The role of elastin in arterial mechanics Structure function relationships in soft tissues

1. The role of elastin in arterial mechanicsStructure 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 Roach, M.R. et al, Can. J. Biochem. & Physiol., 35: 181-190 (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 mechanicsBiaxial 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 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 = 73.96 ± 22.51 kPa, c1 = 1.18 ±1.79 kPa c2 = 0.8 ±1.26 kPa

29. Mechanical properties of arteries Roach, M.R. et al, Can. J. Biochem. & Physiol., 35: 181-190 (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 CMS0106010 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 • 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. Conclusions • neo-Hookean term dominant. • elastin modulus is 522.71 kPa • From Holzapfel1 and Zulliger2 models (obtained by fitting experimental data on arteries), we get elastin modulus of 308.2 kPa and 337.32 kPa respectively which is lower than those experimentally determined. * Gundiah, N. et al, J. Biomech. v40 (2007) 586-594 1 Holzapfel, GA et al, 1996, Comm. Num. Meth. Engg, v12 n8 (1996) 507-517. 2 Zulliger, MA et al, J Biomech, v37 (2004) 989-1000