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

CLINICAL AND BIOENGINEERING

APPLICATIONS

THE CONTINUUM MODEL Governing Equations

THE CONTINUUM MODEL Governing Equations

MODEL IMPLEMENTATION

AND SOLUTION

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

•  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)

The Finite Element 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

Almqvist, Biophys J, 2004

• 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

Cytoskeleton

Mel-c melanocyte treated with cytochalasin and stained for F-actin (blue), microtubules (red), and the melanosome marker TRP-1 (green). John Hammer, NIH

Major components of a cytoskelelaton are: • Actin filaments

• Microtubules

• Intermediate filaments

• Network of fibrous proteins • Establish and maintain the cell structure • Allows cells to move • Provides mechanical strength and integrity • Intracellular transport • Cell division/death

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

Cell Architecture

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

K. Chein 1999.

Myosin I –

Important in cell motility and intracellular transport (assisted diffusion)

Myosin II –

Important in muscle contraction (forms thick filaments)

Each individual myosin type is polarized and only travels 1 direction. Different myosins can move different directions along the actin filament.

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

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)

Lecture 18, 3-12-13

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.

Where Δh: hydrostatic head height = 2.5 µm

ΔP: suction pressure

The force F on a static cell in a micropipette F = P x

Blood Cell mechanics with Optical tweezers

Traction Force Microscopy

Wang et al. 1999

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

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) = E0ε(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).

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).

Gradient force

scattering force

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.

Figure 5: Qualitative ray optics model

• The  light  momentum  reflected  at  the  first  interface  is  pi1,    • the  light  momentum  that  exits  from  the  sphere  a9er  refrac:on  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,  Δpi  =  pi1+  pi2  -­‐  pi    

By  represen:ng  the  light  beam  as  a  collec:on  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

• 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.