De / Guilak / Mofrad Computational Modeling in Biomechanics

1. Auflage 2010
ISBN: 978-90-481-3575-2
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 581 Seiten

Availability of advanced computational technology has fundamentally altered the investigative paradigm in the field of biomechanics. Armed with sophisticated computational tools, researchers are seeking answers to fundamental questions by exploring complex biomechanical phenomena at the molecular, cellular, tissue and organ levels. The computational armamentarium includes such diverse tools as the ab initio quantum mechanical and molecular dynamics methods at the atomistic scales and the finite element, boundary element, meshfree as well as immersed boundary and lattice-Boltzmann methods at the continuum scales. Multiscale methods that link various scales are also being developed. While most applications require forward analysis, e.g., finding deformations and stresses as a result of loading, others involve determination of constitutive parameters based on tissue imaging and inverse analysis. This book provides a glimpse of the diverse and important roles that modern computational technology is playing in various areas of biomechanics including biofluids and mass transfer, cardiovascular mechanics, musculoskeletal mechanics, soft tissue mechanics, and biomolecular mechanics.

Autoren/Hrsg.

De, Suvranu

Guilak, Farshid

Mofrad, Mohammad

Inhaltsverzeichnis

1;Preface: Computational Modeling in Biomechanics;5
2;Contents;7
3;Section I Biofluids and Mass Transport;9
3.1;1 Immersed Boundary/Continuum Methods;10
3.1.1;1 Introduction;11
3.1.1.1;1.1 Immersed Boundary; Nodal Forces; Incompressible Continuum;13
3.1.1.2;1.2 Mapping and Kernel;24
3.1.1.3;1.3 Fictitious Domain Method; Immersed Continuum Method;36
3.1.1.4;1.4 Implicit/Compressible Solver;44
3.1.2;2 Discussions;52
3.1.3;References;53
3.2;2 Computational Modeling of ATP/ADP Concentration at the Vascular Surface;56
3.2.1;1 Introduction;56
3.2.2;2 Mathematical Models;58
3.2.2.1;2.1 History of ATP/ADP Mathematical Models;58
3.2.2.2;2.2 Essential Physics;58
3.2.2.3;2.3 Governing Equations;59
3.2.2.4;2.4 Boundary Conditions;60
3.2.2.4.1;2.4.1 Modeling Flow-Induced ATP Release;61
3.2.2.4.2;2.4.2 Flow Conditions;62
3.2.2.5;2.5 Relevant Dimensionless Parameters;63
3.2.3;3 Model Results: Flow-Mediated Nucleotide Concentration at the EC Surface;64
3.2.3.1;3.1 Synopsis of Model Results and Implications;64
3.2.3.2;3.2 Understanding the Results of the Models: Contributions of Individual Flow Features;66
3.2.3.2.1;3.2.1 Steady Undisturbed Flow;66
3.2.3.2.2;3.2.2 Pulsatile Undisturbed Flow;67
3.2.3.2.3;3.2.3 Disturbed Flow: Flow Recirculation and Beyond;68
3.2.3.2.4;3.2.4 Impact of ATP-Free Perfusion on Nucleotide Concentration at the EC Surface;70
3.2.4;4 General Perspectives and Critical Future Directions;70
3.2.5;References;72
3.3;3 Development of a Lattice-Boltzmann Method for Multiscale Transport and Absorption with Application to Intestinal Function;75
3.3.1;1 Introduction;76
3.3.2;2 Numerical Methods;78
3.3.2.1;2.1 Basic Lattice-Boltzmann Algorithm for Momentum and Pressure;78
3.3.2.2;2.2 Passive Scalar in the Lattice–Boltzmann Method;80
3.3.2.3;2.3 Moving Boundary Conditions for Momentum;82
3.3.2.4;2.4 Scalar Concentration Boundary Conditions;83
3.3.2.4.1;2.4.1 Fixed-Scalar Boundary Condition;84
3.3.2.4.2;2.4.2 Fixed-Flux Boundary Condition;86
3.3.2.5;2.5 Multi Grid Algorithm;86
3.3.2.6;2.6 Implementation of Multigrid Strategy in the Momentum Propagation Method;89
3.3.3;3 Validation of Algorithm with A Multiscale Model of Macro-to-Micro Scale Transport;89
3.3.3.1;3.1 Continuity Between Coarse and Fine Grids;93
3.3.3.2;3.2 Validation of the Multi Grid Strategy;93
3.3.3.3;3.3 Validation of Moving Boundary Conditions;98
3.3.4;4 Concluding Remarks;101
3.3.5;References;102
4;Section II Cardiovascular Biomechanics;103
4.1;4 Computational Models of Vascular Mechanics;104
4.1.1;1 Introduction;104
4.1.2;2 Healthy Vessels;105
4.1.2.1;2.1 Conducting Arteries;106
4.1.2.2;2.2 Distributing Arteries;107
4.1.3;3 Healthy Arterial Mechanical Response and Constitutive Relations;108
4.1.4;4 Mechanics Studies of Non-Atheromatous Arteries;121
4.1.4.1;4.1 Healthy Geometry, Healthy Material;122
4.1.4.2;4.2 High Pressure Response;123
4.1.5;5 Fluid-Structure Interaction;124
4.1.5.1;5.1 Stenotic Geometry, Healthy Material;128
4.1.6;6 Carotid Bifurcation;130
4.1.6.1;6.1 Healthy Carotid Bifurcation, Measurement-Based;131
4.1.7;7 Patient-Specific Studies;137
4.1.8;8 Imaging-Derived Geometry and Flow Boundary Conditions;137
4.1.8.1;8.1 Computed Tomography;138
4.1.8.2;8.2 Magnetic Resonance Imaging;138
4.1.8.3;8.3 Time-Of-Flight (TOF) Methods;139
4.1.8.4;8.4 2-D TOF Methods;140
4.1.8.5;8.5 3-D TOF Methods;140
4.1.8.6;8.6 Phase Contrast MRA/MRI;141
4.1.8.7;8.7 Contrast-Enhanced MRA (CE-MRA);141
4.1.8.8;8.8 Black Blood MRI;142
4.1.8.9;8.9 Ultrasound;143
4.1.8.10;8.10 Image Segmentation;144
4.1.9;9 Image-Based Modeling of Healthy Vessels;144
4.1.10;10 The Atherosclerotic Artery Wall;147
4.1.11;11 Solid Mechanics of Idealized Plaque Lesions;149
4.1.11.1;11.1 Microcalcifications;151
4.1.12;12 2-D Patient-Specific Plaque Studies;152
4.1.13;13 3-D Patient-Specific Plaque Studies;157
4.1.13.1;13.1 Plaque Lesion Fracture, Dissection, Stenting, and Angioplasty;160
4.1.13.2;13.2 Stenting;163
4.1.14;14 Current Developments;166
4.1.15;References;168
4.2;5 Computational Modeling of Vascular Hemodynamics;176
4.2.1;1 Hemodynamics and Vascular Disease;176
4.2.1.1;1.1 A Brief Description of Atherosclerotic and Aneurismal Diseases;176
4.2.1.2;1.2 Hemodynamics in the Initiation and Progression of Atherosclerotic Lesions;178
4.2.1.3;1.3 Hemodynamics in Aneurismal Blood Vessels;180
4.2.2;2 Computational Fluid Dynamics;181
4.2.2.1;2.1 Numerical Methods;181
4.2.2.1.1;2.1.1 Governing Equations and Modeling Assumptions;181
4.2.2.1.2;2.1.2 Numerical Solution of the Flow Equations;182
4.2.2.2;2.2 Flow Boundary Conditions;183
4.2.2.2.1;2.2.1 Typical Assumptions for the Inlet and Outlet Boundary Conditions;183
4.2.2.2.2;2.2.2 Accounting for the Proximal and Distal Circulation;185
4.2.2.3;2.3 Post-processing and Visualization of Numerical and Experimental Results;186
4.2.2.3.1;2.3.1 Various Flow Descriptors;186
4.2.2.3.2;2.3.2 Flow Characterization and Lagrangian Particle Tracking;187
4.2.2.4;2.4 Non-Newtonian Blood Behavior;188
4.2.2.4.1;2.4.1 Shear Thinning and Yield Stress Properties;188
4.2.2.4.2;2.4.2 Commonly Used Non-Newtonian Viscosity Models;188
4.2.2.5;2.5 Transitional and Turbulent Flow;190
4.2.2.6;2.6 Compliant Arterial Wall;191
4.2.2.6.1;2.6.1 CFD Simulations with Moving Boundaries;191
4.2.3;3 Patient-Specific Computational Modeling;192
4.2.3.1;3.1 Medical Imaging Modalities;193
4.2.3.2;3.2 Patient-Specific Lumenal Geometries;194
4.2.3.2.1;3.2.1 Geometry Reconstruction;194
4.2.3.2.2;3.2.2 Longitudinal Studies;196
4.2.3.3;3.3 Patient-Specific Flow Measurements;197
4.2.3.3.1;3.3.1 MR Velocimetry;197
4.2.4;4 Modeling of the Flow in Diseased Blood Vessels;197
4.2.4.1;4.1 Flow in Atherosclerotic Carotid Bifurcations;197
4.2.4.2;4.2 Flow in Aneurysmal Arteries;199
4.2.4.3;4.3 Accuracy and Reliability of CFD Results;200
4.2.4.3.1;4.3.1 Validation with Clinical Data;200
4.2.4.3.2;4.3.2 Validation with Experimental Results;201
4.2.5;5 Current Developments;203
4.2.5.1;5.1 Prediction of Disease Progression for a Given Patient and Guidance for Interventions;203
4.2.6;References;204
4.3;6 Computational Modeling of Coronary Stents;212
4.3.1;1 Introduction;212
4.3.1.1;1.1 Computational Simulations;213
4.3.1.1.1;1.1.1 Governing Equations;213
4.3.1.1.2;1.1.2 Material Models;214
4.3.1.1.3;1.1.3 Geometries;214
4.3.1.2;1.2 Spatial and Temporal Gradients of Shear Stress;216
4.3.1.3;1.3 Arbitrary Lagrange Eulerian (ALE) Method;216
4.3.1.4;1.4 Fluid–Structure Coupling;217
4.3.1.5;1.5 Stent Simulation Results;217
4.3.1.6;1.6 Solid Mechanics Simulations;217
4.3.1.6.1;1.6.1 Under-sizing of Stent;220
4.3.1.6.2;1.6.2 Over-sizing of Stent;222
4.3.1.7;1.7 Mechanical Stresses and Vessel Function;222
4.3.2;2 Summary and Conclusions;223
4.3.3;References;223
4.4;7 Computational Modeling of Aortic Heart Valves;226
4.4.1;1 Introduction;226
4.4.2;2 The Aortic Valve;227
4.4.3;3 Anatomical and Material Properties;229
4.4.4;4 Simulation of the Cardiac Cycle;233
4.4.5;5 Fluid-Structure Interaction;236
4.4.6;6 Multiscale Approach;239
4.4.7;7 Applications;243
4.4.7.1;7.1 Natural Aortic Valve;243
4.4.7.2;7.2 Diseased Aortic Valve;245
4.4.7.3;7.3 Modeling Surgical Repair;248
4.4.7.4;7.4 Optimizing Artificial Heart Valve Design;251
4.4.8;8 Future Directions;253
4.4.9;9 Conclusion;254
4.4.10;References;254
4.5;8 Computational Modeling of Growth and Remodeling in Biological Soft Tissues: Application to Arterial Mechanics;258
4.5.1;1 Introduction;258
4.5.2;2 Towards a Theory of Growth and Remodeling (G&R);259
4.5.3;3 Theoretical Framework for Growth and Remodeling (G&R);263
4.5.4;4 Computational Considerations;270
4.5.5;5 Illustrative Results;272
4.5.6;6 Conclusion;276
4.5.7;References;277
5;Section III Musculoskeletal Biomechanics;280
5.1;9 Computational Modeling of Trabecular Bone Mechanics;281
5.1.1;1 Introduction;282
5.1.2;2 The Finite Element Method;283
5.1.3;3 Idealized Geometric Trabecular Models;285
5.1.3.1;3.1 Historical Context;285
5.1.3.2;3.2 Beam Models;285
5.1.3.3;3.3 Image-Based Beam and Plate Models;287
5.1.4;4 Micro-FEA Modeling;288
5.1.4.1;4.1 Mesh Generation;288
5.1.4.2;4.2 Convergence and Accuracy of Micro-FEA Models;290
5.1.4.3;4.3 Material Property Assignment;292
5.1.4.4;4.4 Calculation of Trabecular Tissue Modulus;292
5.1.4.5;4.5 Homogenization;293
5.1.4.6;4.6 Nonlinear Behavior;293
5.1.4.6.1;4.6.1 Geometric Nonlinearity;294
5.1.4.6.2;4.6.2 Softening Materials;295
5.1.4.6.3;4.6.3 Fracture Simulations;298
5.1.4.7;4.7 Evaluation of Measurement Techniques;298
5.1.4.8;4.8 In vivo Applications;300
5.1.4.9;4.9 Result Post-processing;301
5.1.4.10;4.10 Whole Bone Simulations;302
5.1.5;5 Future Challenges;304
5.1.6;6 Summary;304
5.1.7;References;305
5.2;10 Computational Modeling of Extravascular Flow in Bone;311
5.2.1;1 Introduction;312
5.2.1.1;1.1 Defining ``The System'' Bone;312
5.2.1.2;1.2 Endogeneous Structural and Fluid Components of Bone;315
5.2.1.3;1.3 Role of Fluid Flow and Mass Transport in Bone Physiology;316
5.2.1.4;1.4 Role of Computational Models in Understanding Extravascular Flow in Bone Health and Disease;317
5.2.2;2 Multiscale Models of Extravascular Fluid Flow in Bone;317
5.2.3;3 Parametric Study: Importance of Spatially Defined Material Parameters on Flow Predictions;322
5.2.4;4 Spatially Resolved Permeabilities and Porosities;324
5.2.5;5 Idealization of Geometries at the Cell Scale and Below Results in a Profound Underprediction of Flow Velocities;325
5.2.6;6 Current Hurdles and Future Vision;328
5.2.7;References;329
5.3;11 Computational Modeling of Cell Mechanics in Articular Cartilage;333
5.3.1;1 Introduction;334
5.3.2;2 Continuum Models of Cell Mechanics;334
5.3.2.1;2.1 Single Phase Models;336
5.3.2.2;2.2 Biphasic (Solid-Fluid) Model;336
5.3.2.3;2.3 Biot Poroelastic Model;337
5.3.3;3 Computational Methods;338
5.3.3.1;3.1 Boundary Element Methods;338
5.3.3.1.1;3.1.1 Axisymmetric Elastic BEM;338
5.3.3.1.2;3.1.2 Axisymmetric Incompressible Viscoelastic BEM;341
5.3.3.1.3;3.1.3 Biphasic (Poroelastic) BEM;341
5.3.3.2;3.2 Finite Element Methods;344
5.3.4;4 Applications;345
5.3.4.1;4.1 Micropipette Aspiration;346
5.3.4.1.1;4.1.1 Boundary Element Models of Micropipette Aspiration;347
5.3.4.1.2;4.1.2 Multiphasic Finite Element Models of Micropipette Aspiration;348
5.3.4.2;4.2 Multiphasic Models of Mechanical Cell-Matrix Interactions;349
5.3.5;5 Summary;353
5.3.6;References;353
5.4;12 Computational Models of Tissue Differentiation;357
5.4.1;1 Introduction;357
5.4.2;2 Approaches to Modelling;359
5.4.3;3 Simulation Architecture;360
5.4.3.1;3.1 Overview;360
5.4.3.2;3.2 Algorithms as Building Blocks of the Simulation;361
5.4.3.2.1;3.2.1 Cell Movement;361
5.4.3.2.2;3.2.2 Cell Proliferation and Cell Apoptosis;362
5.4.3.2.3;3.2.3 Determination of Stem Cell Fate and Stem Cell Differentiation;364
5.4.3.2.4;3.2.4 Angiogenesis;366
5.4.3.2.5;3.2.5 Synthesis of Matrix;367
5.4.3.3;3.3 Implementation Using Finite Element Analysis;369
5.4.3.4;3.4 Applications in Tissue Engineering;370
5.4.4;4 Possibilities for Scaffold/Bioreactor Modelling;371
5.4.5;5 Discussion and Conclusion;373
5.4.6;References;375
6;Section IV Soft Tissue Biomechanics;377
6.1;13 A Review of the Mathematical and Computational Foundations of Biomechanical Imaging;378
6.1.1;1 Introduction;378
6.1.2;2 Background;379
6.1.2.1;2.1 Motivation;379
6.1.2.2;2.2 Imaging Tissue Deformation;380
6.1.2.2.1;2.2.1 Ultrasound Imaging of Quasi-Static Compression;381
6.1.2.2.2;2.2.2 Magnetic Resonance Imaging of Time-Harmonic Excitation;381
6.1.2.2.3;2.2.3 Intravascular Ultrasound Imaging of Coronary Plaques;382
6.1.2.2.4;2.2.4 Radiation Force Imaging;382
6.1.2.3;2.3 The Inverse Problem;382
6.1.2.3.1;2.3.1 Quasi-Static Displacement Data;383
6.1.2.3.2;2.3.2 Transient Displacement Data;384
6.1.3;3 Direct Formulation of Inverse Problem ;385
6.1.3.1;3.1 Formulation;386
6.1.3.2;3.2 Uniqueness and Existence;387
6.1.3.2.1;3.2.1 One Dimensional Linear Elasticity;387
6.1.3.2.2;3.2.2 Plane Stress Linear Elasticity;387
6.1.3.2.3;3.2.3 Plane Strain Linear Elasticity;388
6.1.3.2.4;3.2.4 Three Dimensional Linear Elasticity;390
6.1.3.3;3.3 Direct Computational Solution for (x);390
6.1.3.3.1;3.3.1 Exact Solution in Plane Stress;390
6.1.3.3.2;3.3.2 Least Squares;391
6.1.3.3.3;3.3.3 Adjoint Weighted Variational Equation;392
6.1.3.4;3.4 Issues and Opportunities;393
6.1.3.4.1;3.4.1 Full Vector Displacement Data;393
6.1.3.4.2;3.4.2 Traction Data and Other A Priori Information;394
6.1.3.4.3;3.4.3 Discontinuous Modulus Distributions;394
6.1.3.4.4;3.4.4 Uniqueness;394
6.1.3.4.5;3.4.5 Nonlinear Elasticity;394
6.1.4;4 Optimization Formulation ;395
6.1.4.1;4.1 Optimization Methods;396
6.1.4.2;4.2 Gradient Calculation;397
6.1.4.3;4.3 Sample Reconstructions;398
6.1.4.3.1;4.3.1 3D Tissue Mimicking Phantom;399
6.1.4.3.2;4.3.2 Nonlinear Clinical Example;401
6.1.4.4;4.4 Issues and Opportunities;401
6.1.4.4.1;4.4.1 Analysis of the Constrained Optimization Problem;402
6.1.4.4.2;4.4.2 Boundary Conditions for the Forward Problem;403
6.1.4.4.3;4.4.3 Hessian Estimate in BFGS;404
6.1.4.4.4;4.4.4 Incompressibility in nonlinear reconstructions;405
6.1.4.4.5;4.4.5 Systematic Choices of Computational Parameters;405
6.1.5;5 Concluding Remarks;406
6.1.6;References;406
6.2;14 Interactive Surgical Simulation Using a Meshfree Computational Method;412
6.2.1;1 Background;412
6.2.2;2 The Point-Associated Finite Field (PAFF) Approach;416
6.2.2.1;2.1 Real Time Global PAFF (g-PAFF);420
6.2.2.2;2.2 Real Time Local PAFF (l-PAFF);421
6.2.3;3 Real Time Nonlinear PAFF Analysis;423
6.2.3.1;3.1 Fast Localized Solution;425
6.2.4;4 Reduced Order Modeling of Viscoelastic Tissue Response Using PAFF;427
6.2.4.1;4.1 The Elastodynamic Initial/Boundary Value Problem;428
6.2.4.2;4.2 Model Order Reduction Methods;430
6.2.5;5 Discussions;433
6.2.6;References;434
6.3;15 Computational Biomechanics of the Human Cornea;437
6.3.1;1 Introduction;437
6.3.2;2 A Model for the Human Cornea;438
6.3.2.1;2.1 Microscopic Structure;439
6.3.2.2;2.2 Geometry;441
6.3.3;3 Mechanical Properties;444
6.3.4;4 Material Models;446
6.3.5;5 Computational Models;449
6.3.6;6 A Model of the Human Cornea;451
6.3.7;7 Applications and Results;456
6.3.7.1;7.1 Inflation Tests;456
6.3.7.2;7.2 Refractive Surgery;458
6.3.7.3;7.3 Parametric Analysis;460
6.3.8;8 Conclusions;464
6.3.9;References;465
7;Section V Biomolecular Mechanics and Multiscale Modeling;469
7.1;16 Identifying the Reaction Mechanisms of Inteins with QM/MM Multiscale Methods;470
7.1.1;1 Introduction;470
7.1.1.1;1.1 Computational Background;470
7.1.1.2;1.2 Intein Background;471
7.1.2;2 Methods;472
7.1.2.1;2.1 Computational Methodology;472
7.1.2.2;2.2 Quantum Mechanical (QM) Methods;472
7.1.2.2.1;2.2.1 Implicit Solvent;473
7.1.2.3;2.3 Classical Methods;474
7.1.2.4;2.4 Multiscale (QM/MM) Methods;474
7.1.2.4.1;2.4.1 Charge Embedding;475
7.1.2.5;2.5 Geometry Minimization;475
7.1.3;3 Results;476
7.1.3.1;3.1 Non-essential Mutation;476
7.1.3.2;3.2 Classical Protein System;478
7.1.3.3;3.3 Tripeptide Subsystem;478
7.1.3.3.1;3.3.1 Description of Model System;478
7.1.3.3.2;3.3.2 Energetic Results;480
7.1.3.3.3;3.3.3 Charge Analysis;481
7.1.3.4;3.4 Single Amino Acid Molecules;481
7.1.3.4.1;3.4.1 Electron Affinity and Ionization Potential Analysis;481
7.1.3.4.2;3.4.2 Energetic Analysis of Molecular Orbitals near the Fermi Energy;483
7.1.3.4.3;3.4.3 Tripeptide Analysis;485
7.1.4;4 Reaction Analysis with QM/MM Calculations;486
7.1.4.1;4.1 Effect of Mutation on Energy Barriers;486
7.1.4.2;4.2 Effect of Mutation on Electron Occupation;488
7.1.5;5 Conclusions;488
7.1.6;References;489
7.2;17 Computational Scale Linking in Biological Protein Materials;491
7.2.1;1 Introduction;491
7.2.2;2 Computational Materials Science of Biological Protein Materials;493
7.2.2.1;2.1 Mechanical Properties of Biological Protein Materials;494
7.2.2.2;2.2 Strategies of Investigation;496
7.2.2.3;2.3 Linking the Scales: Cross-Scale Interactions;496
7.2.2.4;2.4 Materiomics;500
7.2.3;3 Computational Approaches;502
7.2.3.1;3.1 Molecular Dynamics Simulation at the Atomistic Scale;502
7.2.3.1.1;3.1.1 Conventional Charmm-Type Force Fields and Related Models;505
7.2.3.1.2;3.1.2 ReaxFF Reactive Force Fields;506
7.2.3.2;3.2 Mesoscale Simulation – Coarse-Graining;510
7.2.3.3;3.3 Complementary Experimental Analysis Techniques;515
7.2.4;4 Case Studies;516
7.2.4.1;4.1 Size Effects of Strength of Clusters of H-Bonds;517
7.2.4.2;4.2 Deformation and Failure Behavior of Alpha-Helical Protein Networks;520
7.2.5;5 Future Directions, Challenges and Impact;525
7.2.6;References;526
7.3;18 How to Measure Biomolecular Forces: A ``Tug-of-War'' Approach;532
7.3.1;1 Motivation;532
7.3.2;2 Theory;535
7.3.3;3 Application to Brownian Dynamics Simulation;538
7.3.3.1;3.1 Calibrating the Method in a Force-Free Case;540
7.3.3.2;3.2 Random Walk Over a Gaussian Potential Barrier;541
7.3.4;4 Real-World Example: Dihedral Transition;542
7.3.5;5 Comparison with Other Approaches;545
7.3.6;6 Conclusion;546
7.3.7;References;546
7.4;19 Mechanics of Cellular Membranes;548
7.4.1;1 Lipid Membranes;548
7.4.2;2 Equilibrium Equations;550
7.4.3;3 Axisymmetric Solutions;553
7.4.4;4 Membrane–Protein Interactions and Endocytosis;554
7.4.5;5 Edge Conditions;557
7.4.6;6 Adhesion;559
7.4.7;7 Coexistent Phases;561
7.4.8;8 Conclusion;563
7.4.9;References;563
8;Index;566