Biofluids Modeling –

Methods, Perspectives and Solutions

 

 

 

by

 

Wilson C. Chin and Jamie A. Chin

Stratamagnetic Software, LLC

Houston and Beijing

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents

 

Preface, xi

Acknowledgements, xiv

Dedication, xv

 

1. Fluid Physics in Circulatory Systems – Problems,

Analogies and Methods, 1

Presentation philosophy, 1

1.1 Basic Biological Notions and Fluid-Dynamical

Ideas, 3

Conduit flow examples, 3

Basic continuous flow concepts, 6

Eulerian versus Lagrangian descriptions, 9

Steady versus transient models, 10

Newtonian versus non-Newtonian flows, 10

Porous media continuum flow models, 12

Darcy flows in human and animal tissue, 14

Objectives in conduit and Darcy flow modeling, 15

1.2 Quantitative Modeling Perspectives, 16

1.2.1 Rheology considerations in conduit flows, 16

Better arterial flow models needed, 17

1.2.2 Darcy flow model in continuous media, 19

Temperature diffusion, 19

Darcy flow pressure diffusion, 20

Important porous media approach, 22

Relevance of Darcy flows to biofluids, 22

    1. Preview of Complicated but Simple Boundary Value

Problem Solutions, 24

Closing remarks, 27

1.4 References, 27

2. Math Models, Differential Equations and

Numerical Methods, 29

2.1 Presentation Approach, 31

What we won’t do, 31

Pursuing studies that uncover the physics, 32

Examples on presentation approach, 33

 

2.2 Diffusion Processes, Partial Differential Equations and

Formulation Development, 34

2.2.1 Heat transfer applications, 34

2.2.2 Heat equation derivation, 35

2.2.3 Pressure diffusion in porous media, 37

2.2.4 Dynamically coupled heat and pressure diffusion, 40

 

2.3 Boundary-Conforming Curvilinear Grid Generation, 41

2.3.1 Comments on classical coordinate transforms and conformal

mapping, 41

2.3.2 Curvilinear gridding method for irregular domains, 44

2.3.2.1 Grid generation for eccentric annular flow, 45

Mapping formalism and key ideas, 46

Thompson’s mapping, 48

Some reciprocity relations, 49

Relation to conformal mapping, finally, 51

Solutions to mesh generation equations, 53

Boundary conditions, 55

Fast iterative solutions, 56

On Laplacian transformations, 58

2.3.2.2 Grid generation for singly-connected conduit flow, 60

 

2.4 Finite Difference Solutions Made Easy – Iterative

Methods, Programming and Source Code Details, 63

2.4.1 Basic ideas in finite differences, 63

A simple differential equation, 64

Variable coefficients and grids, 67

2.4.2 Formulating steady flow problems, 68

Direct versus iterative solutions, 69

Iterative methods, 69

Convergence acceleration, 76

Wells and internal boundaries, 78

Peaceman well corrections, 79

Derivative discontinuities, 81

Point relaxation methods, 81

Observations on relaxation methods, 85

Minimal computing resources, 91

Good numerical stability, 91

Fast convergence, 92

Why relaxation methods converge, 93

Over-relaxation, 94

Line and point relaxation, 95

Curvilinear grid generation and relaxation solutions, 96

Coupled equations on curvilinear meshes, 97

 

2.5 References, 98

 

3. Hagen-Poiseuille Extensions – Real Flow Effects

and General Bifurcations, 100

    1. Blood Rheology and Overview, 101

      1. Hagen-Poiseuille – Misunderstandings and limitations, 101

      2. Ideal versus non-Newtonian rheology, 102

      3. Some conventional rheological models, 105

3.1.4 Perfect concentric flow velocity, pressure and flow

rate relations, 106

Newtonian flow solution, 106

Bingham Plastic pipe flow, 107

Power Law fluid pipe flow, 108

Herschel-Bulkley pipe flow, 108

Ellis fluid pipe flow, 108

3.1.5 Example solutions for imperfect arteries with

stenosis and book presentation outline, 109

Book presentation outline, 111

3.2 Newtonian Flow in Simple Bifurcations, 112

3.2.1 Theory – Two uneven bifurcated blood vessels with Q1

specified, 112

Case 1. Flow rate Q1 prescribed, 114

Case 2. Inlet pressure Pi prescribed, 115

Case 3. Identical outlet pressures Po,2 and Po,3 given, 115

3.2.2 Software – Two uneven bifurcated arteries with Q1

specified (Reference, CODE-1), 116

An example computation, 116

An additional validation, 117

3.2.3 Theory – Two uneven bifurcated arteries with Pi

specified, 118

3.2.4 Software – Two uneven bifurcated arteries with Pi

specified (Reference, CODE-2), 118

A practical example, 119

3.3 Theory – Complicated Arteries with Chained

Bifurcations, 120

3.4 Network with Arbitrary Number of Bifurcations, 122

3.5 Bifurcated Newtonian Flow in Noncircular Clogged

Blood Vessels, 123

3.6 References, 125

 

4. Non-Newtonian Flow in Circular Conduits and

Networks, 127

Bifurcation model and analytical approach, 127

Different rheological applications, 128

Validation procedures, 129

4.1 Power Law Fluids with Inlet Flow Rate Prescribed, 130

Iterative "half-step" solution for Pa and Pi, 131

Shear stress, 132

Typical parameters, 132

Example calculations, 134

4.2 Herschel-Bulkley Fluids and Yield Stress, 141

4.2.1 Analytical and numerical approach, 141

Yield stress modeling, 141

4.2.2 BIFURC-6 runs assuming ty = 0 psi (Power Law limit), 143

4.2.3 BIFURC-6 runs assuming ty = 0.00001 psi, 146

4.3 Newtonian and Herschel-Bulkley Examples, 149

Power Law limitations, 153

4.4 References, 154

5. Flows in Clogged Arteries and Veins, 155

5.1 Hagen-Poiseuille Revisited – Rectangular

Coordinates, 157

Newtonian pipe flow recapitulation, 157

A physical description, 157

Detailed assessments, 160

Recapitulation, 163

5.2 Non-Newtonian Power Law Circular Pipe Flow in

Rectangular Coordinates, 164

5.3 Clinical Implications for Pressure Gradient and Viscous

Shear Stress, 167

5.4 Evolutionary Approaches for Complicated

Geometries, 168

Static versus evolutionary approaches, 169

5.5 A Detailed Clog Flow Computation, 175

Simulation 1 – Newtonian flow in perfect circle, 175

Simulation 2 – Power Law flow in perfect circle, 177

Simulation 3 – Power Law flow in a clogged

blood vessel, 179

5.6 References, 182

 

6. Square Stents, Centrifugal Effects, Pulsatile Flow,

Clogged Bifurcations and Axial Variations, 183

6.1 Stent Geometry Effects on Volume Flow Rate, 183

6.1.1 Conventional stents, analytical flow model, 184

Stent detailed function, 184

Analytical modeling, 185

Exact analytical Hagen-Poiseuille solution, 186

6.1.2 Finite difference method, 188

6.1.3 Square stent designs, analytical and numerical models, 192

Exact analytical solution for rectangular stents, 194

Finite difference solution, 196

Example calculation, 199

6.2 General Formulations and Solutions for Complicated

Geometries and Arbitrary Fluids, 200

Recapitulation, 203

6.3 Centrifugal Force Influence on Volume Flow Rate, 204

Straight, closed ducts, 206

Hagen-Poiseuille flow between planes, 207

Flow between concentric plates, 207

Typical calculations, 209

Flows in closed curved ducts, 211

6.4 Unsteady Pulsatile Flow Model for Complicated

Duct Cross-Sections, 214

6.5 Bifurcated Conduits with Newtonian Flow in Clogged

Geometric Cross-sections, 220

6.6 Modeling Axial Variations with Pseudo-Three-

Dimensional Method, 221

6.7 Modeling Transient Wall Effects, 223

6.8 References, 225

 

7. Tissue Properties from Steady and Transient Syringe

Pressure Analysis, 226

7.1 Importance of Compressibility, Permeability, Anisotropy,

Pressure and Porosity in Medical Applications, 228

Compressibility, 228

Permeability, 229

Anisotropy, 231

Local pressures, 233

Porosity, 235

Additional highlights, 236

7.2 Geoscience Perspectives and Background, 238

7.3 Formation Testing in Petroleum Well Logging, 241

7.4 Operational Guidelines to Biofluids Pressure Testing, 247

Intelligent syringe concepts, 247

Multiprobe syringe assemblies for anisotropy and heterogeneity

mapping, 250

7. 5 Intelligent Syringe Fundamentals, 255

7.5.1 Background and Motivation, 256

7.5.2 Clinical and Diagnostic Objectives, 258

7.5.3 Syringe Flow Basics and Porous Media Pressure Conventions, 259

7.5.4 Single Intelligent Syringe Basic Layout, 262

Figure 3A description, 262

Figure 3B description, 268

Figure 3C description, 268

Figure 3D description, 269

Figure 3E description, 269

General comments, 270

7.5.5 Syringe Arrays for Heterogeneity Mapping and Biopsy Sampler, 272

Array syringe and biopsy sampler, 273

Array syringe general concept, 275

7.6 Mathematical Models for Porous Media Flow, 278

7.6.1 Transient Isotropic Darcy Flow – Forward Solutions, 279

7.6.2 Transient Transversely Isotropic Darcy Flow – Forward Solutions, 281

7.6.3 Transient Isotropic and Transversely Isotropic Darcy Flow –

Inverse Solutions, 283

7.6.4 Steady Transversely Isotropic Flow – Inverse Solutions, 286

7.6.5 Modeling Notes and Physical Consequences, 288

Geometric factor, 288

Flowline compressibility, 289

Flowline pressure drops, 290

Pressure effects on tissue, 291

7.6.6 Anisotropic Permeabilities from Oscillatory Pressure

Fields, 292

7.6.7 Formulation for Supercharged Damage Zones, 295

7.6.8 General Properties, Calculated Results and Validations, 296

Example 1. Forward and Inverse Simulations in Isotropic Media Using Drawdown Method, 296

Example 2. Forward and Inverse Simulations in Transversely Isotropic Media Using Pure Drawdown (or Pure Buildup) Methods, 301

Example 3. Forward and Inverse Simulations in Transversely Isotropic Media Using Drawdown-Buildup Method, 306

Example 4. Forward and Inverse Simulations in Transversely Isotropic Media Using Drawdown and Phase Delay Method, 308

Example 5. Forward and Inverse Simulations in Transversely Isotropic Media for Flows with Nonzero Dip Angle, 312

7.6.9 Application to Subcutaneous Injection Yorkshire Swine Laboratory Data, 321

Experimental details, 322

Laboratory setup and raw data analysis, 325

Example 1. Needle Gauge Effect on Mobility Predictions, 328

Example 2. Transversely Isotropic (kh = 20 md, kv = 30 md) Prediction of kh and kv from Steady Pressure Drops, 334

Example 3. Transversely Isotropic (kh = 30 md, kv = 20 md) Prediction of kh and kv from Steady Pressure Drops, 338

Example 4. Anisotropic Transient Method for Effective Permeability, 342

Example 5. Effects of Compressibility, 343

7.6.10 Application to Subcutaneous Injection Adult Human Laboratory Data, 346

Example 1. Pressure Analysis for Figure 7-1D, 349

Example 2. Pressure Analysis for Figures 7-1A,B,C, 352

7.6.11 Laboratory Notes for Flowline Geometry and Frictional Effects, 355

7.6.12 Closing Remarks, 364

7.7 References, 366

8. Artery, Capillary and Vein Interactions in Anisotropic

Heterogeneous Porous Tissue Flows, 372

Intuitive physical ideas, 372

Concrete simulation examples, 373

8.1 Qualitative Review of the Circulatory System, 375

8.2 Porous Media Flows in the Geosciences and in Biofluids

Applications, 381

Biofluids applications, 384

8.3 Electrical and Biological Analogies, 385

Series and parallel electrical circuits, 386

Cardiovascular system model, 386

Validating examples, 389

Simulation 1. Baseline run with three active capillary bed

groups (see Figure 8-18a), 390

Simulation 2. Baseline run with two active capillary groups

(middle group, with smaller permeability, is altered), 394

Simulation 3. Calculating flow rate versus pressure

drop, 395

Tissue masses connected in series, 399

8.4 References, 399

 

9. Geoscience Ideas in Biofluids Modeling, 400

Solution strategies and perspectives, 400

Blood vessel assumptions revisited, 404

9.1 Multisim Background and Biofluids Applications, 406

Interesting possibilities, 409

Multisim limitations in our applications, 410

What Multisim does and how it works, 411

9.2 Running Multisim, 413

9.2.1 Simulation 1. Set-up and "Flatman" visual display, 413

Rendering "Flatman" in Multisim, 417

9.2.2 Simulation 2 – Simple aneurysm model, 429

9.2.3 Simulation 3 – Mimicking pressure drops in blood vessels, 431

9.2.4 Simulation 4 – Pressure versus flow rate specifications, 435

Run 1. Pressure-pressure specification, 435

Run 2. Flow rate – flow rate specification, 437

Run 3. Pressure – flow rate specification, 438

9.3 Closing Remarks, 439

9.4 References, 441

Cumulative References, 442

Index, 453

About the Authors, 469