Modeling of physiological flows

Cover......Page 1
Title Page     \r......Page 4
Copyright Page     \r......Page 5
Preface......Page 6
Table of Contents    \r......Page 9
1.1 Overview......Page 11
1.2 Rigid vs. compliant walls......Page 13
1.3 Compliance and flow boundary conditions......Page 16
1.4 Newtonian vs. non-Newtonian rheology......Page 20
1.5 Rheology and turbulence......Page 23
1.6 Conclusions......Page 25
References......Page 26
2.1 Introduction......Page 29
2.2.1 Review of the basic equations......Page 31
2.3.1 Equations......Page 32
2.3.2 Eigenstructure and characteristic fields......Page 34
2.3.3 Generalized Riemann invariants......Page 35
2.4 The Riemann problem......Page 36
2.4.1 Jump across shocks: Rankine-Hugoniot conditions......Page 37
2.4.2 Jump conditions across rarefactions......Page 38
2.4.3 Jump conditions across the stationary contact......Page 39
2.4.4 Solution of Riemann problem......Page 41
2.5 Sample solutions......Page 42
2.6 Collapse of vessels......Page 44
2.7 Conclusions......Page 47
References......Page 48
3 Blood coagulation: a puzzle for biologists, a maze for mathematicians......Page 50
3.1 Introduction......Page 51
3.2 The modern view of blood coagulation......Page 53
The numbered Factor pairs......Page 54
Tissue Factor (TF)......Page 55
3.2.2 Primary hemostasis......Page 56
3.2.3 The cell-based model for the blood clot formation (secondary hemostasis)......Page 57
Initiation......Page 58
Propagation......Page 59
Termination......Page 60
3.2.4 Fibrinolysis......Page 61
3.3.1 Congenital bleeding disorders......Page 62
3.4 Mathematical models and numerical simulations......Page 67
3.4.1 Early models of blood coagulation......Page 69
3.4.2 Biochemical models......Page 70
The role of activated platelets......Page 71
Protein deficiencies and diseases......Page 73
3.4.3 Models incorporating mechanical and biochemical actions......Page 75
Models incorporating the mechanical activation of platelets......Page 76
A model with a platelets activation front and a clotting front......Page 77
3.5 Conclusions......Page 79
References......Page 80
4 Numerical simulation of electrocardiograms......Page 85
4.1 Introduction......Page 86
4.2 Mathematical modelling......Page 87
4.2.1 Isolated heart modelling......Page 88
4.2.2 Coupling with the torso: ECG modelling......Page 89
4.3 Mathematical analysis......Page 91
4.4 ECG simulations......Page 92
4.4.1 Numerical approximation......Page 93
4.4.2 Simulated ECG signals......Page 95
Cell heterogeneity......Page 96
Monodomain approximation......Page 97
4.5.1 Isolated bidomain model......Page 98
Stability analysis......Page 100
4.5.2 Coupled heart-torso system......Page 101
4.6.1 POD in a nutshell......Page 106
4.6.2 Examples of POD simulations with different sets of parameters......Page 107
4.6.3 Examples of POD simulations for several heart beats......Page 108
4.7 Conclusions......Page 109
References......Page 111
5.1 Introduction......Page 115
5.2.1 Ionic current membrane models......Page 117
5.2.2 The anisotropic macroscopic Bidomain model......Page 118
5.2.3 Modelling cardiac cells arrangements......Page 120
5.2.4 Approximate modelling of the cardiac bioelectric activity by reduced models......Page 124
Eikonal models......Page 125
Relaxed linear and nonlinear anisotropic Monodomain model......Page 126
5.3 Numerical approximation of the Monodomain and Bidomain models......Page 128
5.3.1.1 Functional reformulation of the semi-discrete Bidomain system......Page 130
5.3.1.2 The two-level Additive Schwarz preconditioner......Page 131
5.3.1.3 The multilevel Additive Schwarz preconditioner......Page 132
5.4 Numerical results: parallel scalability......Page 133
5.5 Numerical results: Bidomain simulations......Page 135
5.6 Conclusions......Page 141
References......Page 142
6.1 Introduction......Page 150
Structure/function relationship in healthy arteries......Page 152
Elastic fibres in the arterial wall......Page 153
6.3 Structurally motivated constitutive models – elastic regime......Page 154
6.3.1 Isotropic mechanism......Page 155
6.3.2 Kinematics of the anisotropic mechanism......Page 156
6.3.3 Anisotropic mechanism – discrete fibre models (N-fibre models)......Page 157
6.3.4 Anisotropic mechanism – fibre distribution models......Page 159
Integral fibre distribution models......Page 160
Generalized Structure Tensor (GST) models......Page 161
Generalized Structure Tensor for materials with conical splay......Page 162
Comparison of Generalized Structure Tensor (GST) model with integral fibre distribution model......Page 163
Uniaxial Multi-Photon Microscopy system (UA-MPM)......Page 164
Tissue assessment using the UA-MPM system......Page 165
Recruitment stretch......Page 167
Collagen fibre distribution......Page 168
6.4.1 Physical nature of damage......Page 169
Sources of damage......Page 170
6.4.1.2 Medical impact of damage to the IEL......Page 171
6.4.2 Structurally motivated constitutive models – damage regime......Page 172
Damage modes......Page 173
Continuous damage......Page 174
Enzymatic damage......Page 175
6.4.3 Damage assessment using the UA-MPM device......Page 176
6.5 Applications of the damage model to cerebral angioplasty......Page 178
Arterial wall model......Page 179
Loading states......Page 182
Results......Page 183
References......Page 186
7.1.1 Arterial structure......Page 193
7.1.2 Arterial functions and the roles of hemodynamics......Page 195
7.2.1 Thin-wall models......Page 197
7.2.2 Modelling transmural inhomogeneity......Page 198
7.2.3 Growth & remodelling: a multiple-time-scales approach......Page 202
7.3 Results......Page 205
7.4 Discussion......Page 207
References......Page 208
8 The VPH-Physiome Project: standards, tools and databases for multi-scale physiological modelling......Page 210
8.1 Introduction......Page 211
8.2 Infrastructure for the VPH-Physiome Project......Page 212
8.3 Syntax and semantics of VPH/Physiome models......Page 215
8.4 CellML......Page 216
Organising files......Page 219
Editing files......Page 220
8.6 FieldML......Page 221
Regions and fields......Page 222
Rendering......Page 224
Cmgui related tools......Page 225
8.8 GIMIAS......Page 226
Developer’s features......Page 227
8.9 SED-ML15......Page 230
8.10 BioSignalML......Page 231
BioSignal metadata......Page 232
BioSignalML repository......Page 233
Embedded workspaces......Page 234
Model exposures......Page 235
Background to CMISS......Page 236
Design goals......Page 237
Software systems......Page 238
Open source libraries......Page 240
Multi-physics modelling......Page 241
8.13 Physiome standards based multiscale modelling......Page 242
Cardiac Imaging technologies......Page 243
Image segmentation and modelling at different spatial scales......Page 244
Linking spatial scales......Page 245
Building a composite cell model......Page 247
Clinical applications of heart modelling......Page 248
8.15 Population atlases......Page 250
8.16 Discussion and future directions......Page 251
References......Page 253
9.1 Introduction......Page 256
9.2 Abstract Extended Variational Formulation......Page 258
9.2.1 A partition with equal kinematics......Page 260
9.2.2 A partition with different kinematics......Page 262
9.3 Application to the coupling of fluid flow models......Page 267
9.3.1 Coupling 3D and 2D models......Page 268
9.3.2 Coupling 3D and 1D models......Page 269
9.3.3 About fluid-structure interaction......Page 270
9.4.1 On the use of the non-classical extended variational formulation......Page 271
9.4.2 Open/closed-loop representations of the cardiovascular system......Page 272
9.4.3 1D models......Page 275
9.4.4 0D model for the arterioles and capillaries......Page 276
9.4.5 0D model of the venous and pulmonary circulation.......Page 279
9.4.6 Elastance model for the cardiac chambers......Page 280
9.4.7 Non-ideal diode model for the heart valves......Page 282
9.4.8 3D models......Page 283
9.5 Numerical applications......Page 284
9.5.1 Blood flow in an abdominal aneurism......Page 285
9.5.2 Blood flow in a cerebral aneurism......Page 286
References......Page 290
10 Multiscale modelling of hematologic disorders......Page 294
10.1 Introduction......Page 295
10.2.1 Dissipative particle dynamics: original method......Page 298
10.2.2 DPD method for colloidal particles......Page 299
10.2.3 Multiscale Red Blood Cell (MS-RBC) model......Page 301
10.2.3.2 Membrane viscoelasticity......Page 303
10.2.3.3 RBC-solvent boundary conditions......Page 304
10.2.3.4 RBC aggregation interactions......Page 305
10.2.3.5 RBC adhesion interactions......Page 306
10.2.4 Low-Dimensional RBC (LD-RBC) model......Page 307
10.2.4.2 Aggregation model......Page 308
10.2.5 Scaling of model and physical units......Page 309
10.3 Parameter estimation......Page 310
10.3.1 Shear modulus using optical tweezers......Page 311
10.3.2 Membrane rheology from Twisting Torque Cytometry......Page 312
10.3.3 RBC dynamics in shear flow......Page 313
10.4.1 Single RBC: comparison with microfluidic experiments......Page 315
10.4.2.1 Flow resistance......Page 320
10.4.2.2 Aggregation and Rouleaux formation......Page 321
10.5 Application to malaria modelling......Page 323
10.5.1.1 Mechanics......Page 324
10.5.1.2 Microfluidics......Page 325
10.5.1.3 Pf-RBC adhesive dynamics......Page 326
10.5.2 Whole infected blood......Page 329
10.6 Summary......Page 330
References......Page 332
11 Multiscale computational analysis of degradable polymers......Page 337
11.1 Introduction......Page 338
11.2.1 Macroscale governing equations for a polymer mixture......Page 342
11.2.2 Initial and boundary value problem......Page 344
11.2.3 Constitutive laws......Page 345
11.3 Molecular modelling of polymers......Page 347
11.3.1 Generation and equilibration of the atomistic models of PLA......Page 349
11.4.1 Input/Output description of the models......Page 351
11.4.2 Multiscale coupling strategies......Page 353
11.5.1 Computational results of the microscale model......Page 355
11.5.2 Results and validation for the macroscale model......Page 358
11.6 Conclusions and perspectives......Page 362
References......Page 363
12 Applications of variational data assimilation in computational hemodynamics......Page 366
12.1 Introduction......Page 367
12.2 Variational assimilation of velocity data for the incompressible Navier-Stokes equations......Page 369
12.2.1 Mathematical formulation and numerical approximation......Page 370
12.2.1.1 The discrete DA Oseen problem......Page 372
12.2.1.2 The nonlinear Navier-Stokes problem......Page 373
12.2.2.1 A consistency test......Page 374
12.2.2.2 Regularization & interpolation......Page 375
12.2.2.3 Assimilated derived quantities in nontrivial geometries......Page 376
12.2.3 Perspectives......Page 377
12.3 Image assimilation in a moving domain simulation......Page 378
12.3.1 Mathematical and numerical formulation......Page 379
12.3.1.1 Assimilation of segmented vascular surfaces......Page 380
12.3.2 Numerical results......Page 382
12.3.2.1 A practical workaround for reduced data sets......Page 384
12.4 Variational parameter estimation......Page 386
12.4.1 Mathematical and numerical formulation......Page 387
12.4.2 Numerical results......Page 390
12.4.3 Perspectives......Page 391
12.5 Conclusions......Page 392
References......Page 393
13.1 Introduction......Page 398
13.2 Methods......Page 400
13.2.1 Surface remeshing with finite element conformal maps......Page 401
13.2.2 Volume meshes with boundary layers......Page 404
13.3 High quality meshes......Page 405
13.4 Results......Page 408
13.4.1 Unsteady blood flow in a rigid aortic arch......Page 409
13.4.2 Unsteady blood flow in a compliant femoral bypass......Page 412
13.5 Conclusions......Page 413
References......Page 415
Back matter......Page 418