Tissue Mechanics Slides

8/2/2019 Tissue Mechanics Slides

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Molecular, Cellular & Tissue Biomechanics

Goal: Develop a fundamentalunderstanding ofbiomechanics over a wide range of length scales.

20.310, 2.793, 6.024Fall, 2006

20.310, 2.793, 6.024

Biomolecules and intermolecular forcesSingle molecule biopolymer mechanicsFormation and dissolution of bondsMotion at the molecular/macromolecularlevel

MOLECULAR MECHANICS

Structure/function/properties of the cell

BiomembranesThe cytoskeletonCell adhesion and aggregationCell migrationMechanotransduction

CELLULAR MECHANICS

TISSUE MECHANICSMolecular structure --> physical propertiesContinuum, elastic models (stress, strain,constitutive laws)ViscoelasticityPoroelasticityElectrochemical effects on tissue properties

Fall, 2006

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Some Learning Objectives

1. To understand the fundamental concepts of mechanicsand be able to apply them to simple problems in thedeformation of continuous media

2. To understand the underlying basis for the mechanicalproperties of molecules, cells and tissues

3. To be able to model biological materials using methodsappropriate over diverse length scales

4. To be familiar with the wide spectrum of measurementtechniques that are currently used to determinemechanical properties

5. To appreciate the close interconnections betweenmechanics and biology/chemistry of living systems

20.310, 2.793, 6.024Fall, 2006

Biomechanics of tissues

Mechanics BiologyI. Linear elastic behavior I. Biochemical and

molecular biology ofII. Viscoelasticity

ECM moleculesIII. Poroelasticity

A. CollagenIV. Electrochemical and superfamily

physicochemical propertiesB. Proteoglycan

superfamily

C. Other glycoproteinsII. Nanomolecular

structures tissue

III. Mechanobiology20.310, 2.793, 6.024

Fall, 2006

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Some preliminaries

Equilibrium -- balance of forces

concept of a stress tensor

Compatibility -- relations between displacements and

strains or deformation

normal strains; shear strains; strain tensor

Constitutive laws

stress -- strain

Youngs modulus; Poissons ratio, shear modulus

linear, isotropic, elastic materials

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Force balance in a single cellDesprat et al.,

Biophys J., 2005.

Figure by MIT OCW.

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Stress distributions along the basal surfaceof a resting cell

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Several material testing methods

Linear extension Linear shear

Biaxial tension

Cone-and-plate rheometer Confined compression

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Graphical image of stress distributions on a cell surface removed

due to copyright restrictions.

Hu, et al., AJP Cell, 2003


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Tissue properties: common simplifyingassumptions

Linear-- the elastic modulus is constant,indpendent of strain amplitude

Homogeneous -- the material is spatiallyuniform

Isotropic-- the material exhibits the same elasticproperties in all directions

Time-independent-- stresses and strains areuniquely related, independent of rate of strain

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Constitutive laws for a linear elastic,

isotropic material= )+2G1 11 ( 11+22+33 11 (11 = 11 22 +33)

E 22 = 11+22+33)+2G22(

1( ) = )+2G22 =

E22 11 +33 33 ( 11+22+33 331 12 =2G12

33 =

E33 11 +22) 13=2G13 (

12 (1+)12 23=2G2312 = =

2G E

13 (1+)13 2G E = = = =13

2G E 12(1+

)(12

)

23 (1+)23

23 = =

2G E E= Youngs modulus = Lame constant= Poissons ratio G= shear modulus

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20.310, 2.793, 6.024Fall, 2006

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12 DIAMOND

STEEL

BONE CONCRETE

SILK F-ACTINTENDON WOOD

TUBULIN

CONTRACTED SKELETAL MUSCLE

ELASTIN

RELAXED SKELETAL MUSCLECOLLAGEN GELS

LUNG PARENCHYMA FIBROBLAST CELLS

ENDOTHELIAL CELLS NEUTROPHILS LYMPHOCYTES

Values of the elastic or Youngs modulus (E) forvarious materials

20.310, 2.793, 6.024Fall, 2006

Linear? Unidirectional tensile tests

Stress-strain behavior of a peptide hydrogel

Linear behavior up tofracture

Relatively lowtoughness due to smallfracture strain

a bneedle needle

peptide

matrix

peptide

matrix

plastic

mesh

6

Figure by MIT OCW. Adapted from Leon, et. al., 1998

Constant Es11

e11

00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

10

20

30

50

60

40

Stress

(N/m 2)>

Strain


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7

Linear? Elastin is one of

main structuralcomponents of tissue

Linear; little hysteresis.

Provides thestretchiness of tissues.

Combination of single-molecule characteristicsand microscale structure.

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Linear? ACL -- different strain rates

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Figure by MIT OCW.

Figure by MIT OCW.

Slow Medium Fast

Range of linearity

E = ds/de = 109Pa

2 4 6 8 10 120

80

60

40

20

Strain(%)

Strass(MPa)

Control

100

80

60

40

20

05 10 15 20

Specimen

fixed at zero

stretch in 10% formalin

ELASTIN

Lig. Nuchae denatured

% Strain = (L/L0)*100

Stress(kPa)

The stress-strain curve of elastin. Data from Fung and Sobin (1981).


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Linear and Isotropic? Right tibia

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Canine aorta showing elastic fiber content

8

Figure by MIT OCW.

Scanning electron micrographs showing a low-power view of dog's

aorta and a high-power view of the dense network of longitudinally

oriented elastic fibers in the outer layer of the same blood vessel.

Images removed due to copyright restrictions. See Haas, K. S.,

S. J. Phillips, A. J. Comerota, and J. W. White. Anat. Rec.

230 (1991): 86-96.

250

200

150

100

50

00 1 2 3 4

Stress(MPa)

Strain (%)

RA-direction

Sample no.

Sample no.

BA-direction

6

6

1

1

3

2

2

4

4

5

5

Longitudinal

direction

Radial direction


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Collagen fiber arrangement in skin and cornea withalternating directions

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Homogeneous? Only rarely

9

Electron micrograph of a

cross-section of tadpole skin. The

arrangement of collagen fibrils is

plywoodlike, with successive layers

of fibrils laid down nearly at right

angles to each other. Image

removed due to copyright

restrictions.

Electron micrographs of parallel

collagen fibrils in a tendon and the

mesh work of fibrils in skin removed

due to copyright restrictions.

Figure by MIT OCW.

Tendon

Fascicle

Tendon Hierarchy

X r ay EM X ray EM X ray EM

SEM

EM SEM

OM

SEM

OM

Evidence:

X- ray

Reticular

membrane

Fascicular

membrane

Waveform or

crimp structureFibroblasts

Tropocollagen

35 staining

sites

640

periodicity

15 35o

Ao

A

o

A

o

Ao

A

o

A

100-200 500- 5000 50-300 100-500

Size Scale

SubfibrilMicrofibril Fibril


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Histological cross-sectionof a diseased carotid artery

stained for smooth musclecells.

Elastic response

initially, then stiff,collagen response athigh degrees ofextension.

H = hypertensive

High wall stressleads to functional

remodeling.

Time-dependent? Peptide gels Pre- and Post-

100 M KCl1000 M KCl

Storage Modulus (G)

Loss Modulus (G)

Gelation. Properties can change with time.1%wt KFE12

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Figure by MIT OCW.

Figure by MIT OCW.

1000

100

10

1

1

0.1

0.01

Oscillatory Frequency (rad/sec)

5 10

Moduli(Pa)

1000

100

10

1

1

0.1

0.01

Oscillatory Frequency (rad/sec)

5 10

Moduli(Pa)

100 M

1000

Cl

Cl

Storage Modulus (G')

Loss Modulus (G")

Primary data for gel formation of 1wt % KFE12 equilibrated with

(a) 0.1mM NaCl and (b) 1.0 mM NaCl. The storage modulus (squares),

G', and loss modulus (circles), G", are plotted against oscillatory frequency

on log-log scales.

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.00 100 200 300 400

Group NGroup H

Pressure (mm Hg) Pressure (mm Hg)

Relativeradius

10 90 170 250 3300.1

1

10

Incremental modulus:

E = d/d

Incrementalelasticmodulus(MPa)

Stiffening


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Shear Modulus of cartilage. Modulus can be

frequency-dependent (Dynamic @ 0.5Hz, 0.8% strain)

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Other complications:

Surface tension inlung parenchyma

Surface tension effects20.310, 2.793, 6.024Fall, 2006

11

Images removed due to

copyright restrictions.

Figure by MIT OCW.

Figure by MIT OCW.

1

0.75

0.5

0.25

0

0.01 0.1 1

10

12.5

15

17.5

20

2.5

2

1.5

1

0.5

0

Dynamic G

Equilibrium: G

Ionic Concentration, M

Dynamic Phase

Dynamic G

Dynamic G

Magnitude, MPa

Phase Angle (degree)

(Mean

(Mean

+-

+- S.D, n= 4,

S.D, n= 3.6)

f =0.5Hz, g = 0.8% )

Equilibrium G, MPa

q.G

200

150

100

50

5 10 15 20

Liquid Inflation

Liquid

Air

Effect of

surface

tension of

gas-liquid

interface Effect of

collagen at

high degrees

of extension

Nearly linear response

at small extensions

Air Inflation

Relaxation Pressure

LungVolume(ml)

Von Neergaard Experiments


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Striated

Othercomplications:

Activecontraction

Underlying basis for mechanical properties

Extracellular matrix, cartilage, and tendon are largely

comprised of:

collagen

elastin

proteoglycan

(role of constituent cells??)

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Figure by MIT OCW.

Figure by MIT OCW.Figure by MIT OCW.

A

B

CD

A- restingB- max contractionC- active component

0

0

0

5 10 15 20

20

25

25

30 35

50 75 100

100

125 150 175

40

60

80

(mm)

(%)

Muscle length

Tension(mN)

Caption:A,resting tension;B, maximumtension produced by optimumstimulus;C, activetension (=B -A);

D,potentialed tension produced by successivestimuli.

Theslightest resting tension was produced at75% ofthe in situ length (20 mm).*

*

*

*

*

Theoptimum length, atwhich theactive tension becomemaximum, is between 100%and 125%ofthein situ length.

Theactive tension declines almostsymmetrically on eitherside ofthe optimallength.

Themaximumn tension potentialed by thesuccessive stimutiwas attained at75% ofthe insitu length.

Macroscopic view

120

100

Tension(%o

fmaximum)

80

60

40

20

01.0 2.0 3.0 4.0

Striation spacing (mm)

Many features of the length-tension relation are simply explained by the

sliding-filament theory.

The peak of the curve consists of a plateau between sarcomere lengths

of 2.05 and 2.2 mm.

Microscopsarcomere

ic -level


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Typical elastic behavior for tissues containingcollagen and elastin

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Collagen fibril formation

Figure by MIT OCW.

Figure by MIT OCW.

STRAIN, (L/L0)

TENSILESTRESS,

(

F/A)

Young's Modulus = /

Elastin

dominated

Failure

Linear Region

Toe Region

Collagen dominated

Synthesis of pro- chain

Self-assembly OfThree pro- chain

2

1

4

5

3

6

7 8

9

Hydroxylation Of

Selected Prolines

And Lysines

Glycosylation Of

Selected Hydroxylysines

Propeptide

3 pro- chain

Procollagen Triple-helix

Formation

H2N

H2N

OHOH

OH

OH

OH

OH

OH

OH

OHOH

OHOH

OH

OH

OH

Secretory vesicle

ER/Golgi compartment

Collagen molecule

Self-Assembly

Into Fibril

Aggregation Of

Collagen Fibrils To

Form A Collagen Fiber

Collagen fiber

Collagen fibril

0.5-3m

10-300nm

Procollagen molecule

Secretion Plasma membrane

OHOH

OH

OH

OH

OHOHOH

OH

OH

OH

OH OH OHCleavage ofpropeptides

The Intracellular and Extracellular Events Involved in the Formation of a Collagen Fibril


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Collagen -- single moleculecharacteristics

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As of 2003, therewere 28 (!)different types ofcollagenidentified.

14

Figure by MIT OCW.

Figure by MIT OCW.

Major Collagen Molecules

Type Molecule composition Structural features Representative

tissues

Fibrillar collagens

[ 1(I)]2

[ 2(I)]

[ 1(IV)]2

[ 2(IV)]

[ 1(VI)][ 2(VI)]

[ 1(IX)][ 2(IX)][ 3(IX)]

[ 1(II)]3

[ 1(III)]3

[ 1(V)]3

Fibril-associated collagens

300-nm-long fibrils

300-nm-long fibrils

300-nm-long fibrils;

often with type I

390-nm-long fibrils

with globular N-terminal

domain; often with type I

Lateral association with type

I; periodic globular domains

Lateral association with

type II; N-terminal globular

domain; bound glycosami-

noglycan

Two-dimensional network

Skin, tendon, bone,

ligaments, dentin,

interstitial tissues

Skin, muscle, blood

vessels

Similar to type I; also

cell cultures, fetal

tissues

Most interstitial

tissues

Cartilage, vitreous

humor;

All basal laminaes

I

II

III

V

VI

IX

IV

Sheet-forming collagens

Cartilage, vitreous

humor

Figure by MIT OCW.

14

12

10

8

6

4

2

0

0-0

50 100 150 200 250 300 350

The force-extension curve of a single

collagen II.

Cover glass

XY Stage

Trap Center

Laser light

Procollagen

Stretching a procollagen II molecule with optical tweezers.

Figure by MIT OCW.


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Type IX collagen decorated with type II collagen

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Figure by MIT OCW.

Figure by MIT OCW.

Type IX collagen

moleculeFibril of type II

collagen

Stretch Relax

Elastic Fiber

Single elastin

molecule

Cross-link

Short section

of a collagen

fibril

collagen molecule

300 X 1.5 nm

50 nm

1.5 nmCollagen triple

helix

Contrast between elastin and collagen


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Entropic elasticity

Proteoglycans (PGs) and

glycosaminoglycans (GAGs)

a) GLYCOSAMINOGLYCANS (GAGs) form gelsi) polysaccharide chains of disaccharide unitsii) too inflexible and highly charged to fold in a compact wayiii) strongly hydrophiliciv) form extended conformations and gelsv) osmotic swelling (charge repulsion)vi) usually make up less than 10% of ECM by weightvii) fill most of the ECM spaceviii) four main groups

a. hyaluronanb. chondroitin sulfate and dermatin sulfatec. heparin sulfate and heparind. keratin sulfate

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Figure by MIT OCW.

Stretch Relax

Elastic Fiber

Single elastin

molecule

Cross-link


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b) Proteoglycans (PGs)i) form large aggregatesii) aggrecan is a large proteoglycan in cartilageiii) decorin is secreted by fibroblastsiv) PGs have varying amounts of GAGs.v) PGs are very diverse in structure and contentvi) PGs and GAGs can also complex with collagenvii) secreted proteoglycans have multiple functionsviii) PG/GAGs have important roles in cell-cellsignaling

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Electron micrograph of proteoglycans in the extracellular matrix of rat

cartillage removed due to copyright restrictions.

See Hunziker, E. B., and R. K. Schenk. J Cell Biol98 (1985): 277-282.


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Figure by MIT OCW.

Figure by MIT OCW.

100 nm

GAG

Core Protein

Polypeptide chain

Short, branched

oligosaccharide

side chain

Decorin

(MW - 40,000)

Aggrecan

(MW - 3 x 106)

Ribonuclease

(MW - 15,000)

Examples of a large (aggrecan) and a small (decorin) proteoglycan found in the extracellular matrix.

They are compared to a typical secreted glycoprotein molecule (pancreatic ribonuclease B). All are

drawn to scale. The core proteins of both aggrecan and decorin contain oligosaccharide chains as well

as the GAG chains, but these are not shown. Aggrecan typically consists of about 100 chondroitin sulfate

chains and about 30 keratan sulfate chains linked to a serine-rich core protein of almost 3000 amino

acids. Decorin "decorates" the surface of collagen fibrils, hence its name.

Aggrecan Aggregate

Core protein

Link proteins

Chondroitin sulfate

Hyaluronan

molecule

Keratin sulfate

1 mm

Schematic drawing of an aggrecan aggregate. It is composed of 100 aggrecan

monomers noncovalently bound to a single hyaluronan chain through two link

proteins that bind to both the core protein of the proteoglycan and to the

hyaluronan chain, thus stabilizing the aggregate.


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Atomic Force Microscopy of Aggrecan

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19

Figure by MIT OCW.

Electron micrograph of an aggrecan aggregate removed due to copyright

restrictions.

Images removed due to copyright restrictions.

Aggrecan Chondroitin sulfate GAGChains

(Fetal Bovine)

Ave. GAG length: ~36 nmInter-GAG spacing: 4-5 nm

(L. Ng, C. Ortiz, A. Grodzinsky)

Hyaluronan Molecule

Link Protein N-terminal Hyaluronan-

binding domain

Keratan Sulfate

Chondroitin

Sulfate

Linking SugarsAggrecan Core Protein

30

20

10

0

-10

0.5 1 1.5 2 2.5

Single Hyaluronan Molecule

Force(pN)

Displacement (mm)

A typical result of the force-displacement relationship from

the single hyaluronan molecule measurement.

Figure by MIT OCW.


20/27

Like charge repulsion accounts for a large fraction (~50%) ofthe stiffness in tissues with high GAG content.

These effects can be eliminated either by shielding withcounter-ions or neutralization by changing pH.

+ + + + + ++

+ + +

++

++ +

+++ + +

+ ++ ++ +

+ +20.310, 2.793, 6.024Fall, 2006

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Confined compression experiments

20

Figure by MIT OCW.

1.2

1

0.8

0.6

0.4

0.2

0.001 0.01 0.1 1

Electrostatic: GAG

otherEquilibriumModulus,MPa

Ionic Strength, M

Equilibrium Modulus of Adult Bovine Articular Cartilage

in Different Ionic Strengths


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ADHESION PROTEINSa) fibronectin

i) principal adhesion protein of connective tissuesii) fibronectin is a dimeric glycoproteiniii) fibronectin interacts with other molecules

b) laminini) found in basal laminaeii) form mesh-like polymersiii) has various binding sitesiv) assembles networks of crosslinked proteins

c) integrinsi) cell surface receptor, for attachment of cells to ECMii) family of transmembrane proteinsiii) two subunits, alpha and betaiv) about 20 different integrinsv) binding sites for ECM components

vi) binding sites for the cytoskeleton and linkage to ECM

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Figure by MIT OCW.

Some common proteoglycans

Proteoglycan

Approximate

molecular weight

of core protein

Type of GAG

chains

Number of

GAG chainsLocation Functions

Aggrecan

Betaglycan

Decorin

Perlecan

Serglycin

Syndecan-1

210,000

36,000

40,000

600,000

20,000

32,000

Chondroitin

Chondroitin

Chondroitin

Chondroitin

Chondroitin

Sulfate +

Sulfate +

Sulfate/

Sulfate/

Sulfate/

KeratanSulfate

Dermatan

Dermatan

Dermatan

Sulfate

Sulfate

Sulfate

Sulfate

Heparan

Heparan

Sulfate

~130

1

1

2-15

10-15

1-3

Cartilage

Cell surface

and matrix

Widespread

in connective

tissue

Basal

laminae

Secretory vesicles

in white blood

cells

Fibroblast and

epithelial cell

surface

Mechanical support;

forms large aggregates

with hyaluronan

Binds to type 1

collagen fibrils

and binds TGF -

Structural and

filtering function

in basal lamina

Helps to package

and store secretory

molecules

Binds TGF -

Cell adhesion;

binds FGF


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Fiall, 2006

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The bundle thickness, mesh size, elastic modulus, and critical strainas a function of R at cA = 11.9 M

22

Figure by MIT OCW.

Figure by MIT OCW. Adapted from Shin, J. H. et al. Proc Natl Acad Sci USA 101(2004): 9636-9641.

R0.2

R0.3

R2

I/R0.6

100

100

100

10-1

10-1

10-1

101

101

102

102

100

10-210

-2

R

0

G0 (Pa)

D (nm)

(m)

Bundle thickness

Mesh size

Shear modulus, G

Strain (21) at

which material becomes nonlinear

Collagen binding

Cell binding

Heparin

C C

N

N NH2

Arg

Gly

Asp

HOOC

100 nm

The Structure of a Fibronectin Dimer

SSSS

binding


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Fall, 2006

The stress-strain relationships of F-actin networks formed with different FLNmutants

We apply a prestress to the network (Inset, single-headed filled arrow) and measure the deformation(Inset, dashed arrow) in response to an additional oscillatory stress (Inset, double-headed filled arrow)

23

Figure by MIT OCW. Adapted from Gardel, M. L. et. al. Proc Natl Acad Sci USA 103

(2006): 1762-1767.

Figure by MIT OCW. Adapted from Gardel, M. L. et. al. Proc Natl Acad Sci USA 103

(2006): 1762-1767.

FLNa

FLNa

FLNa h(-)

FLNa h(-)

FLNb h(-)

FLNb h(-)

FLNb

FLNb

0.0 0.2 0.4 0.6 0.8 1.0 1.2

12

10

8

6

4

2

0

Strain

Stress(Pa)

Strain,g

S

tress,s

s

g

ds

dg

10-2 10-1 100

100

101

101

102

102

103

103

Prestress(Pa)

Differential

Stiffness (Pa)


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Focal adhesion

complex

24

Figure by MIT OCW.

Figure by MIT OCW.

CYTOSOL

{{Lipidbilayer

Cell coat

(glycocalyx)

Transmembrane

proteoglycanAdsorbed

glycoprotein

Transmembrane

glycoprotein

Glycolipid

Simplified diagram of the cell coat (glycocalyx)

Sugar residue=

Fibronectin Extracellular

matrix

Cell membrane

Cytoplasm

Integrin

Focal adhesion

Actin

-

Actinin Signals that controlcellular activities

Talin

VinculinVinculin

Zyxin

Talin

Paxillin

Tensin

Ras

SOS

Grb2

FAK

Srckinase

Paxillin p130


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Viscoelastic models

k

FF

k FF

1k1

k2FF

u

u

u2u1

a)

b)

c)

Maxwell

Voigt

Standard linearsolid

Stress relaxation

Creep

Steady-state response tooscillatory displacement

F(t)=ku0exp(t/)

u(t)=Fk 1exp t/( )

=/k

E*()=E'+iE"Concept of a complex

elastic or shear modulus

Poroelastic materials

Governing equations:

1. Constitutive law

ijtot =2Gij +ijij pij

11

x1

u1

U k p =

1D forms

2. Fluid-solid viscous interactions (DarcysLaw)

3. Conservation of mass

uU=(vf vs) =vrel vs =

t4. Conservation of momentum

=0

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tot

11 = (2G+)11pU

1=kpx

1U

1= u1 +U0t

11=0

x1

2u

1 u1=Hk

t x1

25


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Poroelasticity -- confined compression

Impose displacements at boundaries:u1(x1, t=0) = 0 11u1(x1=L, t>0) = 0

u1

u1

u1(x1=0, t>0) = u0

2u

1 u1 x1=Hk

t x1

Characteristic time ~ L2/Hk x1

Solution (Fourier series)

tu1(x1, t)=u0 1

x1 Ansin

nx1expL L

n n

L2

=n 2

2Hkn

20.310, 2.793, 6.024Fall, 2006

Finite rates of protein folding and unfolding can

also contribute to time-dependent behavior AFM can be used to measure both

the elastic (k) and viscous (z) properties of a single molecule as a

function of extension Sequential unfolding of the

immunoglobulin (Ig) domains of titinduring oscillations to measureviscoelasticity

Siingle molecule elastic and viscousproperties appear to scale witheach other

Kawakami et al., BJ, 200620.310, 2.793, 6.024Fall, 2006

26

Figure by MIT OCW.

Extension (nm)

k(pN/nm)

Force(pN)

(

10-7k

g/sec)

20

20

40

40

60 80 100

0

0

2

4

6

50

100

150


27/27

20.310, 2.793, 6.024Fall, 2006

Additional factors: Ionic strength

+2GFigure by MIT OCW.

1.2

1

0.8

0.6

0.4

0.2

0.001 0.01 0.1 1

Electrostatic: GAG

otherEquilibriumModulus,MPa

Ionic Strength, M

Equilibrium Modulus of Adult Bovine Articular Cartilage

in Different Ionic Strengths

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