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Tóm tắt nội dung (trích từ tài liệu gốc): Artwork for the cover design was adapted from Littler, J.W. 1986. The finger extensor system. Some approaches to the correction of its disabilities. Orthop. Clin. North Am. Jul;17(3):483-492. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 � 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-0-8493-8534-6 (Hardcover) This boo
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Artwork for the cover design was adapted from Littler, J.W. 1986. The finger extensor system. Some approaches to the
correction of its disabilities. Orthop. Clin. North Am. Jul;17(3):483-492.
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Biomechanics / editors, Donald R. Peterson and Joseph D. Bronzino.
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ISBN 978-0-8493-8534-6 (alk. paper)
1. Biomechanics. I. Peterson, Donald R. II. Bronzino, Joseph D., 1937- III. Title.
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Table of Contents
1 Mechanics of Hard Tissue
J. Lawrence Katz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1
2 Musculoskeletal Soft Tissue Mechanics
Richard L. Lieber, Thomas J. Burkholder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
3 Joint-Articulating Surface Motion
Kenton R. Kaufman, Kai-Nan An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1
4 Joint Lubrication
Michael J. Furey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1
5 Analysis of Gait
Roy B. Davis, III, Sylvia O~ unpuu, Peter A. DeLuca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1
6 Mechanics of Head/Neck
Albert I. King, David C. Viano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1
7 Biomechanics of Chest and Abdomen Impact
David C. Viano, Albert I. King . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1
8 Cardiac Biomechanics
Andrew D. McCulloch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1
9 Heart Valve Dynamics
Ajit P. Yoganathan, Jack D. Lemmon, Jeffrey T. Ellis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1
10 Arterial Macrocirculatory Hemodynamics
Baruch B. Lieber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1
11 Mechanics of Blood Vessels
Thomas R. Canfield, Philip B. Dobrin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1
12 The Venous System
Artin A. Shoukas, Carl F. Rothe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1
v
13 Mechanics, Molecular Transport, and Regulation in the Microcirculation
Aleksander S. Popel, Roland N. Pittman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-1
14 Mechanics and Deformability of Hematocytes
Richard E. Waugh, Robert M. Hochmuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-1
15 Mechanics of Tissue/Lymphatic Transport
Geert W. Schmid-Scho�nbein, Alan R. Hargens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1
16 Modeling in Cellular Biomechanics
Alexander A. Spector, Roger Tran-Son-Tay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1
17 Cochlear Mechanics
Charles R. Steele, Gary J. Baker, Jason A. Tolomeo, Deborah E. Zetes-Tolomeo . . . . . . 17-1
18 Vestibular Mechanics
Wallace Grant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-1
19 Exercise Physiology
Arthur T. Johnson, Cathryn R. Dooly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-1
20 Factors Affecting Mechanical Work in Humans
Ben F. Hurley, Arthur T. Johnson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1
vi
Preface
Engineering is the integration of art and science and involves the use of systematic knowledge based on the
principles of mathematics and the physical sciences to design and develop systems that have direct practical
applicability for the benefit of mankind and society. With this philosophy in mind, the importance of the
engineering sciences becomes obvious, and this is especially true for the biomedical aspects, where the
implications are easily identifiable. Of all the engineering sciences, biomedical engineering is considered
to be the broadest. Its practice frequently involves the direct combination of the core engineering sciences,
such as mechanical, electrical, and chemical engineering, and requires a functional knowledge of other
nonengineering disciplines, such as biology and medicine, to achieve effective solutions. It is a multidis-
ciplinary science with its own core aspects, such as biomechanics, bioinstrumentation, and biomaterials,
which can be further characterized by a triage of subject matter. For example, the study of biomechanics,
or biological mechanics, employs the principles of mechanics, which is a branch of the physical sciences
that investigates the effects of energy and forces on matter or material systems. It often embraces a broad
range of subject matter that may include aspects of classical mechanics, material science, fluid mechanics,
heat transfer, and thermodynamics, in an attempt to model and predict the mechanical behaviors of any
living system. As such, it may be called the "liberal arts" of the biomedical engineering sciences.
Biomechanics is deeply rooted throughout scientific history and has been influenced by the research
work of early mathematicians, engineers, physicists, biologists, and physicians. Not one of these disciplines
can claim sole responsibility for maturing biomechanics to its current state; rather, it has been a conglom-
eration and integration of these disciplines, involving the application of mathematics, physical principles,
and engineering methodologies, that has been responsible for its advancement. Several examinations exist
that offer a historical perspective on biomechanics in dedicated chapters within a variety of biomechanics
textbooks. For this reason, a historical perspective is not presented within this introduction and it is left
to the reader to discover the material within one of these textbooks. As an example, Y.C. Fung (1993)
provides a reasonably detailed synopsis of those who were influential to the progress of biomechanical
understanding. A review of this material and similar material from other authors commonly shows that
biomechanics has occupied the thoughts of some of the most conscientious minds involved in a variety of
the sciences.
Leonardo da Vinci, one of the early pioneers of biomechanics, was the first to introduce the principle of
"cause and effect" in scientific terms as he firmly believed that "there is no result in nature without a cause;
understand the cause and you will have no need of the experiment" (1478�1518). Leonardo understood
that experimentation is an essential tool for developing an understanding of nature's causes and the results
they produce, especially when the cause is not immediately obvious. The contemporary approach to
understand and solve problems in engineering expands upon Leonardo's principle and typically follows a
sequence of fundamental steps that are commonly defined as observation, experimentation, theorization,
validation, and application. These steps are the basis of the engineering methodologies and their significance
is emphasized within a formal engineering education, especially in biomedical engineering. Each step is
considered to be equally important, and an iterative relationship between steps, with mathematics serving
vii
as the common link, is often necessary in order to converge on a practical understanding of the system in
question. An engineering education that ignores these interrelated fundamentals will produce engineers
who are ignorant of the ways in which real-world phenomena differ from mathematical models. Since most
biomechanical systems are inherently complex and cannot be adequately defined using only theory and
mathematics, biomechanics should be considered a discipline whose progress relies heavily on research
and experimentation and the careful implementation of the sequence of steps. When a precise solution
is not obtainable, utilizing this approach will assist with identifying critical physical phenomena and
obtaining approximate solutions that may provide a deeper understanding as well as improvements to the
investigative strategy. Not surprisingly, the need to identify critical phenomena and obtain approximate
solutions seems to be more significant in biomedical engineering than any other engineering discipline,
which can be attributed to the complex biological processes involved.
Applications of biomechanics have traditionally focused on modeling the system-level aspects of the
human body, such as the musculoskeletal system, the respiratory system, and the cardiovascular and
cardiopulmonary systems. Technologically, most of the progress has been made on system-level device
development and implementation, with obvious influences on athletic performance, work environment
interaction, clinical rehabilitation, orthotics, prosthetics, and orthopaedic surgery. However, more recent
biomechanics initiatives are now focusing on the mechanical behaviors of the biological subsystems, such
as tissues, cells, and molecules, in order to relate subsystem functions across all levels by showing how
mechanical function is closely associated with certain cellular and molecular processes. These initiatives
have a direct impact on the development of biological nano- and microtechnologies involving polymer
dynamics, biomembranes, and molecular motors. The integration of system and subsystem models will
advance our overall understanding of human function and performance and further develop the prin-
ciples of biomechanics. Even still, our modern understanding about certain biomechanic processes is
limited, but through ongoing biomechanics research, new information that influences the way we think
about biomechanics is generated and important applications that are essential to the betterment of human
existence are discovered. As a result, our limitations are reduced and our understanding becomes more
refined. Recent advances in biomechanics can also be attributed to advances in experimental methods and
instrumentation, such as computational power and imaging capabilities, which are also subject to constant
progress.
The rapid advance of biomechanics research continues to yield a large amount of literature that exists in
the form of various research and technical papers and specialized reports and textbooks that are only acces-
sible in various journal publications and university libraries. Without access to these resources, collecting
the publications that best describe the current state of the art would be extremely difficult. With this in
mind, this textbook offers a convenient collection of chapters that present current principles and appli-
cations of biomechanics from respected published scientists with diverse backgrounds in biomechanics
research and application. A total of 20 chapters is presented, 12 of which have been substantially updated
and revised to ensure the presentation of modern viewpoints and developments. The chapters within this
text have been organized in an attempt to present the material in a systematic manner. The first group
of chapters is related to musculoskeletal mechanics and includes hard and soft tissue mechanics, joint
mechanics, and applications related to human function. The next group of chapters covers several aspects
of biofluid mechanics and includes a wide range of circulatory dynamics, such as blood vessel and blood
cell mechanics, and transport. It is followed by a chapter that introduces current methods and strategies
for modeling cellular mechanics. The next group consists of two chapters introducing the mechanical
functions and significance of the human ear. Finally, the remaining two chapters introduce performance
characteristics of the human body system during exercise and exertion. It is the overall intention of this
text to serve as a reference to the skilled professional as well as an introduction to the novice or student
of biomechanics. An attempt was made to incorporate material that covers a bulk of the biomechanics
field; however, as biomechanics continues to grow, some topics may be inadvertently omitted causing a
viii
disproportionate presentation of the material. Suggestions and comments from readers are welcomed on
subject matter that should be considered in future editions of this textbook.
Through the rationalization of biomechanics, I find myself appreciating the complexity and beauty of
all living systems. I hope that this textbook helps your understanding of biomechanics and your discovery
of life.
Donald R. Peterson, Ph.D., M.S.
University of Connecticut Health Center
Farmington, Connecticut
References
Fung YC. 1993. Biomechanics: Mechanical Properties of Living Tissues. 2nd ed. New York, Springer�Verlag.
da Vinci L. 1478�1518. Codice Atlantico, 147 v.a.
ix
The Editors
Donald R. Peterson, Ph.D., M.S., an assistant professor in the Schools of Medicine, Dental Medicine,
and Engineering at the University of Connecticut, and director of the Biodynamics Laboratory and the
Bioengineering Facility at the University of Connecticut Health Center, offers graduate-level courses in
biomedical engineering in the fields of biomechanics, biodynamics, biofluid mechanics, and ergonomics,
and teaches in medicine in the subjects of gross anatomy and occupational biomechanics. He earned a
B.S. in both aerospace and biomedical engineering from Worcester Polytechnic Institute, a M.S. in me-
chanical engineering from the University of Connecticut, and a Ph.D. in biomedical engineering also
from the University of Connecticut. Dr. Peterson's current research work is focused on the development
of laboratory and field techniques for accurately assessing and modeling human�device interaction and
human and/or organism performance, exposure, and response. Recent applications of these protocols
model human interactions with existing and developmental devices such as powered and nonpowered
tools, spacesuits and spacetools for NASA, surgical and dental instruments, musical instruments, sports
equipment, and computer input devices. Other research initiatives focus on cell biomechanics, the acous-
tics of hearing protection and communication, hand�arm vibration exposure, advanced physiological
monitoring methods, advanced vascular imaging techniques, and computational biomechanics.
Joseph D. Bronzino received the B.S.E.E. degree from Worcester Polytechnic Institute, Worcester, MA,
in 1959, the M.S.E.E. degree from the Naval Postgraduate School, Monterey, CA, in 1961, and the Ph.D.
degree in electrical engineering from Worcester Polytechnic Institute in 1968. He is presently the Vernon
Roosa Professor of Applied Science, an endowed chair at Trinity College, Hartford, CT, and president
of the Biomedical Engineering Alliance and Consortium (BEACON), which is a nonprofit organization
consisting of academic and medical institutions as well as corporations dedicated to the development and
commercialization of new medical technologies (for details visit www.beaconalliance.org).
He is the author of over 200 articles and 11 books including the following: Technology for Patient
Care (C.V. Mosby, 1977), Computer Applications for Patient Care (Addison-Wesley, 1982), Biomedical
Engineering: Basic Concepts and Instrumentation (PWS Publishing Co., 1986), Expert Systems: Basic Con-
cepts (Research Foundation of State University of New York, 1989), Medical Technology and Society:
An Interdisciplinary Perspective (MIT Press and McGraw-Hill, 1990), Management of Medical Technology
(Butterworth/Heinemann, 1992), The Biomedical Engineering Handbook (CRC Press, 1st ed., 1995; 2nd ed.,
2000; Taylor & Francis, 3rd ed., 2005), Introduction to Biomedical Engineering (Academic Press, 1st ed.,
1999; 2nd ed., 2005).
Dr. Bronzino is a fellow of IEEE and the American Institute of Medical and Biological Engineering
(AIMBE), an honorary member of the Italian Society of Experimental Biology, past chairman of the
Biomedical Engineering Division of the American Society for Engineering Education (ASEE), a charter
member and presently vice president of the Connecticut Academy of Science and Engineering (CASE),
a charter member of the American College of Clinical Engineering (ACCE), and the Association for the
Advancement of Medical Instrumentation (AAMI), past president of the IEEE-Engineering in Medicine
xi
and Biology Society (EMBS), past chairman of the IEEE Health Care Engineering Policy Commit-
tee (HCEPC), past chairman of the IEEE Technical Policy Council in Washington, DC, and
presently editor-in-chief of Elsevier's BME Book Series and Taylor & Francis' Biomedical Engineering
Handbook.
Dr. Bronzino is also the recipient of the Millennium Award from IEEE/EMBS in 2000 and the Goddard
Award from Worcester Polytechnic Institute for Professional Achievement in June 2004.
xii
Contributors
Kai-Nan An Jeffrey T. Ellis Arthur T. Johnson
Biomedical Laboratory Department of Bioengineering Engineering Department
Mayo Clinic Biological Resource
Rochester, Minnesota and Bioscience University of Maryland
Georgia Institute of Technology College Park, Maryland
Gary J. Baker Atlanta, Georgia
Stanford University J. Lawrence Katz
Stanford, California Michael J. Furey School of Dentistry
Mechanical Engineering University of
Thomas J. Burkholder
School of Applied Physiology Department Missouri-Kansas City
Georgia Institute of Technology Virginia Polytechnic Institute Kansas City, Missouri
Atlanta, Georgia
and State University Kenton R. Kaufman
Thomas R. Contield Blacksburg, Virginia Biomedical Laboratory
Argonne National Laboratory Mayo Clinic
Wallace Grant Rochester, Minnesota
Roy B. Davis, III Engineering Science and
Shriner's Hospital for Children Albert I. King
Mechanics Department Biomaterials Engineering
Peter A. DeLuca Virginia Polytechnic Institute
Gait Analysis Laboratory Center
University of Connecticut and State University Wayne State University
Blacksburg, Virginia Detroit, Michigan
Children's Medical Center
Hartford, Connecticut Alan R. Hargens Jack D. Lemmon
Department of Orthopedic Department of Bioengineering
Philip B. Dobrin
Hines VA Hospital and Loyola Surgery and Bioscience
University of Georgia Institute of Technology
University Medical Center Atlanta, Georgia
Hines, Illinois California-San Diego
San Diego, California Baruch B. Lieber
Cathryn R. Dooly Department of Mechanical and
University of Maryland Robert M. Hochmuth
College Park, Maryland Department of Mechanical Aerospace Engineering
State University of
Engineering
Duke University New York-Buffalo
Durham, North Carolina Buffalo, New York
Ben F. Hurley Richard L. Lieber
Department of Kinesiology Departments of Orthopedics
College of Health and Human
and Bioengineering
Performance University of California
University of Maryland La Jolla, California
College Park, Maryland
xiii
Andrew D. McCulloch Carl F. Rothe Jason A. Tolomeo
Department of Bioengineering Department of Physiology Stanford University
University of Stanford, California
Medical Science
California-San Diego Indiana University Roger Tran-Son-Tay
La Jolla, California Indianapolis, Indiana University of Florida
Gainesville, Florida
Sylvia O~ unpuu Geert W. Schmid-Scho�nbein
Center for Motion Analysis Department of Bioengineering David C. Viano
University of Connecticut University of Wayne State University
Detroit, Michigan
Children's Medical Center California-San Diego
West Hartford, Connecticut La Jolla, California Richard E. Waugh
Department of Pharmacology
Donald R. Peterson Artin A. Shoukas
University of Connecticut Department of Biomedical and Physiology
University of Rochester
Health Center Engineering
Biodynamics Laboratory Johns Hopkins University Medical Center
Farmington, Connecticut Rochester, New York
School of Medicine
Roland N. Pittman Baltimore, Maryland Ajit P. Yoganathan
Medical College of Virginia Department of Bioengineering
Richmond, Virginia Alexander A. Spector
Biomedical Engineering and Bioscience
Aleksander S. Popel Johns Hopkins University Georgia Institute of Technology
Department of Biomedical Baltimore, Maryland Atlanta, Georgia
Engineering Charles R. Steele Deborah E. Zetes-Tolomeo
Johns Hopkins University Applied Mechanical Division Stanford University
Baltimore, Maryland Stanford University Stanford, California
Stanford, California
xiv
1
Mechanics of
Hard Tissue
J. Lawrence Katz 1.1 Structure of Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
1.2 Composition of Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
University of Missouri-Kansas City 1.3 Elastic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
1.4 Characterizing Elastic Anisotropy . . . . . . . . . . . . . . . . . . . . . . 1-9
1.5 Modeling Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12
1.6 Viscoelastic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12
1.7 Related Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16
Defining Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17
Further Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-20
Hard tissue, mineralized tissue, and calcified tissue are often used as synonyms for bone when describ-
ing the structure and properties of bone or tooth. The hard is self-evident in comparison with all other
mammalian tissues, which often are referred to as soft tissues. Use of the terms mineralized and calcified
arises from the fact that, in addition to the principle protein, collagen, and other proteins, glycoproteins,
and protein-polysaccherides, comprising about 50% of the volume, the major constituent of bone is a
calcium phosphate (thus the term calcified) in the form of a crystalline carbonate apatite (similar to
naturally occurring minerals, thus the term mineralized). Irrespective of its biological function, bone is
one of the most interesting materials known in terms of structure�property relationships. Bone is an
anisotropic, heterogeneous, inhomogeneous, nonlinear, thermorheologically complex viscoelastic mate-
rial. It exhibits electromechanical effects, presumed to be due to streaming potentials, both in vivo and
in vitro when wet. In the dry state, bone exhibits piezoelectric properties. Because of the complexity of
the structure�property relationships in bone, and the space limitation for this chapter, it is necessary to
concentrate on one aspect of the mechanics. Currey [1984] states unequivocally that he thinks, "the most
important feature of bone material is its stiffness." This is, of course, the premiere consideration for the
weight-bearing long bones. Thus, this chapter will concentrate on the elastic and viscoelastic properties
of compact cortical bone and the elastic properties of trabecular bone as exemplar of mineralized tissue
mechanics.
1-1
1-2 Biomechanics
1.1 Structure of Bone
The complexity of bone's properties arises from the complexity in its structure. Thus it is important to
have an understanding of the structure of mammalian bone in order to appreciate the related properties.
Figure 1.1 is a diagram showing the structure of a human femur at different levels [Park, 1979]. For
convenience, the structures shown in Figure 1.1 will be grouped into four levels. A further subdivision
of structural organization of mammalian bone is shown in Figure 1.2 [Wainwright et al., 1982]. The
individual figures within this diagram can be sorted into one of the appropriate levels of structure shown
on Figure 1.1 as described in the following. At the smallest unit of structure we have the tropocollagen
molecule and the associated apatite crystallites (abbreviated Ap). The former is approximately 1.5 by
280 nm, made up of three individual left-handed helical polypeptide (alpha) chains coiled into a right-
handed triple helix. Ap crystallites have been found to be carbonate-substituted hydroxyapatite, generally
thought to be nonstoichiometric. The crystallites appear to be about 4 � 20 � 60 nm in size. This level is
denoted the molecular. The next level we denote the ultrastructural. Here, the collagen and Ap are intimately
associated and assembled into a microfibrilar composite, several of which are then assembled into fibers
from approximately 3 to 5 m thick. At the next level, the microstructural, these fibers are either randomly
arranged (woven bone) or organized into concentric lamellar groups (osteons) or linear lamellar groups
(plexiform bone). This is the level of structure we usually mean when we talk about bone tissue properties.
In addition to the differences in lamellar organization at this level, there are also two different types of
architectural structure. The dense type of bone found, for example, in the shafts of long bone is known as
compact or cortical bone. A more porous or spongy type of bone is found, for example, at the articulating
ends of long bones. This is called cancellous bone. It is important to note that the material and structural
organization of collagen�Ap making up osteonic or haversian bone and plexiform bone are the same as
the material comprising cancellous bone.
Articular
cartilage
Trabecula
Spongy bone
Compact bone
Osteon Collagen fibers
Periostoeum Haversian Concentric Apatite
canal lamella mineral crystals
Nutrient (3�7 m) (200�400 � long)
artery
Intramedullary
covity
Line of
epiphyseal
fusion
FIGURE 1.1 Hierarchical levels of structure in a human femur [Park, 1979]. (Courtesy of Plenum Press and
Dr. J.B. Park.)
Mechanics of Hard Tissue 1-3
(a)
01 m
(c) (b)
10 m 10 m
(g)
(e) (f)
(d)
0.5 m
(i)
(h)
(h)
(h)
1 m (h)
FIGURE 1.2 Diagram showing the structure of mammalian bone at different levels. Bone at the same level is drawn
at the same magnification. The arrows show what types may contribute to structures at higher levels [Wainwright et al.,
1982]. (Courtesy Princeton University Press.) (a) Collagen fibril with associated mineral crystals. (b) Woven bone. The
collagen fibrils are arranged more or less randomly. Osteocytes are not shown. (c) Lamellar bone. There are separate
lamellae, and the collagen fibrils are arranged in "domains" of preferred fibrillar orientation in each lamella. Osteocytes
are not shown. (d) Woven bone. Blood channels are shown as large black spots. At this level woven bone is indicated
by light dotting. (e) Primary lamellar bone. At this level lamellar bone is indicated by fine dashes. (f) Haversian bone.
A collection of Haversian systems, each with concentric lamellae round a central blood channel. The large black area
represents the cavity formed as a cylinder of bone is eroded away. It will be filled in with concentric lamellae and form
a new Haversian system. (g) Laminar bone. Two blood channel networks are exposed. Note how layers of woven and
lamellar bone alternate. (h) Compact bone of the types shown at the lower levels. (i) Cancellous bone.
Finally, we have the whole bone itself constructed of osteons and portions of older, partially destroyed
osteons (called interstitial lamellae) in the case of humans or of osteons and/or plexiform bone in the
case of mammals. This we denote the macrostructural level. The elastic properties of the whole bone results
from the hierarchical contribution of each of these levels.
1-4 Biomechanics
TABLE 1.1 Composition of Adult Human and Bovine Cortical Bone
% Dry Weight
Species % H2O Ap Collagen GAGa Reference
Bovine 9.1 76.4 21.5 N.D.b Herring, 1977
Human 7.3 67.2 21.2 0.34 Pellagrino and Blitz, 1965; Vejlens, 1971
a Glycosaminoglycan.
b Not determined.
1.2 Composition of Bone
The composition of bone depends on a large number of factors: the species, which bone, the location
from which the sample is taken, and the age, sex, and type of bone tissue, for example, woven, cancellous,
cortical. However, a rough estimate for overall composition by volume is one-third Ap, one-third collagen
and other organic components, and one-third H2O. Some data in the literature for the composition of
adult human and bovine cortical bone are given in Table 1.1.
1.3 Elastic Properties
Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the
ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone
as an anisotropic, linear elastic solid with Hooke's law as the appropriate constitutive equation. Tensor
notation for the equation is written as:
ij = Cijklkl (1.1)
where ij and kl are the second-rank stress and infinitesimal second-rank strain tensors, respectively, and
Cijkl is the fourth-rank elasticity tenor. Using the reduced notation, we can rewrite Equation 1.1 as
i = Cij j i, j = 1 to 6 (1.2)
where Cij are the stiffness coefficients (elastic constants). The inverse of the Cij, the Sij, are known as the
compliance coefficients.
The anisotropy of cortical bone tissue has been described in two symmetry arrangements. Lang [1969],
Katz and Ukraincik [1971], and Yoon and Katz [1976a, b] assumed bone to be transversely isotropic with
the bone axis of symmetry (the 3 direction) as the unique axis of symmetry. Any small difference in elastic
properties between the radial (1 direction) and transverse (2 direction) axes, due to the apparent gradient
in porosity from the periosteal to the endosteal sides of bone, was deemed to be due essentially to the
defect and did not alter the basic symmetry. For a transverse isotropic material, the stiffness matrix [Cij]
is given by
C11 C12 C13 0 0 0
[Cij] = CC00 1132 C11 C13 0 0 0 (1.3)
C13 C33 0 0 0
0 0 C44 0 0
0 0 0 C44 0
0 0 0 0 0 C66
where C66 = 1 (C11 - C12). Of the 12 nonzero coefficients, only 5 are independent.
2
However, Van Buskirk and Ashman [1981] used the small differences in elastic properties between the
radial and tangential directions to postulate that bone is an orthotropic material; this requires that 9 of
Mechanics of Hard Tissue 1-5
the 12 nonzero elastic constants be independent, that is,
C11 C12 C13 0 0 0
[Cij] = CC001123 C22 C23 0 0 0 (1.4)
C23 C33 0 0 0
0 0 C44 0 0
0 0 0 C55 0
0 0 0 0 0 C66
Corresponding matrices can be written for the compliance coefficients, the Sij, based on the inverse
equation to Equation 1.2:
i = Sij j i, j = 1 to 6 (1.5)
where the Sijth compliance is obtained by dividing the [Cij] stiffness matrix, minus the ith row and
jth column, by the full [Cij] matrix and vice versa to obtain the Cij in terms of the Sij. Thus, although
S33 = 1/E 3, where E 3 is Young's modulus in the bone axis direction, E 3 = C33, since C33 and S33, are not
reciprocals of one another even for an isotropic material, let alone for transverse isotropy or orthotropic
symmetry.
The relationship between the compliance matrix and the technical constants such as Young's modulus
(Ei) shear modulus (Gi) and Poisson's ratio (ij) measured in mechanical tests such as uniaxial or pure
shear is expressed in Equation 1.6.
1 -21 -31
E2 E3
[Sij] = E1 1 0 0 0 (1.6)
-12 E2 -32 0
E3 0 0 0
E1 -23 1 0
-13 E2 E3 0 0 0
1 1
E1 0 0 G 23 0
0 1
0 0 0 G 31
0
0 0 0 00 G 12
Again, for an orthotropic material, only 9 of the above 12 nonzero terms are independent, due to the
symmetry of the Sij tensor:
12 = 21 13 = 31 23 = 32 (1.7)
E1 E2 E1 E3 E2 E3
For the transverse isotropic case, Equation 1.5 reduces to only 5 independent coefficients, since
E 1 = E 2 12 = 21 31 = 32 = 13 = 23
G 23 = G 31 G 12 = E1 (1.8)
2(1 + 12)
In addition to the mechanical tests cited above, ultrasonic wave propagation techniques have been used to
measure the anisotropic elastic properties of bone [Lang, 1969; Yoon and Katz, 1976a, b; Van Buskirk and
Ashman, 1981]. This is possible, since combining Hooke's law with Newton's second law results in a wave
---
[Cuối tài liệu]
I-12 Biomechanics
Pressure-volume area (PVA), heart, 8-11 to 12 microvascular blood flow, 13-2, 13-3, 13-4 to 5, 13-6
Pressure-volume relations, ventricular, 8-10 to 12 modeling, interpretation of experiments, 16-2 to 3
Pressure wave velocities, in arteries, 10-4 size and shape, 14-3
Principal extension ratios, 14-2, 14-11 stress and strain in two dimensions, 14-2
Probabilistic approach, cell models, 16-5 stress relaxation and strain hardening, 14-5
Probability function, chest and abdomen injury, 7-11 Reflection coefficient, arterial hemodynamics, 10-5
Protective gear/systems Regional stress and strain, ventricular, 8-18 to 19
Regurgitation, mitral, 9-10
chest and abdomen impacts, 7-2, 7-5 to 6 Relaxation, cell models, 16-2
abdominal impact modeling, 7-9 Relaxation function, tendon and ligament, 2-9, 2-10
chest, lumped-mass model, 7-8 to 9 Relaxation studies
bone, 1-15 to 16
head, helmets, 6-4, 6-7 cardiac muscle, 8-15
Proteoglycan, cartilage structure, 2-1, 2-2 Remodeling
Proteolytic degradation, joint disease, 4-14, 4-15 to 16, blood vessel wall, 11-2; 16-11
microcirculation, 13-6, 13-10 to 11
4-17 Residual stress
Pseudoelasticity, blood vessels, 11-6 blood vessel biomechanics, 11-2
Pulmonary circulation resting myocardial properties, 8-18
Resistance
defined, 10-9 chest and abdomen impacts, 7-1
geometrical parameters, 10-1, 10-2 vascular
hemodynamics, 10-3
Pulmonic valve exercise physiology, 19-3
dynamics, 9-7 to 8 venous, 12-3 to 4
mechanical properties, 9-3 to 5 Resistance vessels, 13-9
ventricular hemodynamics, 8-9 Resonant ultrasound spectroscopy (RUS), 1-11
Pump, skeletal muscle Resonators, cochlear mechanics, 17-4
hemodynamics, 12-1 to 2 Respiration, aerobic, 13-6; 19-8
lymphatic transport, 15-3, 15-4 Respiratory responses, exercise physiology, 19-4, 19-5,
Pump function, heart, 8-8 to 12
ventricular hemodynamics, 8-8 to 9 19-6, 19-7
ventricular pressure-volume relations and energetics, Resynchronization therapy, heart, 8-19
Reticular sheath, tendons, 2-2
8-10 to 12 Reverse flow, aortic valve, 9-5, 9-6
Pumping, lymphatic transport, 13-8 Reynold's equation, 4-4
Reynolds number, arterial blood flow, 10-3
lymph formation and pump mechanisms, 15-9 Rheology
mechanism with primary and secondary
bone, 1-15
valves, 15-12 joint lubrication, 4-19
tissue mechanical motion and, 15-9 to 11, 15-12 Rib fractures, 7-2, 7-10
Ribs, acceleration injury, 7-3
Q Rigid isoviscous lubrication, 4-5
Rigid-viscous lubrication, 4-5
Quasi-linear viscoelastic approach, tendon and Risk assessment, chest and abdomen impacts, 7-9, 7-11
ligament, 2-9 Rolling
cell modeling, 16-4 to 7, 16-8
R joint motion, 3-2
Rotation, joint, 3-2
Radioisotope studies Rotational acceleration, brain injury, 6-2
interstitial fluid transport, 15-4 Rupture, blood vessels, chest and abdomen impacts,
vein capacitance measurement, 12-4
7-1 to 2
Radiopaque markers, regional ventricular mechanics,
8-18 to 19 S
Rarefaction, microcirculation, 13-6 Safety belts, , 7-2, 7-3, 7-9
Reaction load, chest and abdomen impacts, 7-1, 7-7 to 8 Safety standards, vehicle, 6-5, 6-10
Red blood cells, 14-1 Sarcomeres
bending elasticity, 14-6 to 7 cardiac muscle
blood composition, 10-2 to 3 contraction, 8-13, 8-14
cytosol, 14-3 to 4 resting, 8-17 to 18
exercise physiology, 19-7
membrane skeletal muscle, 2-3
force-length relationships, 2-10, 2-11
area dilation, 14-4
constitutive relations, 14-6
shear deformation, 14-4 to 5
Index I-13
force-velocity relationships, 2-11 oxygen and tissue metabolism, 13-6 to 7
morphology, 2-4 pump function
and muscle contraction, 2-5 to 6
normalization of muscle and fiber length, 2-11 hemodynamics, 12-1 to 2
Scintigraphy, vein capacitance measurement, lymphatic transport, 15-3, 15-4
Skull, see Head and neck mechanics
12-4 Sliding motion, joint, 3-2
Screw displacement axis, 3-1 Smooth muscle, vascular
Semicircular canals, 18-9 to 11, 18-12 anatomy, 11-2
arterial wall structure, 10-2
distributed parameter model, 18-9 to 10 and blood volume redistribution, 12-5 to 6
frequency response, 18-10 to 11, 18-12 contraction/relaxation, 12-1
structure and function, 18-1, 18-2 lymphatic networks, 15-6
Septum, ventricular hemodynamics, 8-12 mechanoelectrical transduction, 16-11
Series elasticity, muscle models, 2-10 microvascular hemodynamics
Servo-null method, microcirculatory blood pressure blood flow mechanics, 13-9
nitric oxide synthase in, 13-8
measurement, 13-2 oxygen and tissue metabolism, 13-6 to 7
Severity indices, head and neck injury, 6-5, 6-7 remodeling, 13-10 to 11
Shear deformation, red cell membrane, 14-4 to 5 wall mechanics, 13-3 to 4
Shear modulus nitric oxide and, 13-7
Soft tissue injury, chest and abdomen impacts,
bone, 1-8
membrane, 14-3, 14-4, 14-11; 16-2 to 3 7-1 to 2
Shear rate, cell adhesion, 16-6 to 7 Soft tissue mechanics, musculoskeletal,
Shear strain
cardiac muscle, resting, 8-17, 8-18 see Musculoskeletal soft tissue mechanics
chest and abdomen impacts, 7-2 Solid-type behavior, cell constitutive relations, 16-2
head and neck injury Solutes, transport in microcirculation, 13-8 to 9
Specific tension, skeletal muscle, 2-6
head, 6-2 Speed of deformation, torso, 7-1
neck, 6-7 Speed of impact, see Loading conditions
microvascular blood flow, 13-6 Sphericity, 14-11
Shear stress Spinal cord injury, cervical, 6-10
endothelial remodeling after, 16-11 Spine
vasomotor responses, 13-10
Shoulder joint motion, 3-16 to 19 cervical, head and neck injury, 6-10, 6-11: see also
axes of rotation, 3-19, 3-20, 3-21, 3-22 Head and neck mechanics
geometry of articulating surfaces, 3-16, 3-17
joint contact, 3-17 to 18 chest and abdomen impacts, 7-3, 7-4
Signaling, cell models, 16-11 Spinning motion, joint, 3-2
Single capillary cannulation method, capillary transport Spongiosa, aortic valve, 9-1
Spongy (cancellous) bone, 1-2, 1-8 to 9, 1-16
studies, 13-2 Spring model, cell adhesion, 16-7
Sinuses, heart and blood vessels, 9-2 to 3 Squeeze-film lubrication, 4-8, 4-9, 4-20
Sinus of Valsalva, 9-2 to 3 Standards, safety, 6-5, 6-10
Skeletal muscle Starling-Landis equation, 15-2
Starling pressures, lymphatic transport, 15-2 to 3, 15-4
blood flow, local regulation, 13-9 to 10 Starling's law, 13-8, 13-11
cell models, rolling and adhesion, 16-4 to 5 Starling's law of the heart, 8-11; 19-4
electromyography, 5-2, 5-5, 5-10 State diagram for cell adhesion, 16-5 to 7
exercise biomechanics, factors effecting mechanical Stereophotogrammetric methods, mechanical response
work, 20-1 to 9 of brain, 6-4
age, 20-6 to 7 Stiffness
equilibrium, 20-1 to 2
gender, 20-8 bone, 1-1, 1-4 to 5
genetics, 20-8 to 9 cardiac muscle, resting, 8-17
locomotion, 20-5 to 6 cartilage, 2-4
muscular movement, 20-3 to 4 chest and abdomen impacts, 7-7 to 8
exercise physiology, 19-1 to 9 cochlea, 17-3
cardiovascular system signaling, 19-3 Storage modulus, 1-15
lymph flow rates, 15-11 Strain energy density function, blood vessel,
microcirculation, 13-8
musculoskeletal soft tissue mechanics 11-6 to 12
material properties, 2-5 to 6 anisotropic vessels, 11-10 to 12
modeling, 2-10 to 12 isotropic vessels, 11-7 to 9, 11-10
structure, 2-3 to 4 Strain-energy functions, cardiac muscle, 8-16 to 17
nitric oxide synthase in, 13-8 Strain gauges, heart, 8-18
I-14 Biomechanics
Strain hardening, red blood cells, 14-5 cell modeling, 16-10 to 12
Strain relaxation, cell models, 16-2 cochlear mechanics, 17-1 to 12: see also Cochlear
Strain softening, myocardial, 8-15
Strength training, effects of, 20-8 mechanics
Stressed volume, venous system terminology, 12-2 vestibular hair cells, structure and, 18-12 to 14
Stress relaxation Transfer function, otoliths, 18-7, 18-8
Transport
blood vessel biomechanics, 11-2 lymphatic, 15-1 to 13: see also Lymphatic transport
cell models, 16-2 in microcirculation, 13-6 to 9
red blood cells, 14-5
Stress relaxation function, tendon and ligament, 2-9, gases, 13-6 to 8
measurement methods, 13-2
2-10 solutes and water, 13-8 to 9
Stress response, tendon and ligament, 2-9 Transverse isotropy, bone, 1-5, 1-6, 1-7, 1-10, 1-11, 1-17
Striated muscle, mitral valve, 9-8 Transversely isotropic strain-energy functions, 8-16 to 17
Stride and temporal parameters, gait analysis, 5-3 Transverse strain, myocardium
Surrogates, human contraction, 8-14 to 15
resting, 8-18
abdominal impact modeling, 7-9 Traumatic brain injury, mild (MTBI), 6-4, 6-7, 6-8,
head and neck injury modeling, 6-1, 6-10 to 11
Swann's Lubricating Glycoprotein, 4-11, 4-12, 4-16, 4-18 6-10, 6-11
Sweating, exercise physiology, 19-7, 19-8 Traveling waves, cochlear mechanics, 17-4 to 5
Swimming, cell modeling, 16-10 Tribology, 4-2 to 3
Synovial fluid, 4-7 to 8
Synovial joints, see Joint lubrication friction, 4-2
Synovial lining, whiplash, 6-7 joint disease, 4-16
Systemic arteries, blood flow, 10-3 wear and surface damage, 4-3
Systemic circulation Tricuspid valve, 8-1; 9-8 to 13
arterial hemodynamics, 10-8 dynamics, 9-10 to 13
defined, 10-9 mechanical properties, 9-10
geometrical parameters, 10-1, 10-2 Tropocollagen, 2-2
Systole, see Cardiac cycle Troponin C, 8-13
Two-dimensional fluid with bending resistance, 16-4
T Two-phase continuum model, cell motility, 16-8 to 9
Temperature U
bone thermoelastic effect, 1-16
exercise physiology, 19-7, 19-8, 20-5 Ultrasound, bone studies, 1-5 to 6, 1-12
and red cell viscosity, 14-4, 14-5 Unloading, chest and abdomen impacts, 7-7
Unstressed volume, venous system terminology,
Tendon and ligament
material properties, 2-4 to 5 12-2
modeling, 2-8 to 10 U-P class of models, 2-8
structure, 2-2
V
Tensile strain, chest and abdomen impacts, 7-2
Tensile stress, tendons, 2-5 Valsalva, sinus of, 9-2 to 3
Tension-extension/flexion injuries, neck, 6-3 Valves
Thermal area expansivity, 14-3
Thermal response, exercise physiology, 19-7 to 8 heart, see Heart valves
Thermoelastic effect, bone, 1-16 lymphatic
Thin-film lubrication, 4-6
Thoracic ducts, 15-2 mechanics of, 15-9
Thoracic trauma index (TTI), 7-3 primary, 15-7 to 9
Three-dimensional finite-element methods, mitral valve pumping mechanism with primary and secondary
properties, 9-10 valves, 15-12
Tissue cylinder model, 13-7, 13-11 secondary/intraluminal, 15-6 to 7
Titin, 2-10 vein, 12-1
Tolerance, chest and abdomen impacts, 7-2 Varicose veins, 12-1
Tomography, heart, 8-4, 8-18 Vascular compliance, defined, 12-2
Topology, microvascular networks, 13-6 Vascular endothelial growth factor (VEGF), 13-10
Torsional loads, neck, 6-3 Vascular mechanics, see Blood vessel biomechanics
Trabecular bone, elastic properties, 1-1 to 2, 1-8 to 9 Vascular networks, structure and hemodynamics, 13-6
Transcapillary filtration, lymphatic transport, 15-2 Vascular smooth muscle, see Smooth muscle, vascular
Transduction Vasomotion
defined, 13-11
and lymph formation, 15-10
Index I-15
Vasomotor responses muscle models, 2-10
arteries, 10-2 red cells, 14-1; 16-3
lymphatics, 15-6 torso, 7-1, 7-5
microcirculatory Viscoelastic models
blood flow regulation, 13-9 cell, 16-2
coordination of, 13-10 otholiths, 18-3
nitric oxide and, 13-7 to 8 Viscosity
wall mechanics, 13-4 blood
vasoconstrictors, defined, 10-9
vasodilators, defined, 10-10 apparent, 13-5
microvascular blood flow, 13-3
Veins/venous system, 12-1 to 6 network hemodynamics, 13-6
definitions, 12-2 to 3 plasma, 13-3
dimensions, 10-2 transcapillary fluid shifts, 15-2 to 3
measurement methods, 12-3 to 5 blood cells, 14-11
normal hemodynamics values, 10-3 red cell cytosol, 14-3 to 4
typical values, 12-5 to 6 red cell membrane, 14-3, 14-5
white cells, 14-9 to 10
Velocity, blood, measurement methods, 13-2 bulk, 4-6
Velocity of deformation, chest and abdomen, 7-5 Viscous damping, tendon and ligament, 2-9
Velocity of impact, chest and abdomen, 7-7, 7-10 Viscous injury, chest and abdomen, 7-5 to 6
Velocity profiles Viscous pressure gradient, lymphatic valves, 15-9
Viscous properties
arterial macrocirculatory hemodynamics, arterial hemodynamics, 10-5, 10-6
10-5 to 6, 10-8 chest and abdomen impacts, 7-1, 7-2
dynamic compliance, 7-6, 7-7
heart valve dynamics, 9-5 to 6, 9-11 injury risk assessment, 7-11
Venae cavae, 10-2, 10-3 lumped-mass model, 7-9
Venous circulation, 10-1 Viscous resistance, chest and abdomen impacts, 7-1
Ventricularis, aortic valve, 9-1, 9-3 to 4 Viscous response, chest and abdomen impacts, 7-10
Ventricular wall, see Heart biomechanics Voigt model, 16-2
Ventriculography, contrast, 8-18
Venules W
blood flow mechanics, 13-6 Walking cycle
dimensions, 10-2 gait analysis, 5-1 to 11
lymphatic channels, 15-5 lubrication regimes, 4-19 to 20
Vestibular mechanics, 18-1 to 16
hair cells, 18-12 to 15 Water
capillary transport studies, 13-2
mechanical model, 18-14 to 15 cartilage structure, 2-1, 2-2
transduction, structure and, 18-12 to 14 transport in microcirculation, 13-8 to 9
otholiths, 18-2 to 8
distributed parameter model, 18-2 to 5 Wave equation, bone, 1-5 to 6
frequency response, 18-7 to 8 Waveforms, breathing, 19-7
motion equations, nondimensionalization of, Wave propagation, arterial macrocirculatory
18-5 to 6 hemodynamics, 10-4 to 5, 10-6, 10-7
transfer function, 18-7, 18-8 Wave propagation velocity, 10-4
semicircular canals, 18-9 to 11, 18-12 Wave reflections, arterial hemodynamics, 10-5
distributed parameter model, 18-9 to 10 Wayland-Johnson technique, microcirculatory blood
frequency response, 18-10 to 11, 18-12
structure and function, 18-1 to 2 velocity measurement, 13-2
Videocamera-based systems, gait analysis, 5-4 Wayne State Tolerance Curve for head injury,
Viscoelasticity
arteries, 10-2 6-7, 6-9
arterial circulation, 10-1 Wear
hemodynamics, 10-4
bone, 1-1, 1-12, 1-14 to 16 friction versus, 4-7, 4-9, 4-18, 4-22
cardiac muscle contraction, 8-14 tribology, 4-3
cartilage, 2-4 in vitro studies, 4-11 to 15, 4-18 to 19
cell models Weeping lubrication, 4-8, 4-9, 4-19
mechanotransduction, 16-11 Whalen method, oxygen partial pressure measurement,
transduction, 16-11
defined, 10-10 13-2
endothelial cell cytoskeleton, 16-4 Whiplash, 6-6 to 7, 6-11
heart valves, aortic, 9-3, 9-4 White blood cells, 14-1
microvascular wall mechanics, 13-4
activation, 14-10, 14-11
I-16 Biomechanics
apparent viscosity, 14-9 to 10 Work, mechanical, 20-1 to 9
bending rigidity, 14-8 age, 20-6 to 7
blood composition, 10-3 equilibrium, 20-1 to 2
lymphatic transport, 15-2, 15-11, 15-12 gender, 20-8
mechanical behavior, 14-8 genetics, 20-8 to 9
microvascular blood flow, 13-2 to 3, 13-5 locomotion, 20-5 to 6
modeling muscular movement, 20-3 to 4
cell component properties, 16-4 Woven bone, 1-3
interpretation of experiments, 16-3 Wrist, articulating surface motion, 3-23 to 28
rolling and adhesion, 16-4 to 5, 16-7
nitric oxide and, 13-7 axes of rotation, 3-26 to 27, 3-28
size and shape, 14-7 to 8 geometry of articulating surfaces, 3-24 to 25
stress and strain in two dimensions, 14-2 joint contact, 3-25 to 26
types of, 14-7
Wiederhielm-Intaglietta method, microcirculatory Y
blood pressure measurement, 13-2 Young's modulus
WKB calculations, cochlear mechanics, 17-5, 17-7, 17-8, bone, 1-6, 1-7, 1-14
cartilage, 2-4
17-14 cochlear components, 17-3, 17-4
Womersley number (alpha parameter), 10-3, 10-5, 10-6,
10-8