Download Topic 1: Introduction to Tissue and Cell Biomechanics

Topic 1: Introduction to Tissue and Cell Biomechanics
Continuum Mechanics
• the mechanical behavior of solids and fluids …
• in a material continuum, the densities of mass, momentum and energy can be
defined at a point, e.g.
m
  lim
V 0 V
•
•
The rules are the conservation laws of mechanics – mass, momentum and energy.
Stress, strain, and rate of deformation vary with position and time. The relation
between them is the constitutive law.
Conservation Laws
Conservation of Mass: Lagrangian
Mass of the material in the initial volume element remains constant as element
deforms.
Conservation of Mass: Eulerian (Continuity Equation)
Rate of increase of mass in fixed region R is equal to the rate that mass flows into
region across bounding surface S
Conservation of Linear Momentum:
Rate of change of linear momentum of particles that lie in fixed region R is equal to
resultant of the body forces b per unit mass acting long the particles + resultant of
surface tractions t(n) along surface S
Conservation of Angular Momentum:
Rate of change of angular momentum of particles that lie within fixed region R is
equal to resultant couple about origin of the body forces b per unit mass acting aplog
particles + resultant couple of surface tractions t(n) along surface S
Conservation of Energy:
Rate of change of kinetic + internal Energy in region R = rate of mechanical work by
body forces b and surface tractions t(n) + rate where heat enters region R across S
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The constitutive law describes the properties of a particular material. Therefore, a
major objective of biomechanics is identifying the constitutive law for biological
cells and tissues.
Knowledge of the fundamental conservation laws of continuum mechanics is
essential.
Topic 2: Statics and Dynamics in Biomechanics
Newton’s Laws
Static Equilibrium:
For a body in equilibrium, the sum of all external forces acting on the body is zero
Also the sum of all external moments around any fixed point is zero
Couples and Moments:
• A couple is a pair of forces, f and -f, equal in magnitude, opposite in direction,
and separated by a distance, d. It can cause a rotation
• Moment: M = r x f
• Newton’s laws also hold for angular velocity, moment, and angular momentum
Angular Momentum
Where to use statics:
Skeletal Joints
• Joints provide mobility and stability in varying degrees
• Degree of mobility vs. stability differs:
• Synarthrodial joints – tight fitting, no mobility, e.g skull
• Amphiarthrodial joints – slight motion, intervening cartilaginous or
ligamentous tissue, e.g. vertebrae
• Diarthrodial joints – articulating, most mobility, e.g.
• shoulder (ball and socket), high mobility (triaxial) but reduced
stability and increased vulnerability;
• elbow, less mobility (biaxial) and less prone to injury
• Joint mobility depends on nature of surfaces, structure of capsular ligament,
length of ligaments, arrangement of muscles.
types:
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hinge (elbow/ankle)
pivot (radioulnar)
condyloid (wrist)
saddle (carpometacarpal of
thumb)
ball and socket (shoulder/hip)
gliding (vertebral facets)
To solve:
Balance X forces: none
Balance Y forces: FJ+W+Wo-FM=0
Balance Moments: FMxa-Wxb-Woxc=0
• But the closer the muscle is to the joint, the larger the range of motion and the
faster the hand can move.
• When the forearm is flexed at 90° to the line of action of FM, the muscle tension
has only a normal rotational component
• For other flexed positions of the forearm, the muscle force also has a tangential
component:
• it has a stabilizing effect when flexion angle >90°
• and a dislocating role when the flexion angle <90°
• Synovial fluid is to distribute the forces at the joint over a relatively large surface
area as pressures
• To make the analysis statically determinate, we disregarded other elbow flexors,
e.g. brachialis and brachioradialis
Skeletal Muscle:
• three types of muscles: skeletal, smooth and cardiac
• striated due to organization of contractile filaments as opposed to unstriated
(smooth) muscle
• contraction types
– concentric (shortening)
– isometric (static)
– eccentric (lengthening)
• agonist muscles (cause a motion)
• antagonist muscles (oppose a motion)
Joints:
• The vector characteristics of the system are known (requires anthropometric data)
– anatomical axes of joint rotation
– muscle attachments
– muscle lines of action
– segment weights and centers of mass
• The system can be simplified to be statically determinate:
– ignore dynamics, i.e. no inertial forces
– ignore deformability of muscles, tendons and bones (rigid motions only)
– only one muscle or muscle group acts
– no joint friction
Static Equilibrium
unknown forces (N)
number of equations (M) from the force and moment balances of statics
Indeterminate System
(overdetermined)
Statically Determinate System
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•
•
(N>M)
Additional information
is needed as answer is
infinitely many
possibilities
Unstable System
(underdetermined)
N=M
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(N<M)
It becomes an
unstable mechanism
Topic 3: Constitutive Properties of Tissues
The Constitutive Law:
• describes mechanical properties of a material, which depend on its constituents
• is a mathematical relation for stress as a function of kinematic quantities, such as
strain or strain-rate
• the validity of the idealization depends not only on the material but on the
mechanical conditions
• determined by experiment
• constrained by thermodynamic and other physical conditions, e.g. conservation of
mass and energy
• should be derived from considerations of material microstructure
Solids
• Can support shear stress indefinitely without flowing
• Assume an unloaded natural shape
• Deform with minimal or substantial energy dissipation
• Usually composites
Fluids
• Liquids and gases
• Gases have lower density and higher compressibility than liquids; dependent on
temperature
• Phase transition as function of temperature and pressure
• Support stress as fluid hydrostatic pressure at rest
• Can not resist a shear stress indefinitely without flowing
• No unique unloaded natural state; conform to the shape of their container
• Dissipate energy as heat when they flow
• Usually mixtures
Elastic solids
• Stress depends only on strain, Tij = Tij(ekl)
• Example: Isotropic Hookean elastic solid, Tij = lekkdij + 2meij
• Return to a unique natural state when loads removed
• Work done during loading is stored as potential energy without dissipation (a
reversible process)
• Hookean solids have a constant elastic modulus, E
• In nonlinear solids, Etangent is dependent on strain
Exponential Stress-Strain Relation of Soft Tissues
Plasticity:
Ductile Vs. Brittle
Ductility can be quantified by the fracture strain, which is the strain at which a test
specimen breaks during a uniaxial tensile test
A material is brittle if it is liable to fracture when subjected to stress
Viscous Fluids
• Shear stress depends on the rate of shear strain, Tij = Tij(Dkl)
• Example: Newtonian viscous fluid, Tij = -pδij + 2μDij
• Linear viscous (Newtonian) fluids have constant viscosity μ
• Viscosity measures resistance to shear,
• Work done on flowing viscous fluids is dissipated as heat
• In non-Newtonian fluids, apparent viscosity depends on the shear rate , e.g.
whole blood is shear-thinning
Viscoelastic Solids and Fluids
Have properties of both viscous and elastic materials
• Stress depends on strain and strain-rate, Tij = Tij(ekl,D kl)
• Hysteresis, energy dissipation during loading and unloading
With these viscoelastic models, elastic stress depends on strain (spring)
Viscous stress depends on strain-rate (dashpot)
• Strains add in series, stresses are the same
• Stresses add in parallel, strains are the same
• In an elastic solid, the stress depends only the strain; it returns to its undeformed
natural state when unloaded.
• In a viscous fluid, the shear stress depends only on the shear strain rate.
Creep-deform under constant stress
Relaxation Functions- relieve stress under constant strain
Viscoelasticity in Soft Tissues
Hysteresis-path dependence, the inputs will affect path
Preconditiong –over time will reach same path
Viscoplastic
• behaves like a viscous fluid after shear stress exceeds a finite yield stress (e.g.
whole blood)
Thixotropic
• sol-gel transformation from solid (gel) to fluid (sol) properties
• induced by shear stress such as agitation or stirring (e.g. actin)
Strain softening
• Also known as the Mullin’s effect
• progressive, irreversible reduction in elastic stiffness induced by increased
maximum previous strain
• e.g. elastomers and small intestine
Topic 4: Structure and Mechanical Properties of Bone
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•
Bone is a hard connective tissue
Infinitesimal strain and linear elasticity are appropriate
•
Cortical (compact) bone
• Dense outer layer
• Woven - found in young subjects (<14-16 yr.) or after injury
• Laminar - replaces woven
• Haversian - formed by vascularization of woven bone; proportion
increases with age
Trabecular (cancellous) bone
• Epiphysis, metaphysis and endostium
• Spongy structure
• High surface area e.g. in human pelvis, trabecular surface area = 20x
periostial surface area
Periostium
• Surrounds entire bone except the articulating surfaces
• Osteogenic inner layer
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Fibrous outer layer
Bone Composition
Composite material:
• ~1/3 organic - extracellular collagen fiber matrix running in lamellae,
impregnated with
• ~1/3 mineral - dense inorganic calcium phosphate: 50x50x200 Å crystals of
hydroxyapatite (3Ca3(PO4)2.Ca(OH)2),
• ~1/3 water
• cells - osteoblasts and osteoclasts
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Bone is a composite of collagen fibers & hydroxyapatite
2/3 dry wt (50% volume) is hydroxyapatite crystals
Bone strength is greater than either of its main constituents
Soft collagen prevents hydroxyapatite from brittle fracture
Stiff hydroxyapatite prevents collagen from yielding
Composite properties depend on structure and bonding between the components
Bone density increases with mineral content and has been correlated (partially)
with strength
The most studied aspect of bone mechanics
Bone strength and fracture depend on:
• specimen preparation - wet, dry, embalmed
• orientation - axial, transverse
• region
– axial strength is highest at mid-diaphysis
– transverse strength is highest at ends
• age - decreases with age
• type - strength of Haversian bone is 30% < lamellar
• type of loading:
Human
Femur
Longitudinal
Tensile
strength (MPa)
Compressive
strength (MPa)
Shear
strength (MPa)
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Transverse
132
58
187
132
67
Axial tension failure
• failure surface is perpendicular to load at high strain rate
• at low strain rate surface is rougher because osteons are pulled out
Compression - fracture plane is at 60° to the load axis
Fracture studied by energy needed to propagate crack
Bone mineralization affects strength
Mechanical Properties of Bone
• Hydroxyapatite E = 165 GPa similar to steel (200 GPa)
• Collagen is nonlinear; at high load Etangent = 1.24 GPa
• Bone composite in tension: E=18 GPa
• UTS 140 MPa or 126 MPa
• stress-strain behavior of dry bone is linear to failure at uniaxial strain of 0.4%
• failure occurs at around 1.2% strain and the curve is nonlinear about 0.4%
• properties vary with age, mode of loading, strain rate, testing environment
• Elastic Properties
• Modulus of elasticity E (tension)
17.6 GPa
• Modulus of elasticity E (compression)
4.9 GPa
• Shear Modulus G (Torsion)
3.2 GPa
• Strength Properties
• Ultimate Tensile Strength UTS
124 MPa
• Ultimate Tensile Strain
1.41%
• Ultimate Compressive Strength
170 MPa
• Ultimate percentage contraction
1.85%
• Ultimate bending strength
160 MPa
• Ultimate torsional shear strength
54 MPa
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• for dense cortical region of the diaphysis
• much lower for spongy cancellous bone
Testing Bone
Properties change during storage
• Drying affects composition and mechanical properties
• Typical storage methods
• Saline (briefly)
• Saline and alcohol (50/50)
• Freezing (in plastic bag) after wrapping in moist gauze or leave muscle on
• Embalming (changes properties)
• Density, 
• measured after fat and oil are removed by boiling
• estimated by radiographic densitometry
• trabecular and cortical bone material are about equal:  = 1.85 - 2.00
g/cm3
• but apparent  of trabecular bone is 0.15 - 1.0 g/cm3
• Mineral content = ash weight (700 °C )/dry weight (60°C for 7d)
Uniaxial Tensile Testing
 Uniaxial tensile testing usually done on standardized
specimens
 l is the gauge length
 Results depend on strain-rate
 Other testing methods
 three-point bending
 uniaxial compression
 torsion (shear)
 ultrasonic (transverse wave speed depends on shear
modulus)
Topic 5: Bone Mechanics and Remodeling
• Yield strain is small < 0.01
• Elastic modulus is high (18 GPa) compared with normal working stresses
• Stress-strain relation is linear in elastic range
• Strain-rate dependence of stress is minor in normal conditions
• Bone is frequently approximated as a linear (Hookean) elastic material
Bone is anisotropic
• Bone is a composite
– mineral matrix
– collagen fibers
• Bone has organized microstructure
– lamellar (layered)
– Haversian (tubular)
– trabecular (spongy, fabric-like)
• Elastic moduli vary with type of loading:
– tension – compression
– bending
– shear
• Elastic moduli vary with orientation
– transverse vs. axial
• Bone is anisotropic
– requires more than two elastic constants
Elasticity Tensor, Cijkl
• fourth-order tensor of elastic constants
• 34 = 81 components
• symmetry conditions:
Tij = Tji ekl = elk
 6x6 = 36 independent constants
2W = Cijklεijεkl = Cklijεklεij
 Cijkl = Cklij  leaves 21 independent constants
• simplest special case – Isotropy:
Tij   kk ij  2  ij
• λ and μ are the Lamé constants
Isotropic Hookean Solids: Technical Constants
Isotropic Hookean Solids: Technical Constants
Orthotropy:
• different properties in the three mutually perpendicular directions:
• 3 Young's moduli; 3 shear moduli; 3 independent Poisson ratios
• structural axes of orthotropic symmetry are defined by bone microstructure
• Long bone structural axes
(1) radial
(2) circumferential
(3) longitudinal
• As for isotropy, stiffness matrix has 12 non-zero components, but 9 independent
Transverse Isotropic
Directions 1 and 2 are similar when compared to 3
• Similarly, n31 and n32 are close compared with n21
•  greater differences between axial and transverse directions than between radial
and circumferential
• Transversely isotropic materials
• one preferred (“fiber”) axis, i.e. long axis of the bone
• in long bones, the "fibers" are the osteons
• isotropic properties in plane transverse to fibers
• stiffness matrix simplifies from 9 to 5 independent constants:
c11=c22
E1 = E2 = Et
E3 = Ef
c13=c23
E
c44=c55
n31 = n32 = nf
n13 = n23 = nf t
Ef
c66=0.5(c11-c12)
n12 = n21 = nt
G31 = G32 = Gf
G12 =
Et
2(1+ nt )
Bone Growth and Remodeling
• Bone continually remodels
– growth, reinforcement, resorption
– depends on stress and strain
• There is an optimal range of stress for maximum strength
– understressed or overstressed bone can weaken
– stresses on fractured bone affect healing
– stress-dependent remodeling affects surgical implant and
prosthesis design, e.g. fracture fixation plates, surgical screws,
artificial joints
– 1978: radiographic evidence of bone resorption seen in 70% of
total hip replacement patients
Remodeling is stress dependent
• Osteoclasts - cells responsible for resorption
• Osteoblasts - cells responsible for growth (hypertrophy)
• compressive stress stimulates formation of new bone and is important for fracture
healing
• loss of normal stress  loss of calcium and reduced bone density
• Time scales:
• remodeling - months/years
• fastest remodeling is due to change in mineral content
• healing - weeks
• growth/maturation – years
Types of Bone Remodeling
Two types of remodeling in bone:
1. surface (external) remodeling
– change in bone shape and dimensions
– deposition on to or resorption of bone material from inner or outer
surfaces
2. internal remodeling
change in: bulk density, trabecular size, orientation, osteon size, etc.
Functional Adaptation
• remodeling of structure, geometry and mechanical properties in response to
altered loading
• Theory of Uniform Strength — attempts to produce the same maximum normal
stress (brittle material) or shear stress (ductile material) throughout the body for a
specific loading
• Theory of Trajectorial Architecture — concentrates material in the paths of force
transmission, such as principal stress lines, e.g. fiber reinforcing of composite
(kevlar-mylar) sails
• Principle of Maximum-Minimum Design — maximize strength for minimum
weight or cost
E.g. Principal stress trajectories of Culmann’s Crane are similar to physiological
trabecular bone architecture
Wolff’s Law:
when loads are changed by trauma or change in activity, functional remodeling
reorients bone trabeculae so they align with the new principal stress axes
• “law of bone transformation” (1884): “there is a perfect mathematical
correspondence between the structure of cancellous bone in proximal femur and
Culmann’s trajectories”
• Culmann’s trajectories and other of Wolff’s assertions were suspect, but
photoelastic studies (Pauwels,1954) confirmed Wolff's law
Torsion:
G=shear modulus
J=polar moment of interia
GJ=torsional rigidity
α=twist per unit length
M=GJ α
Topic 6: Collagen and Collagenous Tissues
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Biomechanics of collagenous tissues
– 1D
Ligament/Tendon
– 2D
Intestine/Blood Vessels/Pericardium
– 3D
Skin/Heart
primary structural protein in the body
most prevalent protein comprising ~30% of ALL proteins
highly conserved between species (i.e.not undergone many evolutionary changes)
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Molecules arranged in staggered pattern
Also relatively resistant to enzymatic breakdown
>20 different types have been identified
• characterized by different α-chains
• each α -chain is coded by a different gene
• exons are often 54 bp long
• 3 bp in a codon, 18 amino acids, 6 sets of Gly-X-Y
Homotrimer all 3 same
Type III=(1(III))3
Heterotrimer
Type I=(1(I))22(I)
Type XI=1(XI)2(XI)3(XI)
Classifications
Fibrillar
I
II
III
V
VI
Fibril Associated
XII
XIV
Network Forming
X
VIII
Filamentous VI
Anchoring VII
Examples
Tendon, Skin, Ligament
Cartilage
Skin Vessels, Tendon
Fetal Membranes - Assoc w/ Type I
Cartilage - Assoc w/ Type II
IX
Cartilage, Cornea
Embryonic Tendon
Fetal Skin & Tendon
IV
Basement Membrane
Hypertrophic Cartilage
Descemets Membrane
Vessels, Skin
Anchoring Filaments
Fibrillar Collagen (I (mostly), III) has greatest stiffness
Summary: Cartilage (II, VI, IX, X)
Skin ( I, VI)
Vessels (III, VI)
Tendons (I, III, XII, XIV)
Material
Collagen
Steel
Bone
Elastin
Stiffness
1000 MPa
200 GPa
18000 MPa
500-600 kPa
UTS
100 MPa
1000 MPa
125 MPa
100-500 MPa
But that's not enough information to predict behavior in tissues...
Tissues are composites
Complex organization
Complex boundary conditions
Ligament/Tendon 1D
• connect bones together (Ligament) vs connect bones to muscle (Tendon)
• Transmit forces, Aid in smooth joint motion, Absorb impacts/stresses, Prevent
large displacements such as dislocations, Basically uniaxial loading elements
Structure
• Loading
• Fibers are parallel
to load axis
• Organization
• some fascicular
organization
• Unloaded =
crimped
• loaded = straight
• Composition
• Collagen 75-80%
• Elastin ‹5 %
• PG 1-2%
Knee:
Lateral
Femur
Medial
Quadriceps
Tendon
Patella
Lateral
Collateral
Ligament Menisci
(LCL)
Medial
Collateral
Ligament
(MCL)
Posterior
Cruciate
Ligament
(PCL)
Fibula
Anterior
Cruciate
Ligament
(ACL)
Tibia
Patellar
Tendon
Mechanical Properties of Tendon/Ligament
100
Tensile Strain
Tensile Stress
Stress (MPa)
75
50
100
25
0
Stress (MPa)
0
75
Ta
2
en
ng
tM
ul
od
4
6
Toe Region Linear Region
50
25
us
8
Strain (%)
10
Tensile Strain
Tensile Stress
Yielding and s Catastrophic
u
Microfailures
Failure
ul
n
ge
n
Ta
tM
od
Intestine – 2D
• Allow distension when digesting food
Strain (%)
• Prevent over-stretching
and consequent damage to other internal organs
0
0
2
4
6
8
10
• Crimped
• 2 primary planes
• doesn’t need to be as complex as skin because
deformations
are predictable
Yielding
and Catastrophic
Toeloaded,
Regionthe Linear
Region
• as tissue gets
collagen
fibers become
more
aligned
with
the axis of
Microfailures
Failure
load
Blood Vessels –2D
Allow distension with increasing blood pressure
Prevent damage to endothelial cells and smooth muscle cells
• Typically Blood vessels have more type III collagen (which is more compliant)
• Also have a lot of elastin
• collagen fiber diameter ~50nm
Three layers:
Adventitial – collagen, elastin, nerves, vessels, fibroblasts
Medial—smooth muscle cells
Intimal—endothelial cells
Collagen straightens with distension
• Collagen is organized in tissue in such a way that it allows increased deformations
in the tissue without actual stretching/straining collagen very much
• The organization is usually such that the axes of the fibers are oriented with the
axis of maximal forces
Skin – 3D
• Protect body from invasion
• Withstand repeated in-plane stresses (knee-elbow)
• Transmit impacts into plane stresses
• Problem: not a well defined direction
• Solution: have collagen oriented in random direction
• Collagen 65 - 70 %
(more type III than ligament)
• Elastin
5 - 10 %
• PG
1.5 - 2 %
Mechanical Properties
• More compliant than ligament or tendon;
needs to be for its functions.
• orientation of coiled fibers change with load
• collagen is stiffer that elastin but has greater
hysteresis (absorbs more energy)
Aging
• collagen crimp decreases with age; stiffness increases
• elastin crimp increases with age; decreasing recoil
Heart—3D
• Pump Blood, allow myocytes to stretch, but prevent overstretch
• Keep blood vessels open
Perimysial
Weave Network
• Transmit contractile forces to chamber
• Elastic recoil
• Lateral slipping - shearing deformation
• Composite
• only 5% of HW
Z-line of Sarcomere
• «1% elastin
Myocytes
Coiled
Perimysial
Fiber
Endomysial Weave
Endomysial Struts
Heart: Endomysial Collagen
Link adjacent myocytes at Z-line of sarcomere
Maintain patency of capillaries
Heart: Perimysial Network
Organize myocardium into laminar sheet architecture
Heart: Coiled Perimysial Fibers
Protect myocytes from overstretching
Major contributor to passive stiffness
Heart: Pathology
Infarction:
• Myocytes die
• Collagen increases in density
• Coiled structure looks more 2-dimensional
Tortuosity
T=Fiber length/midline length
Tortuosity vs. Pressure
• Collagen fibers gradually straighten with increasing pressure
Sarcomere Length vs. Pressure
2.25
SL (µm)
2.2
2.15
2.1
2.05
2
1.95
Data from Grimm et al. 1980
0
25
50
Pressure (mmHg)
75
100
•
Collagen fibers reach near maximal straightening coincident with maximal
sarcomere length
Cardiac Collagen Biomechanics:
Elastica Theory
• Assumptions:
• Myocytes and collagen act in parallel
• Collagen fibers are inextensible
Have bending and torsional rigidity
• Forces act at end of fibers (no torsion)
• Collagen fibers are circular cylinders
• Collagen fibers run parallel to myocytes
• Collagen fibers are helices
•
•
Collagen is a key structural protein in the body
By organizing this stiff material in different ways, the body achieves may
different functions
– 1D
ligament; very stiff; less crimp
– 2D
skin - versatile organization
intestine - less versatile
vessel - less versatile
– 3D
heart - complex organization, large deformations