Non linear elasticity

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Non linear elasticity
F
L
F
L
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Balloon in a string bag model of arterial
elasticity
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Effect of protein digestion on elasticity
Einc Nm-2 x 105
Elastase
Intact vessel
Collagenase
2.6
2.2
1.8
1.4
1.0
R/Ro
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Problem
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Plink!
Plink!
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Possible Solutions
F
L
From Brown R.E. et al., (1994) Conn Tiss Res 30
295-308
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Possible Solutions
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Possible Solutions

• Stiffness is a combination of
• Force/length relation of elastin fibres
• Resistance of ground substance to shear
• Force/length relation of collagen fibres

Adapted from Brown R.E. et al., (1994) Conn Tiss
Res 30 295-308
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Gradual recruitment of collagen fibres
With thanks to Dr. Martin Zulliger, Laboratoire
de Genie Médicale, Ecole Polytechnique Fédéral de
Lausanne
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Connection between VSMC and collagen
Muscle cell
Collagen fibres
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Connection between elastic lamellae and VSMC
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At low strains f is zero
Therefore a plot of Eobs against we should give a
straight line of gradient Ee and zero intercept
y m x c
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At high strains f is 1
Dividing by wc gives
Therefore a plot of Eobs/wc against we/wc
should Give a straight line of gradient Ee and
intercept Ec
y m x c
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Relationship between caudal artery systolic
pressure and age
104
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Aortic elastin content and age
Normal
Hypertensive
Ex hypertensive
Thoracic
Abdominal
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Aortic collagen content and age
Normal
Hypertensive
Ex hypertensive
Thoracic
Abdominal
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Relationship between Einc measured at low strain
and the weight fraction of elastin
y 5.944x 0.567 r 0.779
Einc Nm-2 x 105
We (elastin/wet weight)
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Relationship between Einc measured at high strain
and the ratio of elastin to collagen
y 0.441x 0.247 r 0.805
2.0
1.5
1.0
Einc/Wc Nm-2 x 108
0.5
0.0
0.0
1.0
2.0
3.0
We/Wc
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Two linear components in parallel
At low stretch
At high stretch
At intermediate stretches
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Elastic face
Ehlers Danlos syndrome
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Limitations

• No account of VSMC
• Fibre angle
• Nature of connection between various components
• Residual strains

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Models which take into account a range of
collagen fibre lengths and consider fibre angle

• G. A. Holzapfel and T. C. Gasser, Journal of
  Elasticity 61, 1 (2000).
• Characteristic fibre angle.
• M. A. Zulliger et al., J Biomech 37, 989 (2004).
• Characteristic fibre angle
• Elastin and collagen ratio
• Assumes no radial gradient in composition and
  structure
• N. J. Driessen et al., J Theor Biol 226, 53
  (2004).
• Assumes collagen fibre alignment between
  principal stretch directions
• Two layer structure with differing characteristic
  angles
• No account of other components
• T.C. Gasser, R.W. and G.A. Holzapfel. Journal of
  the Royal Society, Interface. 3,15-35 (2006).
• 4 families of fibre angles (adventitial, 2 and
  medial, 2)
• No account yet of RS OR VSMC activity

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Need for quantitative data on fibre orientation
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Non linear microscopy

• Two photon excited fluorescence
• Autofluorescence of elastin collagen
• Second harmonic generation
• Autofluorescence of collagen

Both techniques are confocal
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Combining SHG and TPF images

SHG (Collagen)
TPEF (Elastin ?)
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Rat aorta adventitia inwards
Rat aorta
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Assessment of changes in fibre angleThe mandrel
stretching experiment
1mm
0.55 mm
Fresh rat carotid artery
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Rat carotidcircumferential stretch
60
40
20
0
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Quantification in progressBut, in the carotid
artery and aorta of the cow and pig (at least),
there is a sudden transition in fibre orientation
at about 30µm from the IEL
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Bovine aorta
Transverse
Axial
VSMC
Elastin
Collagen
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Mapping fibre angle
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Mapping fibre angle
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Mapping fibre angle
? circumferential direction ?
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The case of the three layered media
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The media of the pig carotid artery appears to
consist of two distinct layers.
three

• Histology
• Comparison of vessels with 1 and 2 layer
  media
• Quasi static stress strain behavior
• Opening angle as a measure of residual strain
• Inflation tests at various axial loads
• Entire media
• Inner layer
• Outer layer

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Rat aorta (a typical elastic artery)
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Pig carotid (an atypical elastic artery)
Transverse section (elastic stain)
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Longitudinal section
Elastin
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Tangential section

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Pig carotid tangential sections
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TEM
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Pig carotid
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Pig aorta
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Pig carotid
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Sheep
Cow
Horse
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Transverse section of rabbit aorta
With thanks to T. Matsumoto
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• Initially we found the two layered media only in
  animals weighing more that 75 kg
• But not all animals gt 75kg
• We believed that 2nd layer depended on size
• There is currently one exception so far
  (excluding young animals)
• Pigmy goat (25kg)
• Question
• Is there anything that links the animals that
  have the second layer?

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Kangaroos, possums, wombats (Marsupials)
Armadillos, sloths, anteaters
Shrews, moles, hedgehogs
Bats
Tree shrews
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Thickness of the two medial layers
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The outer layer is relatively thicker in thinner
vessels
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2
R
0.91
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P lt 0.01
Data from 6 animals
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outer/total
25
20
15
200
400
600
800
Total medial thickness micron
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Outer layer has constant thickness in the pig
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Phenotype of the VSMC
Inner layer Contractile
Outer layer not fibroblast
Pig left Iliac
Pig left Carotid
With thanks to Sheena Bhadye
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Smooth Muscle Actin
Desmin
With thanks to Chris Evagora Luke Timmins
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2 Layer summary

• Carotid artery in rat, pig and cow has an inner
  axially orientated layer ? 30µm thick
• Carotid artery in large mammals has 2 more medial
  layers.
• Size alone or animal order?
• Ungulates
• Cetartiodactyla (even toed ungulates)
• Pigs, sheep, cows, camels, whales
• Perissodactyla (odd toed ungulates)
• Horses, rhinos, tapirs
• Small ungulates peccary?
• Other mammalian orders
• Primates, Afrotheria, Carnivora, Bats, etc. etc.?
• In the pig, the thickness of the outer layer is
  independent of the total medial thickness
• Static elastic properties of two layers differ
• Inner more compliant than entire media more than
  outer
• Preliminary results only

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Human common carotid
Newborn
Dragendorff, O., Gefaesse des Stammes und der
Gliedmassen, in Handbuch der Anatomie des
Kindes, K. Peter, G. Wetzel and F. Heiderich,
Eds. 1931, Verlag von JF Bergmann Munich. p.
361-363.
28 years
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Future work

• Detailed measurement of elastic properties in the
  circumferential and axial direction.
• Extended models of multilayer elasticity.
• Separate elastic properties for each layer.
• Characterisation of VSMC phenotype
• Synthetic organelles
• Ca, K channels
• Further animal studies
• Small ungulates
• Big carnivores

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Overall summary

• Carotid artery in rat, pig and cow has an inner
  axially orientated layer ? 30µm thick
• Carotid artery in large mammals has 2 more medial
  layers.
• Size alone or animal order?
• Ungulates
• Artodactyla (even toed ungulates)
• Pigs, sheep, cows, camels, whales??
• Perissodactyla (odd toed ungulates)
• Horses, rhinos, tapirs
• Big carnivora?
• Small ungulates peccary?
• Primates?
• In the pig, the thickness of the outer layer is
  independent of the total medial thickness
• Static elastic properties of two layers differ
• Inner more compliant than entire media more than
  outer
• Preliminary results only