Constitutive modeling of soft biological tissues with emphasis on the vasculature
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1 Constitutive modeling of soft biological tissues with emphasis on the vasculature T.Christian Gasser Dept. of Solid Mechanics, School of Engineering Sciences Royal Institute of Technology (KTH) Stockholm, Sweden
2 Acknowledgements Caroline Forsell, Jacopo Biasetti, Giampaolo Martufi, Sara Gallinetti Jesper Swedenborg, Ulf Hedin, Joy Roy, Fausto Labruto Salvatore Federico Martin Auer Young Faculty Grant, VR, VINNOVA, SSF, Sweden FAD, EU 7 th Framework program AWS & SPG, Austria
3 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Applications o Summary/Conclusions
4 Motivation Abdominal Aortic Aneurysm (AAA) Increasing prevalence with age Rupture is mostly terminating (~80%) No effective medication Craig et al., Ann Intern Med (2005)
5 Motivation Elective AAA repair Endovascular Surgical Risks for the patient 4% 30-day mortality Long-term morbidity Costs for patient/community Elective: Emergency: SVS, Repair indication?
6 Motivation Mech. properties Constitutive modeling Histology Intended use
7 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Applications o Summary/Conclusions
8 Collagenous Biological Tissue Collagen fibril basic building block 28 collagen proteins Continuous turn-over Type I, III vascular tissue Suprafibrilar Structures Determine mech. properties biological tissue
9 Collagenous Biological Tissue Hierarchical structure of vascular tissue Adventitia Elbishger et al. (2004) Media Clark & Glagov (1985) Intima Gasser et al. (2006)
10 Collagenous Biological Tissue Hierarchical structure of vascular tissue Adventitia Media Intima Canham et al., Card Res (1989)
11 Collagenous Biological Tissue Hierarchical structure of tendon tissue Provencano & Vanderby (2005) Ottani et al. (2000) 50 μm Kastelic et al. (1978)
12 Collagenous Biological Tissue Hierarchical structure of stroma (cornea) Cornea 10 μm 50 μm Meek & Fullwood (2001) Ottani et al. (2000)
13 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Applications o Summary/Conclusions
14 Mechanical properties - arteries Passive mechanical properties Highly deformable (health) Material nonlinearity Incompressible (physiological deformations) Anisotropic Residual strains in load-free configuration Strain rate dependency Preconditioning Plastic deformations Damage and Failure Humphrey, Springer Verlag (2002) Roy, Phil Trans R Soc Lond (1880)
15 Mechanical properties - arteries Collagen fibers - Tensile testing Micro-manipolators Miyazaki et. al (2000) Micromechanical Systems (MEMS) Zhilei et al. (2008)
16 Mechanical properties - arteries Passive mechanical properties Collagen Elastin Structural organization Active mechanical properties Smooth Muscle Cells (SMC) Structural organization
17 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Applications o Summary/Conclusions
18 Finite Strain Continuum Mechanics Kinematics Motion in Lagrange Description Ogden, Dover (1997) Bonet & Wood, Cambridge Univ Press (1997) Referential particle position Current particle position Referential configuration Current configuration Motion
19 Finite Strain Continuum Mechanics Kinematics Deformation/Strain Measures Deformation Gradient Right Cauchy Green Strain Left Cauchy Green Strain Green Lagrange Strain Euler Almansi Strain
20 Finite Strain Continuum Mechanics Kinematics Deformation/Strain Measures Unit direction vector Material line element Stretch Structural tensor
21 Finite Strain Continuum Mechanics Concept of Stress Cauchy Theorem Cauchy stress tensor Cauchy traction vector Unit normal vector First Piola Kirchhoff stress tensor Second Piola Kirchhoff stress tensor (work conjugate to )
22 Finite Strain Continuum Mechanics Push forward/pull back operation Push forward Pull back Masden and Hughes, Dover (1994)
23 Finite Strain Continuum Mechanics Constitutive Formulation Cauchy Elasticity Symmetric tensor-valued response function
24 Finite Strain Continuum Mechanics Constitutive Formulation Hyperelasticity Coleman-Noll procedure Strain energy function Physical Constraints Polyconvexity Note: Hyperelasticity (Green Elasticity) is a special case of Cauchy Elasticity Coleman, Arch Rat Mech Anal (1964)
25 Finite Strain Continuum Mechanics Constitutive Formulation Objectivity Rigid body motion Rotation tensor Translation vector
26 Finite Strain Continuum Mechanics Constitutive Formulation Material Symmetry Rigid body motion Rotation tensor Translation vector Material Symmetry group
27 Finite Strain Continuum Mechanics Constitutive Formulation Internal constraints Incompressibility Inextensibility Quasi Incompressibility (Numerical implementation) Modified Deformation Gradient Flory, Trans Farday Soc (1961)
28 Finite Strain Continuum Mechanics Constitutive Formulation FE implementation Loop over elements Compute nodal forces and stiffness Stress tensor Elasticity tensor Assemble linearized system of equations Solve the linearized system of equations
29 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Application o Summary/Conclusions
30 Constitutive Modeling Perfect aligned collagen fibers Modified Right Cauchy Green Strain Structural tensor Gasser & Holzapfel, WCBM (1998) Holzapfel & Gasser, Comp Meth Appl Mech Engrg (2000)
31 Constitutive Modeling Perfect aligned collagen fibers - Stress Constitutive parameter Kinematic parameter Hydrostatic pressure (deformation independent)
32 Constitutive Modeling Perfect aligned collagen fibers - Invariants Invariants of Spencer, CISM Courses (1984)
33 Constitutive Modeling Perfect aligned collagen fibers - Particularization Roach & Burton, Can J BiochemPhysiol35 (1957) Matrix (neohookean model) Collagen fiber reinforcement Fung, Am J Physiol (1967) Demiray, J Biomech (1972)
34 Constitutive Modeling Distr. collagen fiber orientations - Background Orientation density function Normalization Eulerian angles Solid angle Lanir, J Biomech (1983)
35 Constitutive Modeling Distr. collagen fiber orientations Structural Tensor Generalized structural tensor Special case: transverse isotropy Dispersion parameter Mean (fiber) orientation Gasser et. al, J R Soc Interface (2006)
36 Constitutive Modeling Distr. collagen fiber orientations Particularization Isochoric part of the strain energy Green Lagrange like strain
37 Constitutive Modeling Distr. collagen fiber orientations Particularization Collagen fiber distribution Mean stretch of the fiber family
38 Constitutive Modeling Distr. collagen fiber orientations Particularization Analytic description of the orientation density
39 Constitutive Modeling Distr. collagen fiber orientations Particularization Dispersion parameter Perfect aligned fiber model Isotropic model (almost Demiray, 1972) Gasser et. al, J R Soc Interface (2006)
40 Elevation angle (degrees) Constitutive Modeling Distr. collagen fiber orientations Light Microscopy Experimental results from AAA tissue Azimuthal angle (degrees)
41 Constitutive Modeling Distr. collagen fiber orientations Microplane model Orientation of stretched fibers Stored energy in a bundle of fibers Solid angle Lanir, J Biomech (1983) Bažant & Oh, J Enrg Mech (1985)
42 Constitutive Modeling Distr. collagen fiber orientations Microplane model Integration with spherical Design, such that holds for polynoms of order. Design Hardin & Sloane, Discrete Comput Geom (1996)
43 Constitutive Modeling Distr. collagen fiber orientations Microplane model
44 Content o Motivation o Collagenous Biological Tissue o Mechanical properties - arteries o Finite Strain Continuum Mechanics o Constitutive modeling o Application o Summary/Conclusions
45 Application Numerical frames Implementations in FEAP (Univ. of California at Berkley) and A4research (VASCOPS GmbH)
46 Application AAA Rupture Risk Assessment (Vasc. Surg. at KI) Ruptured Non ruptured Discrimination of ruptured from non-ruptured AAAs? Degree of model complexity?
47 ruptured ruptured Non-ruptured Non-ruptured ruptured Non-ruptured Application AAA Rupture Risk Assessment (Vasc. Surg. at KI) Results: Wall & Thrombus model
48 Summary / Conclusion o Abdominal Aortic Aneurysms (AAAs) o Constitutive formulations for biol. tissue o Integration of the micro-histology o Finite Strain Continuum Mechanics o FE implementations o AAA Rupture Risk Assessment Modeling is an important step in knowledge development! Thank you!
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