Full-field measurements and identification for biological soft tissues: application to arteries in vitro

Transcription

1 Centre for Health Engineering CNRS UMR 5146 INSERM IFR 143 Prof. Stéphane Avril Full-field measurements and identification for biological soft tissues: application to arteries in vitro

2 using single-gage 2

3 using single-gage Vascular disorders Atherosclerotic plaque Hypertension Vascular reconstruction Numerical simulations Aneurysms Knowledge of artery mechanics is fundamental 3

4 structure and behavior using single-gage A multi-layer material Passive mechanical behavior Multi-layer Matrix + different fibers Intima Media Biologic sensor and filter Smooth muscle cells Elastin fibers Collagen fibers Adventitia Collagen fibers Anisotropy Non linearities Finite strains 4

5 hyperelasticity of arteries using single-gage using single-gage 5

6 for arteries using single-gage Hyperelasticity Strain energy function: 2 nd Piola-Kirchhoff stress: S Anisotropic hyperelasticity ψ = ψ ( E, structure tensors) 1 ψ = ψ( E ) where E = ( F T. F I 2 ) ψ = E f 1 f 1 6

7 for arteries Fung s phenomenological model using single-gage ( 1) Q 2 2 ψ = c e with Q = a11θθ E + 22 a E zz + 2a 12 θθ E zz E 2 [Fung, Biorheology of soft tissues, Biorheology, 1973] e θ e z Holzapfel s histology-based model c k1 ψ = ( I -3 1 ) + 2 2k i = fibre1, fibre2 2 ( ( ) 2 ) kλ 2 - i1 e - 1 isotropic anisotropic f 2 f 1 α matrix fiber families [Gasser, Holzapfel, Ogden, A new constitutive framework for arterial wall mechanics 7 and a comparative study of material models. Journal of Elasticity, 2000]

8 using single-gage 8

9 using single-gage Usual protocol: True stress (MPa) Stress (Mpa) 2,00 1,80 1,60 1,40 1,20 1,00 0,80 0,60 0,40 0,20 Stress Strain curve Stress-Strain curve Maximum Elastic Modulus 0,00 0,00 0,20 0,40 0,60 0,80 1,00 Strain systole diastole True strain Physiological modulus 9 [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]

10 using single-gage Alternative protocol: closer to physiology d P L P - d curve F - L curve Outer diameter (µm) Axial Force (mn) Pressure (mmhg) Axial Stretch F 10 [Dye et. al., Altered biomechanical properties of carotid arteries in two mouse models of muscular dystrophy, J. of Applied Physiology, 2007]

11 Example using single-gage Identification of a mouse carotid artery behavior Experimental : Prof. Sutton (U. of South Carolina) 3D DIC P Biaxial test Local strain measurement 11 [Sutton et. al., Strain field measurements on mouse carotid arteries using microscopic three-dimensional digital image correlation, J. of Biomedical Materials Research, 2007]

12 Inverse identification using single-gage Solving an inverse problem Basic approach: updating Parameters A i Response M j = f(a i ) Model: f Measurements m j Matching optimization 12

13 Example using single-gage Identification of a mouse carotid artery behavior Numerical model: FE Model (Abaqus ) Optimization algorithm: β adventiti a β media Constitutive model: Holzapfel 5 parameters: C 10, k 1, k 2, β media, β adventitia Levenberg-Marquardt with bounds handling 13

14 Example using single-gage Identification of a mouse carotid artery behavior Results: C 10 k 1 = 0,5 kpa = 33 kpa k 2 = 12.8 β med. = 46.5 β adv. = 27.2 Good fit, robust identification, but 14

15 Example using single-gage β adventiti a β media One-point + 7 parameter identification C 10, k 1, k 2, β for media k 1, k 2, β for adventitia At least 7 parameters needed Multiple solutions Richer is required Field 15

16 hyperelasticity of arteries using single-gage 16

17 using single-gage A more sophisticated testing system 1 17 [Genovese, A video-optical system for time-resolved whole-body measurement on vascular segments, Optics and Lasers in Engineering, 2009]

18 using single-gage 18 Reconstruction of displacement field Pre-conditionning 8 cycles pressure Applying pre stretch λ z = 1.1 Applying pressure: mmhg Radial displacement

19 using single-gage Derivation of strain fields Circumferential Green Lagrange strain 19

20 Field Method using single-gage Assuming constitutive parameters c k1 ψ = ( I -3 1 ) + 2 2k i = fibre1, fibre2 ( ( ) 2 ) kλ 2 - i1 e - 1 ψ T σ = ρ F.sym. F + p I E 2 20

21 Field Method using single-gage 21 Reconstruction of Cauchy stress field Circumferential Cauchy stress Axial Cauchy stress

22 Virtual Fields Method using single-gage Are stresses at equilibrium? The following equations should be satisfied: (principle of virtual work) V - σ :ε dv + T u ds = 0 V * * ij ij i i V * * ( E ) - σ, A :ε dv + T u ds = 0 ij ij i i V Equilibrium Actual constitutive properties 22

23 Field Method using single-gage Principle of identification Iterative approach until reconstructed stresses minimize cost function J: J A = - ( ) σ, A :ε dv + T u ds ( E ) * * ij ij i i virtual fields pressure states V V Internal Virtual Work ( IVW ) 2 External Virtual Work ( EVW ) 23

24 Field Method using single-gage Results of optimization Pressure [kpa] Experimental Neo Hookean «Yeoh» 120 Holzapfel Fung exponential Circumferential elongation λ Pressure [mm Hg] Best fitting parameters: (Holzapfel model, 1 layer) 24 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

25 of arteries using single-gage 25

26 Aortic aneurism arch of aorta a local dilation of the aorta due to aortic wall weakening ascending aorta descending aorta (thoracic aorta and abdominal aorta) aneurysm rupture a fatal medical emergency 26

27 Aneurismal aortic tissue Inflation test Optical Full-field measurement ( Full-field displacement) Deformation gradient Lagrange strain Identification of material parameters Constitutive model Inverse procedure Calculation of stress at rupture Application of the special 27

28 Materials cut media adventitia an excised cylindrical aneurismal aortic tissue a square specimen removing loose connective tissue finding an appropriate location to separate diameter: 30mm adventitia media y x specimen is mounted on the inflation test device making a speckle pattern separated layers two layers are pulled each other to separate 28

29 Inflation test cylinder in vivo loading environments (biaxial stress state due to internal pressure) can be generated inflation device pressure gage 29

30 Digital image stereocorrelation Undeformed Deformed y protector x Instron machine camera 30 tracks the gray value pattern in each subset during deformation

31 Measured displacement fields Ux Uy Uz Theory of finite deformation Deformation gradient F right Cauchy-Green tensor C = F T F Green-Lagrange strain tensor E = 1/2(C - I) from the undeformed and deformed coordinates of each measurement point Assumption: plane stress constant thickness incompressibility 31

32 Results Case Type Adventitia Media Media Media Media Adventitia (thickness) (0.64mm) (0.91mm) (1.02mm) (1.09mm) (1.06mm) (0.62mm) sex, age male, 81 years old male, 68 male, 69 male, 76 diameter (both ends) 36, 39 mm 31, 43 mm 32, 34 mm 36, 38 mm f 2 f 1 α k 1 (MPa) k α 57.7 o o o o o o k 2 is much higher aneurismal aortic tissue is stiffer than healthy aortic tissue 32 measured fibre orientation angle α of the media is lower than that of the adventitia

33 Characterization of rupture Rupture is characterized by oblique tears in the circumferential direction y x 33 the failure of aneurismal aortic tissue is caused principally by axial stress σ yy

34 Stress strain curves circumferential direction (σ xx ) axial direction (σ yy ) Adventitia (α>40 o ) stress (MPa) I II stress (MPa) I II strain strain II Media (α<40 o ) stress (MPa) III IV I stress (MPa) III IV II I strain strain 34

35 Stress at rupture Case type adventitia media media media media adventitia Cauchy stress at rupture (MPa) σ R xx σ R yy σ R α (MPa) Stress parameter at rupture the idea: the aneurysm rupture occurs in the direction of weakness (normal to the collagen fibers) R σ α = σ R xx sin 2 ( α) + σ R yy cos 2 ( α) quantifying the stress in the direction normal to both families of collagen fibres 35 shows clearly the failure mechanism of aneurismal aortic tissue

36 Modes of rupture p = 0.02 MPa MPa MPa MPa ε x Rupture mode ε y B A ε xy 36

37 Modes of rupture the failure stress in the axial direction is much higher in the adventitia layer (about three times) compared to that in the media layer means that the adventitia layer plays a very important role in preventing the artery from rupture 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 37

38 using single-gage Full-field approach - Well reproduces experimental trends - Significant local discrepancies Variations of mechanical properties Branches Variable thickness For the future - Full-field strain measures local heterogeneities - Through-thickness multi layer properties - Tomography techniques 38

39 using single-gage MRI measurements 39

40 using single-gage Full-field measurements using optical flow 40

41 using single-gage Student: Ambroise Duprey, Jin Kim, Alexandre Franquet Colleagues: Pierre Badel (Ecole des Mines) Katia Genovese (Univ. Basilicata) Jean-Noël Albertini (Hospital Saint-Etienne) Jean-Pierre Favre (Hospital Saint-Etienne) Institutions and funding partners: 41