ALBERT

Description / Table of Contents: 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 material 100 -- 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 problem 138 -- 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 The vascular 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.