Introductory Biomechanics From Cells to Organisms 1st Edition by Christopher Ross Ethier ISBN 0521841127 9780521841122

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ISBN 10: 0521841127
ISBN 13: 9780521841122
Author: Christopher Ross Ethier

Introductory Biomechanics is a new, integrated text written specifically for engineering students. It provides a broad overview of this important branch of the rapidly growing field of bioengineering. A wide selection of topics is presented, ranging from the mechanics of single cells to the dynamics of human movement. No prior biological knowledge is assumed and in each chapter, the relevant anatomy and physiology are first described. The biological system is then analyzed from a mechanical viewpoint by reducing it to its essential elements, using the laws of mechanics and then tying mechanical insights back to biological function. This integrated approach provides students with a deeper understanding of both the mechanics and the biology than from qualitative study alone. The text is supported by a wealth of illustrations, tables and examples, a large selection of suitable problems and hundreds of current references, making it an essential textbook for any biomechanics course.

Introductory Biomechanics From Cells to Organisms 1st Edition Table of contents:

1.1 A brief history of biomechanics

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

1.2 An outline of this book

Figure 1.6

Box 1.1 Systems biology and the integration of information

Figure 1.7

References

2 Cellular biomechanics

2.1 Introduction to eukaryotic cellular architecture

Figure 2.1

Figure 2.2

2.2 The cell’s energy system

Figure 2.3

2.3 Overview of the cytoskeleton

Figure 2.4

Figure 2.5

2.3.1 Actin filaments

Figure 2.6

Table 2.1. Summary of Young’s modulus values for F-actin measured by various methods. Modified with permission from Janmey et al. [13].

Figure 2.7

Figure 2.8

2.3.2 Intermediate filaments

2.3.3 Microtubules

Figure 2.9

Figure 2.10

2.4 Cell–matrix interactions

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Table 2.2. Typical magnitudes of quantities measured on the cellular scale (last column)

2.5 Methods to measure the mechanical properties of cells and biomolecules

2.5.1 Atomic force microscopy

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

2.5.2 Optical trapping (“optical tweezers”)

Figure 2.22

2.5.3 Magnetic bead microrheometry

Figure 2.23

2.5.4 Micropipette aspiration

Figure 2.24

Figure 2.25

Figure 2.26

Figure 2.27

Figure 2.28

Figure 2.29

Figure 2.30

Figure 2.31

Extensions of the micropipette aspiration technique

Figure 2.32

Figure 2.33

2.6 Models of cellular biomechanical behavior

2.6.1 Lumped parameter viscoelastic model of the cell

Figure 2.34

Figure 2.35

Figure 2.36

Figure 2.37

2.6.2 Tensegrity model of the cytoskeleton

Figure 2.38

Special case: T = 0

Small strain case

Figure 2.39

Figure 2.40

Figure 2.41

Figure 2.42

2.6.3 Modeling actin filaments as a foam

Unit cell model of the cytoskeleton

Figure 2.43

Predictions of actin network modulus

2.6.4 Computational model of a chondrocyte in its matrix

Figure 2.44

Figure 2.45

Figure 2.46

Figure 2.47

2.7 Mechanotransduction: how do cells sense and respond to mechanical events?

2.7.1 Mechanoreceptors

Figure 2.48

Integrins

Stretch-activated ion channels

Cell-surface receptor proteins

2.7.2 Intracellular signal transduction

Figure 2.49

Cytoskeleton-mediated signal transduction

Biochemically mediated signal transduction

Figure 2.50

2.7.3 Cellular response to mechanical signals

Figure 2.51

2.8 Techniques for mechanical stimulation of cells

Figure 2.52

2.8.1 Compressive loading

Hydrostatic compression

Platen compression

2.8.2 Stretching

Figure 2.53

Figure 2.54

Figure 2.55

Uniaxial stretch

Biaxial stretch

2.8.3 Fluid flow

Viscometers

Box 2.1 Proof that the stress is equi-biaxial for the circular membranes in the devices shown in Fig. 2.55

Figure 2.56

Flow chambers

2.9 Summary of mechanobiological effects on cells in selected tissues

Figure 2.57

2.9.1 Endothelial cells in the vascular system

Figure 2.58

2.9.2 Smooth muscle cells in vascular tissue

2.9.3 Chondrocytes in articular cartilage

2.9.4 Osteoblasts and osteocytes in bone

Figure 2.59

2.10 Problems

Figure 2.60

Figure 2.61

Figure 2.62

Figure 2.63

Table 2.3. For Problem 2.9

Figure 2.64

Figure 2.65

Figure 2.66

Figure 2.67

Table 2.4. Critical buckling loads for 10 microtubules stabilized by microtubule-associated protein. Data approximated from graphical data presented in Kurachi et al. [118].

Figure 2.68

Table 2.5. Young’s moduli for a variety of cell types and measurement techniques. Data after Stamenovic and Coughlin [71].

References

3 Hemodynamics

3.1 Blood rheology

Figure 3.1

3.1.1 Blood composition

Figure 3.2

Table 3.1. Constituents of arterial plasma. From Vander et al. [1]. Reproduced with kind permission of the McGraw-Hill companies.

Table 3.2. Formed elements in blood: erythrocytes (red cells) are responsible for oxygen and carbon dioxide transport; leukocytes (white cells) are responsible in part for immune function; and platelets are responsible in part for blood coagulation. From Caro et al. [2]. Reproduced with kind permission of Oxford University Press.

3.1.2 Relationship between blood composition and rheology

Table 3.3. Geometric parameters of normal human red cells. Statistics are from pooled data on 1581 cells taken from 14 subjects; cells were suspended in 300 mOsm buffer. From Tsang [4] as shown in Fung [5] with kind permission of Professor Y. C. Fung.

Table 3.4. Composition of red cells: as a first approximation, the red cell is essentially hemoglobin and water enclosed in a membrane. From Caro et al. [2]. Reproduced with kind permission of Oxford University Press.

Figure 3.3

Figure 3.4

Table 3.5. Summary of red cell interaction effects on blood rheology

Figure 3.5

Figure 3.6

3.1.3 Constitutive equation for blood

Figure 3.7

3.2 Large artery hemodynamics

3.2.1 Physical characteristics of blood flow patterns in vivo

Figure 3.8

Table 3.6. Typical hemodynamic parameters for selected vessels in a 20 kg dog and a 70 kg human (surface area 1.8 m2). α is the Womersley parameter (Section 3.2.3); ReD is the Reynolds number (based on vessel diameter) given as mean with peak value in parentheses. Values collated from a variety of sources by Milnor [17]. Reproduced with kind permission of Lippincott Williams & Wilkins.

Table 3.7. Comparison of hemodynamic parameters in selected arteries in humans. Q, mean (cycle-averaged) flow rate in ml/s; D, cycle-averaged diameter in mm; U0, mean (cycle-averaged) velocity in cm/s; ReD, Reynolds number based on D, U0 and an assumed blood kinematic viscosity of 3.5 cStokes; τ, mean wall shear stress determined from τ = 8µU0/D = 32 µQ/π D2, in dyne/cm2 (this formula for shear stress ignores approximately 10% error from mean flow–pulsatile interactions). Reproduced from Ethier et al. [12] with permission from WIT Press, Southampton, UK.

3.2.2 Steady blood flow at low flow rates

Figure 3.9

Figure 3.10

Figure 3.11

Box 3.1 Using Equation (3.8) to derive Poiseuille’s law for a Newtonian fluid

Figure 3.12

3.2.3 Unsteady flow in large vessels

Figure 3.13

3.3 Blood flow in small vessels

Figure 3.14

3.3.1 Fahraeus–Lindqvist effect

Figure 3.15

Figure 3.16

Figure 3.17

3.3.2 “Inverse” Fahraeus–Lindqvist effect

Figure 3.18

3.4 Problems

Figure 3.19

Table 3.8. For Problems 3.7 and 3.9

Table 3.9. For Problem 3.14

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Figure 3.26

References

4 The circulatory system

4.1 Anatomy of the vasculature

Figure 4.1

Figure 4.2

Table 4.1. Characteristics of the vascular system in a 20 kg dog; this table is useful for giving a sense of the organization of the vascular tree. Similar to the human, it is clear that the majority of the blood volume is contained within the veins, while the capillaries present the largest cross-sectional area. Some entries in the table are based on direct measurements (e.g., the diameters of the largest vessels); some are based on morphometric studies of small tissue samples and extrapolation to the entire body (e.g., the capillary numbers), subject to constraints such as blood volume and percentage of blood in a given vascular region. Class is simply a way of categorizing vessels; all vessels within a given class have diameters ranging from 50 to 150% of the mean value for that class. Compiled by Milnor [4]. Reproduced with permission.

Figure 4.3

Figure 4.4

Figure 4.5

4.2 The heart

Figure 4.6

4.2.1 Gross anatomy of the heart

Figure 4.7

4.2.2 Qualitative description of cardiac pumping

Figure 4.8

Figure 4.9

4.2.3 Cardiac pumping power and ventricular function

Figure 4.10

Figure 4.11

4.3 Arterial pulse propagation

4.3.1 Systolic and diastolic pressure

Figure 4.12

4.3.2 Windkessel model

Figure 4.13

Box 4.1 Solution of Equation (4.10)

4.3.3 Arterial wall structure and elasticity

Figure 4.14

Figure 4.15

Figure 4.16

4.3.4 Elastic waves

Table 4.2. Values of the “pressure–strain modulus”, Ep, and other arterial parameters in humans. From more complete listings in Milnor [4] and Nichols and O’ Rourke [8], except the values for aortic root.

Figure 4.17

Figure 4.18

Box 4.2 A more complete derivation of Equation (4.22)

Figure 4.19

Box 4.3 Derivation of the Korteweg–Moens wave speed

4.3.5 Pressure–flow relationships: purely oscillatory flow

Figure 4.20

Box 4.4 The electrical/fluid flow analogy

4.3.6 Pressure–flow relationships: mean flow effects

4.3.7 Pressure–flow relationships: deviations from ideality

Viscous losses

Variation of arterial properties

Figure 4.21

Effects of branching and taper

Figure 4.22

Figure 4.23

4.4 The capillaries

Figure 4.24

Figure 4.25

Figure 4.26

4.4.1 Capillary filtration: the experiments of Landis

Figure 4.27

4.4.2 Osmotic pressure

Figure 4.28

4.4.3 Quantitative analysis of capillary leakage

Figure 4.29

4.5 The veins

4.6 Scaling of hemodynamic variables

Figure 4.30

Figure 4.31

Table 4.3. Allometric scaling relationships for selected cardiovascular parameters. a and b are coefficients in the equation y = aMb, where M is body mass in kg; LV, left ventricle; RV, right ventricle; ECG, electrocardiogram.

4.7 Problems

Figure 4.32

Figure 4.33

Figure 4.34

Figure 4.35

Figure 4.36

Table 4.4. For Problem 4.12. Data from Tai et al. [51].

Figure 4.37

Figure 4.38

Table 4.5. For Problem 4.14

Figure 4.39

Figure 4.40

Figure 4.41

Figure 4.42

Figure 4.43

Table 4.6. For Problem 4.24. From Vander et al. [10]. Reproduced with kind permission of The McGraw-Hill Companies.

References

5 The interstitium

5.1 Interstitial fluid flow

5.1.1 Darcy’s law

Figure 5.1

5.1.2 Clearance of edema

Figure 5.2

Detailed parameter derivation

Figure 5.3

5.2 Problems

References

6 Ocular biomechanics

6.1 Ocular anatomy

Figure 6.1

6.2 Biomechanics of glaucoma

6.2.1 Tonometry

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

6.2.2 Drainage of aqueous humor in normal and glaucomatous eyes

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

6.2.3 Aqueous humor circulation in the anterior chamber

Figure 6.11

6.2.4 Optic nerve head biomechanics

Figure 6.12

Figure 6.13

Figure 6.14

6.3 Ocular blood flow

Figure 6.15

Figure 6.16

Figure 6.17

6.4 Problems

Figure 6.18

References

7 The respiratory system

7.1 Gross anatomy

7.1.1 The conducting airways and pulmonary vasculature

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

7.1.2 Associated structures

Figure 7.5

Figure 7.6

Figure 7.7

7.2 Biomechanics of breathing

7.3 Lung elasticity and surface tension effects

Figure 7.8

Figure 7.9

Figure 7.10

Box 7.1 Analogy of an air bubble in liquid

Figure 7.11

Figure 7.12

7.4 Mass transfer

Figure 7.13

Figure 7.14

Figure 7.15

7.4.1 Blood-side acinar mass transfer

Figure 7.16

Figure 7.17

Table 7.1. Normalized blood gas partial pressure, p̂, in the alveolar model as a function of dimensionless axial position, x/Lchar

Complexities associated with O2 and CO2 transport

Figure 7.18

Assumptions of the blood-side model, and more sophisticated models

Figure 7.19

7.4.2 Air-side acinar mass transfer

7.4.3 Whole lung mass transfer

Figure 7.20

Figure 7.21

Table 7.2. Air composition in inspired and expired tidal volumes. All volumes in the table are referenced to BTP. The assumed molar fractions are in column two, corresponding to dry ambient air (trace gases are not shown). The corresponding partial pressures and partial volumes (in a 500 ml inspired tidal volume) are shown in columns three and four. The partial volumes in the expired tidal volume are shown in column 5, calculated as described in the text.

Table 7.3. Comparison of calculated and measured tidal volume compositions. Calculated compositions are from column five of Table 7.2, measured values are from Cooney [15]. With kind permission of Taylor & Francis.

7.5 Particle transport in the lung

Figure 7.22

7.6 Problems

Figure 7.23

Figure 7.24

Figure 7.25

Figure 7.26

Figure 7.27

Figure 7.28

Figure 7.29

Figure 7.30

Figure 7.31

Figure 7.32

Table 7.4. For Problem 7.16

Table 7.5. For Problem 7.17

Table 7.6. For Problem 7.18

Table 7.7. For Problem 7.19

References

8 Muscles and movement

Figure 8.1

8.1 Skeletal muscle morphology and physiology

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

8.1.1 Isotonic versus isometric contraction

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

8.2 Muscle constitutive modeling

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

Figure 8.16

Figure 8.17

Figure 8.18

8.3 Whole muscle mechanics

8.3.1 Parallel versus pinnate muscle types

Figure 8.19

Figure 8.20

8.4 Muscle/bone interactions

8.4.1 Foreleg motion in two species

Figure 8.21

8.4.2 Flexion of the elbow

Figure 8.22

Figure 8.23

Table 8.1. Origin and insertion points and other characteristics of the four major muscles participating in elbow flexion. LH is the distance from the effective center of rotation for the elbow to the muscle origin location on the humerus; LF is the corresponding measurement for the insertion location on the forearm. PCSA is the physiologic cross-sectional area of the muscle (see text). θ = tan− 1(LH / LF) is the angle that the muscle makes with respect to the horizontal when the forearm is in the position shown in Fig. 8.22. “Inserts into” refers to which bone the muscle inserts into in the forearm.

Figure 8.24

Figure 8.25

8.4.3 Biomechanics of the knee

Figure 8.26

Figure 8.27

Table 8.2. Measured femoro-patellar contact loads during squatting, for physiological joint angles. Values in the last column are femoro-patellar contact forces, and should be multiplied by three to get in vivo loads. Values are mean for n = 12 knees (±95% confidence intervals). Reprinted from Huberti and Hayes [35]. With kind permission of Elsevier.

Figure 8.28

Table 8.3. Calculated femoro-patellar contact loads during walking, based on the model shown in Fig. 8.28. Values in the last column are femoro-patellar contact forces, and should be multiplied by three to get in vivo loads. Values are mean for n = 12 knees (±95% confidence intervals). See Fig. 8.28 for explanation of symbols. From Maquet [33]. With kind permission of Springer Science and Business Media.

8.5 Problems

Figure 8.29

Figure 8.30

Figure 8.31

Figure 8.32

Figure 8.33

Figure 8.34

Figure 8.35

Figure 8.36

References

9 Skeletal biomechanics

9.1 Introduction to bone

Figure 9.1

Table 9.1. Distribution of bones in the arms and legs: flexibility is conferred by having more bones (and hence more degrees of freedom) as one moves distally; an exception is the spine, which has 33 vertebrae

Table 9.2. Approximate composition of the extracellular matrix of bone tissue based on consensus values from several sources

9.2 Composition and structure of bone

Figure 9.2

9.2.1 Cortical bone

9.2.2 Trabecular bone

Figure 9.3

Figure 9.4

Figure 9.5

9.3 Biomechanical properties of cortical and trabecular bone

Table 9.3. Summary of mechanical properties of human cortical bone based on data from Reilly and Burstein, Journal of Biomechanics, 8(1975), 393–405.

9.3.1 Cortical bone mechanics

Table 9.4. Summary of mean compressive properties of human trabecular bone from different anatomic locations. Values in parentheses are standard deviations. Femur specimens were pooled from both the proximal and distal femur. The specimens from the tibia, distal femur, and spine were tested in the longitudinal (inferior-superior) direction. The proximal femur specimens were oriented along the neck of the femur. Adapted from Keaveny [11]. Copyright 2001 from Bone Mechanics Handbook by Cowin. Reproduced by permission of Routledge/Taylor & Francis Group, LLC.

9.3.2 Trabecular bone mechanics

Figure 9.6

9.3.3 Trabecular bone mechanics: density dependence

Figure 9.7

Figure 9.8

9.3.4 Trabecular bone mechanics: unit cell models

9.4 Bone fracture and failure mechanics

Figure 9.9

9.4.1 Fast fracture

Figure 9.10

Figure 9.11

Figure 9.12

Figure 9.13

Figure 9.14

Figure 9.15

Table 9.5. Fracture toughness (Kc) and toughness (Gc) of cortical bone loaded in the longitudinal and transverse directions. These values are for mode I loading, as shown in Fig. 9.10. The fracture toughness values for human cortical bone are comparable to those of a tough ceramic, but an order of magnitude less those for aluminium and steel. Adapted from Martin et al. [20] with kind permission of Springer Verlag.

9.4.2 Fatigue fracture

Figure 9.16

Figure 9.17

Figure 9.18

Figure 9.19

9.5 Functional adaptation and mechanobiology

Figure 9.20

9.6 The design of bone

Table 9.6. Incidence of fracture of various bones in males aged 20–45 years. Bone nomenclature is given in Fig. 9.1. Adapted from Currey [26] with permission of the American Society for Bone and Mineral Research.

9.7 Introduction to soft connective tissues

Table 9.7. Biochemical constituents of soft connective tissues. The values for ligament refer to ligaments of the extremities; elastic ligaments (e.g., in the spine) have substantially more elastin (see the description of ligaments in Section 9.9.1). Minor non-collagenous proteins are not listed and make up the remainder of the dry weight.

9.8 Structure of collagen

Figure 9.21

Figure 9.22

9.9 Structure of ligament, tendon, and cartilage

9.9.1 Ligament

Figure 9.23

Figure 9.24

9.9.2 Tendon

Figure 9.25

9.9.3 Cartilage

Figure 9.26

9.10 Biomechanical properties of ligament, tendon, and cartilage

9.10.1 Structural properties

Figure 9.27

Figure 9.28

9.10.2 Material properties

Figure 9.29

Figure 9.30

9.10.3 Material properties: tension

Table 9.8. Equilibrium tensile moduli of skeletal connective tissues. Cartilage specimens were harvested from high-weight-bearing areas (HWA) or low-weight-bearing areas (LWA). The higher tensile moduli in the LWA correlated with a higher collagen to proteoglycan ratio in these regions.

9.10.4 Material properties: compression

Figure 9.31

Figure 9.32

9.10.5 Material properties: viscoelasticity

Figure 9.33

Figure 9.34

Figure 9.35

9.10.6 Material properties: biphasic mixture theory of cartilage

9.11 Problems

Figure 9.36

Figure 9.37

Figure 9.38

References

10 Terrestrial locomotion

10.1 Jumping

10.1.1 Standing jump

Figure 10.1

Figure 10.2

Figure 10.3

10.1.2 Running jumps

Pole vault

Running high jump

Figure 10.4

Figure 10.5

10.2 Description of walking and running

10.2.1 Walking

Figure 10.6

Figure 10.7

Figure 10.8

Figure 10.9

Figure 10.10

Figure 10.11

Figure 10.12

Figure 10.13

Figure 10.14

Step frequency

10.2.2 Running

Figure 10.15

Figure 10.16

10.3 Gait analysis

10.3.1 Kinematics

Figure 10.17

Figure 10.18

Table 10.1. Typical kinematic data gathered from gait analysis of normal subject. Marker locations are as shown in Fig. 10.18. and are measured in the sagittal plane from a fixed reference location following the convention in Fig. 10.17. The frequency of measurement was 50 Hz, with each frame corresponding to one measurement. Data courtesy of Dr. Jennifer A. Moore, University of Toronto.

Figure 10.19

Figure 10.20

10.3.2 Anthropometry

Figure 10.21

Table 10.2. Anthropometric data on segment mass, location of center of mass, radius of gyration and density. Segment masses are expressed relative to total body mass; locations of the centers of mass are expressed relative to the limb segment length (see Fig. 10.21 for definitions of limb segments); the radii of gyration are expressed relative to the segment length, about three points, namely the center of gravity (C of G) for the segment, the proximal end of the segment, and the distal end of the segment. Note that the radii of gyration are for rotation about the z axis (see Fig. 10.17). Collected from various sources in Winter [7]. Reprinted with permission of John Wiley & Sons, Inc.

Table 10.3. Correlations expressing the masses of different body segments in terms of overall body mass (M, in lbm). Compare with the first column of Table 10.2. From Miller and Nelson [1]. With kind permission of Lippincott Williams & Wilkins.

10.3.3 Kinetics

Figure 10.22

Link segment model

Figure 10.23

Figure 10.24

Table 10.4. Typical kinetic (force plate) data gathered from gait analysis of a normal subject. This data was gathered at the same time as that in Table 10.1, and frame numbers are identical for the two tables. Fx and Fy are the horizontal and vertical reaction forces on the foot, following the sign conventions in Fig. 10.17. xcop and ycop are the center of pressure of the reaction forces. The frequency of measurement was 50 Hz, and the mass of the subject was 65 kg. Data courtesy of Dr. Jennifer A. Moore, University of Toronto.

Table 10.5. Kinematic data derived from values in Table 10.1. See text for definition of symbols and derivation of the data.

Figure 10.25

Figure 10.26

Figure 10.27

Joint reaction forces versus bone-on-bone forces

10.4 Problems

Figure 10.28

Figure 10.29

Figure 10.30

Figure 10.31

Figure 10.32

Table 10.6. For Problem 10.14.

Figure 10.33

References

Back matter

Appendix The electrocardiogram

Table A.1. Ionic concentrations in frog muscle fibers and in plasma. The ionic composition of plasma is representative of extracellular fluid composition. Note the substantial differences in ionic composition across the muscle cell membrane, which is typical of the situation in all cells. From Aidley [1] with kind permission of the author.

Figure A.1

Figure A.2

Figure A.3

Table A.2. Standard (Einthoven) electrocardiogram electrode placement locations; the limbs act as conductors so that, for example, an electrode at the left wrist measures a potential at the left shoulder

Table A.3. Voltage differences taken to form standard (Einthoven) electrocardiogram leads (see also Fig. A.6 and Table A.2 for abbreviations).

Figure A.4

Figure A.5

Figure A.6

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