Truestress (MPa)

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Identification methods for characterizing the mechanical behaviour

of human arteries in vitro

Dr. Stéphane Avril

Center for Health Engineering

CNRS UMR 5146 – INSERM IFR 143 Ecole Nationale Supérieure des Mines

November 20-23, 2009, Tozeur

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In vitro: rationale

• Availability of powerful computational software allowing for more and more complex numerical simulation

→ need of accurate input for the mechanical properties of the arteries

→ complex characterization because of

anisotropy, hyperelasticity, heterogeneity, poroelasticity, residual stresses

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Rationale

Numerical models require

experiments

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Truestress (MPa)

True strain

diastole systole

Physiological modulus

Traditional protocols for characterizing arteries

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More sophisticated testing system

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Strain field measurement

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INVERSE PROBLEM

I c e F

sym

F ^X  ^t +









∂ Ψ

= . ∂ .

ρ 0

σ

Ψ

_X⁰

= ?

F: deformation gradient tensor e: Green-Lagrange strain tensor σ: Cauchy stress tensor

I: Identity

c: constant for incompressibility

Conservation equation in quasi-static:

div(σ)=0 + boundary conditions Constitutive equations:

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Resolution of an inverse problem Basic approach: updating

Parameters A_i

Model: f

Response M_j=f(A_i)

Measurements m_j

Matching optimization

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Full-field measurements:

alternative approach

Model f is derived from:

- the constitutive equations, - the conservation equations.

g(A_i,M)=0 M=f(A_i)

THE VIRTUAL FIELDS METHOD

0 )

(σ =

div + boundary conditions

σ = C(m,A)

Direct

computation Measurements

m(x,y,z)

A_i g(A_i,m)=0

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Explanation through an example

1

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Measurements throughout the test

Testing:

→ Applying axial pre-stretching

λ^Z ^{= 1.1}

→ Applying pressure

pressure 0 mmHg → 130 mmHg pressure rate 12 mmHg/s

→ Preconditioning

8 pressure cycles 0 → 120 mmHg 8 axial cycles λ^Z ^{= 1.1}^→ λ^Z^{= 1.4}

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Reconstruction of displacement fields across the whole segment

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Derivation of strain fields in the local coordinate system

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Assume constitutive parameters

( ) ( )

[ ]

(^exp ² ⁺ ² ⁻¹)

=

Ψ β α_θθ e_θθ α_zz e_zz α_θ_ze_θθe_zz

Green Lagrange strain

Example of anisotropic hyperelastic potential: Fung exponential

I c e F

sym

F ^X  ^t +









∂ Ψ

= . ∂ .

ρ 0

σ

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Reconstruction of Cauchy stress fields: circumferencial direction

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Reconstruction of Cauchy stress fields: axial direction

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0

: ^* + ^* =

−

∫ ∫

∂V

i i V

ij

ij ε dV T u dS

σ

At each measurement step, the following equation should be satisfied:

0 :

) ,

( ^* + ^* =

−

∫ ∫

∂V

i i V

ij

ij e A ε dV T u dS

σ

Are the stresses at equilibrium

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∑ ∑ ∫ ∫

_^









 − +

=

∂ fields

virtual all steps V

i i V

ij

ij e A dV T u dS

A J

2

*

: *

) ,

( )

( σ ε

Iterative approach for reconstructing the stress fields until cost function J is minimized:

Principle of identification

IVW EVW

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circumferential:

Choice of virtual fields

axial:

ε^*_θθ ε^*

zz

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Optimization => curve fitting

circumferential: axial:

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Results

→ Fung exponential anisotropic

Best fitting model:

012 .

0

75 . 0

3 . 7

10 8

. 4

12 22 11

2

=

×

= ⁻

α α α

β ^MPa

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

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22

Discussion

Z [mm]

Y [mm] Y [mm] Y [mm]

→ Well reproduced experimental trend

→ Significant local discrepancies

Local heterogeneities in mechanical properties, variable thickness, branches

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Perspectives

Applications in vivo

to atheromatous plaque

in the carotid artery

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Atheromatous plaques

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IVUS 25

MRI

In-vivo full-field measurements:

Examples of methods

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In-vivo full-field measurements:

Examples of data

Courtesy of Simon Le Floc’h, Jacques Ohayon, TIMC Grenoble

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Full-field measurement of the cross section deformation

induced by a change of pressure (diastolic to systolic)

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INVERSE PROBLEM

Assume linear elasticity We must find E and ν in each material…

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Resolution of an inverse problem Basic approach: updating

Parameters A_i

Model: f

Response M_j=f(A_i)

Measurements m_j

Matching optimization

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Full-field measurements:

alternative approach

Model f is derived from:

- the constitutive equations, - the conservation equations.

g(A_i,M)=0 M=f(A_i)

THE VIRTUAL FIELDS METHOD

0 )

(σ =

div + boundary conditions

σ = C(M,A)

Direct

computation Measurements

m(x,y,z)

A_i g(A_i,m)=0

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Results and discussion

• The VFM can be applied for real time analysis

• Study under progress, still needs to be applied to real experimental data.

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CONCLUSION

• Novel efficient approaches for

characterizing the elastic properties of arteries

• What about viscoelasticity or poroelasticity?

• What about fracture?

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Acknowledgements

• Students: Ambroise Duprey, Jean-Pierre Vassal, Alexandre Franquet

• Colleagues: Pierre Badel (Ecole des Mines), Katia Genovese (Univ. Basilicata), Jacques Ohayon (TIMC Grenoble), Jean-Noel Albertini and Jean Pierre Favre (CHU St-Etienne)

• Institutions and funding partners: