## Computational Cardiology: Modeling of Anatomy, Electrophysiology, and MechanicsBiomedical research is at a critical point at present. The research has led to an enormous amount of data and models describing these data, but - proachesforapplication,formalizationand integrationof this knowledgefrom the molecular to the system level are still topics of ongoing research and c- tainly far from fully developed. Also in cardiology the di'erent anatomical and physiological constituents as well as the coupling between them are being researchedin increasing detail and areoften described using computer-based models. But for this domain an integrative framework is still missing. The application of computer-based modeling as a research, development and clinical tool often necessitates the coupling of various models from di?- ent levels. Describing the interactions between these models, which are both physically sound and computationally e'cient, determines the applicability of such promising computer-based attempts. Myhopeisthatthisbookwillcontributetothecomprehension,spreadand impact of computer-based modeling in cardiology,both from a teaching point of view and by summarizing knowledge from several, commonly delimited topics relating to the cardiac manifoldness. The book evolved from revision and extension of my professorial disser- tion(Habilitationsschrift)"MathematicalModelingoftheMammalianHeart" written in 2002. This dissertation was based on notes for the lectures "C- putational Biology: Bioelectromagnetism and Biomechanics," "Simulation of Physical Fields in the Human Body," and "Anatomical, Physical and Fu- tional Models of the Human Body," which I gaveat the Universita ̈t Karlsruhe (TH) from 1998 to 2003. Salt Lake City, 1 February 2004 Frank B. Sachse VI Preface Acknowledgement Manypeople meritmy gratitudefor their assistanceandsupportin this work. |

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### Contents

Introduction | 1 |

12 Organization | 3 |

Mathematical and Numerical Foundation | 5 |

22 Einstein Summation Convention | 6 |

24 Numerics of Systems of Linear Equations | 9 |

242 Direct Methods | 10 |

243 Iterative Methods | 12 |

244 Singular Value Decomposition | 20 |

62 Microscopic Structures and Molecular Organization | 121 |

622 Gap Junctions | 131 |

623 Connective Tissue Structures | 133 |

63 Macroscopic Structures | 134 |

632 Atria | 137 |

633 Blood Vessels | 139 |

635 System of Excitation Conduction and Initiation | 140 |

636 Nervous System | 141 |

25 Numerical Integration of Functions | 21 |

252 Trapezoidal Rule | 22 |

254 Gauss Quadrature | 23 |

262 Euler Method | 24 |

263 RungeKutta Method | 25 |

27 Numerics of Partial Differential Equations | 27 |

272 Initial Values and Boundary Conditions | 29 |

273 Finite Element Method | 30 |

274 Finite Differences Method | 43 |

Theory of Electric Fields | 49 |

32 Physical Laws | 50 |

322 Poissons Equation for Stationary Current Fields | 51 |

323 Electromagnetic Properties of Biological Tissues | 52 |

33 Numerical Solution of Poissons Equation | 57 |

332 Finite Differences Method | 63 |

Theory of Continuum Mechanics | 69 |

42 Deﬁnitions and Physical Laws | 70 |

422 Strain Tensors | 73 |

423 Stress Tensors | 74 |

424 Stress Equilibrium | 75 |

425 Constitutive Relationships | 78 |

43 Numerical Solution | 83 |

432 Interpolation via ShapeFunctions | 84 |

433 Determination of Element Equations | 87 |

Digital Image Processing | 91 |

52 Digital Representation of Images | 92 |

53 Preprocessing | 93 |

533 Filtering Methods | 98 |

54 Segmentation Techniques | 102 |

542 Thresholding | 103 |

543 Region Growing | 104 |

544 Watershed | 105 |

545 Deformable Models | 106 |

546 Manual Methods | 112 |

56 Texture Orientation | 115 |

563 Interpolation | 117 |

Cardiac Anatomy | 119 |

64 Modeling of Anatomy | 142 |

643 Imaging Systems and Data Sources | 144 |

644 Modeling of Orientation and Lamination of Myocytes | 145 |

645 Models from the Visible Human Project | 146 |

646 Models from Magnetic Resonance Imaging | 152 |

Cardiac Electrophysiology | 156 |

72 Cellular Electrophysiology | 158 |

722 Modeling of Cellular Components | 164 |

723 Models of Cardiac Myocytes | 171 |

73 Excitation Propagation | 189 |

732 Modeling Approaches | 194 |

733 Cellular Automata | 195 |

734 Reaction Diffusion Systems | 203 |

735 Comparison of Macroscopic Models of Excitation Propagation | 214 |

Cardiac Mechanics | 221 |

82 Mechanical Properties of Myocardium | 222 |

822 Modeling Approaches | 224 |

83 Tension Development | 236 |

832 Experimental Studies | 240 |

833 Mathematical Modeling Approaches | 245 |

84 Mechanics in Anatomical Models | 260 |

Modeling of Cardiac ElectroMechanics | 267 |

92 Electrophysiology and Force Development of Single Cells | 268 |

93 Cellular Automaton of Cardiac Force Development | 273 |

94 ElectroMechanics of the Myocardium | 275 |

942 Simulations | 278 |

943 Limitations and Perspectives | 287 |

A Physical Units and Constants | 290 |

B Differential Operators | 292 |

C1 Model Assumptions | 293 |

C2 General Formulation | 294 |

C3 Restricted Formulation in Material Coordinate System | 296 |

C4 Coordinate System Transformation | 298 |

299 | |

323 | |

### Other editions - View all

Computational Cardiology: Modeling of Anatomy, Electrophysiology, and Mechanics Frank B. Sachse No preview available - 2004 |

Computational Cardiology: Modeling of Anatomy, Electrophysiology, and Mechanics Frank B. Sachse No preview available - 2014 |

### Common terms and phrases

actin anatomical anisotropic approximation atrial atrioventricular node atrium behavior bidomain model C. D. Werner calcium calcium concentration Ca2+]i canine Cardiac Electrophysiology Cell to Bedside cellular automaton coefficients commonly conductivity tensor configuration current source density D. P. Zipes dependent derivatives described determined differential equations displacements domain electro-mechanics excitation propagation Exemplary extracellular space F. B. Sachse fiber filaments filters finite element force development gap junctions heart Hodgkin-Huxley model interpolation intracellular intracellular calcium concentration ions Jalife linear macroscopic matrix measurements mechanical membrane method Molecular myocardium myosin node points node variables numerical papillary muscle parameters Poisson’s equation potassium current potential proteins resulting sarcolemma sarcomere sarcomere length sarcoplasmic reticulum Sect segmentation shape-functions simulations sodium spatial stimulus strain energy density strain tensor stress tensor stretch structure techniques tension three-dimensional tion tissue transformation transmembrane voltage transmembrane voltage Vm tropomyosin troponin vector ventricular myocytes Visible Human Project voxels W. B. Saunders Company