1

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 AneurysmsAortic Dissections

Supravalvular aortic stenosisWilliam’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

Fung, Y.C. (1979)

• Pseudoelastic

How is stress related to strain: Constitutive equations

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 homogeneousAssuming incompressibility

• Equilibrium:

• Constitutive law: 1. Using tissue compressibility and symmetry 2. Phenomenological model

333222111 ;; XxXxXx

1321

0;; 12331

2222

2

1111

TT

HL

FT

HL

FT

14

Measurement of tissue mechanicsBiaxial stretcher design

15

Data from biaxial experiment

16

1. Phenomenological model• Fung strain energy function

1exp2

1 QcW

22111222222

21111 EEcEcEcQ

1: circ 2: long

Eij Green strain; cij material parameters

Cauchy stresses

Best fit parameters obtained using Levenberg-Marquardt algorithm

TFC

CFIT

)(W

p

22221112

2222

221211112111

)exp(

)exp(

EcEcQcT

EcEcQcT

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

21321 ,)1)((),(,)()()( WIIIWWWW CCCCF

18

Elastin Isolation

• Goal: to completely remove collagen, proteoglycans and other contaminants

1. Hot alkali treatment

2. Repeated autoclaving followed by extraction with 6 mol/L guanidine hydrochloride

1 Lansing. (1952)

2 Gosline. JM (1996).

19

Elastin architecture

N. Gundiah et al, J. Biomech (2007)

•Axially oriented fibers towards intima and adventitia •Circumferential elastin fibers in media.

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

29

8

7

6

5

4

3

22

1

)'.()(

.)'.()(

).()(

).()(

).()(

).()(

1)det()(

)(2

1)(

);()(

MMC

CMMMMC

M'CM'C

CM'M'C

MCMC

CMMC

CC

CCC

CC

2

2

2

I

I

I

I

I

I

I

trtrI

trI

,

C=FTF is the right Cauchy Green tensor

=90 for orthogonal fiber families

22

Theoretical considerations

• Deformation: homogeneous

i are the stretches in the three directions

• Unit vectors

• Strain energy function for arterial elastin networks:

• Define subclass

111 Xx 222 Xx 333 Xx

1

2

eM'

eM

),,,,,(ˆ765421 IIIIIIWW

),,(ˆ641 IIIWW

23

Rivlin Saunders protocol

• Perform planar biaxial experiments keeping I1 constant and get dependence of W1, W4 on I4

• Repeat experiments keeping I4 constant

),(~

),,(ˆ41641 IIWIIIWW

• Constant I1 experiments violates pseudoelasticity requirement

22

21

22

21

23

22

211

1

I

24

0

02

2

22

132312

23133

22122

214

21111

TTT

WpT

WpT

WWpT

TFFB

FMFMBIT

41 22 WWp

Left Cauchy Green tensor

For biaxial experiments

22

21

22

21

2222

21

21112

221

22

41

2

11

TT

W

22

21

22

221

12

TW

Experimental design

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

262

24110 113 IcIcIcW

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

641 , IIWIWW anisoiso

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

33

34

Uniaxial Test Results

35

• Use uniaxial stress-strain data

• Mooney-Rivlin Strain energy function:

• Uniaxial tension experiments

• Plot of Vs

Is it a Mooney-Rivlin material?

3210

3101

IcIcW

1

1001

1

2111

12

ccT

1

2111

12/

T

1

1

36

Is Elastin a Mooney-Rivlin material?

01

1

10

1

2111

12 c

cT

Equation:

N. Gundiah et al, J. Biomech (2007)

37

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

Not a Mooney-Rivlin material

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) 586-5941 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