Biomech Model MechanobiolDOI 10.1007/s10237-010-0260-4

ORIGINAL PAPER

The orthotropic viscoelastic behavior of aortic elastin

Yu Zou · Yanhang Zhang

Received: 11 May 2010 / Accepted: 25 September 2010© Springer-Verlag 2010

Abstract In this paper, we studied the viscoelastic behav-iors of isolated aortic elastin using combined modeling andexperimental approaches. Biaxial stress relaxation and creepexperiments were performed to study the time-dependentbehavior of elastin. Experimental results reveal that stressrelaxation preconditioning is necessary in order to obtainrepeatable stress relaxation responses. Elastin exhibits lessstress relaxation than intact or decellularized aorta. The rateof stress relaxation of intact and decellularized aorta is line-arly dependent on the initial stress levels. The rate of stressrelaxation for elastin increases linearly at stress levels belowabout 60 kPa; however, the rate changes very slightly athigher initial stress levels. Experimental results also showthat creep response is negligible for elastin, and the intactor decellularized aorta. A quasi-linear viscoelasticity modelwas incorporated into a statistical mechanics based eight-chain microstructural model at the fiber level to simulate theorthotropic viscoelastic behavior of elastin. A user materialsubroutine was developed for finite element analysis. Resultsdemonstrate that this model is suitable to capture both theorthotropic hyperelasticity and viscoelasticity of elastin.

Keywords Elastin · Viscoelasticity · Orthotropichyperelasticity · Constitutive model · Biaxial tensile test ·Stress relaxation

1 Introduction

As one of the major extracellular matrix (ECM) compo-nents, elastin is essential to accommodate physiological

Y. Zou · Y. Zhang (B)Department of Mechanical Engineering, Boston University,110 Cummington Street, Boston, MA 02215, USAe-mail: yanhang@bu.edu

deformation and provide elastic support for blood vessels.Elastin degradation is greatly associated with loss in com-pliance and stiffening in the arterial wall, which presentsin many cardiovascular diseases including diabetes, hyper-tension, aortic aneurysm, and atherosclerosis (Campa et al.1987; Chung et al. 2009; Cameron et al. 2003; Agabiti-Roseiet al. 2009; Boutouyrie et al. 2008; Diez 2007; Uitto 1979;Satta et al. 1998). Degradation of elastin in the artery hasbeen assumed to be a major cause that alters the stress distri-bution produced by blood pressure (MacSweeney et al. 1992,1994; Silver et al. 2001).

Elastin degradation and failure greatly compromises themechanical properties of blood vessels. The mechanicalproperties of elastin have been studied in the past threedecades owing to its importance on the functionality of manyconnective tissues (Lyerla and Torchia 1975; Gosline 1976;Lillie and Gosline 1990; Daamen et al. 2007; Gundiah et al.2007, 2009; Lillie and Gosline 2007; Zou and Zhang 2009).However, the time-dependent properties of elastin and itsinvolvement in physiological functions and diseases remainto be understood. The viscoelastic properties of elastin areclosely related to its microstructure, hydration level, pres-sure, temperature, and the external mechanical and chemi-cal environments (Lillie and Gosline 1990, 2002; Weinberget al. 1995; Spina et al. 1999). Elastin in the medial layer ofarteries consists of hydrated elastin fibers that are approxi-mately 3 µm in diameter, which are assembled into the lamel-lar structures at the microscopic level (Winlove and Parker1990). In previous work, molecular probe techniques havebeen used to characterize the organization of isolated elastin(Weinberg et al. 1995). The fibers were found to contain anetwork of water-filled pores accessible to solutes withmolecular weights below 1,000 daltons. The water spacesbetween and around fibrils are accessible to much largersolutes. The intra- and extrafibrillar compartments can be

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Y. Zou, Y. Zhang

modified by mechanical stresses, chemical environment,osmotic pressure, and temperature (Weinberg et al. 1995).When subjected to external stress, elastin molecules rear-range. The relaxation time required for the rearrangement isdirectly related to the free volume available for the molecularchains to move (Lillie and Gosline 1996; Knauss and Emri1981).

In a few previous studies on the time-dependentmechanical behavior of elastin, loading has been limited touniaxial stretching (Weinberg et al. 1995; Silver et al. 2001;Lillie and Gosline 1996, 2002), which ignores the multiax-ial loading state under physiological conditions and thus theintrinsic anisotropic properties of soft tissue. Under physi-ological conditions, aortas are subjected to cyclic strains inboth circumferential and longitudinal directions due to thenormal and shear forces exerted by pulsatile blood flow. Pla-nar biaxial tensile test with independent control of load inboth perpendicular directions has been used broadly to studythe mechanical behavior of various soft biological tissues(Sacks and Sun 2003; Geest et al. 2006; Knezevic et al. 2002;Zou and Zhang 2009; Jhun et al. 2009). Although it cannotreplicate the physiological loading conditions, biaxial tensiletest is sufficient on elucidating the anisotropic mechanicalproperties of soft tissues with plane stress assumptions. Suchcapabilities make it a useful tool for in vitro understandingof tissue mechanics and for determining material parametersin sophisticated constitutive models.

It is commonly accepted that for soft biological tissues,the initial loading curves are substantially different from thesubsequent loading curves (Fung 1993), which makes pre-conditioning an important step in order to obtain a repeat-able stress–strain response of soft tissues (Sauren et al. 1983;Carew et al. 2000). However, preconditioning has often beenlimited to studies on the elastic behaviors of soft biologicaltissues. Carew et al. (2004) found that porcine aortic valvecan only exhibit repeatable stress relaxation curves when theyare subjected to at least 5 cycles of repeated stress relaxationtest. A recent study found that ligaments have good repeat-ability for both elastic and stress relaxation tests (Öhman et al.2009). Despite the discrepancy, to the best of our knowledgeno study has been conducted to assess the repeatability ofstress relaxation behavior of blood vessels and ECM.

Incorporation of viscoelasticity into constitutive modelswill introduce additional material parameters, thus model-ing the time-dependent behavior of soft tissue adds an addi-tional level of computational complexity. Several generaltechniques have been proposed in the literature to model theviscoelastic behavior of soft tissue. State variable approachintroduced a set of state variables to model the nonlinearinelastic process (Holzapfel et al. 2002). Biphasic model thatincludes both solid and interstitial fluid phases was devel-oped for modeling the time-dependent behavior of cartilage(Mow et al. 1980; Soulhat et al. 1999). The rheological

network formulation incorporating a discrete number ofelastic and viscous elements in parallel and/or in serieswas applied to model the viscoelastic behavior of elasto-mers (Bergstrom and Boyce 2001) and biological soft tissue(Bischoff et al. 2004). The quasi-linear viscoelasticity (QLV)theory was introduced into the biomechanical literature byFung (1993). This model is characterized by an instanta-neous elastic stress response and a relaxation function. Dueto its relative ease to implement as well as the limited num-ber of material parameters required to model the viscoelasticresponse of soft tissue, the theory of QLV has been widelyimplemented to model the viscoelastic response of soft bio-logical tissues (Kwan et al. 1993; Carew et al. 1999; Sverdlikand Lanir 2002; Doehring et al. 2004; Bischoff 2006; Gileset al. 2007).

In this study, the viscoelastic behaviors of healthy aor-tic elastin were studied using combined experimental andmodeling approaches. Biaxial stress relaxation and creepexperiments were performed to study the time-dependentbehavior of elastin. Preconditioning of stress relaxation onelastin was investigated. QLV model was incorporated intoa statistical mechanics based eight-chain microstructuralmodel to describe the orthotropic viscoelastic behavior ofaortic elastin.

2 Materials and methods

2.1 Sample preparation

Thoracic aortas were harvested from pigs (12–24 months;160–200lb) at a local abattoir and transported to the labora-tory on ice. Before experiments, the aortas were cleaned offadherent tissues and fat and rinsed in distilled water. Thoracicaortas of similar sizes were chosen for experiments. Samplesof about 20 × 20 mm were cut at about the midpoint of thethoracic aorta. All samples were taken from the same lon-gitudinal region of the aorta to avoid changes in mechanicalproperties with aortic longitudinal position. Experiments onaortas were performed within one hour from delivery. Iso-lated elastin was obtained using a cyanogen bromide (CNBr)treatment (Lu et al. 2004), following procedures describedin our previous study (Zou and Zhang 2009). Briefly,aortic samples were treated in 50 mg/mL CNBr (AcrosOrganics) in 70% formic acid (Acros Organics) solution withgentle stirring for 19 h at room temperature. They were thenstirred in the same solution for 1 h at 60◦C, followed by 5- minboiling. Decellularized ECM was obtained using TritonX-100 decellularization process (Bader et al. 1998). Aor-tic samples were subjected to continuous shaking in a solu-tion of 1% Triton X-100 (Bio-Rad) with 0.2% EDTA inhypotonic Tris buffer (10 mM Tris, pH 8.0) for 48 h,

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The orthotropic viscoelastic behavior of aortic elastin

together with RNase A (20 µg/ml) and DNase (0.2 mg/ml).Dimensions of aorta, decellularized ECM, and elastin tis-sue samples were measured and recorded for further exper-iments and stress calculation. The length and width of thesample were measured using a digital caliper. The thicknessof the each tissue sample was measured at six different loca-tions across the sample and averaged. The tissue sampleswere kept in 1X phosphate buffered saline (PBS) for furtherexperiment. All the chemicals were purchased from FisherScientific unless otherwise specified.

2.2 Scanning electron microscopy (SEM)

SEM was performed on the cross sections of tissue samplesto examine the morphology using a JOEL JSM-6100 SEMmachine operated at 5kv. Before SEM, the samples were fixedin Karnovsky’s fixative kit (2% paraformaldehyde, 2.5%glutaraldehyde, and 0.1M sodium phosphate buffer, FisherScientific) for 1 h. The fixed samples were then dehydrated ina series of graded ethanol of 50, 70, 85, and 100%. The sam-ples were then dried using Denton vacuum DCP-1 CO2 crit-ical point dryer. A 20-nm thin layer of platinum was coatedon the cross section of the samples using Cressington 108Sputter Coater.

2.3 Histology

Histology studies were performed to confirm the removal ofcells in the decellularized ECM, and collagen and other ECMcomponents in the isolated elastin. Tissue samples were fixedin 10% formalin buffer and then embedded in paraffin. Thinsections of about 6 µm in thickness were cut for histolog-ical stain. Tissue samples with Movat’s pentachrome stainwere examined for the presence of cells, elastin, and colla-gen. Alcian blue stain (PH = 2.5) was also performed to con-firm the removal of glycosaminoglycans (GAGs) in isolatedelastin.

2.4 Biaxial tensile testing

A biaxial tensile testing device was used to perform biax-ial stress relaxation and creep tests. This device has beenused in our previous study on the anisotropic mechanicalbehavior of elastin (Zou and Zhang 2009). Briefly, tissuesamples of about 2 cm in square were sutured at the sides,and an equi-biaxial membrane tension was applied to thesample. All biaxial tensile tests of elastin were performed in1X PBS at room temperature following test protocols. Theaorta and decellularized ECM were also tested for compar-ison. Before stress relaxation or creep tests, tissue sampleswere preconditioned for 8 cycles of quasi-static tensile testswith 15 s of half cycle time equi-biaxially to obtain repeat-able material response. This preconditioning test began from

an initial 3 g tare load to the peak membrane tension to beapplied in the following stress relaxation or creep tests. Thepeak membrane tension varied from 20 to 160N/m.

For stress relaxation test, equibiaxial tension was appliedto the sample following the preconditioning protocols. Imme-diately after, the sample was quickly stretched to the targetstretch with a rise time of 2 s and held at this constant stretchfor 1,800 s. For creep test, after the last cycle of equibiaxialtension, the tissue sample was quickly loaded, and the ten-sion was kept constant for 1,800 s. The tension and stretchin both loading directions were recorded during the holdingperiod. In repeatability tests, five cycles of stress relaxationand creep tests were performed on each sample to confirmthe repeatability of the viscoelastic behavior. To study thedependence on initial stress level, repeatability tests were firstperformed to achieve repeatable stress relaxation properties,the sample was then tested under different initial stresses.The sequence of initial stress levels is random. Cauchystress σ (Zou and Zhang 2009) and Green-Lagrange strain E(Humphrey 2002) were calculated and used to describe themechanical behavior of elastin.

2.5 Constitutive modeling

In the present study, one-dimensional QLV model wasincorporated into a statistical mechanics based orthotropichyperelastic model (Bischoff et al. 2002) implemented pre-viously by Zhang et al. (2005) for the study of the mechan-ics of aortic elastin (Zou and Zhang 2009). This three-dimensional orthotropic viscoelastic model was used tosimulate the time-dependent mechanics of elastin. Finite ele-ment implementation of the model was realized using dis-crete spectrum approximation (Puso and Weiss 1998).

2.5.1 Incorporation of fiber-level viscoelasticity for thetissue-level time-dependent behavior

The QLV model was firstly incorporated into a single freelyjointed chain. The viscoelastic chain stress was then incorpo-rated into a network composed of eight-chain unit elementsto model the orthotropic viscoelastic behavior at the tissuelevel (Bischoff 2006). According to the QLV theory, the gen-eral form of the second Piola-Kirchhoff (2nd P-K) stress S(t)is given by (Fung 1993):

S(t) =t∫

0

G(t − τ)∂Se

∂τdτ (1)

In Eq. (1), Se is the elastic stress function and G(t) is thereduced relaxation function:

G(t) = 1 + C[E1(t/τ2) − E1(t/τ1)]1 + C ln(τ2/τ1)

(2)

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Y. Zou, Y. Zhang

where C, τ1, and τ2 are material parameters and E1(z)is the exponential integral function defined by E1(z) =∫ ∞

ze−t

t dt (|arg z| < π) .

The macroscopic strain energy function of the eight-chainorthotropic hyperelastic model is given by (Zhang et al. 2005;Bischoff et al. 2002):

W = nk�

4

[N

4∑i=1

(ρi

Nβ i

ρ + lnβ i

ρ

sinh β iρ

)

− βp√N

ln(λa2

a λb2

b λc2

c

)]+ B [cosh(J − 1) − 1] + W0

(3)

where W is the overall strain energy function, n is the chaindensity per unit volume, and N is the number of rigid linkswithin each chain. Parameters a, b, and c, normalized bythe length of rigid links, are dimensions of the unit elementalong the principal material directions a, b, and c, respec-tively. λa, λb, and λc are stretches along these directions.p = 1

2

√a2 + b2 + c2 is the undeformed normalized length

of each chain, and ρi is the normalized deformed length ofthe i-th chain. βp = −1(p/N ), and β i

ρ = −1(ρi/N ) arethe inverse Langevin functions. k is Boltzmann’s constant(k = 1.38 × 10−23J/K), and � is the absolute temperature.B is a parameter controlling the bulk compressibility. It isassumed to be much greater than the initial stiffness nk� inorder to model the nearly incompressibility of soft tissue. Jis the volume ratio. The elastic 2nd P-K stress tensor Se canbe obtained by:

Se = ∂W

∂E= n

4

[4∑

i=1

Sechain

−k�βp√

N

(a2

λ2a

a ⊗ a + b2

λ2b

b ⊗ b + c2

λ2c

c ⊗ c

)]

+B JC−1 sinh(J − 1) (4)

where C is the right Cauchy-Green tensor and C = FTF,

E = 12 (C − I) is the Green-Lagrange strain tensor, and F is

the deformation gradient. Sechain is the elastic 2nd P-K stress

of a single freely jointed chain and is given by:

Sechain = k�

pi ⊗ pi

ρiβρi (5)

where pi is the normalized chain vector describing the unde-formed chain.

To model the tissue-level time-dependent behavior, theelastic chain stress Se

chain in Eq. (4) is replaced by the vis-coelastic chain stress according to the QLV formulation.

The resulting viscoelastic stress tensor S in the macroscopiccontinuum model is then described as:

S = n

4

⎡⎣ 4∑

i=1

t∫

0

G (t − τ)∂

(Se

chain

)∂τ

dτ

−k�βp√

N

(a2

λ2a

a ⊗ a + b2

λ2b

b ⊗ b + c2

λ2c

c ⊗ c

)]

+B JC−1 sinh(J − 1) (6)

2.5.2 Finite element implementation

For finite element implementation, the viscoelastic stress,which is specified by the history of deformation, is eval-uated using numerical integration algorithm. The reducedrelaxation function in Eq. (2) is approximated by a series ofexponential functions and can be written as (Puso and Weiss1998):

G(t) = Ge + G0 − Ge

Nd + 1

Nd∑I=0

exp(−t/10(I+Io)

)(7)

where Ge, G0, Nd , and I0 are material parameters that will beobtained by fitting the simulation results to the experimentalstress relaxation data.

Based on the approximated reduced relaxation function inEq. (7), the viscoelastic stress at time t +�t can be evaluatedbased on the stress at time t , following Eq. (6):

t+�t∫

0

G (t + �t − τ)∂Se

chain,t+�t

∂τdτ

=t+�t∫

0

⎡⎣Ge + K

Nd∑I=0

exp

(−(t + �t − τ)

vi

)⎤⎦ ∂Se

chain,t+�t

∂τdτ

= Ge(Se

chain,t+�t − Sechain,t=0

) + KNd∑

I=0

exp

(−�t

vi

)HI

+KNd∑

I=0

exp

(−�t

vi

) t+�t∫

t

exp

(− (t − τ)

vi

)∂Se

chain,t+�t

∂τdτ

(8)

where vi = 10I+I0 , K = G0−GeNd+1 and HI = ∫ t

0 exp(− (t−τ)

vi

)∂Se

chain,t∂τ

dτ which can be obtained from previous

time step when �t = 0.In Eq. (8), the first term can be obtained from elastic chain

stress; the second term needs to recall the viscoelastic chainstress from the previous time step. An approximation is made

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The orthotropic viscoelastic behavior of aortic elastin

to the third term using the central difference rule so that:

KNd∑

I=0

exp

(−�t

vi

) t+�t∫

t

exp

(− (t − τ)

vi

)∂Se

chain,t+�t

∂τdτ

= KNd∑

I=0

exp

(−�t

vi

) Sechain,t+�t − Se

chain,t

�t

×t+�t∫

t

exp

(− (t − τ)

vi

)dτ (9)

In addition to the stress tensor, the fourth-order elasticity ten-sor H is needed in order to form the stiffness matrix for finiteelement analysis. The time-dependent elasticity tensor H isdefined as H = 2 ∂S(t+�t)

∂C(t+�t) , where C(t + �t) is the rightCauchy-Green deformation tensor at time t + �t . For time-dependent analysis, H(t + �t) can be estimated from (Pusoand Weiss 1998):

H(t + �t) =⎡⎣Ge + K

Nd∑I=0

(1 − exp(−�t/vi )

�t/vi

⎤⎦ × H(0)

(10)

where H(0) is the elasticity tensor at t = 0, i.e., the begin-ning of the holding period. The elasticity tensor H(0) can beobtained from H = 2 ∂Se

∂C , where Se is the elastic 2nd P-Kstress tensor given in (Eq. 4) (Zhang et al. 2005).

The Cauchy stress and elasticity tensor in the deformedconfiguration can then be achieved by applying a push for-ward operation (Holzapfel 2000) on Eqs. (6) and (10). Auser material subroutine (UMAT) was developed to imple-ment the three-dimensional orthotropic viscoelastic consti-tutive model in ABAQUS 6.7. The elastic and viscoelasticchain stresses were recorded and stored at every time stepfor the calculation of viscoelastic chain stress in the nexttime step. For finite element simulation, a quarter of the sam-ple was modeled due to the symmetric loading conditions inbiaxial tensile testing. General-purpose shell elements (S4R)were used with inherent plane stress assumption. Finite ele-ment simulation of stress relaxation reflected the experimen-

tal procedure described earlier. During the first time step,shell edge load was applied on the sides of the sample tosimulate the biaxial tensile loading. In the second time step,displacements were held constant for about 30 min, and thestress was recorded during the stress relaxation process.

2.6 Statistical analysis

Normalized creep results were expressed as mean ± stan-dard error of the mean. Comparisons between elastin, aorta,and decellularized ECM samples were made using two-tailtwo-sample t-Test assuming unequal variances. Differencesare considered statistically significant when p < 0.05.

3 Results

CNBr treatment removes smooth muscle cells, collagen, andother ECM components to obtain the isolated elastin. TritonX-100 decellularization process removes the smooth musclecells. Decellularization and isolation processes significantlydecrease the thickness of the tissue. The average thicknessesfor thoracic aorta, decellularized ECM, and elastin samplesare (mean±SD, n = 6 for each tissue type) 1.70 ± 0.20 mm,1.34 ± 0.12 mm, and 0.78 ± 0.15 mm, respectively. SEMimage in Fig. 1a shows that the cross section of isolatedelastin consists of concentric layers of elastin sheets, whichare less obvious with the presence of collagen fibers andsmooth muscle cells in the intact aorta (Fig. 1c). Decellu-larized ECM (Fig. 1b) preserves the major microstructure ofaorta. Removal of both the smooth muscle cells and colla-gen fibers in elastin is verified by histological images withMovat’s pentachrome stain in Fig. 2a. For the decellular-ized ECM, only cells are completely removed, as shown inFig. 2b. Histology image of the intact porcine aorta in Fig. 2cshows a cross-linked network of collagen and elastin fiberswith embedded cells. Alcian blue stain (pH=2.5) was per-formed to further verify the removal of GAGs in elastin, sinceGAGs play a crucial role in tissue viscoelasticity (Stephenset al. 2008; Liao and Vesely 2004). Figure 2f shows the

100 µm 50 50

(a) (b) (c)

µm µm

Fig. 1 SEM images of (a) isolated elastin, (b) decellularized ECM, and (c) intact aorta

123

Y. Zou, Y. Zhang

(a) (b) (c)

(f)(e)(d)

Fig. 2 Histology images of (a, d) isolated elastin, (b, e) decellularized ECM, and (c, f) intact aorta. Movat’s pentachrome stains elastin purple-black,collagen yellow, and cells red (a–c). Alcian blue stains GAGs blue and cells pink (d–f)

50

60

70

80

0 500 1000 1500 2000

Time t (sec)

Cau

chy

stre

ss

(kP

a)

Cycle 1Cycle 2Cycle 3Cycle 4Cycle 5

Fig. 3 Representative stress relaxation curves obtained from repeatedtesting the same elastin sample for five testing cycles. Stress relaxationresponses in both circumferential and longitudinal directions are shown

presence of smooth muscle cells, GAGS, and collagen andelastin fibers in the aorta sample. The absence of cells inelastin and decellularized ECM is shown again in Fig. 2d,e.GAGs are preserved in the decellularized ECM but are com-pletely removed in the isolated elastin. Both SEM and histol-ogy images of isolated elastin samples exhibit a lower densityof fiber network due to the removal of cells and other ECMcomponents.

Multiple stress relaxation tests on samples of elastin(n = 6), intact aorta (n = 7), and decellularized ECM (n = 6)were performed to examine the repeatability of stress relaxa-tion behavior. Figure 3 shows the representative stress relax-ation curves obtained from five repeated testing of the same

aortic elastin sample. Elastin exhibits significantly morestress relaxation in the first test. The repeatability is highlyimproved in the subsequent cycles of stress relaxation tests.Although not shown here, intact aorta and decellularizedECM exhibits similar behavior with the initial stress relax-ation curves significantly different from subsequent testing.The repeatability ratio between two curves is defined as theminimum of the ratios of stresses from the more relaxed tothe less relaxed curve calculated at 2, 10, 100, 1,000 and1,800 s (Carew et al. 2004). Table 1 shows the repeatabilityof elastin, aorta, and decellularized ECM. The repeatabilityof stress relaxation behavior is improved by applying sub-sequent cycles of stress relaxation tests. The fifth cycle ofstress relaxation has raised the repeatability to about 0.99for all the samples. These results confirm that stress relaxa-tion preconditioning is essential to achieve repeatable stressrelaxation behavior in soft biological tissues. The fifth cycleof stress relaxation from all three tissue types has satisfyingrepeatability and can be used to study the time-dependentmechanical behavior. If not specifically indicated, the stressrelaxation results showed in the present study are referred toresults from the fifth cycle.

Figure 4 shows the representative biaxial stress relaxa-tion curves of the elastin, intact aorta, and decellularizedECM. All samples were tested at the initial stresses of101.03±0.72 kPa to eliminate the effect of initial stress levelson the rate of stress relaxation. In order to compare the time-dependent responses of different tissue samples, the stresseswere normalized to that at the beginning of the holdingperiod. Over the holding period of half an hour, aorta showsmore significant stress relaxation than decellularized ECM

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The orthotropic viscoelastic behavior of aortic elastin

Table 1 Repeatability of stressrelaxation tests of elastin,decellularized ECM, and aortasamples

Cycle 1 vs. 2 Cycle 2 vs. 3 Cycle 3 vs. 4 Cycle 4 vs. 5

Elastin (n = 6) 0.9031±0.0164 0.9792±0.0147 0.9783±0.0171 0.9926±0.0054

ECM (n = 6) 0.9325±0.0345 0.9668±0.0122 0.9861±0.0096 0.9912±0.0076

Aorta (n = 7) 0.9365±0.0257 0.9621±0.0345 0.9790±0.0146 0.9923±0.0089

0.8

0.85

0.9

0.95

1

0 500 1000 1500 2000

Time t (sec)

No

rmal

ized

str

ess

Elastin

ECM

Aorta

L C

Fig. 4 Representative stress relaxation curves of elastin, decellularizedECM, and intact aorta at initial stresses of 101.03±0.72 kPa. Solid anddotted lines represent the stress relaxation behaviors in longitudinal (L)and circumferential (C) directions, respectively

and elastin in both the circumferential and longitudinal direc-tions. Moreover, decellularized ECM shows more stressrelaxation than elastin. For all three tissue types, most ofthe stress relaxation happens during the first 10 min of theholding period.

Representative biaxial creep behavior of elastin is shownin Fig. 5. Under equi-biaxial tension, elastin has higher strainin the longitudinal direction than in the circumferential direc-tion due to the anisotropic mechanical property of the elastin.The results show that the creep behavior is negligible for theelastin. Although not shown here, aorta and decellularizedECM exhibits similar negligible creep behavior as elastin.The normalized maximum creep of elastin, decellularizedECM, and intact aorta (n = 3) is 101.6±1.4%, 101.7±0.5%,and 101.7 ± 1.3%, respectively, with no statistically signifi-cance and is considered negligible.

The dependence of stress relaxation upon initial stresslevels was investigated. Stress relaxation tests were per-formed at varying stress levels by applying different initialequibiaxial tension to the tissue samples. Each sample wastested at multiple initial stress levels. Figure 6 shows therepresentative stress relaxation curves for aorta, ECM, andelastin samples at different initial stress levels. To better com-pare the initial stress level dependence of stress relaxation,the rate of stress relaxation, stress drop rate, at each initialstress level was obtained by taking the slope of the stressesvs. time semi-log plots in Fig. 6. Averaged stress drop rate

0

0.05

0.1

0.15

0.2

0.25

0 500 1000 1500 2000

Time t (sec)G

reen

-Lag

ran

ge

stra

in E

L

C

Elastin

Fig. 5 Representative creep curves of elastin. The circumferential andlongitudinal direction has different initial strains due to the anisotropyof the elastin. Solid and dotted lines represent the creep behaviors inlongitudinal (L) and circumferential (C) directions, respectively

between the circumferential and longitudinal directions wastaken, although most samples exhibit almost identical stressrelaxation responses in both directions. As shown in Fig. 7,multiple stress relaxation tests demonstrate a linear increasein the rate of stress relaxation with higher initial stress forboth aorta and decellularized ECM. The rate of stress relaxa-tion for elastin increases linearly at stress levels below about60 kPa; however, the rate changes very slightly at higherinitial stress levels. Creep tests at different initial strain levelswere also performed. Elastin, decellularized ECM, and aortaall exhibit negligible creep behavior at various strain levels(results not shown).

In the present study, the QLV model was incorporatedinto a statistical mechanics-based orthotropic hyperelasticmodel, which has been developed earlier and demonstratedto be suitable to model the mechanical properties of elastin(Zhang et al. 2005; Zou and Zhang 2009). Figure 8 showsthe simulation results of elastic and viscoelastic behavior ofelastin. Experimental results were also plotted for the pur-pose of comparison. The material parameters can be dividedinto two groups: four independent parameters (a, b, c, andn) describing the orthotropic hyperelastic behavior of elastinand four more parameters (Ge, G0, Nd , and I0) describingthe viscoelastic behavior. Note that the eight-chain networkmodel does not represent the actual organization/alignmentof the elastin fibers. Here, the orthotropic eight-chain unit

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Y. Zou, Y. Zhang

0

40

80

120

160

1 10 100 1000 10000

Time t (sec)

L

C

Elastin

0

40

80

120

160

1 10 100 1000 10000

Time t (sec)

L

C

ECM

Cau

chy

stre

ss

(kP

a)

0

40

80

120

160

1 10 100 1000 10000

Time t (sec)

L

C

Aorta

Cau

chy

stre

ss

(kP

a)C

auch

y st

ress

(k

Pa)

(a)

(b)

(c)

Fig. 6 Representative stress relaxation at different initial stress levelsfor (a) elastin, (b) decellularized ECM, and (c) intact aorta. Solid anddotted lines represent the stress relaxation behaviors in longitudinal (L)and circumferential (C) directions, respectively

element is a functional representative in a mechanical senseof the load bearing fibrous network of elastin tissue. Thephysical meanings of material parameters (a, b, c, and n)

and the individual parametric effect on predicting the ortho-tropic hyperelastic stress–strain responses have been dis-cussed in our previous work (Zou and Zhang 2009). Thesematerial parameters were obtained by fitting the simulation

0

1

2

3

4

0 40 80 120 160

Initial stress level (kPa)

Str

ess

dro

p r

ate

ElastinECMAorta

Fig. 7 Stress drop rate as a function of initial stress levels for aorta,decellularized ECM, and elastin samples. Stress drop rate was obtainedby taking the slope of the stresses vs. time semi-log plots in Fig. 6and averaged between the longitudinal and circumferential directions.Linear trend lines were added for aorta and decellularized ECM to aidviewing. The stress drop rates of two elastin samples were presented tobetter observe the trend

results to the equi-biaxial tensile test data, as shown in Fig. 8a.The viscoelastic material parameters (Ge, G0, Nd , and I0)

were obtained by fitting the simulation results to the experi-mental data from stress relaxation tests, as shown in Fig. 8b.The first and fifth cycles of stress relaxation tests were simu-lated. Material parameters a, b, c, and n for both cycle testswere kept the same for the description of the orthotropichyperelastic behavior of elastin. The viscoelastic materialparameters were changed assuming that stress relaxation pre-conditioning only affects the viscoelasticity of elastin. Thesimulation results fit the experimental data reasonably wellfor both cycles. These results demonstrate that the orthotro-pic viscoelastic constitutive model incorporating fiber-levelviscoelasticity into the eight-chain statistical mechanics-based microstructural model is suitable to describe the stressrelaxation behavior of elastin.

4 Discussions

Our recent study revealed that aortic elastin possesses aniso-tropic mechanical behavior that is comparable to aorta (Zouand Zhang 2009). In the present study, the viscoelastic prop-erties of elastin were studied using biaxial stress relaxa-tion and creep tests. Stress relaxation preconditioning wasperformed to examine the repeatability of elastin stressrelaxation. Dependence of initial stress levels of elastin’sviscoelastic behavior was also investigated. Fiber-level vis-coelasticity of elastin fibers incorporated into an orthotro-pic hyperelastic constitutive model was developed to studythe time-dependent mechanics of elastin at the tissue level.

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The orthotropic viscoelastic behavior of aortic elastin

0

20

40

60

80

100

0 0.05 0.1 0.15 0.2 0.25

Green-Lagrange strain E

L C

40

50

60

70

80

0 500 1000 1500 2000

Time t (sec)

L C

Cycle 5

Cycle 1

Cau

chy

stre

ss (

kPa)

Cau

chy

stre

ss

(kP

a)(a)

(b)

Fig. 8 Simulation results of elastin first subjected to (a) equi-biax-ial tensile test, then (b) stress relaxation tests in the first andfifth cycles, represented by squares in longitudinal (L) and circlesin circumferential (C) direction. The corresponding experiment datawere also shown for the purpose of comparison and represented bysolid lines in longitudinal (L) and dotted lines in circumferential (C)direction. Material parameters in the simulation are n = 1.5 × 1016

(1/mm3), a = 1.386, b = 1.67, c = 1.3, N = 1.6, Ge = 0.84, Go =1.0, Nd = 6, and Io = 2 for the 1st cycle; and n = 1.5 × 1016

(1/mm3), a = 1.386, b = 1.67, c = 1.3, N = 1.6, Ge = 0.91, Go =1.0, Nd = 6, and Io = 2 for the 5th cycle

These results will help us understand how the effects ofmechanical loading translate into changes in the macroscopicviscoelastic properties of elastin, thus to advance our under-standing on the role of microstructural components on vas-cular remodeling.

Preconditioning has long been adopted to obtain a repeat-able stress–strain behavior of soft biological materials. Tra-ditionally, the sample goes through several cycles of loadingand unloading, either uniaxially or biaxially for precondi-tioning (von Maltzahn and Warriyar 1984; Yin et al. 1986;Fung 1993; Carew et al. 2000; Lally et al. 2004). Carew et al.(2004) found that repeatable stress relaxation curves cannotbe obtained from this conventional preconditioning methodfor porcine aortic valve tissues. They suggested at least fivecycles of repeated stress relaxation tests would be essential

and sufficient to obtain the repeatability of stress relaxation.Our results show similar phenomena (Fig. 3). For intact por-cine aorta, decellularized ECM, and isolated elastin, stressrelaxation preconditioning is necessary to ensure repeat-able stress relaxation behavior. The first cycle test showssignificantly more stress relaxation than the subsequentones. Through measuring the repeatability ratio between twocycles, the fifth cycle test can generate well-repeatable stressrelaxation curves (repeatability ratio > 0.991). The presentstudy was based on the repeatable stress relaxation behaviorof elastin, which was expected to be more representative andaccurate of the viscoelastic behavior of the three tissue types.

Biaxial stress relaxation tests reveal some interesting facts.The stress relaxation responses of the intact aorta and decell-ularized ECM are sensitive to the initial stress levels. Ourresults show that the rate of stress relaxation increases line-arly with the initial stress level for intact and decellularizedaorta, while for elastin the responses are only stress depen-dent at stress levels below about 60 kPa. At higher stresslevels, elastin becomes almost independent of initial stresslevels. Dependence of the magnitude of stress relaxation oninitial stress–strain levels has been shown in previous workon the viscoelasticity of ligament (Provenzano et al. 2001;Hingorani et al. 2004). These studies showed that the rateof stress relaxation decreases with higher stress or strainlevel. They speculated that the decrease in relaxation ratewith increasing strain could be the result of the larger strainscausing greater water loss, which makes the tissue moreelastic and less viscous. A recently study on bladder wall(Nagatomi et al. 2008) reported similar phenomena. More-over, they observed that the decellularized ECM of blad-der wall was insensitive to the initial stress level. They sug-gested that at lower stress level, more load is carried by thesmooth muscle component, while at the higher stress levelthe ECM is recruited to carry more of the load. The dis-crepancy between our results and previous findings mightdue to the differences in composition of biological tissues.Unlike ligament which is collagen dominant, the aorta is anelastic artery with elastin composed up to 30.1% of the totalweight (Humphrey 2002). It is well accepted that elastin isresponsible for the linear elastic response of blood vesselsat lower strains. While at higher strains, the aorta becomesmuch stiffer in both directions due to the strain stiffening andthe involvement of collagen fibers (Cox 1978; Fonck et al.2007; Lally et al. 2004). Our previous work (Zou and Zhang2009) further validated that elastin is responsible for the lin-ear elastic response of blood vessels under lower stress. Silveret al. (2001) have suggested that collagen fibers are relaxedat the lower stress state, but at the higher stress state, collagenfibers are uncrimped and start to take load. We speculate thatduring stress relaxation test of aorta at lower stress or strainlevel, more load is carried by elastin fibers, while at higherinitial stresses, collagen fibers become the major mechanical

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component that carry more of the overall load. The shift ofmechanical roles between the two major ECM componentsresults in more stress relaxation at higher initial strains foraorta and decellularized ECM.

Figure 4 compared the stress relaxation behavior of threetissue types at stress level of about 100 kPa, which has closephysiological relevance. Circumferential stresses in pig aor-tas at 100 mmHg are around 95–120 kPa (Guo and Kassab2004 and Stergiopulos et al. 2001). Previous work showedthat collagen fibers are generally recruited only around phys-iological loads (Silver et al. 2001; Lally et al. 2004; Foncket al. 2007). Aorta exhibits the most obvious stress relaxationthan elastin and decellularized ECM due to the involvementof collagen and smooth muscle cells. It is well recognizedthat elastin relaxes much less compared with smooth musclecells or collagen (Fung 1993). In a previous study on urinarybladder wall (Nagatomi et al. 2004), the amount of stressrelaxation has been demonstrated to correlate with collagencontent. Different stress relaxation responses between elastinand collagen were explained that elastin is easily to be dis-tended under small loads and does not dissipate stored energyas readily as collagen and smooth muscle cells. Our resultsfurther demonstrate that elastin relaxes much less than intactaorta or decellularized ECM. Moreover, our study shows thatdecellularized ECM exhibits less stress relaxation than intactaorta due to the absence of smooth muscle cells. Smoothmuscle has a tremendous relaxation though its elastic mod-ulus is low (Fung 1993). Nagatomi et al. (2008) showed thatdecellularized bladder wall tissue relaxes significantly lessthan intact bladder due to the lack of smooth muscle cells.A recent study by Williams et al. (2009) demonstrated thatthe viscoelastic characteristics of intact and decellularizedcarotid arteries are similar. The stress level of stress relaxa-tion was not reported. It is possible that in their study stressrelaxation was performed at low stress level, so that the dif-ference in stress drop rate between intact and decellularizedaorta is small. Also, storage of intact aortas at −20◦C mightcause a loss of smooth muscle cell integrity which mightaccount for the similarity in stress relaxation between theECM and aorta samples in their study.

Compared with the stress relaxation results, the creepbehaviors of aorta, decellularized ECM, and elastin samplesare negligible in both the circumferential and longitudinaldirections (Fig. 5). This phenomenon has also been observedin earlier studies of soft biological tissues including mitralvalve anterior leaflet (Grashow et al. 2006; Liao et al. 2007)and aortic heart valve leaflet (Stella et al. 2007). The pres-ent study is the first to show that isolated aortic elastin hadvery little creep compared with its obvious stress relaxation.Grashow et al. (2006) put forward that complete lack of creepappears to be unique in soft biological tissues and has a func-tionally independent mechanism as stress relaxation behav-ior. Recently, a “locked-up” fibril network explanation was

brought by Liao et al. (2007). They proposed that as a result ofthe stress re-distributed to noncollagenous ECM under con-stant strain conditions, it allows for stress reduction underconstant strain and prevents strain increasing under constantstress conditions. Stella et al. (2007) suggested that fibersare allowed to rotate in a time-dependent manner to achieve astate of reduced fiber stress in the uniaxial creep testing; how-ever, during biaxial creep test, there are no unconstrained seg-ments of the sample due to a lack of time-dependent changesin fiber orientation.

Less creep than stress relaxation has been observed instudies of ligament (Thornton et al. 1997; Provenzano et al.2001; Hingorani et al. 2004) and corneas (Boyce et al. 2007;Nagatomi et al. 2008). Thornton et al. (1997) suggested thatthe paradoxical observation between creep and stress relaxa-tion behavior may be explained by the functionally indepen-dent mechanisms of these time-dependent behaviors. Stressrelaxation response is corresponding to a discrete group offibers being recruited at prescribed constant elongation, whilecreep is corresponding to fibers being progressively recruitedat constant stress. Their later study (Thornton et al. 2001)incorporated fiber recruitment in structural model whichcould predict creep from stress relaxation behavior. Theysuggested that the differences in the stress relaxation andcreep behavior in ligaments were explained as a result fromthe different ways of recruiting relaxing fibers. The gradualrecruitment of crimped fibers provides a protective functionto reduce creep. The lack of creep can also due to the variationof fibril-ECM interactions, such as the proteoglycan–colla-gen interactions (Grashow et al. 2006; Stella et al. 2007).Our biaxial creep results show no creep for aorta, decellu-larized ECM, and elastin samples, which also suggests thatstress relaxation and creep depend on functionally differentmechanisms. However, more researches are needed to fullyunderstand the separate functions and relations of creep andstress relaxation.

QLV theory has been widely used to model the viscoelas-tic response of soft tissue, including ligaments (Provenzanoet al. 2001), tendons (Johnson et al. 1994; Atkinson et al.1999), heart valves (Carew et al. 1999), bladders (Nagatomiet al. 2004), and skeletal muscle (Best et al. 1994). Nagatomiet al. (2004) evaluated the viscoelastic properties of normaland spinal cord injured rat bladders. The reduced relaxationfunction in the circumferential and longitudinal directionswas fitted to the tissue stress relaxation responses to deter-mine the material parameters. Giles et al. (2007) incorporatedQLV into an exponential model to examine the anomalousrate dependence of the preconditioned response of porcinedermis during load controlled deformation. Bischoff (2006)incorporated the QLV model into a fiber-based orthotropichyperelastic constitutive model for the study of tissue-levelviscoelastic behavior. Similar approach has been adopted inthe present study.

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The orthotropic viscoelastic behavior of aortic elastin

0

40

80

120

160

1 10 100 1000 10000

Time t (sec)

L C

Cau

chy

stre

ss

(kP

a)

Fig. 9 Simulation results of stress relaxation responses of elastin atdifferent initial stress levels, represented by squares in longitudinal(L) and circles in circumferential (C) direction. The correspondingexperiment data were also shown for the purpose of comparison andrepresented by solid lines in longitudinal (L) and dotted lines in circum-ferential (C) direction. Material parameters for all the simulations aren = 1.75 × 1016 (1/mm3), a = 1.53, b = 1.54, c = 1.3, N = 1.6,

Ge = 0.89, Go = 1.0, Nd = 6, and Io = 2

Although the simulation results fit the experimental datareasonably well, the deficiencies associated with the orthotro-pic viscoelastic constitutive model incorporating fiber-levelviscoelasticity using a QLV model should be noted. SinceQLV model assumes a separate strain and time dependenceof the stress response, it is incapable to simulate the initialstress dependent behavior of stress relaxation. Therefore, afit for stress relaxation test based on one stress–strain levelwould predict the same stress relaxation rate for all strains.Such characteristics will limit the usage of QLV model instudies where the viscoelasticity of the aorta or decellular-ized ECM is concerned. However for elastin, as shown inFig. 9, since the dependence of the rate of stress relaxationon the initial stress level is small at physiological load, thematerial parameters fitted from one test can be used to sim-ulate the stress relaxation behavior of elastin under differ-ent initial stress levels and provide reasonable predictions.QLV model is also found to be insufficient to characterizethe creep behavior of soft tissues due to its intrinsic assump-tions on separable strain and time dependence of the stressresponse (Thornton et al. 1997; Boyce et al. 2007; Nguyenet al. 2008). Despite the deficiencies of QLV model, the neg-ligible creep behavior and less initial stress dependence ofstress relaxation behavior of elastin make this present modelstill adequate to investigate the mechanics of elastin.

There are several limitations in the current study callingfor further investigations into the issue. Microstructure stud-ies are lacking to relate the effect of mechanical loading onviscoelasticity. The different mechanisms of stress relaxationand creep behavior are not fully understood, which requiresfurther exploration. Microscopy imaging is needed to bet-ter understand the microstructure change of elastin network

during deformation and explore the mechanisms of stressrelaxation and creep. Besides, although the material param-eters describing the orthotropic hyperelasticity in the consti-tutive model have physical meanings and could be related tothe microstructure of elastin, the QLV model for fiber-levelviscoelasticity is phenomenological and lacks structural con-nections. Due to current experimental device limitations, allthe experiments were performed at room temperature and notat 37◦C. The mechanical behaviors of soft tissues are tem-perature dependent (Guinea et al. 2005). However, previousstudy demonstrated weak temperature effect on the passivemechanical behavior of arterial wall below 60◦C (Kang et al.1995). Recent study showed that the viscoelastic properties oftendons are not sensitive to temperature over short durationsbetween room temperature and 37◦C (Huang et al. 2009).Although temperature dependency is not considered in thisstudy, we expect that the relative responses in the three tissuetypes would be the same at 37◦C.

5 Conclusions

To the best of our knowledge, the present study is the firstto investigate the time-dependent behavior of elastin usingbiaxial stress relaxation and creep tests. Our findings providefundamental understandings on the viscoelasticity of aorticelastin. Such understandings shed light on the role of elastinon controlling the mechanical properties of blood vessels andare essential for future studies on vascular remodeling thatinvolves elastin degradation and failure. The time-dependentmechanics of aortic elastin under biaxial stress relaxationand creep testing were studied using a combination of exper-imental and theoretical method. Multiple stress relaxationpreconditioning was performed to obtain repeatable stressrelaxation behavior. Elastin shows less stress relaxationbehavior than intact or decellularized aorta samples. Thestress relaxation response of elastin is stress level depen-dent at stress levels below 60 kPa but becomes nearly inde-pendent at high stress levels. However, the creep responseis negligible for both elastin, and the intact or decellular-ized aorta samples. The orthotropic viscoelastic constitutivemodel incorporating fiber-level viscoelasticity through QLVmodel into an eight-chain statistical mechanics-based micro-structural orthotropic model is suitable to describe both elas-tic and viscoelastic responses of elastin.

Acknowledgments This work was supported in part by funding fromNSF Grants 0700507, CAREER 0954825, and the College of Engineer-ing at Boston University.

References

Agabiti-Rosei E, Portei E, Rizzoni D (2009) Arterial stiffness, hyper-tension, and rational use of nebivolol. Vasc Health Risk Manag5:353–360

123

Y. Zou, Y. Zhang

Atkinson TS, Ewers BJ, Haut RC (1999) The tensile and stress relax-ation responses of human patellar tendon varies with specimencross-sectional area. J Biomec 32:907–914

Bader A, Schilling T, Teebken OE, Brandes G, Herden T, SteinhoffG, Haverich A (1998) Tissue engineering of heart valves-humanendothelial cell seeding of detergent acellularized porcine valves.Eur J Cardiothorac Surg 14:279–284

Bergstrom JS, Boyce MC (2001) Constitutive modeling of the time-dependent and cyclic loading of elastomers and application to softbiological tissues. Mech Mater 33:523–530

Best TM, McElhaney J, Garret WE, Myers BS (1994) Characterizationof the passive responses of live skeletal muscle using the quasilin-ear theory of viscoelasticity. J Biomec 27:413–419

Bischoff JE, Arruda EM, Grosh K (2002) Orthotropic hyperelasticityin terms of an arbitrary molecular chain model. J Appl Mech 69:198–201

Bischoff JE, Arruda EM, Grosh K (2004) A rheological network modelfor the continuum anisotropic and viscoelastic behavior of soft tis-sue. Biomech Model Mechanobiol 3:56–65

Bischoff JE (2006) Reduced parameter formulation for incorporatingfiber level viscoelasticity into tissue level biomechanical models.Ann Biomed Eng 34:1164–1172

Boutouyrie P, Laurent S, Briet M (2008) Importance of arterial stiff-ness as cardiovascular risk factor for future development of newtype of drugs. Fundam Clin Pharmacol 22:241–246

Boyce BL, Jones RE, Nguyen TD, Grazier JM (2007) Stress-controlledviscoelastic tensile response of bovine cornea. J Biomec 40:2367–2376

Cameron JD, Bulpitt CJ, Pinto ES, Rajkumar C (2003) The aging ofelastic and muscular arteries: a comparison of diabetic and nondi-abetic subjects. Diabetes Care 26:2133–2140

Campa JS, Greenhalgh RM, Powell JT (1987) Elastin degradation inabdominal aortic aneurysms. Atherosclerosis 65:13–21

Carew EO, Talman EA, Boughnew DR, Vesely I (1999) Quasi-linearviscoelastic theory applied to internal shearing of porcine aorticvalve leaflets. J Biomech Eng 121:386–392

Carew EO, Barber JE, Vesely I (2000) Role of preconditioning andrecovery time in repeated testing of aortic valve tissues: valida-tion through quasi-linear viscoelastic theory. Ann Biomed Eng28:1093–1100

Carew EO, Garg A, Barber JE, Vesely I (2004) Stress relaxation precon-ditioning of porcine aortic valves. Ann Biomed Eng 32:563–572

Chung AW, Yang HH, Sigrist MK, Brin G, Chum E, Gourlay WA, LevinA (2009) Matrix metalloproteinase-2 and -9 exacerbate arterialstiffening and angiogenesis in diabetes and chronic kidney dis-ease. Cardiovasc Res. doi:10.1093/cvr/cvp242

Cox RH (1978) Passive mechanics and connective tissue compositionof canine arteries. Am J Physiol Heart Circ Physiol 234:H533–H541

Daamen WF, Veerkamp JH, van Hest JCM, van Kuppevelt TH(2007) Elastin as a biomaterial for tissue engineering. Biomate-rials 28:4378–4398

Diez J (2007) Arterial stiffness and extracellular matrix. Adv Cardiol44:76–95

Doehring TC, Carew EO, Vesely I (2004) The effect of strain rate onthe viscoelastic response of aortic valve tissue: a direct-fitapproach. Ann Biomed Eng 32:223–232

Fonck E, Prod’hom G, Roy S, Augsburger L, Rufenacht DA, Stergiopu-los N (2007) Effect of elastin degradation on carotid wall mechan-ics as assessed by a constituent-based biomechanical model. AmJ Physiol Heart Circ Physiol 292:H2754–H2763

Fung YC (1993) Biomechanics: mechanical properties of living tissues.Springer, New York

Geest JPV, Sacks MS, Vorp DA (2006) The effects of aneurysm on thebiaxial mechanical behavior of human abdominal aorta. J Biomec39:1324–1334

Giles JM, Black AE, Bischoff JE (2007) Anomalous rate dependenceof the preconditioned response of soft tissue during load controlleddeformation. J Biomec 40:777–785

Gosline JM (1976) The physical properties of elastic tissue. Int RevConnect Tissue Res 7:211–249

Grashow JS, Sacks MS, Liao J, Yoganathan AP (2006) Planar biaxialcreep and stress relaxation of the mitral valve anterior leaflet. AnnBiomed Eng 34:1509–1518

Guinea GV, Atienza JM, Elices M, Aragoncillo P, Hayashi K(2005) Thermomechanical behavior of human carotid arteries inthe passive state. Am J Physiol Heart Circ Physiol 288:H2940–H2945

Gundiah N, Ratcliffe MB, Pruitt LA (2009) The biomechanics of arte-rial elastin. J Mech Behav Biomed Mater 2:288–296

Gundiah N, Ratcliffe MB, Pruitt LA (2007) Determination of strainenergy function for arterial elastin: Experiments using histologyand mechanical tests. J Biomec 40:586–594

Guo X, Kassab GS (2004) Distribution of stress and strain along theporcine aorta and coronary arterial tree. Am J Physiol Heart CircPhysiol 286:H2361–H2368

Hingorani RV, Provenzano PP, Lakes RS, Escarcega A, Vanderby RJr(2004) Nonlinear viscoelasticity in rabbit medial collateral liga-ment. Ann Biomed Eng 32:306–312

Holzapfel GA (2000) Nonlinear solid mechanics: a continuumapproach for engineering. Wiley, Chichester

Holzapfel GA, Gasser TC, Stadler M (2002) A structural model for theviscoelastic behavior of arterial walls: continuum formulation andfinite element analysis. Eur J Mech A Solids 21:441–463

Huang CY, Wang VM, Flatow EL, Mow VC (2009) Temperaturedependent viscoelastic properties of the human supraspinatus ten-don. J Biomec 42:546–549

Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues,and organs. Springer, New York

Jhun CS, Evans MC, Barocas VH, Tranquillo RT (2009) Planar biaxialmechanical behavior of bioartificial tissues possessing prescribedfiber alignment. J Biomech Eng 131:081006

Johnson GA, Tramaglini DM, Levine RE, Ohno K, Choi NY, WooSL (1994) Tensile and viscoelastic properties of human patellartendon. J Orthop Res 12:796–803

Kang T, Resar J, Humphrey JD (1995) Heat-induced changes in themechanical behavior of passive coronary arteries. J Biomech Eng117:86–93

Knauss WG, Emri IJ (1981) Non-linear viscoelasticity based on freevolume consideration. Comput Struct 13:123–128

Knezevic V, Sim AJ, Borg TK, Holmes JW (2002) Isotonic biaxialloading of fibroblast-populated collagen gels: a versatile, low-cost system for the study of mechanobiology. Biomech ModelMechanobiol 1:59–67

Kwan MK, Li THD, Woo SLY (1993) On the viscoelastic properties ofthe anteromedial bundle of the anterior cruciate ligament. J Biomec26:47–452

Lally C, Reid AJ, Prendergast PJ (2004) Elastic behavior of porcinecoronary artery tissue under uniaxial and equibiaxial tension. AnnBiomed Eng 32:1355–1364

Liao J, Vesely I (2004) Relationship between collagen fibrils, glycos-aminoglycans, and stress relaxation in mitral valve chordae tendi-neae. Ann Biomed Eng 32:977–983

Liao J, Yang L, Grashow J, Sacks MS (2007) The relationship betweencollagen fibril kinematics and mechanical properties in the mitralvalve anterior leaflet. J Biomech Eng 129:78–87

Lillie MA, Gosline JM (1990) The effects of hydration on the dynamicmechanical properties of elastin. Biopolymers 29:1147–1160

Lillie MA, Gosline JM (1996) Swelling and viscoelastic properties ofosmotically stressed elastin. Biopolymers 39:641–693

Lillie MA, Gosline JM (2002) The viscoelastic basis for the tensilestrength of elastin. Int J Biol Macromol 30:119–127

123

The orthotropic viscoelastic behavior of aortic elastin

Lillie MA, Gosline JM (2007) Limits to the durability of arterial elastictissue. Biomaterials 28:2021–2031

Lu Q, Ganesan K, Simionescu DT, Vyavahare NR (2004) Novel porousaortic elastin and collagen scaffolds for tissue engineering. Bio-materials 25:5227–5237

Lyerla JRJr , Torchia DA (1975) Molecular mobility and structure ofelastin deduced from the solvent and temperature dependence of13C magnetic resonance relaxation data. Biochemistry 14:5175–5183

MacSweeney STR, Young G, Greenhalgh RM, Powell JT(1992) Mechanical properties of the aneurismal aorta. Br JSurg 79:1281–1284

MacSweeney STR, Powell JT, Greenhalgh RM (1994) Pathogenesis ofabdominal aortic aneurysms. Br J Surg 81:935–941

Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep andstress relaxation of articular cartilage in compression: theory andexperiments. J Biomech Eng 102:73–84

Nagatomi J, Toosi KK, Chancellor MB, Sacks MS (2008) Contribu-tion of the extracellular matrix to the viscoelastic behavior of theurinary bladder wall. Biomech Model Mechanobiol 7:395–404

Nagatomi J, Gloeckner DC, Chancellor MB, DeGroat WC, SacksMS (2004) Changes in the biaxial viscoelastic response of theurinary bladder following spinal cord injury. Ann Biomed Eng32:1409–1419

Nguyen TD, Jones RE, Boyce BL (2008) A nonlinear anisotropic vis-coelastic model for the tensile behavior of the corneal stroma.J Biomech Eng 130:041020

Öhman C, Baleani M, Viceconti M (2009) Repeatability of experimen-tal procedures to determine mechanical behavior of ligaments.Acta Bioeng Biomech 11:19–23

Provenzano P, Lakes R, Keenan T, Vanderby R Jr (2001) Nonlinear lig-ament viscoelasticity. Ann Biomed Eng 29:908–914

Puso MA, Weiss JA (1998) Finite element implementation of aniso-tropic quasi-linear viscoelasticity using a discrete spectrumapproximation. J Biomech Eng 120:62–70

Sacks MS, Sun W (2003) Multiaxial mechanical behavior of biologicalmaterials. Ann Rev Biomed Eng 5:251–284

Satta J, Laurila A, Pääkkö P, Haukipuro K, Sormunen R, Parkkila S,Juvonen T (1998) Chronic inflammation and elastin degradation inabdominal aortic aneurysm disease: an immunohistochemical andelectron microscopic study. Eur J Vasc Endovasc Surg 15:313–319

Sauren AAHJ, van Hout MC, van Steenhoven AA, Veldpaus FE, Jans-sen JD (1983) The mechanical properteies of porcine aortic valvetissues. J Biomec 16:327–337

Silver FH, Horvath I, Foran DJ (2001) Viscoelasticity of the vesselwall: the role of collagen and elastic fibers. Crit Rev Biomed Eng29:279–301

Soulhat J, Buschmann MD, Shirazi-Adl A (1999) A fibril-network-reinforced biphasic model of cartilage in unconfined compression.J Biomech Eng 121:340–347

Spina M, Friso A, Ewins AR, Parker KH, Winlove CP (1999) Physico-chemical properties of arterial elastin and its associated glycopro-teins. Biopolymers 49:255–265

Stella JA, Liao J, Sacks MS (2007) Time-dependent biaxial mechanicalbehavior of the aortic heart valve leaflet. J Biomec 40:3169–3177

Stephens EH, Chu CK, Grande-Allen KJ (2008) Valve proteoglycancontent and glycosaminoglycan fine structure are unique to micro-structure, mechanical load and age: relevance to an age-specifictissue-engineered heart valve. Acta Biomaterialia 4:1148–1160

Stergiopulos N, Vulliemoz S, Rachev A, Meister J-J, GreenwaldSE (2001) Assessing the homogeneity of the elastic properties andcomposition of the pig aortic media. J Vasc Res 38:237–246

Sverdlik A, Lanir Y (2002) Time-dependent mechanical behavior ofsheep digital tendons, including the effects of preconditioning.J Biomech Eng 124:78–84

Thornton GM, Oliynyk A, Frank CB, Shrive NG (1997) Ligament creepcannot be predicted from stress relaxation at low stress: a biome-chanical study of the rabbit medial collateral ligament. J OrthopRes 15:652–656

Thornton GM, Frank CB, Shrive NG (2001) Ligament creep behav-ior can be predicted from stress relaxation by incorporating fiberrecruitment. J Rheol 45:493–507

Uitto J (1979) Biochemistry of the elastic fibers in normal connectivetissues and its alterations in diseases. J Invest Dermatol 72:1–10

von Maltzahn WW, Warriyar RG (1984) Experimental measurement ofelastic properties of media and adventitia of bovine carotid arteries.J Biomech 17:839–847

Weinberg PD, Winlove CP, Parker KH (1995) The distribution of waterin arterial elastin: effects of mechanical stress, osmotic pressure,and temperature. Biopolymers 35:161–169

Williams C, Liao J, Joyce EM, Wang B, Leach JB, Sacks MS, WongJY (2009) Altered structural and mechanical properties in decell-ularized rabbit carotid arteries. Acta Biomaterialia 5:993–1005

Winlove CP, Parker KH (1990) Connective tissue matrix II. In: HukinsDWL (ed) MacMillan, London, chap 7

Yin FCP, Chew PH, Zeger SL (1986) An approach to quantification ofbiaxial tissue stress-strain data. J Biomec 19:27–37

Zhang Y, Dunn ML, Drexler ES, McCowan CN, Slifka AJ, Ivy DD,Shandas R (2005) A microstructural hyperelastic model of pul-monary arteries under normo- and hypertensive conditions. AnnBiomed Eng 33:1042–1052

Zou Y, Zhang Y (2009) An experimental and theoretical study on theanisotropy of elastin network. Ann Biomed Eng 37:1572–1583

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