Journal of Biomechanics 44 (2011) 2544–2550

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Journal of Biomechanics

0021-92

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E-m

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A constitutive model for vascular tissue that integrates fibril, fiber andcontinuum levels with application to the isotropic and passive properties ofthe infrarenal aorta

Giampaolo Martufi, T. Christian Gasser n

Department of Solid Mechanics, School of Engineering Sciences, Royal Institute of Technology (KTH), Osquars Backe 1, SE-100 44 Stockholm, Sweden

a r t i c l e i n f o

Article history:

Accepted 17 July 2011A fundamental understanding of the mechanical properties of the extracellular matrix (ECM) is

critically important to quantify the amount of macroscopic stress and/or strain transmitted to the

Keywords:

Collagen

Microfiber

Aneurysm

Aorta

Finite Element

Multiscale

90/$ - see front matter & 2011 Elsevier Ltd. A

016/j.jbiomech.2011.07.015

esponding author.

ail addresses: martufi@kth.se (G. Martufi), tg@

: http://www-old.hallf.kth.se/~tg/vascumech

a b s t r a c t

cellular level of vascular tissue. Structural constitutive models integrate histological and mechanical

information, and hence, allocate stress and strain to the different microstructural components of the

vascular wall. The present work proposes a novel multi-scale structural constitutive model of passive

vascular tissue, where collagen fibers are assembled by proteoglycan (PG) cross-linked collagen fibrils

and reinforce an otherwise isotropic matrix material. Multiplicative kinematics account for the

straightening and stretching of collagen fibrils, and an orientation density function captures the spatial

organization of collagen fibers in the tissue. Mechanical and structural assumptions at the collagen

fibril level define a piece-wise analytical stress–stretch response of collagen fibers, which in turn is

integrated over the unit sphere to constitute the tissue’s macroscopic mechanical properties. The

proposed model displays the salient macroscopic features of vascular tissue, and employs the material

and structural parameters of clear physical meaning. Likewise, the constitutive concept renders a highly

efficient multi-scale structural approach that allows for the numerical analysis at the organ level. Model

parameters were estimated from isotropic mean-population data of the normal and aneurysmatic

aortic wall and used to predict in-vivo stress states of patient-specific vascular geometries, thought to

demonstrate the robustness of the particular Finite Element (FE) implementation. The collagen fibril

level of the multi-scale constitutive formulation provided an interface to integrate vascular wall biology

and to account for collagen turnover.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The extracellular matrix (ECM) provides an essential support-ing framework for the structural and functional properties ofvessel walls. The ECM mainly contains elastin, collagen, andproteoglycans (PGs) (Carey, 1991) and their three-dimensionalorganization is vital to accomplish proper physiological functions.The ECM, therefore, rather than being merely a system of scaffold-ing for the surrounding cells, is an active mechanical structure thatcontrols the micro-mechanical and macro-mechanical environ-ments to which vascular tissue is exposed. Specifically, a properunderstanding of the mechanical properties of the ECM is criticallyimportant to estimate and quantify the amount of stress and/or

ll rights reserved.

hallf.kth.se (T.C. Gasser).

(T.C. Gasser).

strain transmitted from the macroscopic to the cellular levels ofvascular tissue.

Constitutive modeling of vascular tissue is an active field ofresearch and numerous descriptions have been reported. How-ever, the phenomenological approaches (Vaishnav et al., 1972;Fung et al., 1979; Chuong and Fung, 1983; Takamizawa andHayashi, 1987; Humphrey, 1995; Delfino et al., 1997) that havebeen successfully used to fit experimental data cannot allocatestress or strain to the different histological constituents in thevascular wall. Structural constitutive descriptions (Lanir, 1983;Wuyts et al., 1995; Holzapfel et al., 2000; Holzapfel et al., 2002;Zulliger et al., 2004; Gasser et al., 2006; Gasser, 2011) overcomethis limitation and integrate histological and mechanical informa-tion of the arterial wall.

Specifically, collagen fibers in the vascular wall have a majorimpact on the mechanical properties at higher loads (Roach andBurton, 1957; Greenwald and Berry, 1980), i.e. the condition experi-enced by the aneurysm wall. In addition to the volume fraction ofcollagen, its spatial arrangement, including the spread in orientations

G. Martufi, T. Christian Gasser / Journal of Biomechanics 44 (2011) 2544–2550 2545

(Finlay et al., 1995), significantly affects the macroscopic mechanicalproperties (Gasser et al., 2006). Collagen fibrils are regarded as thebasic building blocks of collagenous ECMs, and their organization intosuprafibrillar structures determines macroscopic tissue properties(Wess, 2008; Fratzl, 2008).

The present work proposes a novel constitutive model for thevascular wall, which integrates the collagen’s fibril and fiberlevels with the tissue’s continuum level. The proposed constitu-tive concept renders a highly efficient multi-scale structuralapproach that allows for the numerical analysis at the organlevel. Specifically, cross-linked collagen fibrils are thought to formcollagen fibers, which in turn are integrated over the unit sphereto define the tissue’s macroscopic properties (Lanir, 1983;Federico and Gasser, 2010). Material and structural parametersare estimated from macroscopic in-vitro tests of aortic tissue asdescribed earlier in the published literature (Vande Geest et al.,2006). The model’s collagen fibril level provided an interface tointegrate the vascular wall biology in order to study the impact ofthe turnover of collagen on the macroscopic properties of thetissue, and vice versa.

Fig. 1. Multi-scale kinematics of the collagen-fiber-reinforced tissue.

(a) Configurational map, where the intermediate configuration Ost i separates

the straightening and stretching of the i-th collagen fibril. (b) The hierarchical

structure of a collagen fiber built up of statistically-distributed collagen fibrils

interlinked by proteoglycan (PG) bridges.

2. Methods

2.1. Modeling assumptions

The mechanical response of vascular tissue was regarded as a fibrous

collagenous tissue, where fibers of collagen reinforced an otherwise isotropic

matrix material, thought to capture the mechanics of the non-collagenous tissue

components, i.e. mainly elastin. Each collagen fiber is assembled by a bundle of

collagen fibrils mutually interconnected by proteoglycan (PG) bridges (Scott,

2003) that provide interfibrillar load transition, see Fig. 1(b). Stretching a collagen

fiber involves firstly a continuous recruitment of collagen fibrils. As soon as the

particular collagen fibril reaches its straightening stretch lst, it starts bearing a

tension load. Consequently, at small stretches (less than lst) a collagen fibril

becomes aligned with the stretch direction of the collagen fiber, and stretching it

beyond lst involves both storing elastic energy in the collagen fibril itself and in

the PG-rich matrix between the fibrils. We considered the cumulative mechanical

properties of the collagen fibril including the PG-rich matrix that surrounds it, and

this nanoscale structure is denoted as Collagen Fibril PG-complex (CFPG-complex).

The suggested model accounts for the anisotropic structure of the vascular

wall, and an orientation density function of r(N)¼r(�N) defines the spatial

alignment of collagen bundles, i.e. the density of collagen among the direction Nwith 9N9¼1 (Gasser et al., 2006). Although the detailed approach (described later)

can be applied to any orientation density function r(N), an isotropic distribution

of collagen (r(N)¼const) is thought to capture the salient mechanical properties of

healthy and aneurysmatic aortic tissue in the present work. Note that the aortic

wall in particular exhibits isotropic (mean-population) properties (Vande Geest

et al., 2006) and that isotropic constitutive descriptions have been used exten-

sively in AAA biomechanics (Raghavan and Vorp, 2000; Fillinger et al., 2002; Wang

et al., 2002; DiMartino and Vorp, 2003; Fillinger et al., 2003; Venkatasubramaniam

et al., 2004; Gasser et al., 2010; Larsson et al., 2011).

2.1.1. Kinematics

We considered fibrous tissue under finite deformations and the unit direction

vector N denotes the local collagen fiber direction in the reference configuration

O0, see Fig. 1(a). Collagen fibers are assembled by numerous collagen fibrils,

generally with different degrees of waviness in O0.

The deformation Fst i straightens the i-th collagen fibril, i.e. it maps its crimped

referential configuration into a straight but unstressed intermediate configuration

Ost i. In contrast the deformation FCi records deformation relative to Ost i and maps

the fibril into its spatial configuration O. Consequently, the intermediate config-

uration serves as a local reference configuration, with fibril stretch lCi¼1 and fibril

tension TCi¼0, relative to which the fibril deforms elastically. According to

multiplicative kinematics the continuum deformation

F¼ FCiFst i ð1Þ

is related to the introduced sub-deformations.

To define the intermediate configuration Ost i of the i-th collagen fibril, its

assemblage within the fiber needed to be defined, i.e. its stretch lCi must be

related to the stretch l of the collagen fiber. For simplicity and due to the lack of

microstructural data we assumed that the collagen fibrils straightened according

to a symmetric triangular probability distribution (Kotz and van Dorp, 2004).

Consequently, the first and last fibrils within a collagen fiber straighten at the fiber

stretches of lmin and lmax, respectively, see Fig. 1(b).

Finally, we considered an affine deformation between the continuum and the

collagen fiber, i.e. l¼9FN9¼9n9, and assume an incompressible macro-deforma-

tion (det F¼ J¼1), which completely describes the multi-scale kinematics of the

fibrous tissue.

2.1.2. Description of the non-collagenous contribution

At low strains collagen fibers are (mechanically) inactive and the mechanical

properties are determined by the non-collageneous (matrix) contribution. We

used the (classical) isotropic Neo-Hookean strain energy function cnH¼c(I1�3) to

model the properties of the matrix material, where c40 was a stress-like material

parameter that denoted the referential’s shear stiffness and I1¼ J�2/3FTF:I denotes

a strain invariant.

2.1.3. Description of the CFPG-complex

A logarithmic relation between Second Piola–Kirchhoff stress and stretch of

the i-th CFPG-complex was assumed, i.e. SCi¼k log lCi, where k denoted the CFPG-

complex stiffness. Note that the First Piola–Kirchhoff stress TCi¼klCi loglCi shows

a virtually linear response within a reasonable stretch range (see Fig. 2), which has

also been observed from tensile testing of collagen fibrils (Miyazaki and Hayashi,

1999; Shen et al., 2008).

For the following derivation it is convenient to map SCi back to O0. This

mapping undulates the fibril but does not contract it, such that the scalar SCi

remains unaffected. Specifically, it was assumed that the cross-section of the

1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Collagen fiber stretch

Nor

mal

ized

Fir

st P

iola

−K

irch

hoff

str

ess

Fig. 3. Nonlinear mechanical properties of a collagen fiber assembled from

statistically-distributed collagen fibril proteoglycan-complexes (CFPG-complexes).

Normalized First Piola–Kirchhoff stress T/k is shown at collagen fiber stretch lwith lmin¼1 and lmax¼2.

G. Martufi, T. Christian Gasser / Journal of Biomechanics 44 (2011) 2544–25502546

CFPG-complex does not change when undulating the fibril. To derive the

constitutive relation for a bundle of CFPG-complexes, we considered the Second

Piola–Kirchhoff stress increment dSCi¼(k/lCi)dlCi¼(k/l)dl of the i-th CFPG-com-

plex, where the multiplicative kinematics relation lCi¼l/lsti was applied. It is

emphasized that, due to the particular choice of the constitutive law of the CFPG-

complex, the Second Piola–Kirchhoff stress increment was independent from its

intermediate configuration Osti, i.e. the stretch lsti.

Finally, the integration of all CFPG-complexes (fibrils) in tension, i.e. consider-

ing the triangular distribution function introduced in Section 2.1.1, defined a

Second Piola–Kirchhoff stress increment for the collagen fiber of

dSðlÞ ¼ kCFDðlÞdll

, ð3Þ

where CFD(l) denoted the Cumulative Density Function of the triangular

probability distribution. According to the considered incompressible deformation,

the First Piola–Kirchhoff stress increment reads dT(l)¼kCFD(l)dl, which in turn

leads to the constitutive relation for the collagen fiber using work conjugate

variables

TðlÞ ¼ k

Z l

0CDFðlÞdl: ð4Þ

Integrating the cumulative density function CFD(l) of the triangular prob-

ability distribution and using the abbreviations a¼lmax�lmin and

b¼(lmaxþlmin)/2 yields the piece-wise analytical expressions for the First

Piola–Kirchhoff stress

TðlÞ ¼

0, 0olrlmin

2k3a2 ðl�lminÞ

3 , lmin olrb

k l� 2ðl�lmax Þ3

3a2 �bh i

, bolrlmax

kðl�bÞ: lmax olo1

8>>>>><>>>>>:

ð5Þ

This set of equations exhibits the typically observed nonlinear property of soft

biological tissues, as showed in Fig. 3. Note that a spatial numerical implementa-

tion of the model follows (see Section 2.2), which finally considers the Cauchy

stress s(l)¼lT(l) of a collagen fiber.

2.2. Numerical implementation

The proposed constitutive model for vascular tissue was implemented in the

multipurpose Finite Element software FEAP vers. 8.0 (Taylor, 2007) at the Gauss

point level of a Q1P0 mixed Finite Element (Simo and Taylor, 1991) formulation.

To this end, the constitutive description of the collagen fiber was added on top of

the uncoupled Neo-Hookean contribution for the isotropic matrix material.

A quasi-incompressible tissue was considered and the volumetric energy con-

tribution cvol¼k(J�1)2 was added to the constitutive formulation. The parameter

k was adjusted at each time step such that the volume ratio J-1.

Consequently, the additive decomposition of the Cauchy stress read

r¼ rvolþrnHþrfiber , ð6Þ

where rvol and rnH were stress contributions arising from cvol and cnH (Miehe,

1994; Gasser and Holzapfel, 2002), respectively. The superimposed bar indicates

1 1.05 1.1 1.150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

CFPG−complex stretch

Nor

mal

ized

Fir

st P

iola

−K

irch

hoff

str

ess

Fig. 2. Constitutive model for the collagen fibril proteoglycan-complex

(CFPG-complex) revealing a virtually linear First Piola–Kirchhoff stress property.

isochoric stress quantities, and rfiber is specifically detailed in the following

section.

2.2.1. Microfiber model

To compute the macroscopic mechanical properties of the collagenous tissue

at the Gauss point level, integration over the solid angle o (i.e. over the unit

sphere) was required (Lanir, 1983), such that the deviatoric contribution from the

CFPG-complexes read

rfiber ¼

ZorðNÞsðlðNÞÞdevðn� nÞdo: ð7Þ

This integral was numerically approximated by a spherical design (Hardin and

Sloane, 1996; Federico and Gasser, 2010), i.e.

Zoð�Þdo� ð4p=lintÞ

Xlint

l ¼ 1

ð�Þl ,

where lint denoted the total number of integration points. Further details

regarding the numerical implementations are given in Table 1.

2.2.2. Integration order

The integration of Eq. (7) required a sufficiently accurate spherical design,

which was directly linked to the polynomial degree of the function to be

integrated. The order of the spherical design that leads to an (virtually) exact

integration of Eq. (7) is defined by the surface in Fig. 4. Specifically, variations with

respect to stiffness of the CFPG-complex k and the stretch range a¼lmax�lmin,

within which the collagen fibrils are straightened, were studied. To this end the

associated baseline data in Table 2 was scaled, where the factors a and b are

related to a and k, respectively. This figure illustrates that a 8-design up to a

11-design (involving 36 up to 70 integration points over the unit sphere,

respectively) are required to integrate the governing equations accurately. Note

that this study used an isotropic collagen-orientation distribution, and concen-

trated (anisotropic) distributions will naturally require higher-order integration

schemas (Federico and Gasser, 2010).

3. Results

3.1. Macroscopic model response

A single cubic tissue element was used to estimate thematerial parameters from the experimental data, investigate themodel’s macroscopic response and to test the performance of itsnumerical implementation.

3.1.1. Parameter estimation

Typically the Abdominal Aorta (AA) and the Abdominal AorticAneurysm (AAA) displayed nonlinear mechanical properties

Table 1Numerical implementation of the multi-scale vascular tissue model to compute

stress and stiffness at the Gauss-point level of a finite element.

Initialization at time t¼0:

Define spherical design, i.e. directions Ni ,i¼ 1,. . .,lint, w¼4p/lint

Define microstructural data, i.e. lmin and lmax

Define collagen fiber density rAlgorithm at time t:

Given: isochoric part of the deformation gradient F

Initialize stress tensor rfiber’0 and associated elasticity tensor cfiber’O

DO i¼1,lint

Compute deformed fiber direction m¼ FNi and l¼9m9IF (l4lmin) THEN

Compute stress factor a(l)¼T/lCompute stiffness factor b(l)¼l2(dT/dl�T/l)

Map stress and elasticity to continuum metricrfiber i ¼ raðlÞ devðm�mÞ

cfiber i ¼ r½bðlÞ devðm�mÞ � devðm�mÞ�1=6ðI� rfiberþrfiber � IÞ�

Update stress tensor rfiber’rfiberþrfiber i

Update elasticity tensors cfiber’cfiberþcfiber i

ENDIF

ENDDO

Multiply stress and elasticity tensors by w rfiber’wrfiber, cfiber’wcfiber

0.10.2

15

10 0.1

0.2

1

5

107

8

9

10

11

12

βα

Sphe

rica

l des

ign

orde

r

Fig. 4. Required order of the spherical design to integrate the governing Eq. (7).

Surface represents the order of the spherical design and a and b are factors varying

the baseline data given in Table 2.

Table 2Material and structural parameters estimated from mean-population data of the

normal and aneurysmatic abdominal aortic wall. (sr denotes steradian).

Abdominal Aorta (AA)Matrix material

Neo-Hookean parameter c 0.012 MPa

Collagen fibers

Stiffness of the CFPG-complex k 40 MPa/sr

Density rAA 1/4p 1/sr

Lower limit of the straightening stretch lmin 1.034

Higher limit of the straightening stretch lmax 1.276

Abdominal Aortic Aneurysm (AAA)Matrix material

Neo-Hookean parameter c 0.021 MPa

Collagen fibers

Stiffness of the CFPG-complex k 63 MPa/sr

Density rAAA 1.35rAA

Lower limit of the straightening stretch lmin 1.045

Higher limit of the straightening stretch lmax 1.117

G. Martufi, T. Christian Gasser / Journal of Biomechanics 44 (2011) 2544–2550 2547

(Vande Geest et al., 2006), which could be captured by theanisotropic strain energy function

W ¼ b0ðeð1=2Þb1E2

yyþeð1=2Þb2E2LLþeb3EyyELL�3Þ, ð8Þ

originally proposed for canine pericardium (Choi and Vito, 1990).Here, Eyy ¼ ð1=2Þðl2

y�1Þ and ELL ¼ ð1=2Þðl2L�1Þ are (in-plane) Green-

Lagrange strain components in circumferential and longitudinaldirections, with associated stretches ly and lL, respectively.

The material parameters c, k and the structural parameterslmin, lmax, r of our constitutive model were identified from thestrain energy function (8), where parameters b0,y,b3 thatreflected the mean-population properties of the AA and the AAAwere used (Vande Geest et al., 2006). The parameters for the(isotropic) AA wall were estimated from a simple tension test, andthe same set of parameters was used to predict the tissue’s biaxialproperties, see Fig. 5 and Table 2, in order to investigate thepredictive capability of our model. In contrast, simple tensiondata could not provide enough information for an anisotropictissue, and hence, the AAA wall parameters were directly esti-mated from the biaxial data, see Fig. 6 and Table 2. We considereda 35% increase in collagen content in the AAA wall (Menashi et al.,1987; Baxter et al., 1994; Minion et al., 1994), and in all cases theclear physical meaning of the model parameters introducedallowed their straightforward manual estimation. Specifically,the parameters c and k affect the low and high strain responses,whereas lmin and lmax define the transition in between andreflect the degrees of waviness of collagen fibrils.

The applied analytical stress expressions are given in theAppendix.

3.2. Organ-level analysis

In this section the stress fields of an AA and an AAA using theproposed constitutive model for vascular tissue were predicted.The examples aimed to demonstrate the applicability of theconstitutive frame for the analysis of realistic (clinically relevant)problems.

3.2.1. Modeling assumptions

The geometry between the renal arteries and 1.5 cm distal iliacbifurcation was segmented using Computer Tomography-Angio-graphy (CT-A) data (A4research, VASCOPS GmbH), where deform-able (active) contour models provided artifact-insensitive anduser-independent reconstructions (Auer and Gasser, 2010). A wallwith a constant thickness of 2.0 mm, represented by 5650 (AA)and 6209 (AAA) hexahedral elements, was considered. A singleelement across the thickness was used, and hence, the bendingeffects from inhomogeneous stress across the wall were excluded.The intra-luminal thrombus was not considered in the stressanalysis for the AAA model.

A mean arterial blood pressure of 100 mmHg (13.33 kPa) wasapplied and the FE models were fixed at the bottom and top slices(nodal degrees of freedom were locked). No contact with sur-rounding organs was considered.

3.2.2. Numerical predictions

Solving the nonlinear FE problem required approximately 20 minwith a standard PC (Dell Optiplex 760), which is roughly 2–3 timesmore than using a single-scale constitutive model, i.e. describing thestress tensor directly as a function of the strain tensor. To this end thepressure loading was incremented and the quasi-static solution wascomputed using FEAP’s direct solver option. We neither observednumerical instabilities nor required any kind of numerical regulariza-tion to compute the quasi-static solution.

1 1.05 1.1 1.150

0.01

0.02

0.03

0.04

0.05

Stretch λ

Firs

t Pio

la−

Kir

chho

ff s

tres

s [M

Pa]

Mean population dataFE model best fit

AA simple tension

1 1.05 1.1 1.150

0.01

0.02

0.03

0.04

0.05

0.06

Stretch λ

Firs

t Pio

la−

Kir

chho

ff s

tres

s [M

Pa]

Mean population data longitudinalMean population data circumferentialFE model prediction

AA biaxial tension

Fig. 5. Macroscopic constitutive response of the Abdominal Aorta (AA) wall under simple tension (left) and biaxial tension (right). Finite Element (FE) results (points)

computed with the proposed multi-scale constitutive model (left: model best fit; right: model prediction) were compared to the mean-population data (lines) reported in

the literature (Vande Geest et al., 2006) and the FE prediction (right) of the biaxial behavior. Material and structural parameters were estimated from simple tension

properties and are given in Table 2.

1 1.02 1.04 1.06 1.08 1.10

0.05

0.1

0.15

0.2

0.25

0.3

Stretch λ

Firs

t Pio

la−

Kir

chho

ff s

tres

s [M

Pa]

Mean pop. data circumferentialMean pop. data longitudinalFE model best fit

AAA biaxial tension

Fig. 6. Macroscopic constitutive response of the Abdominal Aortic Aneurysm

(AAA) wall under biaxial tension. Best fit Finite Element (FE) results (points)

according to the proposed multi-scale constitutive model were compared to

mean-population data (lines) reported in the literature (Vande Geest et al.,

2006). Material and structural parameters are given in Table 2.

Fig. 7. Finite Element (FE) predictions of the maximum principal Cauchy stress in

the Abdominal Aorta (AA) (left) and the Abdominal Aortic Aneurysm (AAA) (right).

FE calculations are based on the material properties given in Table 2.

G. Martufi, T. Christian Gasser / Journal of Biomechanics 44 (2011) 2544–25502548

The maximum principal Cauchy stress distribution in the AA andAAA is plotted in Fig. 7. As with macroscopic stress patternspredicted by alternative constitutive formulations (e.g. Raghavanand Vorp, 2000), mechanical stress was distributed in a complexmanner. Naturally the Peak Wall Stress (PWS) in an AAA was severaltimes higher than in an AA. Circular spots of low stress, specificallyseen in the AA model, indicated bifurcations of larger arteries.

4. Discussion

Although simple mechanical assumptions and kinematicsdefine the CFPG-complex (collagen fibril level), the model capturesthe macroscopic complexity of vascular tissue, i.e. it replicates thetypical stiffening of vascular tissue at the physiological strain level.The introduction of a probability distribution function that definesthe load-carrying mechanism at the fibril level is most critical tothe model. Similar recruitment mechanisms have been suggestedearlier (Lanir, 1983; Wuyts et al., 1995; Zulliger et al., 2004).However, in contrast to them, our particular approach led toanalytical (piece-wise) relations for the stress–strain properties

of the collagen fiber, and hence, facilitated an efficient numericalimplementation of the model.

The model integrates two mechanical parameters (c, k) andthree structural parameters (lmin, lmax, r) (with clear physicalmeaning) that allow their straightforward estimation fromexperimental data. Specifically, lmin, lmax control the transitionfrom the elastin-driven low stress and collagen-driven high stressresponse of the tissue, and r defines its anisotropy. The developedstress–strain relation for collagen fibers can substitute existingphenomenological descriptions of the collagen reinforcement inthe vascular wall (e.g. in the HGO model (Holzapfel et al., 2000))and reinforce their micro-histological relevance.

The structural parameter lmax, i.e. that defines the upper limitof collagen undulation, was significantly lower in AAA tissue ascompared to the normal aorta. Consequently, collagen in AAAtissue is remarkably less undulated, which can be explained bythe lower elastin collagen ratio (Raghavan et al., 1996). Inaddition, the more than 3 times wider spectrum of straighteningstretches, i.e. lmax�lmin of the normal aorta defines a muchsmoother transition from the low-strain to the high-strain tissueproperties. It is also noted that the stiffening in the CFPG-complex(determined by parameter k) was slightly elevated in AAA tissue,which may point towards an elevated collagen cross-linking in

G. Martufi, T. Christian Gasser / Journal of Biomechanics 44 (2011) 2544–2550 2549

aneurysmatic tissue (Lindeman et al., 2010). Finally, the AAA wallis known to have less and even fragmented elastin (Choke et al.,2005) but its referential stiffness (determined by parameter c)was increased. Consequently, it is unlikely that elastin alonedetermines the low-strain properties of AAA tissue.

The application of the proposed multi-scale constitutive modelrequires the collagen fiber orientation density r of the entiresimulation domain. Unfortunately, no applicable experimentaldata has been reported in the literature, and hence, an isotropiccollagen orientation was used in this study. However, it isemphasized that other collagen distributions can also definemacroscopic isotropic mechanical properties. From a biologicalpoint of view, collagen fiber formation (orientation density)develops according to the tissue’s macroscopic mechanical stressstate. Consequently, collagen undergoes an adaptation process(Humphrey, 2002) and modeling the process may be a promisingapproach to provide the requested information.

The close packing and the cross-linking of collagen fibrilsdefines a virtually inextensible fiber, such that the strain withincollagen fibrils is always much smaller than the macroscopicstrain in collagenous tissue (Folkhard et al., 1987a; Folkhard et al.,1987b; Fratzl et al., 1998). This has been confirmed by recentexperimental data, which suggests that sliding between collagenfibers (inter-fiber sliding) and between collagen fibrils (as it iscaptured by the present model) plays a significant role in tendondeformation (Gupta et al., 2010). If inter-fiber sliding also appliesto vascular tissue, the considered affine deformation betweencollagen fiber and continuum should be relaxed, which directlywill lower collagen straining.

Following earlier attempts (e.g. the HGO model (Holzapfelet al., 2000)), the constitutive model disregards the interactionbetween collagen fibers, and the impact of this has not yet beenthoroughly discussed in the literature.

The proposed constitutive model considers the passive responseof the vascular tissue, and any tissue remodeling has been sup-pressed. Consequently, it covers only a limited time period in whichthe biological process cannot change the mechanical tissue proper-ties (Cowin and Humphrey, 2001).

5. Conclusions

The proposed constitutive model has a strong biologicalmotivation and integrates the fibril and fiber levels of collagenwith the tissue’s macroscopic properties. Such a structural view isimportant to understand the interplay of the tissue’s histology(internal architecture) and its macroscopic mechanical properties.The presented multi-scale structural approach allows for thenumerical analysis at the organ level, and although the modelcould successfully prescribe earlier reported experimental data, arigorous validation is required to thoroughly evaluate the descrip-tive and predictive capability of the constitutive approach.

Conflict of interest statement

None declared.

Acknowledgments

We gratefully acknowledge the support given to this work bythe Young Faculty Grant no. 2006-7568 provided by the SwedishResearch Council, VINNOVA, the Swedish Foundation for StrategicResearch, and the EC Seventh Framework Program; ‘FightingAneurysmal Disease (FAD-200647)’.

Appendix A. Supporting information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jbiomech.2011.07.015.

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