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
gasser@kth.se
Acknowledgements
Jesper Swedenborg, Ulf Hedin, Joy Roy,
Fausto Labruto
Martin Auer
_________________
Young Faculty Grant, VR, VINNOVA, SSF, Sweden
FAD, EU 7
thFramework program
AWS & SPG, Austria
Salvatore Federico
Caroline Forsell, Jacopo Biasetti, Giampaolo Martufi,
Sara Gallinetti
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Applications
Motivation
Increasing prevalence with age
Rupture is mostly terminating (~80%)
No effective medication
Abdominal Aortic Aneurysm (AAA)
Surgical
Endovascular
Elective AAA repair
Risks for the patient
4% 30-day mortality
Long-term morbidity
Costs for patient/community
Elective: € 19 000
Emergency: € 55 000
Intended use
Motivation
Constitutive modeling
Histology
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Applications
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
Hierarchical structure of vascular tissue
Adventitia
Media
Elbishger et al. (2004)
Media
Adventitia
Intima
Canham et al., Card Res (1989)
Collagenous Biological Tissue
Ottani et al. (2000)
Provencano & Vanderby (2005)
50 μm
Collagenous Biological Tissue
Ottani et al. (2000)
Meek & Fullwood (2001)
Cornea
10 μm
50 μm
Collagenous Biological Tissue
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Applications
Mechanical properties - arteries
Passive mechanical properties
Residual strains in load-free configuration
Highly deformable (health)
Incompressible (physiological deformations)
Material nonlinearity
Anisotropic
Strain rate dependency
Preconditioning
Plastic deformations
Damage and Failure
Humphrey, Springer Verlag (2002)
Roy, Phil Trans R Soc Lond (1880)
Mechanical properties - arteries
Collagen fibers - Tensile testing
Micromechanical Systems (MEMS)
Zhilei et al. (2008)
Micro-manipolators
Miyazaki et. al (2000)
Passive mechanical properties
Collagen
Elastin
Structural organization
Active mechanical properties
Smooth Muscle Cells (SMC)
Structural organization
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Applications
Finite Strain Continuum Mechanics
Kinematics – Motion in Lagrange Description
Referential particle position
Current particle position
Referential configuration
Current configuration
Motion
Ogden, Dover (1997)
Kinematics – Deformation/Strain Measures
Deformation Gradient
Right Cauchy Green Strain
Left Cauchy Green Strain
Green Lagrange Strain
Unit direction vector
Material line element
Stretch
Structural tensor
Finite Strain Continuum Mechanics
Concept of Stress
Cauchy Theorem
Cauchy stress tensor
Cauchy traction vector
Unit normal vector
Push forward/Pull back operation
Push forward
Pull back
Masden and Hughes, Dover (1994)
Constitutive Formulation – Cauchy Elasticity
Symmetric tensor-valued response function
Constitutive Formulation – Hyperelasticity
Note: Hyperelasticity (Green Elasticity) is a special case
of Cauchy Elasticity
Coleman-Noll procedure
Coleman, Arch Rat Mech Anal (1964)
Physical Constraints
Polyconvexity
Strain energy function
Constitutive Formulation – Objectivity
Rigid body motion
Rotation tensor
Translation vector
Constitutive Formulation – Material Symmetry
Rigid body motion
Rotation tensor
Translation vector
Material Symmetry
group
Constitutive Formulation – Internal constraints
Incompressibility
Inextensibility
Quasi Incompressibility (Numerical implementation)
Modified Deformation Gradient
Constitutive Formulation – FE implementation
Loop over elements
Compute nodal forces
and stiffness
Assemble linearized
system of equations
Solve the linearized
system of equations
Stress tensor
Elasticity tensor
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Application
Constitutive Modeling
Perfect aligned collagen fibers
Modified Right Cauchy Green Strain
Structural tensor
Gasser & Holzapfel , WCBM (1998)
Constitutive parameter
Kinematic parameter
Constitutive Modeling
Invariants of
Spencer, CISM Courses (1984)
Constitutive Modeling
Matrix (neoHookean model)
Roach & Burton, Can J BiochemPhysiol35 (1957)
Constitutive Modeling
Distr. collagen fiber orientations - Background
Orientation density function
Lanir, J Biomech (1983)
Normalization
Eulerian angles
Solid angle
Generalized structural tensor
Special case: transverse isotropy
Distr. collagen fiber orientations – Structural Tensor
Green Lagrange –like strain
Isochoric part of the strain energy
Distr. collagen fiber orientations – Particularization
Mean stretch of the fiber family
Collagen fiber distribution
Distr. collagen fiber orientations – Particularization
Analytic description of the orientation density
Constitutive Modeling
Dispersion parameter
Perfect aligned fiber model
Isotropic model (almost Demiray, 1972)
Constitutive Modeling
Experimental results from AAA tissue
Azimuthal angle (degrees)
El
evati
on
angle
(de
gr
ee
s)
Distr. collagen fiber orientations – Light Microscopy
Orientation of stretched fibers
Stored energy in a bundle of fibers
Distr. collagen fiber orientations – Microplane model
Integration with spherical Design, such that
holds for polynoms of order .
Hardin & Sloane, Discrete Comput Geom (1996)
Design
Distr. collagen fiber orientations – Microplane model
Distr. collagen fiber orientations – Microplane model
Content
o
Motivation
o
Collagenous Biological Tissue
o
Mechanical properties - arteries
o
Finite Strain Continuum Mechanics
o
Constitutive modeling
o
Application
Application
Numerical frames
Implementations in FEAP (Univ. of California at Berkley)
and A4research (VASCOPS GmbH)
AAA Rupture Risk Assessment (Vasc. Surg. at KI)
Ruptured
Non ruptured
Discrimination of ruptured from non-ruptured AAAs?
Degree of model complexity?
Results: Wall & Thrombus model
ur
ed
ed
AAA Rupture Risk Assessment (Vasc. Surg. at KI)
Application
ed
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!
