Transcript Document

Tuesday, February 15, 2005
• Mechanical Testing (continued)
Types of mechanical analysis
•
•
•
•
•
Kinematics - just the connections
Statics- forces without motion
Dynamics- forces with motion
Rigid versus deformable body
FBDs
FEL
FBL
FBR
FER
Mechanics of rigid versus
deformable body
• Rigid body: Sum of forces in all directions
• Deformable body: Sum of differential
stresses in all directions
• Continuum mechanics describes
equilibrium
Loading Types
•
•
•
•
•
Tension- compression
Uniaxial/bi-axial
Bending
Torsion
Shear
• Reaction
• Traction
• Friction
Cytomechanical forces:
•
•
•
•
•
Gravitational:
Muscle contraction:
Contact:
Buoyant:
Hydraulic: (Static or
dynamic)
• Pneumatic
• Fluid shear
Uniaxial
Shear
Pressure
Biaxial
Tension or
Compression
Bending
Twisting
Cell Deformation
• Most cells are constantly deformed in vivo
by both internal and external forces.
• Experimental deformations can be done by
poking, squishing, osmotic swelling,
electrical/magnetic fields, drugs, etc.
• Comparative strain tolerance
• Unit : microstrain (me)
Elasticity (Stiffness)
•
•
•
•
“ut tensio sic vis”
Young’s Modulus: Stress over strain
Shear Modulus: Related to Poisson
Cells have both area and shear stiffness,
mostly due to the cytoskeleton, although
lipids contribute some.
• Comparative Stiffnesses
• Related to polymer cross-linking
Material Parameters
• Moduli: Young’s (E, KV) area (KA ) shear (G),
bending (kf, flexural, energy*length) (also lp)
• Stiff versus compliant (E versus Y)
• Strength (UTS); Failure point
• Brittle versus ductile (Area under stress/strain)
• Incompressible/Compressible (Poisson, n)
• Hardness: Moh’s scale: Talc= 1; Diamond = 10.
To characterize cells- how do they respond to
forces in their environment?
Comparative Mechanical
Properties
Steel
Wood
Bone
Steel Wood Bone
Cells
Strain e
Cellular
‘pre-stress’
Cells
Comparative Stiffness
10000
1200
210
21
14
1
0.007
0.01
0.0002
di
am
on
d
l
st
ee
ne
bo
d
wo
o
er
ru
bb
su
e
0.0001
tis
Modulus (GPa)
100
Material
Elastic Behaviours
Unixaxial stress
Pressure
n<1
E = s/e
n< 0
KA = P/DA/A
1
2
Poisson’s Effect
For most engineering materials, n < 0.3
Materials w ith n = 0.5 are "Incompressible."
Some materials have n> 1
Cauchy Strain
lx  l.xo
ex
v =-(.7-1)/1 = 0.3
l.xo
Y
1
ly  lyo
ey
0.7
0
X
Incompressible
Means no volume change
2
lyo
Poisson's Ratio
n
swelling

ey
ex
Tension
a. Uniaxial tension, b. Flexure
Both with orthogonal strain. Cells
Are in nutrient broth and attached
To substrate.
b. Radial and biaxial tensions
Cell testing methods
Stretching
• Out-of-plane distension of
circular substrates:
• A and B are kinematically
driven, I.e. surface strain of
culture ~ friction between
platen & substrate.
• C and D are kinetically
driven: surface strain ~
fluid interaction with
substrate.
Compressing
Hydrostatic loading (a) and
‘platen abutment’ (b), with a 3D
Cell arrangement, can be either
Confined or without side support
Hydrostatic
Porous
High pO2
Anisotropic strain
Friction:
Nutrient block
Shear
F/A
E
G
2(1  n )
a
Shear Strain = tan(a)
tG tan(a)
Shear due to fluid flow
du
t 
dl
i.e.,  for water = 0.01 Poise
Shear stress from flow in a pipe
Shear rate
dU
t 
dr
DP 2 2
U (r ) 
(a  r )
4 L
DP
t 
r
2L
P1
P2
Shear stimuli to cells
u (r )  r
u (r )
t  [
]
A cone-plate flow chamber, where
 kinematically controls shear rate
l
(
r
)
(dU/dl). Fully developed viscous flows
exist (thin) atop the culture surface:
homogeneous shear stress.
Shear Stimuli
Parallel plate flow chamber,
Kinetically controlling shear
Rate by D P.
DI distribution in a single cell grouped by height
for consecutive 3 min intervals with no flow, and immediately
after flow onset. DI in individual 3D subimages increased
Magnetic tweezers
Wang et al, Science
Pulling on CSK
• Force produced is
proportional to
deflection of a stiff
beam
• Tends to sink into cell.
• AFM best for pure
elastic materials.
Ferromagnetic Bead Integrin/matrix
• Beads can be ‘functionalized’ by coating with RGD
or ‘de-functionalized’ by coating with AcLDL.
• Then beads can be put in with cells, allowed to
attach.
• Cells are then fixed, then decorated with stained
Ab’s for CSK proteins.
• Then compare stain intensity on cells
• Area of contact is uncontrolled
Proteins binding to RGD beads
Optical Tweezer
Large strains to RBCs with
Optical Tweezers
•
•
•
•
High resolution
Refractivity of bead
Trapping in the beam
Limited force
Dao, Lim & Suresh. J. of Mech. & Physics of solids
Ordinary versus phase-contrast
microscopy
Fluid shear and pressure: Blood flow
forces
Microspheres
DIC overlaid with Fluorescence
•Images from confocal laser-scanning microscope optical cross-section
• microspheres with dark red-fluorescent ring stain with a
•green-fluorescent stain throughout the bead.
• Left panel provides represents poor instrument alignment.
•Correct image registration has been achieved in the right panel,
•where the dark red ring is aligned with the green disk.
Microspheres in cells
Particle Tracking
Test both structure and function
5 nM, 33 ms resolution
Heidemann: Trends in Cell Biology 14:160, 2004
Like a flock
Of birds
Stiffness from particle tracking
• Network stiffness by
particle tracking
• Metamorph Software from
Universal Imaging
Dr 2 (t )  4 Dt
Dr (t )  [ x(t  t )  x(t )]  [ y (t  t )  y (t )]
2
kT
D
6a
2
2
• In an ideal elastic material,
the K.E. imparted by KT,
moves the msphere , that is
then subject to restoring
force back to its original
position. MSD = C,
therefore D = C/t.
• For a VE material, D not
constant.
Nuclear lamin
• For a 1 micron sphere in lamin-poor
regions, D ~ 0.21 mm2/s, corresponding to 
= 2 X 10-3 Pa-s.. In water, D ~ 0.44 mm2/s,
corresponding to  = 1 X 10-3 Pa-s
Actin red, microtubules green
• Heterogeneous
distribution: the
polymer solution is
main determinant of
mechanics.
Stiffness from thermal motion
(a)-(c) Serial images of a 23 mm
long relatively stiff fiber. There
is little visible bending,
consistent with a long
persistence length, = 12.0 mm.
(d)-(f) Serial images of a 20 mm
long ¯flexible fiber. There is
marked bending and a short
persistence length, =0.28 mm.
The fibers undergo diffusional
motion and are not adhering to
a glass surface, rather are free
in solution, a necessary
condition for using statistical
mechanics to obtain
persistence lengths. The width
of each frame is 25 mm.
0
0
22
52
42 Seconds
62 S
Video Tracking
Contraction Measurement
contracting
30 mm
resting
TTP
90%
R
0.5
sec
Resting
contraction
Max relaxation
velocity
Max shortening
velocity
control
Swelling
RBCs
Hypo
Ca++ pore
Hypo+ EGTA
•
Necturus erythrocytes loaded with fluo-4 (10 µM) and exposed to UV light emitted from a mercury
vapor bulb and filtered through a FITC cube (400x). (A) Cells display little fluorescence under
isosmotic conditions (n=6). (B) Addition of A23187 (0.5 µM) to the extracellular medium
increased fluorescence under isosmotic conditions (n=6). (C) Exposure to a hypotonic
(0.5x) Ringer solution increased fluorescence compared to basal conditions (n=6). (D) A
low Ca2+ hypotonic Ringer solution (5 mM EGTA) did not display the level of fluorescence
normally observed following hypotonic swelling (n=6).
•
Light et al.
A. Whole Patched Cell
Micropipette
Pp
Qpp
Micropipette
Stretch
Tension
Pi
Mesangial
Cell
C i
Q m (K w)
C o
 C i =constant
B. Isolated Cell
Stretch
Solutes
Pi
Solutes
C i(t)
Qm (Kw )
Mesangial
Cell
C o
Stimulation Protocols
Impulse
Step
Sinusoid
Ramp
Magnitude
TIME
Figure 4.2 Modes (top) and timing protocols
(lower) of force application
Harmonic motion (undamped)
Gel motion follows simple rules
Model will predict dynamic and
Static equilibrium.

m x  2 PAu(t )  k ( x  x0 )

m x  k ( x )

x  2 x  0
Natural Frequency
Damped Spring

c  k
x  x x  0
m
m
Viscosity & Elasticity
• A complex material can be modeled as a purely
viscous material combined with a purely elastic
material, thus mathematically separating the
viscosity of a material from its elasticity. A purely
viscous component is a Newtonian fluid- it has no
memory and no elasticity; it cannot deform as a
solid. Cells generally behave as solid-liquid
composites. V-E tools can quantify their
behaviour, since the models separate viscosity
from elasticity in a kind of finite element model.
Maxwell Model: Differential
method e  e  e
T
E

s T  s E  s
e
s
1/E

For step input:
de/dt=0
s
e  
E

de 1 ds s


dt E dt 

1  s
s  (e  )
E

Maxwell model: Laplace Method
eV
1/E

R
C
1 1
Z (s)  
E s
s
e
e
Compliance
+
Slipperiness
Z
eo
s
For a step input
eo
Viscosity: Pascal-sec
Mechanical
Impedance.
s
s
1 / E  1 / s
t=/E
Transform pairs
f(t)
F(s)
Impulse (t)
1
Step
1/s
e-at
1/(s+a)
1
[e a1t  e a2t ]
a2  a
1
( s  a1 )(s  a 2 )
Maxwell model: Simulink method
• Implicit parameters
Gel/cell Model
• Make a complete model and label all
parameters
• Describe the output, relating what happens
and why.
• What is the time constant?
• State the assumptions and simplifications
Classwork/Homework
• Add damping to your model of cytogel
• Describe how you can model thermal
fluctuations in cell diameter, and list all the
elements. List assumptions.
• Write the model equation for the above.
• Complete a simulink model of the above,
and do all labelling, including all parameter
values.
6
r0   1 0
P 1   2 00
P 0   3 00
7
r1   5 1 0
7
 5 .5 1 0
7
.. 9 .99 991
 0
2
 r12 P 1
r0  P 0  r P 1  r P 0

fr ( r1)   

2 h
 ( h )  ( r0  r1) h  ( r0  r1) 
0
200
fr( r1) 400
600
800
7
5 .10
6 .10
7
7 .10
8 .10
7
r1
7
9 .10
7