Vascular Biomechanics: Concepts, Models, and Applications

✍ Scribed by T. Christian Gasser

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 622
• Category: Library

✦ Synopsis

This textbook serves as a modern introduction to vascular biomechanics and provides the comprehensive overview of the entire vascular system that is needed to run successful vascular biomechanics simulations. It aims to provide the reader with a holistic analysis of the vascular system towards its biomechanical description and includes numerous fully through-calculated examples. Various topics covered include vascular system descriptions, vascular exchange, blood vessel mechanics, vessel tissue characterization, blood flow mechanics, and vascular tissue growth and remodeling.

This textbook is ideally suited for students and researchers studying and working in classical and computational vascular biomechanics. The book could also be of interest to developers of vascular devices and experts working with the regulatory approval of biomedical simulations.

• Follows the principle of “learning by doing” and provides numerous fully through-calculated examples for active learning, immediate recall, and self-examination;
• Provides a holistic understanding of vascular functioning and the integration of information from different disciplines to enable students to use sophisticated numerical methods to simulate the response of the vascular system;
• Includes several case studies that integrate the presented material. Case studies address problems, such as the biomechanical rupture risk assessment of Abdominal Aortic Aneurysms, Finite Element analysis of structural and blood flow problems, the computation of wall stress and wall shear stress in the aorta.

✦ Table of Contents

Foreword
Preface
Acknowledgments
Contents
Acronyms
1 Modeling in Bioengineering
1.1 Introduction
1.1.1 Bottom-Up Approach
1.1.2 Top-Down Approach
1.1.3 Opportunities and Challenges
1.2 Model Design
1.2.1 Simplifications
1.2.2 Strategies
1.2.2.1 White-Box Model
1.2.2.2 Black-Box Model
1.2.2.3 Gray-Box Model
1.2.2.4 Surrogate Model
1.3 Model Development and Testing
1.3.1 Sensitivity Analysis
1.3.1.1 Sobol's Variance-Based Sensitivity Analysis
1.3.2 Verification
1.3.3 Validation
1.3.3.1 Study Design
1.4 Statistics-Based Modeling
1.4.1 Correlation Amongst Variables
1.4.1.1 Pearson's Product-Moment Correlation Coefficient
1.4.1.2 Spearman's Rank Correlation Coefficient
1.4.2 Regression Modeling
1.4.2.1 Significance of a Regression
1.4.2.2 Non-linear Regression
1.4.2.3 Multiple Regression
1.4.2.4 Multivariant Regression
1.4.3 Hypothesis Testing
1.4.3.1 Error and Power of a Hypothesis Test
1.4.4 Mean Difference Test
1.4.4.1 One-Sample t-test
1.4.4.2 Two-Sample t-test of Independent Samples
1.5 Artificial Intelligence
1.5.1 Learning and Prediction
1.5.2 Artificial Neural Network
1.5.3 Bayesian Network
1.5.4 Decision Tree
1.6 Case Study: Biomechanical Rupture Risk Assessment
1.6.1 Shortcomings of the Current AAA Risk Assessment
1.6.2 Intended Model Application
1.6.3 Failure Hypothesis
1.6.4 Work Flow and Diagnostic Information
1.6.5 Key Modeling Assumptions
1.6.5.1 Organ-Level Model
1.6.5.2 Vascular Tissue Model
1.6.6 Clinical Validation
1.7 Summary and Conclusion
2 The Circulatory System
2.1 Introduction
2.1.1 Vascular System
2.1.2 Key Concepts
2.1.2.1 Form Follows Function
2.1.2.2 Blood Flows in Closed Loops
2.1.2.3 Vascular Network Is ``Space Filling''
2.1.3 Cells in the Vascular System
2.1.3.1 Endothelium Cell
2.1.3.2 Smooth Muscle Cell
2.1.3.3 Pericyte
2.1.3.4 FibroBlast
2.1.3.5 Erythrocyte
2.1.3.6 Leukocyte
2.1.3.7 Platelets
2.1.3.8 Dendritic Cell
2.1.4 Macrocirculation
2.1.4.1 Blood Vessel Structure and Function
2.1.4.2 The Intima and the Endothelium
2.1.4.3 The Media
2.1.4.4 The Adventitia
2.1.4.5 Wall Shear Stress (WSS)
2.1.5 Lymphatic System
2.1.5.1 Lymphatic Vessels
2.1.6 Microcirculation
2.1.6.1 Exchange
2.1.6.2 Colloid Osmotic Pressure and the Role of Albumin
2.1.6.3 Functional Adaptation of Capillaries
2.1.6.4 The Glycocalyx
2.1.6.5 Controlling Blood Pressure and the Role of Resistance Vessels
2.1.7 Hemodynamic Regulation
2.1.7.1 Autoregulation Mechanisms
2.1.7.2 Short-Term Nervous Control of the Blood Pressure
2.1.7.3 Long-Term Control of the Blood Pressure
2.2 Mechanical System Properties
2.2.1 Waves in the Vascular System
2.2.2 Vascular Pressure
2.2.2.1 Pressure Waveform
2.2.3 Vascular Capacity
2.2.4 Vascular Flow
2.2.4.1 Venous Return
2.2.5 The Pressure–Velocity Loop
2.2.6 Vascular Resistance
2.2.7 Transcapillary Transport
2.3 Modeling the Macrocirculation
2.3.1 WindKessel Models
2.3.1.1 Two-Element WindKessel Model
2.3.1.2 Homogeneous Solution
2.3.1.3 Impedance
2.3.1.4 Parameter Identification
2.3.1.5 Three-Element WindKessel Model
2.3.1.6 Homogeneous Solution
2.3.1.7 Four-Element WindKessel Model
2.3.2 Vessel Network Modeling
2.3.2.1 Vessel Segment Resistance
2.3.2.2 Vessel Segment Capacity
2.3.2.3 Blood Inertance
2.3.2.4 Governing Equation
2.3.2.5 Assembly of Vessel Networks
2.4 Modeling the Microcirculation
2.4.1 Transcapillary Concentration Difference
2.4.2 Filtration
2.4.2.1 Starling's Filtration Model
2.4.2.2 Predicted Exchange
2.4.2.3 Current Understanding of Microvascular Exchange
2.4.2.4 Colloid Osmotic Pressure
2.4.2.5 The Non-linear Filtration Law
2.4.2.6 Two-Pore Models
2.5 Summary and Conclusion
3 Continuum Mechanics
3.1 Introduction
3.2 Kinematics
3.2.1 Deformation Gradient
3.2.2 Multiplicative Decomposition
3.2.3 Polar Decomposition
3.2.4 Deformation of the Line Element
3.2.5 Deformation of the Volume Element
3.2.6 Deformation of the Area Element
3.2.7 Concept of Strain
3.2.7.1 Engineering, or Linear Strain
3.2.7.2 Non-linear Strain Measures
3.2.7.3 Particular Strain States
3.3 Concept of Stress
3.3.1 Cauchy Stress Theorem
3.3.2 Principal Stresses
3.3.3 Coordinate Rotation and Stress Components
3.3.4 Isochoric and Volumetric Stress
3.3.5 Octahedral Stress and von Mises Stress
3.3.6 Cauchy Stress in Rotated Coordinates
3.3.7 First Piola–Kirchhoff Stress
3.3.8 Second Piola–Kirchhoff Stress
3.3.9 Implication of Material Incompressibility on the Stress State
3.4 Material Time Derivatives
3.4.1 Kinematic Variables
3.4.1.1 Velocity Gradient
3.4.1.2 Rate of Deformation
3.4.1.3 Spin Tensor
3.4.1.4 Rate of Volume Change
3.4.2 Stress Rates
3.4.3 Power-Conjugate Stress and Strain rates
3.5 Constitutive Modeling
3.5.1 Some Mechanical Properties of Materials
3.5.1.1 Incompressibility
3.5.1.2 Isotropy and Anisotropy
3.5.1.3 Strain Energy in Solids
3.5.1.4 Dissipation in Isothermal Solids
3.5.1.5 Dissipation in Isothermal Fluids
3.5.1.6 Stiffness
3.5.1.7 Strength
3.5.1.8 Fracture Toughness
3.5.2 Linear-Elastic Material
3.5.2.1 Hooke's Law in Voigt's Notation
3.5.2.2 Hooke's Law in Tensor Notation
3.5.2.3 Hooke's Law with Decoupled Shear and Bulk Contributions
3.5.3 Hyperelasticity
3.5.3.1 Coupled Formulation
3.5.3.2 Volumetric–Isochoric Decoupled Formulation
3.5.3.3 Incompressible Formulation
3.5.4 Viscoelasticity
3.5.4.1 Newtonian Viscosity Model
3.5.4.2 Linear Viscoelasticity
3.5.4.3 Maxwell Rheology Element
3.5.4.4 Kelvin–Voigt Rheology Element
3.5.4.5 Standard Solid Rheology Element
3.5.4.6 Generalized Models
3.5.4.7 Visco-hyperelasticity for Incompressible Materials
3.5.4.8 Stress-Decomposed Visco-hyperelasticity for Incompressible Materials
3.5.5 Multiphasic Continuum Theories
3.5.5.1 Mixture Theory
3.5.5.2 Poroelasticity Theory
3.6 Governing Laws
3.6.1 Mass Balance
3.6.1.1 Lagrangian Description
3.6.1.2 Eulerian Description
3.6.2 Balance of Linear Momentum
3.6.2.1 Lagrangian Description
3.6.2.2 Eulerian Description
3.6.3 Thermodynamic Laws
3.6.3.1 The First Law of Thermodynamics
3.6.3.2 The Second Law of Thermodynamics
3.6.4 The Relation Between the Stress and the Helmholtz Free Energy
3.6.4.1 Stress of an Incompressible Material
3.7 General Principles
3.7.1 Maxwell Transport and Localization
3.7.2 Free Body Diagram
3.7.3 Boundary Value Problem
3.7.4 Principle of Virtual Work
3.7.4.1 Principle of Virtual Work for Small Deformation Problems
3.7.4.2 Principle of Virtual Work for Finite Deformation Problems
3.8 Damage and Failure
3.8.1 Physical Consequences of Damage
3.8.2 Strain Localization
3.8.3 Linear Fracture Mechanics
3.8.4 Non-linear Fracture Mechanics
3.8.4.1 J-Integral
3.8.4.2 Cohesive Zone Modeling
3.9 Summary and Conclusion
4 The Finite Element Method
4.1 Introduction
4.2 Spatial Discretization
4.2.1 Shape Function
4.2.1.1 Shape Functions for 1D Problems
4.2.1.2 Shape Functions for 2D Problems
4.2.1.3 Shape Functions for 3D Problems
4.2.2 Gradient Interpolation
4.2.3 Mixed and Hybrid Finite Elements
4.3 Calculus of Variations
4.3.1 Diffusion Boundary Value Problem
4.3.2 Advection–Diffusion Boundary Value Problem
4.3.3 Linear Solid Mechanics Boundary Value Problem
4.3.4 Non-linear Solid Mechanics Boundary Value Problem
4.3.5 Incompressible Flow Boundary Value Problem
4.4 Finite Element Equations
4.4.1 Diffusion Problems
4.4.2 Advection–Diffusion Problems
4.4.3 Linear Solid Mechanics Problems
4.4.3.1 The Linear Truss Finite Element Equations
4.4.4 Non-linear Solid Mechanics Problems
4.4.4.1 Pressure Boundary Condition
4.4.5 Incompressible Flow Problems
4.4.6 Numerical Quadrature
4.5 Constrained Problems
4.5.1 Penalty Constraint
4.5.2 Lagrange Constraint
4.5.3 Augmented-Lagrange Constraint
4.6 Globalization
4.7 Stabilization
4.7.1 Positive Definiteness of the Finite Element Stiffness
4.7.2 Stabilization of the Advection–Diffusion Finite Element
4.7.2.1 Full Upwind Stabilization
4.7.2.2 Petrov–Galerkin Finite Elements
4.7.2.3 Generalization for Multi-dimensional Problems
4.7.2.4 Petrov–Galerkin Formulations
4.8 Solving the System of Finite Element Equations
4.8.1 Solving Sparse Linear Systems
4.8.1.1 Direct Solution Methods
4.8.1.2 Iterative Solution Methods
4.8.2 Time Integration
4.8.3 Non-linear Formulations
4.8.4 Incremental Formulation
4.8.5 Explicit Solution
4.8.6 Implicit Solution
4.8.6.1 Newton–Raphson Method
4.8.6.2 Load Incrementation
4.8.6.3 Dirichlet Versus Neumann Boundary Conditions
4.8.6.4 Arc-Length, or Continuation Methods
4.8.6.5 Incompressible Flow Equation
4.9 Case Study: Planar Biaxial Testing
4.9.1 Mechanical Problem Description
4.9.2 Results and Verification
4.10 Case Study: Inflated Cylindrical Vessel
4.10.1 Results and Verification
4.11 Case Study: Bulge Inflation
4.11.1 Mechanical Problem Description
4.11.2 Analytical Problem Assessment
4.11.3 Solution Strategy and Results
4.12 Case Study: Flow Through a Network of Conduit Arteries
4.13 Case Study: Flow Through the Cylindrical Tube
4.13.1 Steady-State Analysis
4.13.2 Steady-Pulsatile Analysis
4.13.3 Transient Analysis with WindKessel Outlet Boundary Condition
4.14 Summary and Conclusion
5 Conduit Vessels
5.1 Introduction
5.2 Histology and Morphology of the Vessel Wall
5.2.1 Layered Vessel Wall Organization
5.2.2 Differences Between Arteries and Veins
5.2.3 Extra Cellular Matrix (ECM)
5.2.3.1 Collagen Structure
5.2.3.2 Elastin Structure
5.2.4 Cells
5.3 Mechanical Properties and Experimental Observations
5.3.1 Aorta
5.3.2 Carotid Artery
5.3.3 Coronary Artery
5.3.4 Iliac and Femoral Artery
5.4 Vascular Diseases
5.4.1 Diagnostic Examinations
5.4.2 Atherosclerosis
5.4.3 Biomechanical Factors in Atherosclerosis
5.4.3.1 Atherosclerosis Development
5.4.3.2 Plaque Stress and Strain
5.4.3.3 Clinical Relevance of Atherosclerosis
5.4.4 Carotid Artery Disease
5.4.5 Coronary Heart Disease
5.4.6 Aneurysm Disease
5.4.6.1 Aneurysm Pathophysiology
5.4.6.2 The Elastic Properties of the Infrarenal Aorta
5.4.6.3 Strength of Aorta Tissue
5.5 Constitutive Descriptions
5.5.1 Capacity of a Vessel Segment
5.5.2 Hyperelasticity for Incompressible Solids
5.5.2.1 Stress and Strain Analysis in Principal Directions
5.5.3 Purely Phenomenological Descriptions of the Vessel Wall
5.5.3.1 Isotropic Models
5.5.4 Inflated and Axially Stretched Two-Layered Vessel
5.5.5 Inflated and Axially Stretched Thick-Walled Vessel
5.5.5.1 Anisotropic Models
5.5.6 Histo-mechanical Descriptions
5.5.7 General Theory of Fibrous Connective Tissue
5.5.7.1 Collagen Fiber Models
5.5.7.2 Description of Collagen Fiber Orientations
5.5.8 Residual Stress and Load-Free Configuration
5.5.9 Visco-hyperelastic Descriptions
5.5.10 Cyclic Deformation of the Visco-hyperelastic Thin-Walled Tube
5.5.11 Damage and Failure Descriptions
5.5.11.1 Modeling Irreversible Properties of Collagen Fibers
5.5.11.2 Failure Represented by Cohesive Zone Models
5.6 Identification of Constitutive Parameters
5.6.1 Analytical Vessel Wall Models
5.6.1.1 Yeoh Model
5.6.1.2 Fung Model
5.6.1.3 GOH Model
5.6.2 Optimization Problem
5.6.2.1 Yeoh Model
5.6.2.2 Fung Model
5.6.2.3 GOH Model
5.6.2.4 Influence of Noise in the Experimental Data
5.7 Case Study: Structural Analysis of the Aneurysmatic Infrarenal Aorta
5.7.1 Modeling Assumptions
5.7.2 Results and Interpretation of the Computed Wall Stress
5.8 Summary and Conclusion
6 Hemodynamics
6.1 Introduction
6.2 Blood Composition
6.2.1 Erythrocyte
6.2.2 Leukocyte
6.2.3 Thrombocyte
6.2.4 Plasma
6.3 Forces Acting at Blood Particles
6.3.1 Drag Force
6.3.2 Gravitational and Inertia Forces
6.3.3 Forces Related to Fluid Pressure
6.3.4 Forces Related to Fluid Velocity and Shear Stress
6.3.5 Forces from Collisions
6.3.6 Chemical and Electrical Forces
6.3.7 Segregation of Blood Particles
6.4 Blood Rheology Modeling
6.4.1 Shear Rate-Dependent Changes of Blood Microstructure
6.4.2 Modeling Generalized Newtonian Fluids
6.4.3 Single-Phase Viscosity Models for Blood
6.4.3.1 Power Law Model
6.4.3.2 Carreau–Yasuda Model
6.4.3.3 Casson Model
6.4.4 Composition-Based Viscosity Models for Blood
6.4.4.1 Walburn–Schneck Model
6.4.4.2 Quemada Model
6.4.4.3 Krieger-Based Model
6.5 Blood Damage
6.5.1 Hemolysis
6.5.1.1 Eulerian Implementation of the Hemolysis Power Law Model
6.5.1.2 Lagrangian Implementation of the Hemolysis Power Law Model
6.5.2 Thrombocyte Activation
6.6 Description of Incompressible Flows
6.6.1 Energy Conservation
6.6.1.1 Blood Flow Under the Action of Gravitation
6.6.2 Linear Momentum Conservation
6.6.2.1 Governing Equation for 1D Flows in Cartesian Coordinates
6.6.2.2 Fluid Flow Down the Inclined Plane
6.6.2.3 Fluid Layer at Oscillating Pressure Load
6.6.2.4 Governing Equation for 1D Flows in Cylindrical Coordinates
6.6.2.5 Steady-State Flow of a Newtonian Fluid Througha Circular Tube
6.6.2.6 Resistance of a Vessel Segment
6.6.2.7 Pulsatile Newtonian Fluid Flow Through a Circular Tube
6.6.3 Flow in the Elastic Tube
6.7 Blood Flow Phenomena
6.7.1 Laminar, Transitional, and Turbulent Flow
6.7.2 Boundary Layer Flow
6.7.3 Blood Flow Through Circular Tubes
6.7.4 Multi-dimensional Flow Phenomena
6.7.4.1 Secondary Flow
6.7.4.2 Vortex Flow and Vortex Structures
6.7.4.3 Jet Flow
6.7.4.4 Creep Flow
6.7.4.5 Inviscid Flow
6.8 Description of Blood Flow Properties
6.8.1 Wall Shear Stress-Based Parameters
6.8.2 Transport-Based Parameters
6.9 Case Study: Blood Flow in the Aneurysmatic Infrarenal Aorta
6.9.1 Modeling Assumptions
6.9.2 Results
6.10 Summary and Conclusion
7 The Vascular Wall, an Active Entity
7.1 Introduction
7.2 Vasoreactivity
7.2.1 SMC Phenotypic Modulation
7.2.2 Structure and Contraction Machinery of Contractile SMC
7.2.2.1 Contraction and Relaxation
7.2.2.2 Regulation of SMC Contraction and Relaxation
7.2.3 SMC Tone and Vessel Wall Properties
7.3 Arteriogenesis
7.3.1 Interplay of Endothelium Cells and Smooth Muscle Cells
7.3.2 WSS Profile and Inflammation
7.3.3 Collagen Synthesis
7.3.3.1 Intracellular Processes
7.3.3.2 Extracellular Processes
7.3.4 Elastin Synthesis
7.4 Angiogenesis
7.5 Modeling Vasoreactivity
7.5.1 Hill's Three-Parameter Muscle Model
7.5.2 Phenomenological Descriptions
7.5.3 Structural-Based Descriptions
7.5.4 Calcium Concentration-Based Descriptions
7.5.5 Thermodynamics of SMC Contraction
7.5.6 A Chemo-mechanical Description of SMC
7.5.6.1 Stress Versus Stretch Properties
7.5.6.2 Stress versus Stretch Rate Properties
7.5.6.3 Quick-Release Experiment
7.6 Modeling Arteriogenesis
7.6.1 Open System Governing Laws
7.6.1.1 Mass Balance
7.6.1.2 Balance of Linear and Angular Momentum
7.6.1.3 The First Law of Thermodynamics
7.6.1.4 The Second Law of Thermodynamics
7.6.2 Kinematics-Based Growth Description
7.6.3 Spatial Distribution of Volume Growth
7.6.3.1 Constant-Volume Growth
7.6.3.2 Constant-Density Growth
7.6.3.3 Homeostatic Growth
7.6.4 The Thick-Walled Elastic Tube at Homeostatic Growth
7.6.5 Continuous Turnover-Based Growth Description
7.6.6 Tissue at Simple Tension and Continuous Mass Turnover
7.6.7 Multiphasic and Miscellaneous Descriptions
7.7 Summary and Conclusion
Correction to: Vascular Biomechanics: Concepts, Models, and Applications
A Mathematical Preliminaries
A.1 Statistics
A.1.1 Definitions and Terminology
A.1.2 Probability Distributions
A.1.3 Data Distribution Testing
A.1.4 Confidence Interval
A.2 Complex Numbers
A.3 Fourier Series Approximation
A.4 Laplace and Fourier Transforms
A.5 Matrix Algebra
A.5.1 Trace of a Matrix
A.5.2 Identity Matrix
A.5.3 Determinant of a Matrix
A.5.4 Inverse and Orthogonal Matrix
A.5.5 Linear Vector Transform
A.5.6 Eigenvalue Problem
A.5.7 Relation Between the Trace and the Eigenvalues of a Matrix
A.5.8 Cayley–Hamilton Theorem
A.6 Vector Algebra
A.6.1 Basic Vector Operations
A.6.2 Coordinate Transformation
A.6.2.1 Vector Components undergoing Coordinate Transformation
A.7 Tensor Algebra
A.7.1 Spherical Tensor
A.7.2 Tensor Operations
A.7.3 Invariants of Second-Order Tensors
A.8 Vector and Tensor Calculus
A.8.1 Local Changes of Field Variables
A.8.1.1 Gradient
A.8.1.2 Divergence
A.8.2 Divergence Theorem
B Some Useful Laplace and Fourier Transforms
B.1 Laplace Transforms
Dirac Delta Function δ(t)
Heaviside Function H(t)
The Function tH(t)
The Function sin(at)H(t)
The Function cos(at)H(t)
B.2 Fourier Transforms
Dirac Delta Function δ(t)
The Function exp(-at)H(t)
The Function sin(at)H(t)
The Function cos(at)H(t)
C Some Useful Tensor Relations
The Identity (∂C/∂F):F-1=2I
The Time Derivative of C-1
The Identity ∂J/∂C=JC-1/2
The Identity ∂I1/∂C=I
The Identity ∂I2/∂C=I1I-C
The Identity ∂I3/∂C=J2C-1
The Identity ∂C/∂C=J-2/3[I-(CC-1)/3]
The Identity C-1:=trd
The Identity dev[FAFT]=F{Dev[A]}FT
D Some Useful Variations and Directional Derivatives
The Relations DuF= Gradu and δF=Gradδu
The Relations DuE=sym(FT Gradu) and δE=sym(FT Gradδu)
The Relation DuδE=sym(GradTu Gradδu)
The Relations Due=gradsu and δe=gradsδu
The Relations DuF-1=-F-1 grad u and δF-1=-F-1 grad δu
The Relations Dugradδu=-gradδugradu
The Relations DuJ=J divu and δJ=J divδu
E Physical Preliminaries
E.1 Basic Circuit Elements
E.1.1 Resistor Element
E.1.2 Capacitor Element
E.1.3 Inductor Element
E.2 Transport Mechanisms
E.2.1 Diffusion
E.2.2 Flow Through Porous Media
E.2.3 Advection
E.3 Osmosis
E.3.1 Osmotic Pressure
E.3.2 Transport Across Semipermeable Membranes
F Biaxial Experimental Vessel Wall Testing
F.1 Tissue Harvesting and Sample Preparation
F.2 Test Protocol Definition and Data Recording
F.3 Acquired Test Data
G Definitions of Symbols, Functions, and Operators
Solutions
References
Index