Biomechanics

Mechanical Properties of Living Tissues

Second Edition

Other titles by the same author:

Biomechanics: Motion, Flow, Stress and Growth (1990)

Biomechanics: Circulation, second edition (1996)

y.c. Fung

Biomechanics Mechanical Properties of Living Tissues

Second Edition

With 282 Illustrations

~ Springer

Y.C. Fung Department of Bioengineering University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093-0412 USA

Cover illustration: Branch point of capillary blood vessel (see Fig. 5.7:2).

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

Biomechanics: mechanical properties ofliving tissues / Yuan­Cheng Fung. - 2nd ed.

p. cm. Includes bibliographical references and index. ISBN 978-1-4419-3104-7 ISBN 978-1-4757-2257-4 (eBook) DOI 10.1007/978-1-4757-2257-4 1. Tissues. 2. Biomechanics. 3. Rheology (Biology). I. Title.

QP88.F87 1993 612'.014-dc20 92-33749

ISBN 978-1-4419-3104-7 Printed on acid-free paper.

© 1993 Springer Science+Business Media New York Originally published by Springer Science+Business Media, Inc. in 1993 Softcover reprint of the hardcover 2nd edition 1993 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

15 14 13 12 11 10

springeronline.com

Dedicated to Chia-Shun Yih,

and Luna, Conrad, and

Brenda Fung

Preface to the Second Edition

The objective of this book remains the same as that stated in the first edition: to present a comprehensive perspective of biomechanics from the stand point of bioengineering, physiology, and medical science, and to develop mechanics through a sequence of problems and examples. My three-volume set of Bio­mechanics has been completed. They are entitled: Biomechanics: Mechanical Properties of Living Tissues; Biodynamics: Circulation; and Biomechanics: Motion, Flow, Stress, and Growth; and this is the first volume. The mechanics prerequisite for all three volumes remains at the level of my book A First Course in Continuum Mechanics (3rd edition, Prentice-Hall, Inc., 1993).

In the decade of the 1980s the field of Biomechanics expanded tremen­dously. New advances have been made in all fronts. Those that affect the basic understanding of the mechanical properties of living tissues are described in detail in this revision. The references are brought up to date. Among the new topics added are the following: the coagulation of blood, thrombus formation and dissolution, cellular mechanics, deformability of passive leukocytes, me­chanics of the endothelial cells in a continuum, news about new types of collagen, new methods of testing mechanical properties of soft tissues, the relationship between continuum mechanics and the structure and ultrastruc­ture of tissues, the cross-bridge theory of muscle contraction, experimental evidences for sliding elements in muscle cells, the constitutive equation of myocardium, the residual stresses in organs, the constitutive equations of soft tissues based on their zero-stress states, the constitutive equation of the individual layers of a multilayered tissue such as the blood vessel wall, the influence of stress and strain on the remodeling of living tissues, the tensorial Wolff's law, the triphasic theory of cartilage, tissue engineering, and a perspec­tive of biomechanics of the future. And, in keeping with our tradition of

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Vlll Preface to the Second Edition

emphasizing problem-formulation and problem-solving as a means to learn a subject, many new problems are added.

For this edition, I wish to record my thanks to Drs. Mohan D. Deshpande, J.P. du Plessis, L.J.M.G. Dortmans, A.A.F. van de Ven, A.A.H.J. Sauren, and R. Ponnalagar Samy for sending me errata and discussions. To Professors Aydin Tozeren, Richard Skalak, Shu Chien, Geert Schmid-Schonbein, John Pinto, Andrew McCulloch, Robert Nerem, Van Mow, Michael Lai, and Savio Woo, I am grateful for frequent discussions and advices. I have enjoyed working with my former students, Drs. Paul Zupkas, Shu Qian Liu, Ghassan Kassab, Jianbo Zhou, Jack Debes, and Hai Chao Han, and Visiting Professors Qi Lian Yu, Jia Ping Xie, Rui Fang Yang, ShanXi Deng, and Jun Tomioka whose results are referred to here. To them, and to Perne Whaley who has collaborated with me over twenty years on manuscript preparation, I am very grateful.

La Jolla, California Yuan-Cheng Fung

Preface to the First Edition

The motivation for writing a series of books on biomechanics is to bring this rapidly developing subject to students of bioengineering, physiology, and mechanics. In the last decade biomechanics has become a recognized disci­pline offered in virtually all universities. Yet there is no adequate textbook for instruction; neither is there a treatise with sufficiently broad coverage. A few books bearing the title of biomechanics are too elementary, others are too specialized. I have long felt a need for a set of books that will inform students of the physiological and medical applications of biomechanics, and at the same time develop their training in mechanics. We cannot assume that all students come to biomechanics already fully trained in fluid and solid mechanics; their knowledge in these subjects has to be developed as the course proceeds. The scheme adopted in the present series is as follows. First, some basic training in mechanics, to a level about equivalent to the first seven chapters of the author's A First Course in Continuum Mechanics (Prentice-Hall, Inc. 1977), is assumed. We then present some essential parts of biomechanics from the point of view of bioengineering, physiology, and medical applications. In the meantime, 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.

To strike a balance between biological and physical topics in a single course is not easy. Biology contains a great deal of descriptive material, whereas mechanics aims at quantitative analysis. The need to unify these topics sometimes renders the text nonuniform in style, stressing a mathemati­cal detail here and describing an anatomy there. This nonuniformity is more

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x Preface to the First Edition

pronounced at the beginning, when the necessary background material has to be introduced.

A special word needs to be said about the exercises. Students of mechanics thrive on exercises. We must constantly try to formulate and solve problems. Only through such practice can we make biomechanics a living subject. I do not wish to present this book as a collection of solved problems. I wish to present it as a way of thinking about problems. I wish to illustrate the use of mechanics as a simple, quantitative tool. For this reason many problems for solution are proposed in the text; some are used as a vehicle to inform the readers of some published results, others are intended to lead the reader to new paths of investigation. I followed this philosophy even at the very beginning by presenting some problems and solutions in the Introductory Chapter 1. I think colleagues who use this as a textbook would appreciate this, because then they can assign some problems to the students after the first lecture.

With our limited objective, this book does not claim to be a compendium or handbook of current information on the selected topics, nor a review of literature. For those purposes a much larger volume will be needed. In this volume we develop only a few topics that seem related and important. A comprehensive bibliography is not provided; the list of references is limited to items quoted in the text. Though the author can be accused of quoting 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 more about his own views than the views of others. I have tried, however, never to forget mentioning the existence of other points of view.

Biomechanics is a young subject. Our understanding of the subject is yet imperfect. Many needed pieces of information have not yet been obtained; many potentially important applications have not yet been made. There are many weaknesses in our present position. For example, the soft tissue mechanics developed in Chapter 7, based on the concept of quasilinear viscoelasticity and pseudo-elasticity, may someday be replaced by constitu­tive equations that are fully nonlinear but not too complex. The blood vessel mechanics developed in Chapter 8 is based on a two-dimensional average. Our discussion of the muscle mechanics in Chapter 9-11 points out the deficiency in our present knowledge on this subject.

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 Professors Sidney Sobin, Evan Evans, Harry Goldsmith, Jen-shih Lee, Wally Frasher, Richard Skalak, Andrew Somlyo, Salvatore Sutera, Andrus Viidik, Joel Price, Savio Woo, and Benjamin Zweifach who supplied original photographs for reproduction.

This book grew out of my lecture notes used at the University of California, San Diego over the past ten years. To the students of these classes I am grateful for discussions. Much of the results presented here are the work of my colleagues, friends, and former students. Professors Sidney Sobin,

Preface to the First Edition xi

Benjamin Zweifach, Marcos Intaglietta, Arnost and Kitty Fronek, Wally Frasher, Paul Johnson, and Savio Woo provided the initial and continued collaboration with me on this subject. Drs. Jen-shi Lee, Pin Tong, Frank Yin, John Pinto, Evan Evans, Yoram Lanir, Hyland Chen, Michael Yen, Donald Vawter, Geert Schmid-Schoenbein, Peter Chen, Larry Malcom, Joel Price, Nadine Sidrick, Paul Sobin, Winston Tsang, and Paul Zupkas contributed much of the material presented here. The contribution of Paul Patitucci to the numerical handling of data must be especially acknowledged. Dr. Yuji Matsuzaki contributed much to my understanding of flow separation and stability. Professor Zhuong Feng-Yuan read the proofs and made many useful suggestions. Eugene Mead kept the laboratory going. Rose Cataldi and Virginia Stephens typed the manuscript. To all of them I am thankful.

Finally, I wish to thank the editorial and production staffs of Springer­Verlag for their care and cooperation in producing this book.

La Jolla, California Yuan-Cheng Fung

Contents

Preface to the Second Edition Preface to the First Edition

Chapter 1 Introduction: A Sketch of the History and Scope of the Field

1.1 What Is Biomechanics? 1.2 Historical Background 1.3 What's in a Name? 1.4 Mechanics in Physiology 1.5 What Contributions Has Biomechanics Made to Health Science? 1.6 Our Method of Approach 1.7 Tools ofInvestigation 1.8 What Contributions Has Biomechanics Made to Mechanics? 1.9 On the Law of Laplace

Problems References

Chapter 2 The Meaning of the Constitutive Equation

2.1 Introduction 2.2 Stress 2.3 Strain 2.4 Strain Rate 2.5 Constitutive Equations

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1 2 6 7

10 11 12 14 14 17 22

23

23 25 29 34 35

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xiv Contents

2.6 The Nonviscous Fluid 35 2.7 The Newtonian Viscous Fluid 36 2.8 The Hookean Elastic Solid 38 2.9 The Effect of Temperature 40 2.10 Materials with More Complex Mechanical Behavior 40 2.11 Viscoelasticity 41 2.12 Response of a Viscoelastic Body to Harmonic Excitation 48 2.13 Use of Viscoelastic Models 50 2.14 Methods of Testing 52 2.15 Mathematical Development of Constitutive Equations 57

Problems 58 References 65

Chapter 3 The Flow Properties of Blood 66

3.1 Blood Rheology: An Outline 66 3.2 The Constitutive Equation of Blood Based on Viscometric Data and

Casson's Equation 72 3.3 Laminar Flow of Blood in a Tube 76 3.4 Speculation on Why Blood Viscosity Is the Way It Is 82 3.5 Fluid-Mechanical Interaction of Red Blood Cells with a Solid Wall 91 3.6 Thrombus Formation and Dissolution 93 3.7 Medical Applications of Blood Rheology 96

Problems 99 References 105

Chapter 4 Mechanics of Erythrocytes, Leukocytes, and Other Cells 109

4.1 Introduction 109 4.2 Human Red Cell Dimensions and Shape 112 4.3 The Extreme-Value Distribution 117 4.4 The Deformability of Red Blood Cells (RBC) 120 4.5 Theoretical Considerations of the Elasticity of Red Cells 122 4.6 Cell Membrane Experiments 128 4.7 Elasticity of the Red Cell Membrane 140 4.8 The Red Cell Membrane Model 144 4.9 The Effects of Red Cell Deformability on Turbulence in Blood Flow 146 4.10 Passive Deformation of Leukocytes 147 4.11 Cell Adhesion: Multipipets Experiments 151 4.12 Topics of Cell Mechanics 151

Problems 156 References to Erythrocytes 158 References to Leukocytes and Other Cells 162

Contents XV

Chapter 5 Interaction of Red Cells with Vessel Wall, and Wall Shear with Endothelium 165

5.1 Introduction 165 5.2 Apparent Viscosity and Relative Viscosity 166 5.3 Effect of Size of the Blood Vessel on the Apparent Viscosity of Blood:

The Fahraeus-Lindqvist Effect 172 5.4 The Distribution of Suspended Particles in Fairly Narrow Rigid

Tubes 176 5.5 The Motion of Red Cells in Tightly Fitting Tubes 176 5.6 Inversion of the Fahraeus-Lindqvist Effect in Very Narrow Tubes 182 5.7 Hematocrit in Very Narrow Tubes 186 5.8 Theoretical Investigations 194 5.9 The Vascular Endothelium 196 5.10 Blood Shear Load Acting on the Endothelium 198 5.11 Tension Field in Endothelial Cell Membranes Under the Fluid Interior

Hypothesis 199 5.12 The Shape of Endothelial Cell Nucleus Under the Fluid Interior

Hypothesis 201 5.13 Transmission of the Tension in the Upper Endothelial Cell Membrane

to the Basal Lamina through the Sidewalls 203 5.14 The Hypothesis of a Solid-Like Cell Content 210 5.15 The Effect of Turbulent Flow on Cell Stress 211

Problems 213 References to Blood Cells in Microcirculation 215 References to Endothelial Cells 217

Chapter 6 Bioviscoelastic Fluids 220

6.1 Introduction 220 6.2 Methods of Testing and Data Presentation 222 6.3 Protoplasm 226 6.4 Mucus from the Respiratory Tract 227 6.5 Saliva 231 6.6 Cervical Mucus and Semen 232 6.7 Synovial Fluid 233

Problems 238 References 240

Chapter 7 Bioviscoelastic Solids 242

7.1 Introduction 242 7.2 Some Elastic Materials 243

xvi Contents

7.3 Collagen 251 7.4 Thermodynamics of Elastic Deformation 265 7.5 Behavior of Soft Tissues Under Uniaxial Loading 269 7.6 Quasi-Linear Viscoelasticity of Soft Tissues 277 7.7 Incremental Laws 292 7.8 The Concept of Pseudo-Elasticity 293 7.9 Biaxial Loading Experiments on Soft Tissues 295 7.10 Description of Three-Dimensional Stress and Strain States 298 7.11 Strain-Energy Function 300 7.12 An Example: The Constitutive Equation of Skin 302 7.13 Generalized Viscoelastic Relations 306 7.14 The Complementary Energy Function: Inversion of the Stress-Strain

Relationship 307 7.15 Constitutive Equation Derived According to Microstructure 310

Problems 311 References 314

Chapter 8 Mechanical Properties and Active Remodeling of Blood Vessels 321

8.1 Introduction 321 8.2 Structure and Composition of Blood Vessels 322 8.3 Arterial Wall as a Membrane: Behavior Under Uniaxial Loading 326 8.4 Arterial Wall as a Membrane: Biaxial Loading and Torsion

Experiments 336 8.5 Arterial Wall as a Membrane: Dynamic Modulus of Elasticity from

Flexural Wave Propagation Measurements 343 8.6 Mathematical Representation of the Pseudo-Elastic Stress-Strain

Relationship 345 8.7 Blood Vessel Wall as a Three-Dimensional Body: The Zero Stress

State 349 8.8 Blood Vessel Wall as a Three-Dimensional Body: Stress and Strain,

and Mechanical Properties of the Intima, Media, and Adventitia Layers 352

8.9 Arterioles. Mean Stress-Mean Diameter Relationship 357 8.10 Capillary Blood Vessels 360 8.11 Veins 363 8.12 Effect of Stress on Tissue Growth 369 8.13 Morphological and Structural Remodeling of Blood Vessels Due to

Change of Blood Pressure 370 8.14 Remodeling the Zero Stress State of a Blood Vessel 373 8.15 Remodeling of Mechanical Properties 374 8.16 A Unified Interpretation of the Morphological, Structural, Zero Stress

State, and Mechanical Properties Remodeling 376 Problems 377 References 384

Contents xvii

Chapter 9 Skeletal Muscle 392

9.1 Introduction 392 9.2 The Functional Arrangement of Muscles 393 9.3 The Structure of Skeletal Muscle 394 9.4 The Sliding Element Theory of Muscle Action 397 9.5 Single Twitch and Wave Summation 397 9.6 Contraction of Skeletal Muscle Bundles 398 9.7 Hill's Equation for Tetanized Muscle 399 9.8 Hill's Three-Element Model 405 9.9 Hypotheses of Cross-Bridge Theory 413 9.10 Evidences in Support of the Cross-Bridge Hypotheses 415 9.11 Mathematical Development of the Cross-Bridge Theory 418 9.12 Constitutive Equation ofthe Muscle as a Three-Dimensional

Continuum 420 9.13 Partial Activation 422

Problems 423 References 424

Chapter 10 Heart Muscle 427

10.1 Introduction: The Difference Between Myocardial and Skeletal Muscle Cells 427

10.2 Use of the Papillary or Trabecular Muscles as Testing Specimens 431 10.3 Use of the Whole Ventricle to Determine Material Properties of the

Heart Muscle 433 10.4 Properties of Unstimulated Heart Muscle 433 10.5 Force, Length, Velocity of Shortening, and Calcium Concentration

Relationship for the Cardiac Muscle 441 10.6 The Behavior of Active Myocardium According to Hill's Equation and

Its Modification 445 10.7 Pinto's Method 453 10.8 Micromechanical Derivation of the Constitutive Law for the Passive

Myocardium 10.9 Other Topics

Problems References

Chapter 11 Smooth Muscles

ILl Types of Smooth Muscles 11.2 The Contractile Machinery 11.3 Rhythmic Contraction of Smooth Muscle 11.4 The Property of a Resting Smooth Muscle: Ureter

455 457 460 462

466

466 468 470 475

x viii Contents

11.5 Active Contraction of Ureteral Segments 481 11.6 Resting Smooth Muscle: Taenia Coli 487 11.7 Other Smooth Muscle Organs 495

Problems 495 References 497

Chapter 12 Bone and Cartilage 500

12.1 Introduction 500 12.2 Bone as a Living Organ 504 12.3 Blood Circulation in Bone 507 12.4 Elasticity and Strength of Bone 510 12.5 Viscoelastic Properties of Bone 513 12.6 Functional Adaptation of Bone 514 12.7 Cartilage 519 12.8 Viscoelastic Properties of Articular Cartilage 520 12.9 The Lubrication Quality of Articular Cartilage Surfaces 525 12.10 Constitutive Equations of Cartilage According to a Triphasic Theory 531 12.11 Tendons and Ligaments 535

Problems 536 References 538

Author Index 545

Subject Index 559