incremental modulus E (kPa)

(1)

Centre for Centre for Centre for

Centre for Health Health Health Health Engineering Engineering Engineering Engineering CNRS UMR 5146

StSt

StStééééphane Avril and coll.phane Avril and coll.phane Avril and coll.phane Avril and coll.

Elastic and fracture properties of aneurismal aortic tissue:

novel results using imaging techniques

(2)

Grenoble - 2011/11/18 - Fed3G - Stéphane AVRIL

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION

(3)

Motivations

VVVVascularascularascularascular disordersdisordersdisordersdisorders

Atherosclerotic plaque

Hypertension

Vascular reconstruction

……

THE WALL MECHANICS IS ESSENTIALTHE WALL MECHANICS IS ESSENTIALTHE WALL MECHANICS IS ESSENTIALTHE WALL MECHANICS IS ESSENTIAL

Aneurysms

(4)

ascending aorta

descending aorta

(thoracic aorta and abdominal aorta)

arch of aorta ▶ a local dilation of the aorta

due to aortic wall weakening

a fatal medical emergency aneurysm rupture

Aortic aneurisms

(5)

Dissection is a tear that develops in the intima of the aorta, the blood enters at the site of the tear, separates the layers of the aorta, and spreads the dissection

Pathologies affecting the aortic arch

(6)

Numerical modeling

(7)

Design of vascular biomaterials

(8)

NumericalNumerical simulations NumericalNumerical simulations simulations aimedsimulations aimedaimedaimed atatatat supportingsupportingsupportingsupporting the the the the surgical

surgicalsurgical

surgical decisiondecisiondecisiondecision

Towards predictive models?

(9)

State of the art about the State of the art about the State of the art about the State of the art about the mechanicalmechanicalmechanical propertiesmechanical propertiespropertiesproperties of of of of the

the the

the aorticaorticaorticaortic wallwallwallwall

(10)

DC h R R

R h p R

E /

/

/ =

∆ ∆

=

0 200 400 600 800 1000 1200

Isnard et al, 2001, normal

Muhs et al, 2006 and Lu et al,

Lénàrd et al., 1999, < 35

Lénàrd et al., 1999, >35

Isnard et al, 2001,

incremental modulus E (kPa)

10% volume variation between diastole and systole

→ Windkessel effect

Elastic properties

(11)

Experimental considerations

UsualUsualUsualUsual protocolprotocolprotocolprotocol::::

Truestress (MPa)

True strain

diastole systole

Physiological modulus

Stress – Strain curve

[Duprey et. al., In-vitro characterisation of physiological and maximum elastic modulus of ascending thoracic aortic aneurysms using uniaxial

(12)

Haskett D, Johnson G, Zhou A, Utzinger U, van de Geest J. Microstructural and biomechanical

Comparison with other locations

(13)

longitudinal

circumferential

σ = F/S =1.7 MPa

Vorp DA, Schiro BJ, Ehrlich MP, Juvonen TS, Ergin MA, Griffith BP. Effect of aneurysm on the tensile strength and biomechanical behaviour of the ascending thoracic aorta. Ann Thorac Surg 2003; 75(4):1210-4.

Failure properties

(14)

W < 10 mJ/cm²

Sommer G, Gasser TC, Regitnig P, Auer M., Holzapfel G.A. Dissection properties of the human aortic media: an experimental study. ASME J Biomech Eng 2008; 130:021007.

Fracture properties

(15)

RelativelyRelativelyRelativelyRelatively good good good good knowledgeknowledgeknowledgeknowledge of of of of averageaverageaverageaverage propertiespropertiespropertiesproperties ((((elasticityelasticityelasticityelasticity, fracture), fracture), fracture), fracture)

Few data about Few data about Few data about Few data about hyperelastichyperelastichyperelastichyperelastic modelsmodelsmodels basedmodels basedbasedbased on the on the on the on the local microstructure (Holzapfel)

local microstructure (Holzapfel) local microstructure (Holzapfel) local microstructure (Holzapfel)

Bad Bad Bad Bad knowledgeknowledgeknowledge about knowledge about about residualabout residualresidualresidual stresses and local stresses and local stresses and local stresses and local variations due to

variations due to variations due to

variations due to curvaturecurvaturecurvaturecurvature and branchesand branchesand branchesand branches

No No No No knowledgeknowledgeknowledge about how knowledge about how about how about how remodellingremodellingremodellingremodelling affects the affects the affects the affects the biomechanical

biomechanical biomechanical

biomechanical propertiespropertiespropertiesproperties of the of the of the of the aorticaorticaorticaortic archarcharch wallarch wallwallwall....

NeedNeedNeedNeed of of of of locallocallocal mechanicallocal mechanicalmechanical characterizationmechanical characterizationcharacterizationcharacterization withwithwithwith advanced

advanced advanced

advanced measurementmeasurementmeasurement techniquesmeasurement techniquestechniquestechniques

Summary of data

(16)

Local Local Local Local approachapproachapproachapproach of the of the of the of the mechanicalmechanicalmechanical behaviourmechanical behaviourbehaviourbehaviour of the of the of the of the arterial

arterial arterial

arterial wallwallwallwall

(17)

A multiA multiA multiA multi----layer layer layer layer materialmaterialmaterialmaterial

Passive Passive Passive Passive mechanicalmechanicalmechanical behaviormechanical behaviorbehaviorbehavior

Multi-layer

Matrix + different fibers

Arteries: a complex structure and behavior

Intima

Media

Smooth muscle cells Elastin

Elastin Elastin

Elastin fibersfibersfibersfibers Collagen Collagen Collagen

Collagen fibersfibersfibersfibers

Biologic sensor and filter

Adventitia CollagenCollagenCollagenCollagen fibersfibersfibersfibers

Anisotropy – Non linearities – Finite strains

(18)

Anisotropic hyperelastic models for arteries

HyperelasticityHyperelasticityHyperelasticityHyperelasticity

Strain energy function:

2^nd Piola-Kirchhoff stress:

AnisotropicAnisotropicAnisotropicAnisotropic hyperelasticityhyperelasticityhyperelasticityhyperelasticity

( )

ψ = ψ E ^where ^E⁼¹₂

(

^{F F}^T^. ⁻^I

)

= ∂ψ S ∂

E

(

^,

)

ψ = ψ E structure tensors

f₁ f₁

(19)

Anisotropic hyperelastic models for arteries

FungFungFungFung’’’’s s s s phenomenologicalphenomenologicalphenomenologicalphenomenological modelmodelmodelmodel

MultilayeredMultilayeredMultilayeredMultilayered HolzapfelHolzapfelHolzapfelHolzapfel’’’’ssss histologyhistologyhistologyhistology----basedbasedbased modelbased modelmodelmodel

ez

eθ

( )

¹ ¹

(

²⁽ ⁱ ⁾²

)

2

k λ - 1 i = fibre1,

fibre2

k ψ = c I -3 +

2

∑

2k e - 1

(

¹

)

2

Q 2 2

11 θθ 22 zz 12 θθ zz

ψ = c

e − ^with Q = a E + a E + 2a E E

[Fung, Biorheology of soft tissues, Biorheology, 1973]

[Gasser, Holzapfel, Ogden, A new constitutive framework for arterial wall mechanics

isotropic anisotropic

matrix fiber families

f1

f2

α

(20)

Example

Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotidcarotidcarotidcarotid arteryarteryarteryartery behaviorbehaviorbehaviorbehavior

Experimental data: Prof. Sutton (U. of South Carolina)

P

3D DIC

Biaxial test

Local strain measurement

(21)

Inverse identification

SolvingSolvingSolvingSolving an inverse an inverse an inverse an inverse problemproblemproblemproblem Basic

Basic Basic

Basic approachapproachapproachapproach: : : updating: updatingupdatingupdating

Parameters A_i

Response M_j = f(A_i)

Measurements m_j

Model: f Matching

optimization

P. Badel, S. Avril, S. Lessner, M. Sutton. Mechanical identification of layer-specific properties of mouse carotid arteries using 3D-DIC and a hyperelastic anisotropic constitutive model, . Computer Methods in Biomechanics and Biomedical Engineering,

(22)

Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotidcarotidcarotidcarotid arteryarteryarteryartery behaviorbehaviorbehaviorbehavior Numerical model:

Optimization algorithm:

Example

FE Model (Abaqus ®) Constitutive model: Holzapfel

β_media β_adventitia

5 parameters: C₁₀, k₁, k₂, β_media, β_adventitia

(23)

Example

Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotidcarotidcarotidcarotid arteryarteryarteryartery behaviorbehaviorbehaviorbehavior Results:

C₁₀ = 0,5 kPa

k₁ = 33 kPa

k₂ = 12.8

β_med. = 46.5°

β_adv. = 27.2°

Good fit, robust identification, but…

(24)

Example

C₁₀,

k₁, k₂, β for media k₁, k₂, β for adventitia

At least 7 parameters needed

One-point data +

7 parameter identification

Multiple solutions

β_media β_adventitia

(25)

CharacterizingCharacterizingCharacterizingCharacterizing hyperelasticityhyperelasticityhyperelasticity of hyperelasticity of of of arteriesarteriesarteriesarteries usingusingusingusing full

full full

full----fieldfieldfieldfield datadatadatadata

Presentation of the virtual fields method

(26)

Experimental method

A more A more A more A more sophisticatedsophisticatedsophisticatedsophisticated testingtestingtestingtesting systemsystemsystemsystem

1

(27)

Experimental method

Reconstruction of Reconstruction of Reconstruction of Reconstruction of displacementdisplacementdisplacementdisplacement fieldfieldfieldfield

Radial displacement

→ Pre-conditionning

8 cycles pressure

→ Applying pre stretch

λ_z = 1.1

→ Applying pressure:

(28)

Experimental method

DerivationDerivationDerivationDerivation of of of of strainstrainstrainstrain fieldsfieldsfieldsfields

(29)

Identification by Virtual Field Method

AssumingAssumingAssumingAssuming constitutive constitutive constitutive constitutive parametersparametersparametersparameters

ψ T

= ρ .sym ∂∂ . + p

σ F F I

E

( )

¹ ¹

(

²⁽ ⁱ ⁾²

)

2

k λ - 1 i = fibre1,

fibre2

c k

ψ = I -3 +

2

∑

2k e - 1

(30)

Reconstruction of Cauchy stress Reconstruction of Cauchy stress Reconstruction of Cauchy stress Reconstruction of Cauchy stress fieldfieldfieldfield

Circumferential Axial

(31)

Identification by the Virtual Fields Method

Are stresses Are stresses Are stresses Are stresses atatatat equilibriumequilibriumequilibriumequilibrium????

The following equations should be satisfied:

(principle of virtual work)

* *

ij ij i i

V V

- σ :ε dV + T u dS = 0

∫

∂

∫

( )

^* ^*

ij ij i i

V V

- σ , A :ε dV + T u dS = 0

∫

^E ∂

∫

Equilibrium ⇔ Actual constitutive properties

(32)

PrinciplePrinciplePrinciplePrinciple of identificationof identificationof identificationof identification

Iterative approach until reconstructed stresses minimize cost function J:

( )

^ij

( )

^*^ij ⁱ ^*ⁱ ²

virtual fields pressure states V V

J A = - σ , A :ε dV + T u dS

∂

 

 

 

∑ ∑ ∫

^E

∫

Internal Virtual Work

( IVW )

External Virtual Work

( EVW )

(33)

ResultsResultsResultsResults of of of of optimizationoptimizationoptimizationoptimization

1 1.05 1.1 1.15 1.2 1.25

0 2 4 6 8 10 12 14 15 18

20 150

15 30 45 60 75 90 120 135

105

Pressure [mm Hg]

0

Pressure [kPa]

λλλλ

Circumferential elongationλλλλ

Circumferential elongation Experimental data

Neo Hookean

«Yeoh »

Fung exponential

Best fitting parameters:

(Holzapfel model, 1 layer)

Avril S, Badel P, Duprey A., Anisotropic and hyperelastic identification of in vitro human arteries from full-field optical measurements, Journal of Biomechanics, Volume 43, Issue

Holzapfel

(34)

CharacterizingCharacterizingCharacterizingCharacterizing aneurismalaneurismalaneurismal tissue up to rupture aneurismal tissue up to rupture tissue up to rupture tissue up to rupture using

using using

using fullfullfullfull---field-fieldfield datafield datadatadata

(35)

ascending aorta

descending aorta

(thoracic aorta and abdominal aorta)

arch of aorta ▶ a local dilation of the aorta

due to aortic wall weakening

a fatal medical emergency aneurysm rupture

Aortic aneurisms

(36)

Deformation gradient Lagrange strain

Aneurismal

aortic tissue Inflation test Optical Full-field measurement ( Full-field displacement)

Inverse procedure

Application of the special Virtual Fields Method Identification of

material parameters Constitutive model

Calculation of

Methodology

(37)

an excised cylindrical aneurismal aortic tissue

a square specimen removing loose connective tissue

finding an appropriate location to separate

specimen is mounted on the inflation test device

making a speckle pattern separated layers two layers are pulled each other to separate cut

adventitia media

media

adventitia

x y

diameter: 30mm

Materials

(38)

inflation device cylinder

pressure gage

in vivo loading environments

(biaxial stress state due to internal pressure) can be generated

Inflation test

(39)

camera

Instron machine protector

Undeformed Deformed

x y

tracks the gray value pattern Digital image

stereocorrelation

(40)

Theory of finite deformation

Deformation gradient F

right Cauchy-Green tensor C = F^TF

Ux Uy Uz

from the undeformed and deformed

coordinates of each measurement data point

Assumption: plane stress

homogeneous initial thickness incompressibility

Measured displacement fields

(41)

PrinciplePrinciplePrinciplePrinciple of identificationof identificationof identificationof identification

Iterative approach until reconstructed stresses minimize cost function J:

( )

^ij

( )

^*^ij ⁱ ^*ⁱ ²

virtual fields pressure states V V

J A = - σ , A :ε dV + T u dS

∂

 

 

 

∑ ∑ ∫

^E

∫

Internal Virtual Work

( IVW )

External Virtual Work

( EVW )

S. Avril, P. Badel, A Duprey. Anisotropic and hyperelastic identification of in vitro human arteries from full-field measurements. Journal of Biomechanics -2010, vol 43, N°15, pp 2978-2985.

(42)

43.74 43.74 43.74 43.74ôôôô 37.35

37.3537.35 37.35ôôôô 37.12

37.12 37.12 37.12ôôôô 23.79

23.7923.79 23.79ôôôô 40.15

40.15 40.15 40.15ôôôô 57.7

57.7 57.7 57.7ôôôô αααα

2.3701 2.3701 2.3701 2.3701 5.175

5.175 5.175 5.175 5.1182

5.1182 5.1182 5.1182 9.8838

9.88389.8838 9.8838 1.963

1.9631.963 1.963 6.7701

6.77016.7701 6.7701 k

k k k₂₂₂₂

0.1186 0.1186 0.1186 0.1186 0.126

0.126 0.126 0.126 0.1744

0.1744 0.1744 0.1744 0.3072

0.30720.3072 0.3072 0.1333

0.1333 0.1333 0.1333 0.2858

0.28580.2858 0.2858 kk

kk₁₁₁₁((((MPaMPaMPaMPa))))

36, 38 mm 36, 38 mm36, 38 mm 36, 38 mm 32, 34 mm

32, 34 mm32, 34 mm 32, 34 mm 31, 43 mm

31, 43 mm 31, 43 mm 31, 43 mm 36, 39 mm

36, 39 mm36, 39 mm 36, 39 mm diameter

diameter diameter diameter (both ends) (both ends) (both ends) (both ends)

male, 76 male, 76 male, 76 male, 76 male, 69

male, 69 male, 69 male, 69 male, 68

male, 68 male, 68 male, 68 male, 81 years old

male, 81 years old male, 81 years old male, 81 years old sex, age

sex, age sex, age sex, age

(0.62mm) (0.62mm) (0.62mm) (0.62mm) (1.06mm)

(1.06mm)(1.06mm) (1.06mm) (1.09mm)

(1.09mm) (1.09mm) (1.09mm) (1.02mm)

(1.02mm)(1.02mm) (1.02mm) (0.91mm)

(0.91mm) (0.91mm) (0.91mm) (0.64mm)

(0.64mm)(0.64mm) (0.64mm) (thickness)

(thickness) (thickness) (thickness)

Adventitia Adventitia Adventitia Adventitia Media

Media Media Media Media

Media Media Media Adventitia

Adventitia Adventitia Adventitia Type

Type Type Type

66 66 555

5 44

44 333

3 22

22 111

1 CaseCase

CaseCase

▶k₂ is much higher aneurismal aortic tissue is stiffer than healthy aortic tissue Results

f1

f2

α

(43)

the failure of aneurismal aortic tissue is oriented along preferred directions!

x y

Rupture is characterized by oblique tears in the circumferential direction

Characterization of rupture

(44)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

0 0.2 0.4 0.6 0.8 1 1.2

0 0.1 0.2 0.3 0.4

strain

stress (MPa)

I

II

I

II

I II

III IV

I II

III IV

Media (α^<40^o⁾ Adventitia

(α^>40^o⁾

circumferential direction (σσσσxx) axial direction (σσσσyy)

Stress strain curves

( )

¹ ^k¹

(

^k²⁽^{λ - 1}ⁱ ⁾²

)

ψ = c I -3 +

∑

e - 1 _Ψ_{’ = (1-D)}_Ψ

(45)

p = 0.02 MPa 0.029 MPa 0.038 MPa 0.047 MPa

Rupture mode

A B

ε

x

ε

xy

ε

y

Modes of rupture

(46)

Stress parameter at rupture

) ( cos )

(

sin

²

α σ

²

α σ

σ

_α^R

=

_xx^R

+

_yy^R

1.0522 0.3483

0.4107 0.3686

0.3719 0.6257

σ σσ

σ^R^R^R^Rα((((MPaMPaMPaMPa))))

1.0933 1.09331.0933 1.0933 0.2958

0.29580.2958 0.2958 0.327

0.327 0.327 0.327 0.2163

0.21630.2163 0.2163 0.3398

0.33980.3398 0.3398 1.143

1.143 1.143 1.143 σσ

σσ^R^R^R^R_yy_yy_yy_yy

1.0073 1.00731.0073 1.0073 0.4384

0.43840.4384 0.4384 0.5568

0.55680.5568 0.5568 1.1524

1.15241.1524 1.1524 0.417

0.417 0.417 0.417 0.4189

0.41890.4189 0.4189 σ

σ σ σ^R^R^R^Rxxxxxxxx

Cauchy stress at Cauchy stress at Cauchy stress at Cauchy stress at

rupture ( rupture (rupture ( rupture (MPaMPaMPaMPa))))

adventitia adventitiaadventitia adventitia media

mediamedia media media

mediamedia media adventitia

adventitiaadventitia adventitia type

type type type

6 6 6 6 5

5 5 5 4

4 4 4 3

3 3 3 2

2 2 2 1

1 1 1 Case

Case Case Case

the idea: the aneurysm rupture occurs in a preferred direction Stress at rupture

J. Kim, S. Avril, A Duprey, JP Favre. Experimental characterization of rupture in human aortic

(47)

▶ the failure stress in the axial direction is much higher

in the adventitia layer (about three times) compared to that in the media layer

▶ the failure in the aneurismal aortic tissue may initiate in the media layer

▶ inflation test for the whole layer

even though the media ruptured,

only small hole or no damage was found in the adventitia

▶ means that the adventitia layer plays a very important role in preventing the artery from rupture

Modes of rupture

(48)

Future Future Future Future workworkworkwork

(49)

Tissue engineering

(50)

In vivo imaging

MRI MRI MRI MRI measurementsmeasurementsmeasurementsmeasurements

S. Avril, F. Schneider, C. Boissier, ZY Li. In vivo velocity vector imaging and time-resolved strain rate measurements in the wall of blood vessels using MRI. Journal of Biomechanics, 2010, 44(5) pp 979-983.

(51)

StudentsStudentsStudentsStudents: : : Ambroise Duprey, Jin Kim, Alexandre Franquet, Nicolas : Demanget, Aaron Romo

ColleaguesColleaguesColleaguesColleagues::::

Dr Pierre Badel (Ecole des Mines Saint-Etienne) Dr Katia Genovese (Univ. Basilicata)

Prof Jean-Noël Albertini (Univ Hospital Saint-Etienne) Prof Jean-Pierre Favre (Univ Hospital Saint-Etienne) Dr Laurent Orgéas (Grenoble University)

Prof Christian Geindreau (Grenoble University)

Institutions and Institutions and Institutions and Institutions and fundingfundingfundingfunding partnerspartnerspartners::::partners

Acknowledgements

(52)

Invitation

incremental modulus E (kPa)

Centre for Centre for Centre for

Centre for Health Health Health Health Engineering Engineering Engineering Engineering CNRS UMR 5146

DC h R R

R h p R

E /

/

/ =

∆ ∆

=

( )

(

)

(

)

( )

(

)

∑

(

)

( )

(

)

∑

∫

∫

( )

∫

∫

( )

( )

∑ ∑ ∫

∫

Theory of finite deformation

( )

( )

∑ ∑ ∫

∫

Rupture is characterized by oblique tears in the circumferential direction

( )

(

)

∑

ε

ε

ε

) ( cos )

(

sin

α σ

α σ

σ

=

+

http://euromech534.emse.fr