C W Cryer P P Lunkenheimer Mathematical Mo delling of the Cardiovascular System Mathematical Mo delli ng of the Cardiovascular System Version Septemb er Colin Walker Cryer Paul Peter Lunkenheimer Abstract We describ e the problem of mo delling the cardiovascular system and give current refer ences AMS Sub ject Classication Z C C C Keywords Mo delling heart ventricular mechanics biomechanics cardiovascular system mathematics Institut fur Numerische und instrumentelle Mathematik Westfalische WilhelmsUniversitat Einste instrae Munster email cryermathunimuensterde Exp erimentelle Thorax Herz und Gefachirurgie Domagkstr Munster Contents Prologue I Biomechanics The Human Machine Energetics Mechanics II The Heart Intro duction The Circulatory System The Heart Visualization Disease and Pathology ComparativePhysiology History of Cardiology The Cardiac Cycle Intro duction The Cardiac Cycle Ventricular lling Isovolumetric contraction Ejection Isovolumetric Relaxation Global Motion of the Heart The Pericardium Heart Sounds and Murm urs Concluding Remarks The Structure of the Myo cardium Intro duction Connective Tissue Previous Mathematical Mo dels The Pericardium Endo cardium and Epicardium Exp erimental Observations The Ventricular Function Curve In Vivo Stress Measurements In Vivo Strain Measurements Coronary Perfusion Intramyo cardial Pressure Other Exp erimental Observations Pericardial Eusion Hyp ervolemia Oligemia Arterial Hyp ertension Aortic Coarctation Aortic Stenosis Pulmonary Stenosis Idoventricular Rhythms Coronary Occlusion Myo cardial Ischemia Cardiac Valves Aortic Regurgitation Mitral Insuciency Mitral Stenosis III Mathematical Mo delling of the Heart In tro duction The Field Equations The Constitutive Equations The Goal of Mo delling Diastolic Expansion Ventricular Blo o d Pressure Myo cardial stress Elastic Recoil Residual Stress Ventricular Geometry Muscle Mechanics Intro duction The Phenomological Approach The Fenn eect Feng eect Classic Results Molecular Muscle Mechanics Ventricular Interaction PassiveMyo cardium Exp erimental Metho ds and Results Constitutive Equations StrainEnergy Functions Bovendeerd Smaill and Hunter Inverse Metho ds for Parameter Estimation Cardiac Mechanics Intro duction The Comp onents of the Hearts Wall Exp erimental StressStrain Measurements ConstitutiveLaws Hydraulic Eects Myo cytes Mo delling the Fibre Architecture of the Myo cardium Foliations Top ology Geo desics Mo dels with Spherical Geometry The Law of Laplace Spherical Mo dels Analytical Work Comparison of Mo dels Mo dels with Cylindrical Geometry Mo dels with Spheroidal Geometry Mo dels with Additional Features The Pericardium Fibres Hydrostatic Skeletons and Muscular Hydrostats FluidFibre Mo dels Systolic Contraction Fibres Treatment of the Band Structure Poro elastic Mo dels Other Factors Numerical Mo dels Numerical Metho ds Prop erties of Blo o d Fluid Flow in the heart Intro duction Numerical Metho ds The ImmersedBoundary Metho d EulerLagrangian Formulation Vortex and whirl ows Mo delling the Heart Valves Intro duction The Mitral Valve The Aortic Valve Architecture of the Aortic Valve Leaets Articial Valves Valve Vibration Mo delling other Ventricular Pro cesses Electrical and Electrophysiological Activity Oxygen Diusion Heat Flow ComparativePhysiology of the Heart Intro duction Sources of Data Allometric Equations Dierences Between Mammalian Hearts Concluding Remarks Mo delling the Global Motion of the Heart The Heart as a Mechanical Pump Cardiac Energetics Growth Miscellaneous Topics Heart Sounds The IntervalForce Relationship Assistance and Replacement Anrep Eect Gregg Eect IV Mathematical Mo delling of the Circulatory System Intro duction The Arterial System Intro duction The Pulmonary Circulation Intro duction The V enous System The Coronary Circulation The Micro circulation V Mathematical Mo delling of the Entire Cardiovascular Sys tem Intro duction Mo dels of the Cardiovascular System Control Design Principles of the Cardiovascular System Optimization of Biological Structures A The SI System A Physical Units A Units A Useful conversion factors A Physical Quantities B Dimensional Analysis B Dimensional Analysis C Elementary Concepts from Physical Chemistry DPhysiology E Cardiovascular Data E Man E Woman E Dog F Mathematical Tools F Dierential geometry F Geo desics F Geometric Measure Theory F Top ology F Knots F Partial Dierential Equations F Singularities F Chaos G Mechanics G Applied Mechanics G Thermo dynamics G Contiuum Mechanics G Elasticity G Fibrereinforced Materials G Elastic Solids with Microstructure H Biomechanics H General References H Biomaterials H Mathematical Biology H Pattern Formation H Bio energetics H Kinesiology I Computer Graphics J Numerical Metho ds J Finite Elements K Sources K Sources of Preserved Hearts K Bo oks Rep orts Theses K Journals K So cieties K Deutsche Literatur K DeutschEnglischVerzeichnis K Diplomthemen L Worksheets L Questions L Availability of Journals etc in Munster L Literature Searched L PhD Theses L Corresp ondance M Biomathematics Prologue App endix References Concepts List of Figures The simplied circulatory system The plumbing of the normal heart The lo cation of the heart Crosssections of the heart The cardiac cycle Simplied diagram of isolated dog heartlung preparation of Starling The ventricular function curve Horts Mo del for elastic recoil Contractile b ehaviour of isolated myo cardial muscle Stressextension relations for left ventricular midwall sp ecimen during st th and th cycles of equibiaxial loading Cycle p erio d sec and sp eci men thickness mm The solid lines indicate the order of loading in the crossb er direction for clarity these are omitted in the b er direction The Law of Laplace Blo o d ow a Blo o d ow b Chapter Prologue The human heart is a pump which converts metab olic energy to mechanical work and generates the pressure that drives the blo o d throughout the b o dys circulation system It can do so for years or more with a rate varying from ab out litresmin in an adult human male at rest to litresmin in Olympic athletes For an organ which is ab out the size of a st this is a remarkable achievement In we b egan a longterm pro ject to examine ways of mo delling the heart sp ecically its mechanical prop erties This requires several steps The relevant features of the heart must b e formulated Quantitative exp erimental data must b e obtained Appropriate mathematical mo dels must b e constructed As far as p ossible the mo dels should b e analyzed mathematically Numerical metho ds.