Vascular Biomechanics

1 Modeling in Biomechanics 1.- 1.1 The different perspectives 2.- 1.1.1 The engineering approach 2.- 1.1.2 The clinical approach 2.- 1.1.3 The pre- clinical approaches 2.- 1.2 Opportunities and challenges 2.- 1.3 Statistical analysis 3.- 1.3.1 Probability distributions 4.- 1.3.2 Hypothesis testing 7.- 1.3.3 Correlation amongst variables 9.- 1.3.4 Regression modeling 10.- 1.3.5 Mean difference test 13.- 1.3.6 Study design 14.- 1.4 Model definition 16.- 1.5 Model development and testing 17.- 1.5.1 Sensitivity analysis 17.- 1.5.3 Validation 21.- 1.6 Case study: Biomechanical Rupture Risk Assessment (BRRA) 21.- 1.6.1 Short comings of the current AAA risk assessment 21.- 1.6.2 Intended Model Application (IMA) 21.- 1.6.3 Failure hypothesis 22.- 1.6.4 Work flow and diagnostic information 22.- 1.6.5 Key modeling assumptions 23.- 1.6.6 Clinical validation 24.- 1.7 Summary and conclusion 25.- Appendix: Biomechanics Modeling 27.- A.1 Definitions and terminology in statistics 27.- 2 The circulatory system 29.- 2.1 Physiology 29.- 2.1.1 Vascular system 29.- 2.1.2 Key concepts 31.- 2.1.3 Cells in the vascular system 32.- 2.1.4 Macrocirculation 33.- 2.1.5 Lymphatic system 37.- 2.1.6 Microcirculation 38.- 2.1.7 Hemodynamic regulation 41.- 2.2 Mechanical system properties 42.- 2.2.1 Vascular pressure 43.- 2.2.2 Vascular flow 44.- 2.2.3 Vascular resistance 45.- 2.2.4 Transcapillary transport 45.- 2.3 Modeling the macrocirculation 45.- 2.3.1 Windkessel (WK) models 46.- 2.3.2 Vessel network modeling 57.- 2.4 Modeling the Microcirculation 63.- 2.4.1 Transcapillary concentration difference 63.- 2.4.2 Filtration 65.- 2.5 Summary and conclusion 70.- Appendix: Mathematical preliminaries 72.- A.1 Complex numbers 72.- A.2 Fourier series approximation 72.- Appendix: Basic circuit elements 73.- B.1 Resistor element 73.- B.2 Capacitor element 73.- B.3 Inductor element 74.- Appendix: Transport mechanisms 74.- C.1 Diffusion 74.- C.2 Advection 75.- Appendix: Osmosis 75.- D.1 Osmotic pressure 75.- D.2 Transport across semipermeable membranes 76.- 3 Continuum Mechanics 77.- 3.1 Kinematics 78.- 3.1.1 Deformation gradient 78.- 3.1.2 Multiplicative decomposition 79.- 3.1.3 Polar decomposition 79.- 3.1.4 Deformation of the line element 79.- 3.1.5 Deformation of the volume element 80.- 3.1.6 Deformation of the area element 80.- 3.1.7 Concept of strain 81.- 3.2 Concept of stress 85.- 3.2.1 Cauchy stress theorem 86.- 3.2.2 Principal stresses 87.- 3.2.3 Isochoric and volumetric stress 89.- 3.2.4 Octahedral stress and von Mises stress 89.- 3.2.5 Cauchy stress in rotated coordinates 91.- 3.2.6 First Piola-Kirchhoff stress 91.- 3.2.7 Second Piola-Kirchhoff stress 92.- 3.2.8 Implication of material incompressibility on the stress state 93.- 3.3 Material time derivatives 94.- 3.3.1 Kinematic variables 94.- 3.3.2 Stress rates 95.- 3.3.3 Power-conjugate stress and strain rates 96.- 3.4 Constitutive modeling 97.- 3.4.1 Some mechanical properties of materials 97.- 3.4.2 Linear elastic material100.- 3.4.3 Hyperelasticity 102.- 3.4.4 Viscoelasticity 105.- 3.5 Governing laws 113.- 3.5.1 Mass balance 114.- 3.5.2 Balance of linear momentum 116.- 3.5.3 Maxwell transport and localization 118.- 3.5.4 Thermodynamic principles 119.- 3.6 General principles 125.- 3.6.1 Free body diagram 125.- 3.6.2 Initial Boundary Value Problem 126.- 3.6.3 Principle of Virtual.- 3.7 Damage and failure 129.- 3.7.1 Physical consequences 129.- 3.7.2 Strain localization 130.- 3.7.3 Linear Fracture Mechanics 132.- 3.7.4 J.- Integral 133.- 3.7.5 Cohesive zone modeling 133.- 3.8 Multiphasic continuum theories 134.- 3.8.1 Mixture theory 134.- 3.8.2 Poroelasticity theory 134.- 3.9 Summary and conclusion 135.- Appendix: Mathematical preliminaries 136.- A.1 Laplace and Fourier transforms 136.- A.2 Matrix algebra 136.- A.2.1 Trace of a matrix 137.- A.2.2 Identity matrix 137.- A.2.3 Determinant of a matrix 137.- A.2.4 Inverse and orthogonal matrix 138.- A.2.5 Linear vector transform 138.- A.2.6 Eigenvalue problem138.- A.2.7 Relation between the trace and the eigenvalues 139.- A.2.8 Cayley-Hamilton theorem 139.- A.3 Vector algebra 140.- A.3.1 Basic vector operations 140.- A.3.2 Coordinate transformation 142.- A.4 Tensor algebra 144.- A.4.1 Spherical tensor 144.- A.4.2 Tensor operations 145.- A.4.3 Invariants of second-order tensors 145.- A.5 Vector and tensor calculus 146.- A.5.1 Local changes of field variables 146.- A.5.2 Divergence theorem 147.- Appendix: Some useful Laplace and Fourier transforms 148.- B.1 Laplace transforms 148.- B.2 Fourier transforms 150.- Appendix: Some useful tensor relations 151.- 4 Conduit vessels 153.- 4.1 Histology and morphology of the vessel wall 154.- 4.1.1 Layered vessel wall organization 154.- 4.1.2 Differences between arteries and veins 155.- 4.1.3 Extra Cellular Matrix (ECM) 156.- 4.1.4 Cells 157.- 4.2 Mechanical properties and experimental observations 158.- 4.2.1 Aorta 160.- 4.2.2 Carotid artery 161.- 4.2.3 Coronary artery 162.- 4.2.4 Iliac artery 163.- 4.3 Vascular diseases 163.- 4.3.1 Diagnostic examinations 164.- 4.3.2 Atherosclerosis 165.- 4.3.3 Biomechanical factors in atherosclerosis 167.- 4.3.4 Carotid artery disease 169.- 4.3.5 Coronary heart disease 171.- 4.3.6 Aneurysm disease 172.- 4.4 Vascular adaptation 174.- 4.5 Constitutive descriptions 175.- 4.5.1 Capacity of a vessel segment 176.- 4.5.2 Hyperelasticity for incompressible solids 177.- 4.5.3 Purely phenomenological descriptions 178.- 4.5.4 Histo-mechanical descriptions 183.- 4.5.5 General theory of fibrous connective tissue 185.- 4.5.6 Residual stress and load.- free configuration 188.- 4.5.7 Visco-elastic descriptions 189.- 4.5.8 Damage and failure descriptions 191.- 4.5.9 Non-passive vessel wall properties 194.- 4.6 Identification of constitutive parameters 194.- 4.6.1 Analytical vessel wall models 197.- 4.6.2 Optimization problem 199.- 4.7 Case study: Wall stress analysis of the normal and aneurysmatic.- infrarenal aorta 205.- 4.7.1 the analysis type 205.- 4.7.2 Setting the boundary conditions- Dirichlet boundary 205.- 4.7.3 Setting the loading conditions - Neuman boundary 205.- 4.7.4 Setting the vascular wall properties 206.- 4.7.5 Setting the output options 206.- 4.8 Summary and Conclusion 206.- Appendix: Protocol experimental vessel wall testing 208.- A.1 Tissue harvesting and sample preparation 208.- A.2 Test protocol definition and data recording 208.- A.3 Acquired.- x CONTENTS.- 5 Blood flow 211.- 5.1 Blood composition 211.- 5.1.1 Erythrocyte (or red blood cell) 212.- 5.1.2 Leukocyte (or white blood cell) 212.- 5.1.3 Thrombocyte (or platelet) 213.- 5.1.4 Plasma 213.- 5.2 Forces acting at blood particles 214.- 5.2.1 Drag force 214.- 5.2.2 Gravitational and inertia forces 214.- 5.2.3 Forces related to fluid pressure 214.- 5.2.4 Forces related to fluid velocity and shear stress 215.- 5.2.5 Forces arising from collisions 216.- 5.2.6 Chemical and electrical forces 216.- 5.2.7 Segregation of blood particles 218.- 5.3 Blood rheology modeling 218.- 5.3.1 Alteration of blood microstructure with the shear rate 218.- 5.3.2 Modeling generalized Newtonian fluids 219.- 5.3.3 Single-phase viscosity models for blood 220.- 5.3.4 Composition-based viscosity models for blood 221.- 5.4 Blood damage 224.- 5.5 Description of incompressible flows 224.- 5.5.1 Energy conservation 224.- 5.5.2 Linear momentum conservation 226.- 5.6 Blood flow phenomena 232.- 5.6.1 Laminar and turbulent flow 232.- 5.6.2 Boundary layer flow 233.- 5.6.3 Blood flow through circular tubes 233.- 5.6.4 Multi-dimensional flow phenomena 234.- 5.7 Case study: Wall Shear Stress (WSS) analysis of the normal and.- aneurysmatic infrarenal aorta 236.- 5.7.1 Setting the analysis type 236.- 5.7.2 Setting the boundary conditions -Dirichlet boundary 236.- 5.7.3 Setting the loading conditions -Neuman boundary 237.- 5.7.4 Setting the blood rheological properties 237.- 5.7.5 Setting the output options 237.- 5.8 Summary and conclusion 238.- Appendix: Mathematical preliminaries 239.- 6 Thevascular wall, an active entity 241.- 6.1 Vasoreactivity 242.- 6.1.1 Structure of contractile SMC 242.- 6.1.2 SMC contraction regulation 243.- 6.2 Arteriogenesis 243.- 6.3 Angiogenesis 244.- 6.4 Damage, healing and failure 244.- 6.5 Modeling frameworks 244.- 6.5.1 Open system governing laws 245.- 6.5.2 Kinematics-based growth description 246.- 6.5.3 Tensorial distribution of volume growth 248.- 6.5.4 Homeostatic growth 249.- 6.5.5 Continues turnover-based growth description 252.- 6.5.6 Other formulations 256.- 6.5.7 Applications of growth descriptions 257.- 6.6 Conclusion and Discussion 258.- 6.7 Applications 259.- 6.7.1 Tensile testing the passive and active vessel wall 259.- 6.7.2 Biaxially loaded vessel wall patch 260.- 6.7.3 Ring testing of vessel segments 262.- References 265.- Problem Solutions 287.- Index 373.