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

Introduction

Arterial microstructure

Arterial microstructure Intima Endothelial cells

Arterial microstructure Media Smooth muscle cells, collagen & elastin

Arterial microstructure Adventitia Collagen fibers

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

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

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: 181-190 (1957).

How do you study the mechanics of materials?

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)

Continuum mechanical framework

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

Measurement of tissue mechanics Biaxial stretcher design

Data from biaxial experiment

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

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

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).

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

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

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

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

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

Experimental design Left Cauchy Green tensor For biaxial experiments

Results from biaxial experiments

Constant I4 experiments: W1 and W4 dependence Gundiah et al, unpublished

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

Fits to new Strain Energy Function c0 = 73.96 ± 22.51 kPa, c1 = 1.18 ±1.79 kPa c2 = 0.8 ±1.26 kPa

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: 181-190 (1957).

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

How do elastin & collagen influence arterial behavior?

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

Uniaxial Test Results

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

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

Mooney-Rivlin material? Not a Mooney-Rivlin material c01 kPa c10 kPa Autoclaving 162.57 ±115.44 -234.62 ± 166.23 Hot Alkali 76.94 ±27.76 -24.89 ± 35.11 Baker-Ericksen inequalities c01, c10 ≥0 Greater principal stress occurs always in the direction of the greater principal stretch

Constant I1: W1 and W4 dependence

neo-Hookean term dominant. elastin modulus is 522.71 kPa 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