Contents

Preface xiii Acknowledgments xv

Chapter 1. Review of Basic Fluid Mechanics Concepts 1

1.1 A Brief History of Biomedical Fluid Mechanics 1

1.2 Fluid Characteristics and Viscosity 6

1.2.1 Displacement and velocity 7

1.2.2 Shear stress and viscosity 8

1.2.3 Example problem: shear stress 10

1.2.4 Viscosity 11

1.2.5 Clinical feature: polycythemia 13

1.3 Fundamental Method for Measuring Viscosity 14

1.3.1 Example problem: viscosity measurement 16

1.4 Introduction to Pipe Flow 16

1.4.1 Reynolds number 17

1.4.2 Example problem: Reynolds number 19

1.4.3 Poiseuille's law 19

1.4.4 Flow rate 23

1.5 Bernoulli Equation 24

1.6 Conservation of Mass 24

1.6.1 Venturi meter example 26

1.7 Fluid Statics 27

1.7.1 Example problem: fluid statics 28

1.8 The Womersley Number a: A Frequency Parameter for Pulsatile Flow 29

1.8.1 Example problem: Womersley number 30

Problems 31

Bibliography 33

Chapter 2. Cardiovascular Structure and Function 35

2.1 Introduction 35

2.2 Clinical Features 36

2.3 Functional Anatomy 37

2.4 The Heart as a Pump 38

2.5 Cardiac Muscle 39

2.5.1 Biopotential in myocardium 40

2.5.2 Excitability 41

2.5.3 Automaticity 43

2.6 Electrocardiograms 43

2.6.1 Electrocardiogram leads 44

2.6.2 Mean electrical axis 45

2.6.3 Example problem: mean electrical axis 47

2.6.4 Unipolar versus bipolar and augmented leads 48

2.6.5 Electrocardiogram interpretations 49

2.6.6 Clinical feature: near maximal exercise stress test 50

2.7 Heart Valves 51

2.7.1 Clinical features 52

2.8 Cardiac Cycle 52

2.8.1 Pressure-volume diagrams 55

2.8.2 Changes in contractility 57

2.8.3 Ventricular performance 58

2.8.4 Clinical feature: congestive heart failure 58

2.8.5 Pulsatility index 59

2.8.6 Example problem: pulsatility index 59

2.9 Heart Sounds 60

2.9.1 Clinical features 61

2.9.2 Factors influencing flow and pressure 61

2.10 Coronary Circulation 63

2.10.1 Control of the coronary circulation 64

2.10.2 Clinical features 65

2.11 Microcirculation 65

2.11.1 Capillary structure 65

2.11.2 Capillary wall structure 66

2.11.3 Pressure control in the microvasculature 67

2.11.4 Diffusion in capillaries 68

2.11.5 Venules 68

2.12 Lymphatic Circulation 69 Problems 69 Bibliography 75

Chapter 3. Pulmonary Anatomy, Pulmonary Physiology, and Respiration 77

3.1 Introduction 77

3.1.1 Clinical features: hyperventilation 78

3.2 Alveolar Ventilation 79

3.2.1 Tidal volume 79

3.2.2 Residual volume 79

3.2.3 Expiratory reserve volume 80

3.2.4 Inspiratory reserve volume 80

3.2.5 Functional residual capacity 80

3.2.6 Inspiratory capacity 80

3.2.7 Total lung capacity 80

3.2.8 Vital capacity 81

3.3 Ventilation-Perfusion Relationships 81

3.4 Mechanics of Breathing 81

3.4.1 Muscles of inspiration 82

3.4.2 Muscles of expiration 83

3.4.3 Compliance of the lung and chest wall 83

3.4.4 Elasticity, elastance, and elastic recoil 83

3.4.5 Example problem: compliance 84

3.5 Work of Breathing 85

3.5.1 Clinical features: respiratory failure 87

3.6 Airway Resistance 88

3.6.1 Example problem: Reynolds number 91

3.7 Gas Exchange and Transport 91

3.7.1 Diffusion 92

3.7.2 Diffusing capacity 92

3.7.3 Oxygen dissociation curve 94

3.7.4 Example problem: oxygen content 95

3.7.5 Clinical feature 96

3.8 Pulmonary Pathophysiology 96

3.8.1 Bronchitis 96

3.8.2 Emphysema 96

3.8.3 Asthma 97

3.8.4 Pulmonary fibrosis 98

3.8.5 Chronics obstructive pulmonary disease (COPD) 98

3.8.6 Heart disease 98

3.8.7 Comparison of pulmonary pathologies 98

3.9 Respiration in Extreme Environments 99

3.9.1 Barometric pressure 100

3.9.2 Partial pressure of oxygen 101

3.9.3 Hyperventilation and the alveolar gas equation 102

3.9.4 Alkalosis 103

3.9.5 Acute mountain sickness 103

3.9.6 High-altitude pulmonary edema 104

3.9.7 High-altitude cerebral edema 104

3.9.8 Acclimatization 104

3.9.9 Drugs stimulating red blood cell production 105 3.9.10 Example problem: alveolar gas equation 106

Review Problems 106

Bibliography 109

Chapter 4. Hematology and Blood Rheology 111

4.1 Introduction 111

4.2 Elements of Blood 111

4.3 Blood Characteristics 111

4.3.1 Types of fluids 112

4.3.2 Viscosity of blood 113

4.3.3 Fahr^us-Lindqvist effect 114

4.3.4 Einstein's equation 116

4.4 Viscosity Measurement 116

4.4.1 Rotating cylinder viscometer 116

4.4.2 Measuring viscosity using Poiseuille's law 118

4.4.3 Viscosity measurement by a cone and plate viscometer 119

4.5 Erythrocytes 121

4.5.1 Hemoglobin 123

4.5.2 Clinical features—sickle cell anemia 125

4.5.3 Erythrocyte indices 125

4.5.4 Abnormalities of the blood 126

4.5.5 Clinical feature—thalassemia 127

4.6 Leukocytes 127

4.6.1 Neutrophils 128

4.6.2 Lymphocytes 129

4.6.3 Monocytes 131

4.6.4 Eosinophils 131

4.6.5 Basophils 131

4.6.6 Leukemia 131

4.6.7 Thrombocytes 132

4.7 Blood Types 132

4.7.1 Rh blood groups 134

4.7.2 M and N blood group system 135

4.8 Plasma 135

4.8.1 Plasma viscosity 136

4.8.2 Electrolyte composition of plasma 136

4.8.3 Blood pH 137

4.8.4 Clinical features—acid-base imbalance 137 Review Problems 138 Bibliography 139

Chapter 5. Anatomy and Physiology of Blood Vessels 141

5.1 Introduction 141

5.2 General Structure of Arteries 141

5.2.1 Tunica intima 142

5.2.2 Tunica media 142

5.2.3 Tunica externa 143

5.3 Types of Arteries 144

5.3.1 Elastic arteries 144

5.3.2 Muscular arteries 144

5.3.3 Arterioles 144

5.4 Mechanics of Arterial Walls 144

5.5 Compliance 147

5.5.1 Compliance example 151

5.5.2 Clinical feature—arterial compliance and hypertension 152

5.6 Pulse Wave Velocity and the Moens-Korteweg Equation 153

5.6.1 Applications box—fabrication of arterial models 153

5.6.2 Pressure-strain modulus 153

5.6.3 Example problem—modulus of elasticity 154

5.7 Vascular Pathologies 155

5.7.1 Atherosclerosis 155

5.7.2 Stenosis 155

5.7.3 Aneurysm 156

5.7.4 Clinical feature—endovascular aneurysm repair 156

5.7.5 Thrombosis 157

5.8 Stents 157

5.8.1 Clinical feature—"Stent Wars" 158

5.9 Coronary Artery Bypass Grafting 159

5.9.1 Arterial grafts 160

Review Problems 161

Bibliography 162

Chapter 6. Mechanics of Heart Valves 165

6.1 Introduction 165

6.2 Aortic and Pulmonic Valves 165

6.2.1 Clinical feature—percutaneous aortic valve implantation 169

6.3 Mitral and Tricuspid Valves 171

6.4 Pressure Gradients across a Stenotic Heart Valve 172

6.4.1 The Gorlin equation 173

6.4.2 Example problem—Gorlin equation 175

6.4.3 Energy loss across a stenotic valve 175

6.4.4 Example problem—energy loss method 178

6.4.5 Clinical features 178

6.5 Prosthetic Mechanical Valves 178

6.5.1 Clinical feature—performance of the On-X valve 180

6.5.2 Case study—the Bjork-Shiley convexo-concave heart valve 180

6.6 Prosthetic Tissue Valves 184 Review Problems 184 Bibliography 185

Chapter 7. Pulsatile Flow in Large Arteries 187

7.1 Introduction 187

7.2 Fluid Kinematics 188

7.3 Continuity 189

7.4 Complex Numbers 190

7.5 Fourier Series Representation 192

7.6 Navier-Stokes Equations 198

7.7 Pulsatile Flow in Rigid Tubes—Womersley Solution 202

7.8 Pulsatile Flow in Rigid Tubes—Fry Solution 214

7.9 Instability in Pulsatile Flow 221 Review Problems 222 Bibliography 227

Chapter 8. Flow and Pressure Measurement 229

8.1 Introduction 229

8.2 Indirect Pressure Measurements 229

8.2.1 Indirect pressure gradient measurements using

Doppler ultrasound 230

8.3 Direct Pressure Measurement 231

8.3.1 Intravascular—strain gauge tipped pressure transducer 231

8.3.2 Extravascular—catheter-transducer measuring system 237

8.3.3 Electrical analog of the catheter measuring system 238

8.3.4 Characteristics for an extravascular pressure measuring system 240

8.3.5 Example problem—characteristics of an extravascular measuring system 241

8.3.6 Case 1: the undamped catheter measurement system 243

8.3.7 Case 2: the undriven, damped catheter measurement system 244

8.3.8 Pop test—measurement of transient step response 248

8.4 Flow Measurement 249

8.4.1 Indicator dilution method 249

8.4.2 Fick technique for measuring cardiac output 250

8.4.3 Fick technique example 250

8.4.4 Rapid injection indicator-dilution method—

dye dilution technique 250

8.4.5 Thermodilution 251

8.4.6 Electromagnetic flowmeters 252

8.4.7 Continuous wave ultrasonic flowmeters 253

8.4.8 Example problem—continuous wave Doppler ultrasound 254

8.5 Summary and Clinical Applications 255 Review Problems 256 Bibliography 258

Chapter 9. Modeling 259

9.1 Introduction 259

9.2 Theory of Models 260

9.2.1 Dimensional analysis and the Buckingham Pi theorem 260

9.2.2 Synthesizing Pi terms 262

9.3 Geometric Similarity 264

9.4 Dynamic Similarity 265

9.5 Kinematic Similarity 265

9.6 Common Dimensionless Parameters in Fluid Mechanics 266

9.7 Modeling Example 1—Does the Flea Model the Man? 266

9.8 Modeling Example 2 268

9.9 Modeling Example 3 269 Review Problems 271 Bibliography 273

Chapter 10. Lumped Parameter Mathematical Models 275

10.1 Introduction 275

10.2 Electrical Analog Model of Flow in a Tube 276

10.2.1 Nodes and the equations at each node 277

10.2.2 Terminal load 278

10.2.3 Summary of the lumped parameter electrical analog model 288

10.3 Modeling of Flow through the Mitral Valve 288

10.3.1 Model description 289

10.3.2 Active ventricular relaxation 292

10.3.3 Meaning of convective resistance 292

10.3.4 Variable area mitral valve model description 292

10.3.5 Variable area mitral valve model parameters 293

10.3.6 Solving the system of differential equations 294

10.3.7 Model trials 294

10.3.8 Results 294

10.4 Summary 296 Review Problems 297 Bibliography 297

Index 299

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