Download BE112A Topic 1: Introduction to Biomechanics

BENG 112A
Biomechanics
Review
Final Exam
• 3 hours
• Closed book and notes. No calculators or
cell phones. Bring blank blue books.
• 45% of total grade
• 6 questions, 30 points (minutes) each
• 2 “problems”
• 2 short answers
• 2 “essays”
Biomechanics: MechanicsPhysiology
Continuum Mechanics
Physiology
Geometry and structure
Anatomy and morphology
Boundary conditions
Environmental influences
Conservation laws
Biological principles
• mass
• mass transport, growth
• energy
• metabolism and
energetics
• momentum
• motion, flow, equilibrium
Constitutive equations
Structure-function relations
Therefore, continuum mechanics provides a
mathematical framework for integrating the structure
of the cell and tissue to the mechanical function of
the whole organ
MEASURE
MODEL INPUTS
PHYSIOLOGICAL
TESTING
THE CONTINUUM MODEL
Governing Equations
MODEL
IMPLEMENTATION
AND SOLUTION
CLINICAL AND
BIOENGINEERING
APPLICATIONS
Introduction
• Biomechanics is mechanics applied to biology; our specific
focus is continuum mechanics applied to physiology.
• Continuum mechanics is based on the conservation of mass,
momentum and energy at a spatial scale where these
quantities can be approximated as continuous functions.
• 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.
• Biomechanics involves the interplay of experimental
measurement in living tissues and theoretical analysis based
on physical foundations
• Biomechanics has numerous applications in biomedical
engineering, biophysics, medicine, and other fields.
• Knowledge of the fundamental conservation laws of continuum
mechanics is essential.
Constitutive Properties
• The constitutive law describes the mechanical properties
of a material, which depend on its constituents
• Unlike fluids, solids can support a shear stress
indefinitely without flowing
• 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.
• Stress depends on strain and strain rate in viscoelastic
materials; they exhibit creep, relaxation, hysteresis.
• Viscoelastic properties can be modeled by combinations
of springs and dashpots.
Bone
• Bone is a hard and can be approximated as linearly elastic
• The shaft (diaphysis) of long bone consists of compact
cortical bone.
• The epiphyses at the ends of long bone contain spongy
trabecular bone, and are capped with articular cartilage.
• The basic unit of compact bone is the osteon, which forms
the Haversion canal system.
• Bone is a composite of water, hydroxyapatite and collagen.
• Typical compact bone under standard uniaxial testing, has
an elastic modulus of ~ 18 GPa, an ultimate tensile stress
of ~ 140 MPa, an ultimate tensile strain of ~1.5%, and a
yield strain of ~0.08%. Trabecular bone is less stiff, less
dense and less strong.
• Bone strength and stiffness vary with density, mineral
content, and structure
Bone Mechanics: Key Points
• Under physiological loads, bone can be assumed Hookean
elastic with a high elastic modulus (10-20 GPa)
• The microstructure of the bone composite makes the
material response anisotropic.
• Compared with an isotropic Hookean elastic solid which has
two independent technical constants, transversely isotropic
linearly elastic solids have five independent elastic constants
and orthotropic Hookean solids have nine.
• For human cortical bone orthotropy is a somewhat better
assumption than transverse isotropy, but transverse isotropy
is a much better approximation than isotropy.
• The equilibrium equations, together with the constitutive
equation for linear elasticity and the strain-displacement
relation give us Navier’s equations of linear elastostatics.
• They are used to solve boundary value problems for bone.
Bone growth and remodeling:
Summary of key points
• Historical principles
• Wolff’s Law
• Functional adaptation (stress-adaptive remodeling)
• Types of bone remodeling
• internal remodeling
– changes of bone density (and hence strength and stiffness)
– changes of trabecular architecture
• external remodeling
– changes of bone geometry
Collagenous Tissues
• Collagen is a ubiquitous structural protein with
many types all having a triple helix structure that is
cross-linked in a staggered array.
• Some of the most common collagen types are
fibrillar and the collagen can be organized in 1-D,
2-D or 3-D in different tissues to confer different
material properties.
• The 1-D hierarchical arrangement of stiff collagen
fibers in ligaments and tendons gives these
tissues very high tensile stiffness
• The 2-D arrangement of collagen fibers in tissues
such as skin is often quite wavy or disordered to
permit higher strains
Collagenous Tissues (continued)
• Crimping, coiling and waviness of collagen matrix
gives the tissue nonlinear properties in tension.
• Collagen structure in tissues changes with
disease and ageing.
• Different tissue types require different testing
configurations
Soft Tissues
• Soft tissues are structurally complex, hydrated
composites of cells and extracellular matrices
• Their characteristic mechanical properties include:
– Finite deformations, nonlinearity, anisotropy,
inhomogeneity
– Viscoelastic properties including creep, stress
relaxation and hysteresis
– Other anelastic properties such as strain
softening
• Because soft tissues exhibit load-history dependent
behavior, mechanical tests must be repeated until
the tissue is “preconditioned”.
Kinematics
• Deformation Gradient tensor
• Polar decomposition theorem
• Stretch and rotation.
• Lagrangian finite Green’s strain
• Eulerian finite Almansi strain
• Volume and area change
The Finite Element Method
• Evolved first from the matrix methods of structural analysis
in the early 1960’s
• Uses the algorithms of linear algebra
• Later found to have a more fundamental mathematical
foundation
• The essential features are in the formulation
• There are two main formulations that are mostly equivalent
– Variational formulations, e.g. the Rayleigh-Ritz method
– Weighted Residual Formulations, e.g. Galerkin’s method
• Both approaches lead to integral equations (the weak form)
instead of differential equations (the strong form)
• Thus when we discretize the integral we get sums instead
of differences (as we do in the finite difference method)
Nonlinear Elasticity
•
•
•
•
•
•
•
•
Soft tissues have nonlinear material properties
Because strain-rate effects are modest, soft tissues can be
approximated as elastic: pseudoelasticity
Strain energy W relates stress to strain in a hyperelastic
material; it arises from changes in internal energy or
entropy with loading
For finite deformations it is more convenient to use the
Lagrangian Second Piola-Kirchhoff stress
Exponential strain-energy functions are common for soft
tissues
For isotropic materials, W is a function of the principal
strain invariants
Transverse isotropy and orthotropy introduce additional
invariants
For incompressible materials an additional pressure enters
Blood Vessels
• Blood vessels form arterial and venous networks in the
systemic and pulmonary circulations
• Vessel walls have an intima, media and adventitia
• Composite tissue structure affects vessel properties
• Vessels are nonlinear, anisotropic, viscoelastic and exhibit
preconditioning behavior
• Biaxial testing is used to measure anisotropic properties
• Blood vessels have residual stress in the no-load state
• Blood vessel structure, mechanics and residual stress can
change (remodel) with changes in blood pressure
Cell Mechanics
• Cell cytoskeleton composed of actin filaments,
microtubules and intermediate filaments
• Actin filaments resist tension, are polarized and can
catastrophically extend and collapse
• Microtubules resist compression, are polarized and show
treadmilling behavior
• Atomic force microscopy uses low-force indentation of
the cell membrane to study cell mechanics.
• Traction force microscopy observes a cell’s ability to
deform its surroundings to compute shear stress and,
indirectly, cell force.
• Micropost deflection and optical traps
•Cells are dynamic,
•constantly reorganizing their cytoskeleton as a function of their
• Activity and
• Environmental conditions
• Cell mechanics plays important roles in
•
•
•
•
Growth
Development
Tissue remodeling
Tissue homeostasis
Example: endothelial cell
cytoskeleton reorganizes after
adding antibody that clusters
growth factor receptors and
induce signal transduction
necessary to grow the cells
Almqvist, Biophys J, 2004
Cytoskeleton
•Network of fibrous proteins
•Establish and maintain the cell structure
•Allows cells to move
•Provides mechanical strength and integrity
•Intracellular transport
•Cell division/death
Major components of
a cytoskelelaton are:
•Actin filaments
•Microtubules
•Intermediate
filaments
Mel-c melanocyte treated with cytochalasin and stained for F-actin (blue), microtubules
(red), and the melanosome marker TRP-1 (green). John Hammer, NIH
Cell Architecture
Actin filament
-7 nm in diameter
-Elastic fibers
-Polarized
1. Cortical/apical filament:
Anchor transmembrane/cytoskeletal
proteins
2, Basal filament:
Long bundle/stress fibers that transport
organnelles and help cellular loomotion
3. In Skeletal muscles:
Actin interacts with myosin filament to
form actin-myosin complex for muscle
contraction
Microtubule
-25 in diameter (hollow cylinder)
-Brittle in compression
-Polarized
1. Key role in molecular motor and
transport
-Kinesin at the + end
-and dynein at the - end
Intermediate filament
-8-10 nm in diameter
-Plastic deformation
-Nonpolarized
6 types (e.g.,)
Keratin: epithelial cells
Neurofilaments: neurons
Vimenti: endothelial cells
http://www.yellowtang.org/images/three_protein_fi_be_c_la_784.jpg
Actin resists tensile
stretching and generates
internal tension
Purves et al., Life: The Science of Biology, 4th Edition, by Sinauer Associates ; Gimzewski, UCLA
Myosin moves along actin in ATP dependent manner
In skeletal muscle cells for muscle contraction
Each individual myosin type is
polarized and only travels 1 direction.
Different myosins can move different
directions along the actin filament.
Myosin I –
Important in cell motility
and intracellular
transport (assisted
diffusion)
Myosin II –
Important in muscle
contraction (forms thick
filaments)
K. Chein 1999.
Motor proteins transport loads
across cells and move filaments
relative to each other
Schliwa and Woehlke. Nature 422, 759-765 (17 April 2003)
Attachment to ECM
http://www.steve.gb.com/images/science/cell_adhesion_summary.png
Lecture 18, 3-12-13
Today, we will study data obtained using three techniques:
•Micropipette aspiration
•Force range: 10 pN – 1000 nN
•soft cells
•hard cells
•Optical tweezers
•Force range: 0 pN – 200 pN
• human blood cells
•Atomic force microscopy (AFM)
•Force range: 1 pN – 1000 nN
•atrial myocytes and endothelial cells (soft cells)
•hair fibers (composite hard materials)
Where Dh: hydrostatic head height = 2.5 mm
DP: suction pressure
The force F on a static cell in a micropipette
F=Px
Two micropipettes in a chamber. A pneumatic micromanipulator controls the
movement of a micropipette along three orthogonal axes. (a) A spherical cell
being aspirated into a micropipette with a suction pressure *P. (b) An attached
cell being aspirated into a pipette. (c) A closely fitting cell or bead moving
freely in a pipette like a piston in a cylinder. When static, the suction pressure
times the cross-sectional area of the pipette equals the attachment force F.
Blood Cell mechanics with Optical tweezers
Traction Force Microscopy
The basic principle of traction force microscopy
A cell (gray) adheres to an elastic substrate predominantly at
specific locations (black ellipses).
Mechanical forces (solid arrows) result in deformations of the
substrate (dotted arrows) that decay in normal direction (dotted
lines) and in tangential direction (not shown).
The elastic film is supported by a rigid substrate, in general a
microscope coverslip (black).
Merkel, Biophys J 2007
Wang et al. 1999
Traction Force Microscopy
• Measure Displacements
• Create matrix relating shear stress to
displacements
• Solve inverse problem (d=AT for T) to find
traction stress (often called “traction
force”) in units of force/area (pressure)
• Integrate stresses over an area to
compute total force
Micropost Array
Chen lab, University of Pennsylvania
The force on a post can be
calculated knowing the post’s
radius, length and deflection
Linear Viscoelasticity: Summary
of Key Points
•
•
•
•
•
•
In viscoelastic materials stress depends on strain and
strain-rate
They exhibit creep, relaxation and hysteresis
Viscoelastic models can be derived by combining springs
with syringes
3-parameter linear models (e.g. Kelvin Solid) have
exponentially decaying creep and relaxation functions;
time constants are the ratio of elasticity to damping
The instantaneous elastic modulus is the stress:strain
ratio at t=0
The asymptotic elastic modulus is the stress:strain ratio
as t→∞
Quasilinear Viscoelasticity:
Summary of Key Points
•
•
•
•
•
•
The stress-strain relation is not unique, it depends on the
load history.
The elastic modulus depends on the load history, e.g. the
instantaneous elastic modulus E0 at t=0 is not, in general,
equal to the asymptotic elastic modulus E0 at t=0.
The instantaneous elastic response T(e)(t) = E0(t).
Creep, relaxation and recovery are all properties of linear
viscoelastic models.
Creep solution can be normalized by the initial strain to
give the reduced creep function J(t). J(0)=1.
Relaxation solution can be normalized by the initial stress
to give the reduced relaxation function G(t). G(0)=1.
Skeletal Muscle
• Skeletal muscle is striated and voluntary
• It has a hierarchical organization of myofilaments forming
myofibrils forming myofibers (cells) forming fascicles
(bundles) that form the whole muscle
• Overlapping parallel thick (myosin) and thin (actin)
contractile myofilaments are organized into sarcomeres in
series
• Thick filaments bind to thin filaments at crossbridges which
cycle on and off during contraction in the presence of ATP
• Nerve impulses trigger muscle contraction via the
neuromuscular junction
• The parallel and/or pennate architecture of muscle fibers
and tendons affects muscle performance
Muscle Mechanics
•
•
•
•
•
•
•
Skeletal muscle contractions can be twitches or
tetani, isometric or isotonic, eccentric or concentric
Twitch duration varies 10-fold with muscle fiber type
Tetanic contraction is achieved by twitch summation
The isometric length-tension curve is explained by
the sliding filament theory
Isotonic shortening velocity is inversely related to
force in Hill’s force-velocity equation
Hill’s three-element model assume passive and
active stresses combine additively
The series elastance is Hill’s model is probably
experimental artifact, but crossbridges themselves
are elastic
Cardiac Muscle
•
•
•
•
•
•
•
Cardiac muscle fibers (cells) are short and rod-shaped but
are connected by intercalated disks and collagen matrix
into a spiral-wound laminar fibrous architecture
The cardiac sarcomere is similar to the skeletal muscle
sarcomere
Cardiac muscle has a very slow twitch but it can not be
tetanized because the cardiac action potential has a
refractory period
Calcium is the intracellular trigger for cardiac muscle
contraction
Cardiac muscle testing is much more difficult than skeletal
muscle: laser diffraction has been used in trabeculae
Cardiac muscle has relatively high resting stiffness (titin?)
The cardiac muscle isometric length-tension curve has no
real descending limb
Ventricular Function
•
•
•
•
•
•
•
Ventricular geometry is 3-D and complex
Ventricular shape is similar across mammalian species
and prolate spheroidal coordinates provide a useful
approximation
Fiber angles vary smoothly across the wall
Systole consists of isovolumic contraction and ejection;
diastole consists of isovolumic relaxation and filling
Area of the pressure-volume loop is ventricular stroke
work which increases with filling (Starling’s Law)
Ventricles behave like time-varying elastances
The slope of the end-systolic pressure volume relation is
a load-independent measure of contractility or inotropic
state.
Basic Principle
•
For a dielectric particle trapped using an optical tweezer, the main optical forces can be divided into two categories:
•
•
i. Nonconservative absorption and scattering forces and
Ii. Conservative gradient forces.
•
(Absorption forces can be minimized by choosing a trapping frequency that is off-resonance. Hence, only the
scattering force and the gradient force are considered significant for optical tweezers.)
•
The scattering force
1.
arises due to the direct scattering of photons due to incoherent interaction of light with matter.
2. acts in the same direction as incident light and is proportional to the intensity of incident light.
•
The gradient force
1. occurs whenever a transparent material with a refractive index greater than its surrounding medium is placed
within a light gradient.
2. acts in the direction of increasing light intensity and is proportional to the gradient of light intensity.
•
If a dielectric particle is placed within the narrow waist of a sharply focused beam of
light, the scattering force will have a tendency to push the particle away, while the
gradient force will have a tendency to hold the particle within the waist (Figure 4).
Gradient force
scattering force
Figure 4: Optical forces.
Stable trapping occurs when the gradient force is strong enough to overcome the
scattering force.
(A strong gradient force can be achieved by using a high numerical aperture2 (NA) lens to
focus a laser beam to a diffraction-limited spot).
Physical explanation
Interaction of photons and matters can occur in two boundary conditions
A. Rayleigh Regime (D << λ)
In the Rayleigh regime, the particle is very small compared to the wavelength (D << λ). The distinction between the
components of reflection, refraction and diffraction can be ignored. Since the perturbation of the incident wave front is minimal, the
particle can be viewed as an induced dipole behaving according to simple electromagnetic laws.
NOT COMMONLY USED FOR BIOLOGICAL SYSTEMS and hence ignored
B. Ray Optics Regime (D >> λ)
In the ray optics regime, the size of the object is much larger than the wave lenght of the light, and a single beam can
be tracked throughout the particle. (This situation is for example when whole cells are trapped using infrared light while suspended in
solution. The incident laser beam can be decomposed into individual rays with appropriate intensity, momentum, and direction. These rays
propagate in a straight line in uniform, nondispersive media and can be described by geometrical optics)
According to this model, “the basic operation of optical tweezers can be explained by the momentum transfer associated
with the redirection of light at a dielectric interface”.
When light hits a dielectric interface, part of the light is refracted and part of it is reflected. Figure 5 shows a light ray
with momentum ~pi being incident upon a dielectric sphere with an index of refraction higher than the medium surrounding it.
•The light momentum reflected at the first interface is pi1,
•the light momentum that exits from the sphere after refraction at the second interface is pi2.
(In reality, a small fraction of the light ray will be reflected back into the sphere, causing an
infinite number of internal reflections, but this can be ignored during a first approximation)
The net change of momentum of the single ray of light, Dpi = pi1+ pi2 - pi
By representing the light beam as a collection of light rays, the total change of light momentum is
From Newton’s Second Law, the resulting force acting on the light is given by the rate of change of light
momentum
According to Newton’s Third Law, the dielectric sphere will experience an equal and opposite trapping force
Figure 5: Qualitative ray optics model
•The above equations ignore internal reflections and polarization effects.
•The net effect of internal reflections is to add to the scattering force, making the trap weaker.
•In practice, the equilibrium position of the sphere lies slightly beyond the focal point of the beam.
•In fact, ray optics theory predicts that the exact equilibrium location of the trap should be approximately 3•5% of the sphere diameter beyond the laser focus.
•Polystyrene (C8H8) beads are commonly used for trapping.
•Polystyrene has a density of 1040-1070 kg/m3,
•dielectric constant of 2-2.8,
•electric resistivity of 1013-1015 Ώm,
•heat capacity of 1200-2100 J/kg.K,
•thermal conductivity of 0.12-0.193 W/m.K,
•and visible transmission of 80-90%.
•Since water and polystyrene have almost identical densities, the net force due to gravity can be neglected.