Computational Modeling in Biomechanics (eBook)

Preface: Computational Modeling in Biomechanics 5
Contents 7
Section I Biofluids and Mass Transport 9
1 Immersed Boundary/Continuum Methods 10
1 Introduction 11
1.1 Immersed Boundary Nodal Forces
1.2 Mapping and Kernel 24
1.3 Fictitious Domain Method Immersed Continuum Method
1.4 Implicit/Compressible Solver 44
2 Discussions 52
References 53
2 Computational Modeling of ATP/ADP Concentration at the Vascular Surface 56
1 Introduction 56
2 Mathematical Models 58
2.1 History of ATP/ADP Mathematical Models 58
2.2 Essential Physics 58
2.3 Governing Equations 59
2.4 Boundary Conditions 60
2.4.1 Modeling Flow-Induced ATP Release 61
2.4.2 Flow Conditions 62
2.5 Relevant Dimensionless Parameters 63
3 Model Results: Flow-Mediated Nucleotide Concentration at the EC Surface 64
3.1 Synopsis of Model Results and Implications 64
3.2 Understanding the Results of the Models: Contributions of Individual Flow Features 66
3.2.1 Steady Undisturbed Flow 66
3.2.2 Pulsatile Undisturbed Flow 67
3.2.3 Disturbed Flow: Flow Recirculation and Beyond 68
3.2.4 Impact of ATP-Free Perfusion on Nucleotide Concentration at the EC Surface 70
4 General Perspectives and Critical Future Directions 70
References 72
3 Development of a Lattice-Boltzmann Method for Multiscale Transport and Absorption with Application to Intestinal Function 75
1 Introduction 76
2 Numerical Methods 78
2.1 Basic Lattice-Boltzmann Algorithm for Momentum and Pressure 78
2.2 Passive Scalar in the Lattice–Boltzmann Method 80
2.3 Moving Boundary Conditions for Momentum 82
2.4 Scalar Concentration Boundary Conditions 83
2.4.1 Fixed-Scalar Boundary Condition 84
2.4.2 Fixed-Flux Boundary Condition 86
2.5 Multi Grid Algorithm 86
2.6 Implementation of Multigrid Strategy in the Momentum Propagation Method 89
3 Validation of Algorithm with A Multiscale Model of Macro-to-Micro Scale Transport 89
3.1 Continuity Between Coarse and Fine Grids 93
3.2 Validation of the Multi Grid Strategy 93
3.3 Validation of Moving Boundary Conditions 98
4 Concluding Remarks 101
References 102
Section II Cardiovascular Biomechanics 103
4 Computational Models of Vascular Mechanics 104
1 Introduction 104
2 Healthy Vessels 105
2.1 Conducting Arteries 106
2.2 Distributing Arteries 107
3 Healthy Arterial Mechanical Response and Constitutive Relations 108
4 Mechanics Studies of Non-Atheromatous Arteries 121
4.1 Healthy Geometry, Healthy Material 122
4.2 High Pressure Response 123
5 Fluid-Structure Interaction 124
5.1 Stenotic Geometry, Healthy Material 128
6 Carotid Bifurcation 130
6.1 Healthy Carotid Bifurcation, Measurement-Based 131
7 Patient-Specific Studies 137
8 Imaging-Derived Geometry and Flow Boundary Conditions 137
8.1 Computed Tomography 138
8.2 Magnetic Resonance Imaging 138
8.3 Time-Of-Flight (TOF) Methods 139
8.4 2-D TOF Methods 140
8.5 3-D TOF Methods 140
8.6 Phase Contrast MRA/MRI 141
8.7 Contrast-Enhanced MRA (CE-MRA) 141
8.8 Black Blood MRI 142
8.9 Ultrasound 143
8.10 Image Segmentation 144
9 Image-Based Modeling of Healthy Vessels 144
10 The Atherosclerotic Artery Wall 147
11 Solid Mechanics of Idealized Plaque Lesions 149
11.1 Microcalcifications 151
12 2-D Patient-Specific Plaque Studies 152
13 3-D Patient-Specific Plaque Studies 157
13.1 Plaque Lesion Fracture, Dissection, Stenting, and Angioplasty 160
13.2 Stenting 163
14 Current Developments 166
References 168
5 Computational Modeling of Vascular Hemodynamics 176
1 Hemodynamics and Vascular Disease 176
1.1 A Brief Description of Atherosclerotic and Aneurismal Diseases 176
1.2 Hemodynamics in the Initiation and Progression of Atherosclerotic Lesions 178
1.3 Hemodynamics in Aneurismal Blood Vessels 180
2 Computational Fluid Dynamics 181
2.1 Numerical Methods 181
2.1.1 Governing Equations and Modeling Assumptions 181
2.1.2 Numerical Solution of the Flow Equations 182
2.2 Flow Boundary Conditions 183
2.2.1 Typical Assumptions for the Inlet and Outlet Boundary Conditions 183
2.2.2 Accounting for the Proximal and Distal Circulation 185
2.3 Post-processing and Visualization of Numerical and Experimental Results 186
2.3.1 Various Flow Descriptors 186
2.3.2 Flow Characterization and Lagrangian Particle Tracking 187
2.4 Non-Newtonian Blood Behavior 188
2.4.1 Shear Thinning and Yield Stress Properties 188
2.4.2 Commonly Used Non-Newtonian Viscosity Models 188
2.5 Transitional and Turbulent Flow 190
2.6 Compliant Arterial Wall 191
2.6.1 CFD Simulations with Moving Boundaries 191
3 Patient-Specific Computational Modeling 192
3.1 Medical Imaging Modalities 193
3.2 Patient-Specific Lumenal Geometries 194
3.2.1 Geometry Reconstruction 194
3.2.2 Longitudinal Studies 196
3.3 Patient-Specific Flow Measurements 197
3.3.1 MR Velocimetry 197
4 Modeling of the Flow in Diseased Blood Vessels 197
4.1 Flow in Atherosclerotic Carotid Bifurcations 197
4.2 Flow in Aneurysmal Arteries 199
4.3 Accuracy and Reliability of CFD Results 200
4.3.1 Validation with Clinical Data 200
4.3.2 Validation with Experimental Results 201
5 Current Developments 203
5.1 Prediction of Disease Progression for a Given Patient and Guidance for Interventions 203
References 204
6 Computational Modeling of Coronary Stents 212
1 Introduction 212
1.1 Computational Simulations 213
1.1.1 Governing Equations 213
1.1.2 Material Models 214
1.1.3 Geometries 214
1.2 Spatial and Temporal Gradients of Shear Stress 216
1.3 Arbitrary Lagrange Eulerian (ALE) Method 216
1.4 Fluid–Structure Coupling 217
1.5 Stent Simulation Results 217
1.6 Solid Mechanics Simulations 217
1.6.1 Under-sizing of Stent 220
1.6.2 Over-sizing of Stent 222
1.7 Mechanical Stresses and Vessel Function 222
2 Summary and Conclusions 223
References 223
7 Computational Modeling of Aortic Heart Valves 226
1 Introduction 226
2 The Aortic Valve 227
3 Anatomical and Material Properties 229
4 Simulation of the Cardiac Cycle 233
5 Fluid-Structure Interaction 236
6 Multiscale Approach 239
7 Applications 243
7.1 Natural Aortic Valve 243
7.2 Diseased Aortic Valve 245
7.3 Modeling Surgical Repair 248
7.4 Optimizing Artificial Heart Valve Design 251
8 Future Directions 253
9 Conclusion 254
References 254
8 Computational Modeling of Growth and Remodeling in Biological Soft Tissues: Application to Arterial Mechanics 258
1 Introduction 258
2 Towards a Theory of Growth and Remodeling (G& R)
3 Theoretical Framework for Growth and Remodeling (G& R)
4 Computational Considerations 270
5 Illustrative Results 272
6 Conclusion 276
References 277
Section III Musculoskeletal Biomechanics 280
9 Computational Modeling of Trabecular Bone Mechanics 281
1 Introduction 282
2 The Finite Element Method 283
3 Idealized Geometric Trabecular Models 285
3.1 Historical Context 285
3.2 Beam Models 285
3.3 Image-Based Beam and Plate Models 287
4 Micro-FEA Modeling 288
4.1 Mesh Generation 288
4.2 Convergence and Accuracy of Micro-FEA Models 290
4.3 Material Property Assignment 292
4.4 Calculation of Trabecular Tissue Modulus 292
4.5 Homogenization 293
4.6 Nonlinear Behavior 293
4.6.1 Geometric Nonlinearity 294
4.6.2 Softening Materials 295
4.6.3 Fracture Simulations 298
4.7 Evaluation of Measurement Techniques 298
4.8 In vivo Applications 300
4.9 Result Post-processing 301
4.10 Whole Bone Simulations 302
5 Future Challenges 304
6 Summary 304
References 305
10 Computational Modeling of Extravascular Flow in Bone 311
1 Introduction 312
1.1 Defining ``The System'' Bone 312
1.2 Endogeneous Structural and Fluid Components of Bone 315
1.3 Role of Fluid Flow and Mass Transport in Bone Physiology 316
1.4 Role of Computational Models in Understanding Extravascular Flow in Bone Health and Disease 317
2 Multiscale Models of Extravascular Fluid Flow in Bone 317
3 Parametric Study: Importance of Spatially Defined Material Parameters on Flow Predictions 322
4 Spatially Resolved Permeabilities and Porosities 324
5 Idealization of Geometries at the Cell Scale and Below Results in a Profound Underprediction of Flow Velocities 325
6 Current Hurdles and Future Vision 328
References 329
11 Computational Modeling of Cell Mechanics in Articular Cartilage 333
1 Introduction 334
2 Continuum Models of Cell Mechanics 334
2.1 Single Phase Models 336
2.2 Biphasic (Solid-Fluid) Model 336
2.3 Biot Poroelastic Model 337
3 Computational Methods 338
3.1 Boundary Element Methods 338
3.1.1 Axisymmetric Elastic BEM 338
3.1.2 Axisymmetric Incompressible Viscoelastic BEM 341
3.1.3 Biphasic (Poroelastic) BEM 341
3.2 Finite Element Methods 344
4 Applications 345
4.1 Micropipette Aspiration 346
4.1.1 Boundary Element Models of Micropipette Aspiration 347
4.1.2 Multiphasic Finite Element Models of Micropipette Aspiration 348
4.2 Multiphasic Models of Mechanical Cell-Matrix Interactions 349
5 Summary 353
References 353
12 Computational Models of Tissue Differentiation 357
1 Introduction 357
2 Approaches to Modelling 359
3 Simulation Architecture 360
3.1 Overview 360
3.2 Algorithms as Building Blocks of the Simulation 361
3.2.1 Cell Movement 361
3.2.2 Cell Proliferation and Cell Apoptosis 362
3.2.3 Determination of Stem Cell Fate and Stem Cell Differentiation 364
3.2.4 Angiogenesis 366
3.2.5 Synthesis of Matrix 367
3.3 Implementation Using Finite Element Analysis 369
3.4 Applications in Tissue Engineering 370
4 Possibilities for Scaffold/Bioreactor Modelling 371
5 Discussion and Conclusion 373
References 375
Section IV Soft Tissue Biomechanics 377
13 A Review of the Mathematical and Computational Foundations of Biomechanical Imaging 378
1 Introduction 378
2 Background 379
2.1 Motivation 379
2.2 Imaging Tissue Deformation 380
2.2.1 Ultrasound Imaging of Quasi-Static Compression 381
2.2.2 Magnetic Resonance Imaging of Time-Harmonic Excitation 381
2.2.3 Intravascular Ultrasound Imaging of Coronary Plaques 382
2.2.4 Radiation Force Imaging 382
2.3 The Inverse Problem 382
2.3.1 Quasi-Static Displacement Data 383
2.3.2 Transient Displacement Data 384
3 Direct Formulation of Inverse Problem 385
3.1 Formulation 386
3.2 Uniqueness and Existence 387
3.2.1 One Dimensional Linear Elasticity 387
3.2.2 Plane Stress Linear Elasticity 387
3.2.3 Plane Strain Linear Elasticity 388
3.2.4 Three Dimensional Linear Elasticity 390
3.3 Direct Computational Solution for (x) 390
3.3.1 Exact Solution in Plane Stress 390
3.3.2 Least Squares 391
3.3.3 Adjoint Weighted Variational Equation 392
3.4 Issues and Opportunities 393
3.4.1 Full Vector Displacement Data 393
3.4.2 Traction Data and Other A Priori Information 394
3.4.3 Discontinuous Modulus Distributions 394
3.4.4 Uniqueness 394
3.4.5 Nonlinear Elasticity 394
4 Optimization Formulation 395
4.1 Optimization Methods 396
4.2 Gradient Calculation 397
4.3 Sample Reconstructions 398
4.3.1 3D Tissue Mimicking Phantom 399
4.3.2 Nonlinear Clinical Example 401
4.4 Issues and Opportunities 401
4.4.1 Analysis of the Constrained Optimization Problem 402
4.4.2 Boundary Conditions for the Forward Problem 403
4.4.3 Hessian Estimate in BFGS 404
4.4.4 Incompressibility in nonlinear reconstructions 405
4.4.5 Systematic Choices of Computational Parameters 405
5 Concluding Remarks 406
References 406
14 Interactive Surgical Simulation Using a Meshfree Computational Method 412
1 Background 412
2 The Point-Associated Finite Field (PAFF) Approach 416
2.1 Real Time Global PAFF (g-PAFF) 420
2.2 Real Time Local PAFF (l-PAFF) 421
3 Real Time Nonlinear PAFF Analysis 423
3.1 Fast Localized Solution 425
4 Reduced Order Modeling of Viscoelastic Tissue Response Using PAFF 427
4.1 The Elastodynamic Initial/Boundary Value Problem 428
4.2 Model Order Reduction Methods 430
5 Discussions 433
References 434
15 Computational Biomechanics of the Human Cornea 437
1 Introduction 437
2 A Model for the Human Cornea 438
2.1 Microscopic Structure 439
2.2 Geometry 441
3 Mechanical Properties 444
4 Material Models 446
5 Computational Models 449
6 A Model of the Human Cornea 451
7 Applications and Results 456
7.1 Inflation Tests 456
7.2 Refractive Surgery 458
7.3 Parametric Analysis 460
8 Conclusions 464
References 465
Section V Biomolecular Mechanics and Multiscale Modeling 469
16 Identifying the Reaction Mechanisms of Inteins with QM/MM Multiscale Methods 470
1 Introduction 470
1.1 Computational Background 470
1.2 Intein Background 471
2 Methods 472
2.1 Computational Methodology 472
2.2 Quantum Mechanical (QM) Methods 472
2.2.1 Implicit Solvent 473
2.3 Classical Methods 474
2.4 Multiscale (QM/MM) Methods 474
2.4.1 Charge Embedding 475
2.5 Geometry Minimization 475
3 Results 476
3.1 Non-essential Mutation 476
3.2 Classical Protein System 478
3.3 Tripeptide Subsystem 478
3.3.1 Description of Model System 478
3.3.2 Energetic Results 480
3.3.3 Charge Analysis 481
3.4 Single Amino Acid Molecules 481
3.4.1 Electron Affinity and Ionization Potential Analysis 481
3.4.2 Energetic Analysis of Molecular Orbitals near the Fermi Energy 483
3.4.3 Tripeptide Analysis 485
4 Reaction Analysis with QM/MM Calculations 486
4.1 Effect of Mutation on Energy Barriers 486
4.2 Effect of Mutation on Electron Occupation 488
5 Conclusions 488
References 489
17 Computational Scale Linking in Biological Protein Materials 491
1 Introduction 491
2 Computational Materials Science of Biological Protein Materials 493
2.1 Mechanical Properties of Biological Protein Materials 494
2.2 Strategies of Investigation 496
2.3 Linking the Scales: Cross-Scale Interactions 496
2.4 Materiomics 500
3 Computational Approaches 502
3.1 Molecular Dynamics Simulation at the Atomistic Scale 502
3.1.1 Conventional Charmm-Type Force Fields and Related Models 505
3.1.2 ReaxFF Reactive Force Fields 506
3.2 Mesoscale Simulation – Coarse-Graining 510
3.3 Complementary Experimental Analysis Techniques 515
4 Case Studies 516
4.1 Size Effects of Strength of Clusters of H-Bonds 517
4.2 Deformation and Failure Behavior of Alpha-Helical Protein Networks 520
5 Future Directions, Challenges and Impact 525
References 526
18 How to Measure Biomolecular Forces: A ``Tug-of-War'' Approach 532
1 Motivation 532
2 Theory 535
3 Application to Brownian Dynamics Simulation 538
3.1 Calibrating the Method in a Force-Free Case 540
3.2 Random Walk Over a Gaussian Potential Barrier 541
4 Real-World Example: Dihedral Transition 542
5 Comparison with Other Approaches 545
6 Conclusion 546
References 546
19 Mechanics of Cellular Membranes 548
1 Lipid Membranes 548
2 Equilibrium Equations 550
3 Axisymmetric Solutions 553
4 Membrane–Protein Interactions and Endocytosis 554
5 Edge Conditions 557
6 Adhesion 559
7 Coexistent Phases 561
8 Conclusion 563
References 563
Index 566