behavior of arteries in vitro

(1)

St St

St Sté é éphane Avril é phane Avril phane Avril phane Avril

Inverse methods for characterizing the anisotropic hyperelastic

behavior of arteries in vitro

(2)

UTAD - 2011/04/27 - Stéphane AVRIL

2

Introduction

Introduction Motivations

Arteries: a complex structure and behavior

1. Characterizing hyperelasticity of arteries using single-gage data

Anisotropic hyperelastic models for arteries

Identification procedures Experimental considerations

2. Characterizing hyperelasticity of arteries using full-field data

Experimental method Identification by Virtual Field Method

Identification by the Virtual Fields Method

Identification by Virtual Field Method

3. Characterizing rupture of arteries using full-field data

Methodology

4. Conclusion & Prospects In vivo applications

(3)

3

V V V Vascular ascular ascular ascular disorders disorders disorders disorders

Atherosclerotic plaque

Hypertension

Vascular reconstruction

……

Numerical Numerical Numerical Numerical simulations simulations simulations simulations

Aneurysms

Knowledge of artery mechanics is fundamental

Methodology

(4)

4

A multi A multi A multi A multi- - -layer - layer layer material layer material material material

Passive Passive Passive Passive mechanical mechanical mechanical mechanical behavior behavior behavior behavior

Multi-layer

Matrix + different fibers Arteries: a complex structure

and behavior

Intima

Media

Smooth muscle cells Elastin

Elastin Elastin

Elastin fibers fibers fibers fibers Collagen Collagen Collagen

Collagen fibers fibers fibers fibers

Biologic sensor and filter

Adventitia Collagen Collagen fibers Collagen Collagen fibers fibers fibers

Anisotropy – Non linearities – Finite strains

Methodology

(5)

5

Methodology

(6)

6

Anisotropic hyperelastic models for arteries

Hyperelasticity Hyperelasticity Hyperelasticity Hyperelasticity

Strain energy function:

2 ^nd Piola-Kirchhoff stress:

Anisotropic Anisotropic Anisotropic Anisotropic hyperelasticity hyperelasticity hyperelasticity hyperelasticity

( )

ψ = ψ E ^where ^E ⁼ ¹ ₂ ( ^{F F} ^T ^. ⁻ ^I )

= ∂ ψ S ∂

E

( ^, )

ψ = ψ E structure tensors

f

₁

f

₁

Methodology

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7

Fung Fung Fung Fung’’’’s s s s phenomenological phenomenological phenomenological model phenomenological model model model

Holzapfel Holzapfel Holzapfel Holzapfel’’’’s s s s histology histology histology histology- - -based - based based model based model model model

e

z

e

θ

( )

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 and a comparative study of material models. Journal of Elasticity, 2000]

isotropic anisotropic

matrix fiber families

f

1

f

2

α

Methodology

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8

Identification procedures

Methodology

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9

Usual Usual Usual Usual protocol protocol protocol protocol::::

T ru e s tr e s s ( M P a )

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 tensile testing, Eur. J. Vascular & Endovascular Surgery, 2010]

Methodology

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10

Experimental considerations

Alternative protocol: closer to physiology Alternative protocol: closer to physiology Alternative protocol: closer to physiology Alternative protocol: closer to physiology

P - d curve d

L

F

F - L curve

Pressure (mmHg)

O u te r d ia m e te r ( m ) A x ia l F o rc e ( m N )

Axial Stretch

[Dye et. al., Altered biomechanical properties of carotid arteries in two mouse models of muscular dystrophy, J. of Applied Physiology, 2007]

P

Methodology

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11

Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotid carotid carotid carotid artery artery artery artery behavior behavior behavior behavior

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

[Sutton et. al., Strain field measurements on mouse carotid arteries using microscopic three-dimensional digital image correlation, J. of Biomedical Materials Research, 2007]

P

3D DIC

Biaxial test

Local strain measurement

Methodology

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Inverse identification

Solving Solving Solving Solving an inverse an inverse an inverse an inverse problem problem problem problem Basic

Basic Basic

Basic approach approach approach approach: : : updating : updating updating updating

Parameters A _i

Response M _j = f(A _i )

Measurements m _j

Model: f Matching

optimization

Methodology

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Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotid carotid carotid carotid artery artery artery artery behavior behavior behavior behavior Numerical model:

Optimization algorithm:

FE Model (Abaqus ®) Constitutive model: Holzapfel

Levenberg-Marquardt with bounds handling

β

_media

β

_adventitia

5 parameters: C

₁₀

, k

₁

, k

₂

, β

_media

, β

_adventitia

Methodology

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Example

Identification of a mouse Identification of a mouse Identification of a mouse Identification of a mouse carotid carotid carotid carotid artery artery artery artery behavior behavior behavior behavior Results:

C

₁₀

= 0,5 kPa

k

₁

= 33 kPa

k

₂

= 12.8

β

_med.

= 46.5°

β

_adv.

= 27.2°

Good fit, robust identification, but…

Methodology

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15

C

₁₀

,

k

₁

, k

₂

, β for media k

₁

, k

₂

, β for adventitia

At least 7 parameters needed

One-point data +

7 parameter identification

Multiple solutions

… Field data

β

_media

β

_adventitia

Richer data is required…

Methodology

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16

2. Characterizing

hyperelasticity of arteries using full-field data

Methodology

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A more A more A more A more sophisticated sophisticated sophisticated sophisticated testing testing testing testing system system system system

1

[Genovese, A video-optical system for time-resolved whole-body measurement on vascular segments, Optics and Lasers in Engineering, 2009]

Methodology

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Experimental method

Reconstruction of Reconstruction of Reconstruction of Reconstruction of displacement displacement displacement displacement field field field field

Radial displacement

→ Pre-conditionning

8 cycles pressure

→ Applying pre stretch

λ

_z

= 1.1

→ Applying pressure:

0 130 mmHg

Methodology

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Derivation Derivation Derivation Derivation of of of of strain strain strain strain fields fields fields fields

Circumferential Green Lagrange strain

Methodology

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Identification by Virtual Field Method

Assuming Assuming Assuming Assuming constitutive constitutive constitutive constitutive parameters parameters parameters parameters

ψ T

= ρ .sym     ∂ ∂     . + p

σ F F I

E

Methodology

( )

¹ ¹

(

²

⁽

ⁱ

⁾

²

)

2

k λ - 1 i = fibre1,

fibre2

c k

ψ = I -3 +

2 ∑ 2k e - 1

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Reconstruction of Cauchy stress Reconstruction of Cauchy stress Reconstruction of Cauchy stress Reconstruction of Cauchy stress field field field field

Circumferential Cauchy stress

Axial

Cauchy stress

Methodology

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Identification by the Virtual Fields Method

Are stresses Are stresses Are stresses Are stresses at at at at equilibrium equilibrium equilibrium equilibrium????

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

Methodology

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Principle Principle Principle Principle of identification of identification of identification of 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 )

Methodology

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Identification by Virtual Field Method

Results Results Results Results of of of of optimization optimization optimization optimization

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

P re s s u re [ m m H g ]

0

P re s s u re [ k P a ]

λλλλ

Circumferential elongation λλλλ

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 15, Pages 2978-2985

Methodology

Holzapfel

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25

Methodology

(26)

26

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 aneurism

(27)

27

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

stress at rupture

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28

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

(29)

29

inflation device cylinder

pressure gage

in vivo loading environments

(biaxial stress state due to internal pressure)

can be generated

(30)

30

camera

Instron machine protector

Undeformed Deformed

x y

tracks the gray value pattern

in each subset during deformation Digital image

stereocorrelation

(31)

31

Theory of finite deformation

Deformation gradient F

Green-Lagrange strain tensor E = 1/2(C - I) right Cauchy-Green tensor C = F ^T F

Ux Uy Uz

from the undeformed and deformed

coordinates of each measurement data point

Assumption: plane stress

constant thickness

incompressibility

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32

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

▶ k ₂ is much higher aneurismal aortic tissue is stiffer than healthy aortic tissue

▶ measured fibre orientation angle α of the media is lower than that of the adventitia Results

f

1

f

2

α

(33)

33

the failure of aneurismal aortic tissue is caused principally by axial stress σσσσ yy

x y

Rupture is characterized by oblique tears in the

circumferential direction

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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 ( σσσσ

xx

) axial direction ( σσσσ

yy

)

Stress strain curves

(35)

35

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 σ σ

σ

^R^R^R^Rα

((((MPa MPa MPa MPa))))

1.0933 1.0933 1.0933 1.0933 0.2958

0.2958 0.2958 0.2958 0.327

0.327 0.327 0.327 0.2163

0.2163 0.2163 0.2163 0.3398

0.3398 0.3398 0.3398 1.143

1.143 1.143 1.143 σ σ

σ σ

^R^R^R^R_yy_yy_yy_yy

1.0073 1.0073 1.0073 1.0073 0.4384

0.4384 0.4384 0.4384 0.5568

0.5568 0.5568 0.5568 1.1524

1.1524 1.1524 1.1524 0.417

0.417 0.417 0.417 0.4189

0.4189 0.4189 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 (MPa MPa MPa MPa))))

adventitia adventitia adventitia adventitia media

media media media media

media media media adventitia

adventitia adventitia 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

quantifying the stress in the direction normal to both families of collagen fibres shows clearly the failure mechanism of aneurismal aortic tissue

the idea: the aneurysm rupture occurs in the direction of weakness (normal to the

collagen fibers)

(36)

36

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

(37)

37

▶ 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

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38

4. Conclusion & Prospects

Full Full Full Full- - -field - field field field approach approach approach approach

- Well reproduces experimental trends

- Significant local discrepancies

Variations of mechanical properties Branches

Variable thickness

For the future For the future For the future For the future… … … …

- Full-field strain measures → local heterogeneities - Through-thickness data → multi layer properties - Tomography techniques…

Methodology

(39)

39

MRI MRI MRI MRI measurements measurements measurements measurements

Methodology

(40)

40

In vivo applications

Full Full Full Full- - -field - field field field measurements measurements measurements using measurements using using using optical optical optical optical flow flow flow flow

Methodology

Introduction

(41)

41

behavior of arteries in vitro

St St

St Sté é éphane Avril é phane Avril phane Avril phane Avril

Inverse methods for characterizing the anisotropic hyperelastic

behavior of arteries in vitro

Introduction

V V V Vascular ascular ascular ascular disorders disorders disorders disorders

Atherosclerotic plaque

Hypertension

Vascular reconstruction

……

Numerical Numerical Numerical Numerical simulations simulations simulations simulations

Aneurysms

Knowledge of artery mechanics is fundamental

A multi A multi A multi A multi- - -layer - layer layer material layer material material material

Passive Passive Passive Passive mechanical mechanical mechanical mechanical behavior behavior behavior behavior

Multi-layer

Matrix + different fibers Arteries: a complex structure

and behavior

Intima

Media

Smooth muscle cells Elastin

Elastin Elastin

Elastin fibers fibers fibers fibers Collagen Collagen Collagen

Collagen fibers fibers fibers fibers

Biologic sensor and filter

Adventitia Collagen Collagen fibers Collagen Collagen fibers fibers fibers

Anisotropy – Non linearities – Finite strains

Anisotropic hyperelastic models for arteries

Hyperelasticity Hyperelasticity Hyperelasticity Hyperelasticity

Strain energy function:

2 nd Piola-Kirchhoff stress:

Anisotropic Anisotropic Anisotropic Anisotropic hyperelasticity hyperelasticity hyperelasticity hyperelasticity

( )

ψ = ψ E where E = 1 2 ( F F T . − I )

= ∂ ψ S ∂

E

( , )

ψ = ψ E structure tensors

f

f

Fung Fung Fung Fung’’’’s s s s phenomenological phenomenological phenomenological model phenomenological model model model

Holzapfel Holzapfel Holzapfel Holzapfel’’’’s s s s histology histology histology histology- - -based - based based model based model model model

e

e

( )

(

(

)

)

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

isotropic anisotropic

matrix fiber families

f

f

α

Identification procedures

Usual Usual Usual Usual protocol protocol protocol protocol::::

T ru e s tr e s s ( M P a )

True strain

Physiological modulus

Stress – Strain curve

Experimental considerations

Alternative protocol: closer to physiology Alternative protocol: closer to physiology Alternative protocol: closer to physiology Alternative protocol: closer to physiology

P - d curve d

L

F

F - L curve

Pressure (mmHg)

O u te r d ia m e te r ( m ) A x ia l F o rc e ( m N )

Axial Stretch

2 ^nd Piola-Kirchhoff stress:

ψ = ψ E ^where ^E ⁼ ¹ ₂ ( ^{F F} ^T ^. ⁻ ^I )

( ^, )

⁽

⁾

( ¹ )

e − ^with Q = a E + a E + 2a E E

O u te r d ia m e te r ( m ) A x ia l F o rc e ( m N )

Parameters A _i

Response M _j = f(A _i )

Measurements m _j