Biomechanics

Circulation

Second Edition

Springer Science+ Business Media, LLC

Y.C.Fung

Biomechanics Circulation

Second Edition

With 289 Illustrations

Springer

Y.c. Fung Department of Bioengineering University of California, San Diego La JoHa, CA 92093-0419 USA

Cover illustration by Frank DeLano for the First World Congress of Biomechanics.

Library of Congress Cataloging in Publication Data Fung, Y.C. (Yuan-cheng), 1919-

Biomechanics : circulation / Y.c. Fung. - 2nd ed. p. cm.

Inc1udes bibliographica1 references and index.

ISBN 978-1-4419-2842-9 ISBN 978-1-4757-2696-1 (eBook) DOI 10.1007/978-1-4757-2696-1 1. Hemodynamics. 1. Title.

QP105.F85 1996 612.1-DC20

Printed on acid-free paper.

96-11887

The first edition of this book was published as Biodynamics: Circu[ation.

© 1984, 1997 Springer Science+8usiness Media New York Originally published by Springer-Verlag New York, Inc. in 1997 Softcover reprint of the hardcover 2nd edition 1997

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SPIN 10480464

Preface to the Second Edition

The theory of blood circulation is the oldest and most advanced branch of biomechanics, with roots extending back to Huangti and Aristotle, and with contributions from Galileo, Santori, Descartes, Borelli, Harvey, Euler, Hales, Poiseuille, Helmholtz, and many others. It represents a major part of humanity's concept of itself. This book presents selected topics of this great body of ideas from a historical perspective, binding important experiments together with mathematical threads.

The objectives and scope of this book remain the same as in the first edition: to present a treatment of circulatory biomechanics from the stand­points of engineering, physiology, and medical science, and to develop the subject through a sequence of problems and examples. The name is changed from Biodynamics: Circulation to Biomechanics: Circulation to unify the book with its sister volumes, Biomechanics: Mechanical Properties of Living Tissues, and Biomechanics: Motion, Flow, Stress, and Growth. The major changes made in the new edition are the following: When the first edition went to press in 1984, the question of residual stress in the heart was raised for the first time, and the lung was the only organ analyzed on the basis of solid morphologic data and constitutive equations. The detailed analysis of blood flow in the lung had been done, but the physiological validation experiments had not yet been completed. Now, the residual stress is well understood, the zero stress states of the heart and blood vessels are well documented, and the morphometry of the blood vessels of the heart and skeletal muscles has been advanced sufficiently to allow an analysis of the blood flow in these organs on the basis of realistic geometric descriptions. Thus, two new chapters were added to discuss coronary blood flow and skeletal muscle microcirculation.

Chapters 6, 7, and 8 together illustrate a biomechanical approach to cir­culatory physiology, with emphasis on formulating questions in the form of boundary-value problems, and predicting outcome by solving the problems. In these chapters, the mechanics of vascular smooth muscles stands out as particularly important. In 1984, the mechanical properties of the vascular smooth muscle were mysterious. The autoregulation phenomenon was

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vi Preface to the Second Edition

known, as was the phenomenon of hyperemia. The importance of local control of blood flow was appreciated, but there was an unresolved debate about whether the mechanism was myogenic, neurogenic, or metabolic. Now, however, we have extracted a length-tension relationship of the vas­cular smooth muscle from the results of a series of outstanding experiments on coronary arterioles by Kuo et al., who measured the diameters of the vessels in response to pressure and flow. We now know how the length­tension curve is shifted to the right by the shear stress acting on the endo­thelial cells. The length-tension curves of the vascular smooth muscle are found to be arch-like and quite similar to those of the heart and skeletal muscles. However, whereas the working range of the lengths of the sar­comeres of the heart and skeletal muscles lie on the left leg of the length­tension arch, that of the vascular smooth muscle lies on the right leg. This difference is revealed in the contrasting behavior of the muscles: while the heart muscle obeys Starling's law, the vascular smooth muscle exhibits the Bayliss phenomenon. Thus the mechanical properties of the vascular smooth muscle are no longer so strange. I believe that these differences are due to the different patterns of actin-myosin relationship in these muscles. Chapter 2, on the heart, was revised extensively, with the addition of new results and methods on the analysis of strain distribution in the ventricles, especially in association with in vivo experiments. Chapter 3 now includes a long section on fluid mechanics and solid mechanics of atherogenesis. It explains why atherogenesis is a problem of biomech­anics, and discusses contemporary thinking on the subject. The significance of the shear stress acting on the endothelial cells due to blood flow, and the tensile stress in the blood vessel wall due to blood pressure are discussed from the point of view of gene expression and tissue remodeling of the vessel.

Chapter 4, on veins, presents advances on the stability of flow in col­lapsible tubes, with an emphasis on the nonlinear effects of longitudinal tension. Forced oscillations in collapsible tubes due to flow separation or turbulence are discussed. In Chapter 5, on microcirculation, a classical solution of a sheet-flow model of capillaries has been added. Chapter 6, on pulmonary circulation, presents the details of an analysis of the waterfall phenomenon in the lung, along with experimental validations. These are given because the causes of waterfall in different organs are different. For example, the waterfall in the lung occurs at the junctions of the capillaries and venules. When the pressure at the end of the capillaries is equal to the alveolar gas pressure, the pressure in the venule is below alveolar gas pressure, and the sluicing gates are kept open by the tension in the inter­alveolar septa. On the other hand, in the heart, the waterfalls are believed to be located in the coronary capillaries that are squeezed by the heart muscle. The waterfalls in the skeletal muscle are similar to those in the heart, but the muscle squeeze is milder. The interaction between the muscle cells and the capillary blood vessels is a subject of major interest.

Preface to the Second Edition vii

Many other advances have been made in the field of circulation in the last ten years. To survey all of the field is beyond my ability. I discussed only those topics familiar to me. Even in these, I may have missed some impor­tant references. To those authors, I apologize. To people who are looking for a handbook, a compendium of solved problems, a review of the current literature, or a record of computational methods, I also apologize, because this book is not designed to serve those functions. But if a reader finds the book interesting, lucid, and useful, then I shall be very grateful. I wish to thank friends and readers who have offered suggestions for improving this book. I want to thank Dr. Geert Schmid-Schonbein for providing most of the materials in the last chapter; Dr. Ghasson Kassab for providing the mor­phological data of the coronary vasculature; Dr. Michael Yen for new results on pulmonary circulation; and Drs. Andrew McCulloch and Lew Waldman for advances in the analysis and experimentation on the mechanics of the heart and for materials used in the last three sections of Chapter 2. I want to express my pleasure to Dr. Shu Qian Liu for our close working rela­tionship during the past nine years, and to Eugene Mead for our coopera­tive work during the past 31 years. I also enjoyed and benefited from working with Drs. Paul Zupkas, Yasuyuki Seguchi, Yuji Matsuzaki, Mit­sumasa Matsuda, Takaaki Nakagawa, Jun Tomioka, Maw Chang Lee, Jeffrey Omens, Jack Debes, Jainbo Zhou, Shanxi Deng, Qilian Yu, Zong Jie Li, Zong Lai Jiang, Hai Chao Han, Gong Rui Wong, Win Peng Wu, Rui Fang Yang, Yun Qin Gao, Kegan Dai, Hao Xue, Rong Zhu Gan, Wei Huang, and Hans Gregersen, whose work is mentioned in this book. Finally, to my friends and colleagues Drs. Geert Schmid-SchOnbein, Sidney Sobin, Shu Chien, Richard Skalak, David Gough, John Frangos, Andrew McCulloch, Paul and Amy Sung, Benjamin Zweifach, and Marcos Intaglietta, I want to offer sincere thanks for creating a most remarkable, stimulating, and pleasant environment, in which I was surrounded by bright students. I am extremely lucky to have good friends and a warm family. Luna has always supported me and my work. Conrad and Brenda make me proud of their character, achievements, and strength. Chia Shun Yih delights me with fre­quent letters announcing his exact solutions of difficult problems in fluid mechanics.

I dedicate this book to Luna.

YUAN-CHENG FUNG

Preface to the First Edition

This book is a continuation of my Biomechanics. The first volume deals with the mechanical properties of living tissues. The present volume deals with the mechanics of circulation. A third volume will deal with respiration, fluid balance, locomotion, growth, and strength. This volume is called Biody­namics in order to distinguish it from the first volume. The same style is fol­lowed. My objective is to present the mechanical aspects of physiology in precise terms of mechanics so that the subject can become as lucid as physics.

The motivation of writing this series of books is, as I have said in the preface to the first volume, to bring biomechanics to students of bioengi­neering, physiology, medicine, and mechanics. I have long felt a need for a set of books that will inform the students of the physiological and medical applications of biomechanics, and at the same time develop their training in mechanics. In writing these books I have assumed that the reader already has some basic training in mechanics, to a level about equivalent to the first seven chapters of my First Course in Continuum Mechanics (Prentice Hall, 1977). The subject is then presented from the point of view of life science while mechanics is developed through a sequence of problems and exam­ples. The main text reads like physiology, while the exercises are planned like a mechanics textbook. The instructor may fill a dual role: teaching an essential branch of life science, and gradually developing the student's knowledge in mechanics.

The style of one's scientific approach is decided by the way one looks at a problem. In this book I try to emphasize the mathematical threads in the study of each physical problem. Experimental exploration, data collection, model experiments, in vivo observations, and theoretical ideas can be wrapped together by mathematical threads. The way problems are formu­lated, the kind of questions that are asked, are molded by this basic thought. Much of the book can be read, however, with little mathematics. Those pas­sages in which mathematics is essential are presented with sufficient details to make the reading easy.

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x Preface to the First Edition

This book begins with a discussion of the physics of blood flow. This is followed by the mechanics of the heart, arteries, veins, microcirculation, and pulmonary blood flow. The coupling of fluids and solids in these organs is the central feature. How morphology and rheology are brought to bear on the analysis of blood flow in organs is illustrated in every occasion. The basic equations of fluid and solid mechanics are presented in the Appendix. The subject of mass transfer, the exchange of water, oxygen, carbon dioxide, and other substances between plasma and red cells and between capillary blood vessels and extravascular space, is deferred to the third volume, Biody­namics: Flow, Motion, and Stress, in order to keep the three volumes at approximately the same size.

Circulation is a many-sided subject. What we offer here is an under­standing of the mechanics of circulation. We present methods and basic equations very carefully. The strengths and weaknesses of various methods and unanswered questions are discussed fully. To apply these methods to a specific organ, we need a data base. We must have a complete set of mor­phometric data on the anatomy, and rheological data on the materials of the organ. Unfortunately, such a data base does not exist for any organ of any animal. A reasonably complete set has been obtained for the lungs of the cat. Hence the analysis of the blood flow in the lung is presented in detail in Chapter 6. We hope that a systematic collection of the anatomical and rheological data on all organs of man and animals will be done in the near future so that organ physiology can be elevated to a higher level.

Blood circulation has a vast literature. The material presented here is nec­essarily limited in scope. Furthermore, there are still more things unknown than known. Progress is very rapid. Aiming at greater permanency, I have limited my scope to a few fundamental aspects of biomechanics. For hand­book information and literature survey, the reader must look elsewhere. Many exercises are proposed to encourage the students to formulate and solve new problems. The book is not offered as a collection of solved prob­lems, but as a way of thinking about problems. I wish to illustrate the use of mechanics as a simple, reliable tool in life science, and no more. A rea­sonably extensive bibliography is given at the end of each chapter, some with annotations from which further references can be found. Perhaps the author can be accused of quoting frequently papers and people familiar to him; he apologizes for this personal limitation and hopes that he can be for­given because it is only natural that an author should talk about his own views. I have tried, however, never to forget mentioning the existence of other points of view.

I wish to express my thanks to many authors and publishers who per­mitted me to quote their publications and reproduce their figures and data in this book. I wish to mention especially Drs. Michael Yen, Sidney Sobin, Jen-Shih Lee, Benjamin Zweifach, Paul Patitucci, Geert Schmid-Schoen­bein, William Conrad, Lawrence Talbot, H. Werle, John Maloney, Paul Stein, and John Hardy who supplied original photographs for reproduction. I wish

Preface to the First Edition xi

also to thank many of my colleagues, friends, and former students who read parts of the manuscripts and offered valuable suggestions. To Virginia Stephens I am grateful for typing the manuscript. Finally, I wish to thank the editorial and production staff of Springer-Verlag for their care and cooperation in producing this book.

In spite of great care and effort on my part, I am sure that many mistakes and defects remain in the book. I hope you will bring these to my attention so that I can improve in the future.

YUAN-CHENG FUNG

La Jolla, California

Contents

Preface to the Second Edition Preface to the First Edition

Chapter 1 Physical Principles of Circulation 1.1 Conservation Laws 1.2 Forces That Drive or Resist Blood Flow 1.3 Newton's Law of Motion Applied to a Fluid 1.4 Importance of Turbulence 1.5 Deceleration as a Generator of Pressure Gradient 1.6 Pressure and Flow in Blood Vessels-Generalized

Bernoulli's Equation 1.7 Analysis of Total Peripheral Flow Resistance 1.8 Importance of Blood Rheology 1.9 Mechanics of Circulation 1.10 A Little Bit of History 1.11 Energy Balance Equation

References

Chapter 2 The Heart 2.1 Introduction 2.2 Geometry and Materials of the Heart 2.3 Electric System 2.4 Mechanical Events in a Cardiac Cycle 2.5 How Are the Heart Valves Operated? 2.6 Equations of Heart Mechanics 2. 7 Active Contraction of Heart Muscle 2.8 Fluid Mechanics of the Heart 2.9 Solid Mechanics of the Heart

v IX

1 1 1 3 6 8

9 10 13 14 14 16 22

23 23 27 30 34 42 49 65 69 72

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xiv Contents

2.10 Experimental Strain Analysis 81 2.11 Constitutive Equations of the Materials of the Heart 86 2.12 Stress Analysis 88

References 101

Chapter 3 Blood Flow in Arteries 108 3.1 Introduction 108 3.2 Laminar Flow in a Channel or Tube 114 3.3 Applications of Poiseuille's Formula: Optimum Design of

Blood Vessel Bifurcation 118 3.4 Steady Laminar Flow in an Elastic Tube 125 3.5 Dynamic Similarity. Reynolds and Womersley Numbers.

Boundary Layers 130 3.6 Turbulent Flow in a Tube 134 3.7 Turbulence in Pulsatile Blood Flow 136 3.8 Wave Propagation in Blood Vessels 140 3.9 Progressive Waves Superposed on a Steady Flow 151 3.10 Nonlinear Wave Propagation 154 3.11 Reflection and Transmission of Waves at Junctions

of Large Arteries 155 3.12 Effect of Frequency on the Pressure-Flow Relationship at any

Point in an Arterial Tree 164 3.13 Pressure and Velocity Waves in Large Arteries 170 3.14 Effect of Taper 172 3.15 Effects of Viscosity of the Fluid and Viscoelasticity

of the Wall 174 3.16 Influence of Nonlinearities 178 3.17 Flow Separation from the Wall 180 3.18 Flow in the Entrance Region 182 3.19 Curved Vessel 189 3.20 Messages Carried in the Arterial Pulse Waves and

Clinical Applications 191 3.21 Biofluid and Biosolid Mechanics of Arterial Disease 192

References 200

Chapter 4 The Veins 206 4.1 Introduction 206 4.2 Concept of Elastic Instability 208 4.3 Instability of a Circular Cylindrical Tube Subjected to

External Pressure 214 4.4 Vessels of Naturally Elliptic Cross Section 223

Contents xv

4.5 Steady Flow in Collapsible Tubes 227 4.6 Unsteady Flow in Veins 235 4.7 Effect of Muscle Action on Venous Flow 241 4.8 Self-Excited Oscillations 243 4.9 Forced Oscillation of Veins and Arteries Due to Unsteady

Flow, Turbulence, Separation, or Reattachment 247 4.10 Patency of Pulmonary Veins When the Blood Pressure Is

Exceeded by Airway Pressure 252 4.11 Waterfall Condition in the Lung 261

References 262

Chapter 5 Microcirculation 266 5.1 Introduction 266 5.2 Anatomy of Microvascular Beds 267 5.3 Pressure Distribution in Microvessels 274 5.4 Pressure in the Interstitial Space 278 5.5 Velocity Distribution in Microvessels 279 5.6 Velocity-Hematocrit Relationship 282 5.7 Mechanics of Flow at Very Low Reynolds Numbers 291 5.8 Oseen's Approximation and Other Developments 298 5.9 Entry Flow, Bolus Flow, and Other Examples 300 5.10 Interaction Between Particles and Tube Wall 306 5.11 Stokes Flow in Pulmonary Capillaries: Sheet Flow Around

a Circular Post 309 5.12 Force of Interaction of Leukocytes and Vascular Endothelium 316 5.13 Local Control of Blood Flow 324

References 328

Chapter 6 Blood Flow in the Lung 333 6.1 Introduction 333 6.2 Pulmonary Blood Vessels 335 6.3 Pulmonary Capillaries 340 6.4 Spatial Distribution of Pulmonary Capillaries: Shape of the

Alveoli and Alveolar Ducts 348 6.5 Spatial Distribution of Pulmonary Arterioles and Venules 350 6.6 Relative Positions of Pulmonary Arterioles and Venules,

and Alveolar Ducts 354 6.7 Elasticity of Pulmonary Arteries and Veins 357 6.8 Elasticity of Pulmonary Alveolar Sheet 366 6.9 Apparent Viscosity of Blood in Pulmonary Capillaries 369 6.10 Formulation of the Analytical Problems 372 6.11 An Elementary Analog of the Theory 378 6.12 General Features of Sheet Flow 382

xvi Contents

6.13 Pressure-Flow Relationship of Pulmonary Alveolar Blood Flow 389

6.14 Blood Flow in the Whole Lung 393 6.15 Regional Difference of Pulmonary Blood Flow 404 6.16 Patchy Flow in the Lung 408 6.17 Analysis of Flow Through a Pulmonary Sluicing Gate 409 6.18 Stability of a Collapsing Pulmonary Alveolar Sheet 417 6.19 Hysteresis in the Pressure-Flow Relationship of Pulmonary

Blood Flow in Zone-2 Condition 423 6.20 Distribution of Transit Time in the Lung 429 6.21 Pulmonary Blood Volume 434 6.22 Pulsatile Blood Flow in the Lung 435 6.23 Fluid Movement in the Interstitial Space of the Pulmonary

Alveolar Sheet 438 References 438

Chapter 7 Coronary Blood Flow 446 7.1 Introduction 446 7.2 Morphometry of Coronary Arteries 446 7.3 Coronary Veins 458 7.4 Coronary Capillaries 467 7.5 Analysis of Coronary Diastolic Arterial Blood Flow with

Detailed Anatomical Data 472 7.6 Morphometry of Vascular Remodeling 478 7.7 In Vivo Measurements of the Dimensions of Coronary

Blood Vessels 480 7.8 Mechanical Properties of Coronary Blood Vessels 485 7.9 Mechanical Properties of Heart Muscle Cells in Directions

Orthogonal to Active Tensile Force 485 7.10 Coronary Arteriolar Myogenic Response: Length-Tension

Relationship of Vascular Smooth Muscle 489 7.11 Vessel Dilation Due to Flow: Effect of Shear Stress on

the Endothelium on Smooth Muscle Length-Tension Relationship 495

7.12 Regulation and Autoregulation of Coronary Blood Flow 501 7.13 Pressure-Flow Relationship of Coronary Circulation 503 7.14 Model of Coronary Waterfall 507

References 509

Chapter 8 Blood Flow in Skeletal Muscle 514 8.1 Introduction 514 8.2 Topology of Skeletal Muscle Vasculature 514

8.3 Skeletal Muscle Arterioles and Venules 8.4 Capillary Blood Vessels in Skeletal Muscle 8.5 Resistance to Flow in Capillaries 8.6 Mechanical Properties of Muscle Capillaries 8.7 Constitutive and Hemodynamic Equations of

Skeletal Muscle Vasculature 8.8 Pulsatile Flow in Single Vessel 8.9 Blood Flow in Whole Skeletal Muscle 8.10 Finite Zero-Flow Arterial Pressure Gradient in

Skeletal Muscle 8.11 Fluid Pump Mechanism in Initial Lymphatics

References

Author Index Subject Index

Contents xvii

519 522 527 527

532 533 533

538 540 543

547 557