6 The Inverse problem of Electrocardiology -- 7 Numerical methods for the Bidomain and reduced models -- 8 Parallel solvers for the Bidomain system -- 9 Simulation studies of cardiac bioelectrical activity -- 10 Appendix A: Cardiac simulation projects, software, libraries
880-01/(S Machine generated contents note: 1. Basic Cardiac Anatomy and Electrocardiology -- 1.1. Conduction System: SA and AV Node, Purkinje Network -- 1.2. Cardiac Tissue Organization -- 1.3. Fiber and Laminar Architecture of Ventricular Myocardium -- 1.4. Cardiac Action Potentials -- 1.4.1. Action Potential Phases -- 1.4.2. Action Potential Heterogeneity -- 1.5. Electrocardiogram (ECG) -- 1.5.1. ECG Leads -- 1.5.2. ECG Deflections and Intervals -- 1.5.3. ECG Diagnosis -- 1.6. Cardiac Imaging -- 2. Mathematical Models of Cellular Bioelectrical Activity -- 2.1. Excitable Cellular Membranes -- 2.2. Nernst-Planck Equation -- 2.3. Goldman-Hodgkin-Katz (GHK) Current-Voltage Relation -- 2.4. Nernst Equilibrium Potential -- 2.5. Thermodynamical Derivation of the Nernst Potential -- 2.6. Electrodiffusion Models: The Poisson-Nernst-Planck (PNP) Equation -- 2.6.1. PNP: The Short Channel or Low Concentrations Limit -- 2.6.2. PNP: The Long Channel or High Concentrations Limit -- 2.6.3. Equilibrium Potential for Multi-ion Fluxes -- 2.7. Electrical Circuit Model of the Cellular Membrane -- 2.8. Ion Channel Gating -- 2.9. Cardiac Action Potential Models -- 2.9.1. Hodgkin-Huxley Model -- 2.9.2. General Structure of Cardiac Cellular Membrane Models -- 2.9.3. Ionic Models of Purkinje Fibers, Sinoatrial Node (SAN), Atria -- 2.9.4. Ventricular Models -- 2.9.5. Charge Conservation in Ionic Models -- 2.9.6. Action Potential Duration Restitution Curve -- 2.9.7. Reduced Ionic Models -- 2.9.8. Phase-Plane Analysis of the FitzHugh-Nagumo (FHN) Model -- 2.9.9. Bifurcation Diagrams -- 3. Mathematical Models of Cardiac Cells Arrangements: The Bidomain Model -- 3.1. Models of Cardiac Fibers -- 3.1.1. Cable Equation -- 3.1.2. Homogenization -- 3.1.3. Traveling Waves -- 3.1.4. Conduction Velocity Restitution Curve -- 3.2. Models of Cardiac Tissue -- 3.2.1. Dimensionless Cellular Model Pe -- 3.2.2. Formal Two-Scale Homogenization -- 3.2.3. Dimensionless Averaged Model P -- 3.2.4. Theoretical Results for the Cellular and Averaged Models -- 3.2.5. Γ-Convergence Result for the Averaged Model with FHN Dynamics -- 3.3. Macroscopic Anisotropic Bidomain Model -- 3.4. Well-Posedness Results Based on Semi-discretization in Time -- 3.5. Well-Posedness Results Based on Faedo-Galerkin Techniques -- 3.6. Well-Posedness Results Based on Fixed Point Arguments -- 3.7. Semi-discrete Approximation of the Bidomain Model with FHN Dynamics -- 4. Reduced Macroscopic Models: The Monodomain and Eikonal Models -- 4.1. Linear Anisotropic Monodomain Model -- 4.2. Eikonal Models -- 4.3. Relaxed Non-linear Anisotropic Monodomain Model -- 4.4. Dimensional Form of the Reduced Models -- 4.4.1. Well-Posedness Results for Reduced Models -- 4.4.2. Frank and Wulff Diagrams -- 4.5. Numerical Comparison -- 5. Anisotropic Cardiac Sources -- 5.1. Differential Formulation of the Potential Field -- 5.2. Integral Formulation of the Potential Field -- 5.3. Approximate Representation of Cardiac Sources -- 5.3.1. Heart Surface Source Model -- 5.3.2. Oblique Dipole Source Model -- 5.4. Cardiac Source Splitting -- 5.4.1. Axially Symmetric Media -- 5.4.2. Orthotropic Media -- 5.5. Interpretation of the Field Components -- 5.6. Limit Case: Oblique Dipole Layer Model -- 5.7. Experimental and Simulation Results -- 6. Inverse Problem of Electrocardiology -- 6.1. Inverse Problem in Terms of Potential Alone -- 6.2. Macroscopic Equivalent Excitation Cardiac Sources -- 6.3. Boundedness of the Potential -- 6.4. Numerical Approximation of the Integral Representation of the Potential -- 6.5. Inverse Problem in Terms of Wavefront -- 7. Numerical Methods for the Bidomain and Reduced Models -- 7.1. Space Discretization of Monodomain and Bidomain Models -- 7.2. Time Discretization of Monodomain and Bidomain Models -- 7.2.1. Fully Implicit Methods -- 7.2.2. Decoupled Implicit Methods -- 7.2.3. Decoupled Semi-implicit Methods -- 7.2.4. Operator Splitting Methods: Splitting ODEs and PDEs -- 7.3. Numerical Approximation of the Eikonal-Diffusion Equation -- 8. Parallel Solvers for the Bidomain System -- 8.1. Bidomain Variational Setting -- 8.2. Abstract Convergence Theory for Schwarz Methods -- 8.3. Two-Level Additive Schwarz Methods for the Bidomain System -- 8.4. Multilevel Additive Schwarz Methods for the Bidomain System -- 8.5. Numerical Results for Multilevel Schwarz Preconditioners -- 8.5.1. Additive Preconditioner -- 8.5.2. Multiplicative and Hybrid Preconditioners -- 8.6. Block Preconditioners for the Bidomain System -- 8.6.1. Block-Diagonal and Block-Factorized Bidomain Preconditioners -- 8.6.2. Numerical Results with Block Preconditioners -- 9. Simulation Studies of Cardiac Bioelectrical Activity -- 9.1. Cardiac Excitation and Virtual Electrode Phenomena -- 9.1.1. Methods and Parameter Calibration -- 9.1.2. Anode and Cathode Make Mechanisms -- 9.1.3. Anode and Cathode Break Mechanisms -- 9.1.4. Cathodal and Anodal Strength-Interval S-I Curves -- 9.2. Anisotropic Propagation of Excitation and Recovery Fronts -- 9.2.1. Excitation and Repolarization Sequences -- 9.2.2. Discussion on APD Distribution and Dispersion -- 9.3. Heterogeneous Cardiac Tissue -- 9.3.1. Transmural Heterogeneity in 3-D Cardiac Slabs -- 9.3.2. Transmural Heterogeneity in 3-D Ellipsoids -- 9.3.3. Transmural and Apex-Base Heterogeneity in 3-D Ellipsoids -- 9.4. QRS Complex and T Wave Morphology in Electrograms -- 9.4.1. Methods and Parameter Calibration -- 9.4.2. Unipolar and Bipolar ECG Simulations -- 9.5. Extracellular Markers of Excitation and Repolarization Times -- 9.5.1. Waveform Postprocessing and Repolarization Time Markers -- 9.5.2. Parameter Calibrations for the Model Simulations -- 9.5.3. Global Quantitative Analysis of RT Markers -- 9.6. Subendocardial Ischemia, ST Depression and Elevation -- 9.6.1. Mechanisms for the ST Segment Potential Patterns -- 9.6.2. Ischemic Simulations -- 9.7. Reentry Phenomena -- 9.7.1. Stable Scroll Waves -- 9.7.2. Scroll Waves Breakup -- 9.7.3. Scroll Waves in Ellipsoidal Geometry -- A. Cardiac Simulation Projects, Software, Libraries -- A.1. IUPS Physiome Project -- A.2. Virtual Physiological Human (VPH) -- A.3. NSR Physiome -- A.4. Other Simulation Software and Modeling Environments -- A.5. Some Related Monographs -- A.6. Physical Units and Constants
Summary This book covers the main mathematical and numerical models in computational electrocardiology, ranging from microscopic membrane models of cardiac ionic channels to macroscopic bidomain, monodomain, eikonal models and cardiac source representations. These advanced multiscale and nonlinear models describe the cardiac bioelectrical activity from the cell level to the body surface and are employed in both the direct and inverse problems of electrocardiology. The book also covers advanced numerical techniques needed to efficiently carry out large-scale cardiac simulations, including time and space discretizations, decoupling and operator splitting techniques, parallel finite element solvers. These techniques are employed in 3D cardiac simulations illustrating the excitation mechanisms, the anisotropic effects on excitation and repolarization wavefronts, the morphology of electrograms in normal and pathological tissue and some reentry phenomena. The overall aim of the book is to present rigorously the mathematical and numerical foundations of computational electrocardiology, illustrating the current research developments in this fast-growing field lying at the intersection of mathematical physiology, bioengineering and computational biomedicine. This book is addressed to graduate student and researchers in the field of applied mathematics, scientific computing, bioengineering, electrophysiology and cardiology
Analysis wiskunde
mathematics
biomedische techniek
biomedical engineering
computerwiskunde
computational mathematics
numerieke wiskunde
numerical mathematics
biomedische wetenschappen
biomedicine
Mathematics (General)
Wiskunde (algemeen)
Bibliography Includes bibliographical references and index
Notes English
Online resource; title from PDF title page (SpringerLink, viewed December 1, 2014)
Subject Heart -- Electric properties -- Mathematical models
Heart -- physiology
Electrophysiological Phenomena
MEDICAL -- Physiology.
SCIENCE -- Life Sciences -- Human Anatomy & Physiology.
Heart -- Electric properties -- Mathematical models
Form Electronic book
Author Pavarino, Luca F., author.
Scacchi, Simone, author
ISBN 9783319048017
3319048015