1. Namrata Gundiah University of California, San Francisco
The role of elastin in arterial mechanics Structure function relationships in soft tissues
Namrata Gundiah
University of California, San Francisco

2. Introduction

3. Arterial microstructure

4. Arterial microstructure
Intima Endothelial cells

5. Arterial microstructure
Media Smooth muscle cells, collagen & elastin

6. Arterial microstructure
Adventitia Collagen fibers

7. Complex tissue architecture
Masson’s trichrome
Collagen: blue
Verhoeff’s Elastic
Elastin: black

8. Diseases affecting arterial mechanics
Atherosclerosis
Abdominal Aortic Aneurysms
Aortic Dissections
Supravalvular aortic stenosis
William’s syndrome
Marfan’s syndrome
Cutis laxa
etc.

9. Mechanical properties of arteries
The complex geometry of the arterial walls results in its nonlinear mechanical properties. For most part, this nonlinearity depends on the ratio of elastin and collagen in the vessel wall. The characteristic rubbery behavior of vessels at low extensions is dominated by the presence of elastin while collagen dictates the response at higher extensions and prevents the wall from ballooning. The physiological blood pressure of an organism usually lies in the region of transition from elastin to collagen dominated responses.
Roach, M.R. et al, Can. J. Biochem. & Physiol., 35: (1957).

10. How do you study the mechanics of materials?

11. Arterial Behavior Arteries are composite structures
Rubbery protein elastin and high strength collagen
Nonlinear elastic structures undergoing large deformations
Anisotropic
Viscoelastic
Pseudoelastic
How is stress related to strain: Constitutive equations
Fung, Y.C. (1979)

12. Continuum mechanical framework

13. Biaxial test : Preliminaries
Material is sufficiently thin such that plane stress exists in samples and top and bottom of sample is traction free
Kinematics: Deformations are homogeneous
Assuming incompressibility
Equilibrium:
Constitutive law: 1. Using tissue compressibility and symmetry
2. Phenomenological model

14. Measurement of tissue mechanics Biaxial stretcher design

15. Data from biaxial experiment

16. 1. Phenomenological model
Fung strain energy function
1: circ 2: long
Eij Green strain; cij material parameters
Cauchy stresses
Arterial tissues are pseudoelastic, hyperelastic, homogeneous, incompressible and orthotropic to obtain strain energy functions.
Data from load cells were converted to Cauchy stress.
Marker displacements were converted to Green strains.
Data for the loading cycle was fit to the Fung strain energy function.
Constitutive parameters were fit to experimental data using a MATLAB program employing the Levenberg-Marquardt optimization method for multivariate nonlinear regression.
The solution was constrained to nonzero values to both ensure a physically meaningful solution and guarantee convexity of the strain energy function.
Best fit parameters obtained
using Levenberg-Marquardt algorithm

17. 2. Function using material symmetry
Define strain invariants
For isotropic and incompressible material
Need to know symmetry in the underlying microstructure.
Transverse isotropy: 5 parameters
Orthotropy: 9 parameters

18. Elastin Isolation
Goal: to completely remove collagen, proteoglycans and other contaminants
Hot alkali treatment
Repeated autoclaving followed by extraction with 6 mol/L guanidine hydrochloride
1 Lansing. (1952)
2 Gosline. JM (1996).

19. Elastin architecture
Axially oriented fibers towards intima and adventitia
Circumferential elastin fibers in media.
N. Gundiah et al, J. Biomech (2007)

20. Histology Results Circumferential sections:
Elastin fibers in concentric circles in the media
Transverse sections:
Elastin in adventitia and intima is axially-oriented.
Elastin in media is circumferentially-oriented.
Elastin microstructure in porcine arteries can be described using orthotropic symmetry

21. Orthotropic material Assume orthotropic,
f=90 for orthogonal fiber families
C=FTF is the right Cauchy Green tensor

22. Theoretical considerations
Deformation: homogeneous
li are the stretches in the three directions
Unit vectors
Strain energy function for arterial elastin networks:
Define subclass

23. Rivlin Saunders protocol
Perform planar biaxial experiments keeping I1 constant and get dependence of W1, W4 on I4
Repeat experiments keeping I4 constant
Constant I1 experiments violates pseudoelasticity requirement

24. Experimental design
Left Cauchy Green tensor
For biaxial experiments

25. Results from biaxial experiments

26. Constant I4 experiments: W1 and W4 dependence
Gundiah et al, unpublished

27. W4 dependence on I4 SEF has second order dependence on I4, hence on I6
We propose semi-empirical form, similar to standard reinforcing model
Coefficients c0, c1 and c2 determined by fitting equibiaxial data to new SEF using the Levenberg-Marquardt optimization

28. Fits to new Strain Energy Function
c0 = ± kPa,
c1 = 1.18 ±1.79 kPa
c2 = 0.8 ±1.26 kPa

29. Mechanical properties of arteries
The complex geometry of the arterial walls results in its nonlinear mechanical properties. For most part, this nonlinearity depends on the ratio of elastin and collagen in the vessel wall. The characteristic rubbery behavior of vessels at low extensions is dominated by the presence of elastin while collagen dictates the response at higher extensions and prevents the wall from ballooning. The physiological blood pressure of an organism usually lies in the region of transition from elastin to collagen dominated responses.
Roach, M.R. et al, Can. J. Biochem. & Physiol., 35: (1957).

30. Mechanical Test Results
Strain energy function for arteries
Isotropic contribution mainly due to elastin
Anisotropic contribution due to collagen fiber layout

31. How do elastin & collagen influence arterial behavior?

32. Acknowledgements Prof Lisa Pruitt, UC Berkeley/ UC San Francisco
Dr Mark Ratcliffe UCSF/ VAMC for use of biaxial stretcher
Jesse Woo & Debby Chang for help with histology
NSF grant CMS to UC Berkeley

34. Uniaxial Test Results

35. Is it a Mooney-Rivlin material?
Use uniaxial stress-strain data
Mooney-Rivlin Strain energy function:
Uniaxial tension experiments
Plot of Vs

36. Is Elastin a Mooney-Rivlin material?
Equation:
N. Gundiah et al, J. Biomech (2007)

37. Mooney-Rivlin material?
Not a Mooney-Rivlin material
c01 kPa
c10 kPa
Autoclaving
±115.44 ±
Hot Alkali
76.94 ±27.76
± 35.11
Baker-Ericksen inequalities
c01, c10 ≥0
Greater principal stress occurs always in the direction of the greater principal stretch

38. Constant I1: W1 and W4 dependence

39. neo-Hookean term dominant. elastin modulus is 522.71 kPa
Conclusions
neo-Hookean term dominant.
elastin modulus is kPa
From Holzapfel1 and Zulliger2 models (obtained by fitting experimental data on arteries), we get elastin modulus of kPa and kPa respectively which is lower than those experimentally determined.
* Gundiah, N. et al, J. Biomech. v40 (2007)
1 Holzapfel, GA et al, 1996, Comm. Num. Meth. Engg, v12 n8 (1996)
2 Zulliger, MA et al, J Biomech, v37 (2004)