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A model for saccular cerebral aneurysm growth by collagen fibre remodelling

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Journal of Theoretical Biology 247 (2007) 775–787
A model for saccular cerebral aneurysm growthby collagen ﬁbre remodelling
Martin Kroon
a
, Gerhard A. Holzapfel
a,b,
a
Department of Solid Mechanics, Royal Institute of Technology (KTH), School of Engineering Sciences, 100 44 Stockholm, Sweden
b
Center for Biomedical Engineering, Institute for Biomechanics, Graz University of Technology, Kronesgasse 5, 8010 Graz, Austria
Received 16 December 2006; received in revised form 6 February 2007; accepted 7 March 2007Available online 15 March 2007
Abstract
The ﬁrst structural model for saccular cerebral aneurysm growth is proposed. It is assumed that the development of the aneurysm isaccompanied by a loss of the media, and that only collagen ﬁbres provide load-bearing capacity to the aneurysm wall. The aneurysm ismodelled as an axisymmetric multi-layered membrane, exposed to an inﬂation pressure. Each layer is characterized by an orientationangle, which changes between different layers. The collagen ﬁbres and ﬁbroblasts within a speciﬁc layer are perfectly aligned. The growthand the morphological changes of the aneurysm are accomplished by the turnover of collagen. Fibroblasts are responsible for collagenproduction, and the related deformations are assumed to govern the collagen production rate. There are four key parameters in themodel: a normalized pressure, the number of layers in the wall, an exponent in the collagen mass production rate law, and the pre-stretchunder which the collagen is deposited. The inﬂuence of the model parameters on the aneurysmal response is investigated, and a stabilityanalysis is performed. The model is able to predict clinical observations and mechanical test results, for example, in terms of predictedaneurysm size, shape, wall stress and wall thickness.
r
2007 Elsevier Ltd. All rights reserved.
Keywords:
Aneurysm; Saccular; Cerebral; Collagen; Membrane; Artery
1. Introduction
Aneurysms are abnormal dilatations of vessels in thevascular system, and they exist in two major forms:fusiform and saccular. Fusiform aneurysms are foundfrequently in the human abdominal aorta, but also in thebasilar artery of the brain. As for intracranial aneurysms,the saccular type dominates. Saccular cerebral aneurysmsare found in less than 5% of the human population, theyare usually diagnosed in elderly people (Kim and Cervos-Navarro, 1991), and they are more prevalent in females(Mettinger, 1982). The saccular brain artery aneurysm is asphere-like expansion from the branching region of a majorbrain artery. A fully developed saccular aneurysm has thecharacteristic geometry of a thin-walled, balloon-likestructure with a relatively narrow neck region commu-nicating to the lumen of the parent artery. Most bifurca-tions of the cerebral vasculature are structurally stable, buta small number develop a weakness that causes the wall toexpand outwards in the region near the ﬂow divider of thebranching artery. In total there are about 20 arterialbranches of the human cerebral circulation that accountfor all saccular aneurysms (Canham et al., 2006). Hemo-dynamic stresses at arterial bifurcations, congenital defects,and degenerative wall changes are believed to contribute toaneurysm development (Pentimalli et al., 2004).Rupture of these cerebrovascular lesions does not occurvery frequently, but when it does it has severe consequencesfor the individual. Thus, there is an urgent need tounderstand the etiology of these lesions and to establishreliable criteria by which surgeons can predict the risk of rupture and thus the need for operation. Clinical decisionsare usually based on the size of the aneurysm, and
ARTICLE IN PRESS
www.elsevier.com/locate/yjtbi0022-5193/$-see front matter
r
2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.jtbi.2007.03.009
Corresponding author. Center for Biomedical Engineering, Institutefor Biomechanics, Graz University of Technology, Kronesgasse 5, 8010Graz, Austria. Tel.: +433168731625; fax: +433168731615.
E-mail address:
gh@biomech.tu-graz.ac.at (G.A. Holzapfel).
experience shows that these lesions usually do not ruptureif they are less than 10mm in diameter (Austin et al., 1993).However, more reliable criteria are needed.It is difﬁcult to elucidate the process of aneurysmformation by use of human specimens, because the changesof these lesions (once detected) are already advanced orhave been modiﬁed by other factors such as atherosclerosis(Kondo et al., 1998). Therefore, numerous animal modelshave been developed to study aneurysm pathogenesis (e.g.,AAssar et al., 2003; Espinosa et al., 1984; Hashimoto et al.,1984; Kamphorst et al., 1991; Zhang et al., 2003), and thesemodels have provided useful insights into the formationprocess of cerebral aneurysms.In order to be able to predict the risk of rupture of ananeurysm it is necessary to understand the structuraldevelopment of these lesions, not only in terms of biology,but also from a mechanical point of view. Previousattempts to model aneurysms (e.g., Baek et al., 2006;Canham and Ferguson, 1985; Driessen et al., 2004;Humphrey and Rajagopal, 2002; Watton et al., 2004)have provided useful information about the underlyingmechanics.In this article, a model for saccular cerebral aneurysmgrowth is proposed. The model, which seems to be the ﬁrstone, is based on the assumption that a collagen fabric is theonly load-bearing constituent in the aneurysm wall, andthe turnover of collagen is responsible for the growth of theaneurysm. The aneurysm is modelled as an axisymmetric,multi-layered membrane, exposed to a constant pressure.The general behaviour and the stability properties of themodel are investigated. Physically relevant entities areapplied to the model, and the results are in excellentagreement with clinical observations and mechanical testresults for aneurysmal tissue.
2. Mechanical aspects of saccular cerebral aneurysms
2.1. The healthy cerebral artery wall
Like all arteries cerebral arteries consist of three layers:intima (adjacent to the lumen), media and adventitia.There is a wide range of calibres of brain arteries fromthread-like communicating arteries (0.25mm) to the basilarand internal carotid (3–5mm). In a healthy artery, there areessentially three constituents of interest from a mechanicalpoint of view: elastin, smooth muscle cells, and collagen.Cerebral arteries are of the muscular type, havingsigniﬁcantly less elastin in the media than elastic arteries.The internal elastic lamina separates the intima from themedia, and is essentially a fenestrated sheet of elastin thatallows the transport of water, nutrients, and electrolytesacross the wall. In the media, the smooth muscle cells andcollagen ﬁbres are almost perfectly aligned along thecircumferential direction of the cerebral artery wall(Canham et al., 1991b; Walmsley et al., 1983). The collagenin the media appears to be mainly of type III, and it isstretched at physiological pressure levels (Canham et al.,1991b).The adventitia consists primarily of a dense network of type I collagen ﬁbres, some elastin, nerves, ﬁbroblasts, andthe vasa vasorum. The adventitia is thought to limit acuteoverdistension in all vascular vessels at higher levels of pressure, where the collagen ﬁbers reach their straightenedlengths and the adventitia changes to a stiff ‘jacket-like’tube which prevents the artery from overstretch andrupture (Holzapfel et al., 2000). The collagen in theadventitia is unstretched and appear in a wavy form atphysiological pressure levels (Canham et al., 1992; Smithet al., 1981). In contrast to the media, the collagen of theadventitia has ﬁbre orientations that range from long-itudinal to circumferential. The collagen in the adventitiaappears to change from a longitudinal orientation at theouter layer to a circumferential orientation in the layeradjacent to the media (Finlay et al., 1995; Rowe et al.,2003; Smith et al., 1981).The collagen ﬁbrils in the adventitia are about50–100nm in diameter (Finlay et al., 1998; Merrileeset al., 1987), and they are mainly produced by ﬁbroblasts.Fibroblasts produce, organize, and remove the extracel-lular matrix. The repair and maintenance of connectivetissues are performed predominantly by these cells.Conversely, the extracellular matrix inﬂuences the devel-opment, shape, migration, proliferation, survival, andfunction of the ﬁbroblasts. The collagen matrix is anessential framework, which the ﬁbroblasts use as ascaffolding to crawl along. Thus, the collagen orientationalso inﬂuences the orientation of ﬁbroblasts and theirability to move (Dallon and Sherratt, 1998).There is a continuous process of deposition anddegradation of collagen going on in the artery wall. Inthe process of remodelling the collagen matrix, ﬁbroblastscontract the surrounding matrix, and the new matrixcollagen is deposited in a pre-strained condition (Huanget al., 1993). Collagen molecules are synthesized in minutesand secreted in less than an hour by ﬁbroblasts. They self-assemble with other collagen molecules in the extracellularspace to form ﬁbrils and ultimately cross-linked ﬁbres. Theturnover of collagen is fairly rapid under normal condi-tions with a half-life of 3–90 days (Humphrey, 1999).
2.2. The saccular cerebral aneurysm wall
The mechanical behaviour of saccular cerebral aneur-ysms is strongly related to the strength and organization of the collagen ﬁbres in the aneurysm wall. The elastin and theactivated smooth muscle cells contribute to the mechanicsof the healthy artery wall, but these components are onlypresent in negligible amounts in the aneurysm wall, and,therefore, do not make a signiﬁcant contribution to wallmechanics (Espinosa et al., 1984; Hashimoto et al., 1984;Kamphorst et al., 1991; Kondo et al., 1998; MacDonaldet al., 2000; Miskolczi et al., 1998; Steiger, 1990). Theinternal elastic lamina and the media often terminate
ARTICLE IN PRESS
M. Kroon, G.A. Holzapfel / Journal of Theoretical Biology 247 (2007) 775–787
776
sharply at the neck of the formed aneurysms (Abruzzoet al., 1998; Hassler, 1972; Kamphorst et al., 1991; Zhanget al., 2003). Thus, the aneurysm wall consists mainly of collagen ﬁbres, ﬁbroblasts, and amorphous material(Espinosa et al., 1984), and the collagen in an aneurysmalwall is mainly of type I (Austin et al., 1993; Whittakeret al., 1988). The aneurysm wall appears to be thinned andhomogenized, and the adventitia of the srcinal artery wallappears to supply the basis of the aneurysm wall (Austinet al., 1993; Canham et al., 1999; Hashimoto et al., 1984;Hassler, 1972; Kamphorst et al., 1991). The collagen of theaneurysm wall is a highly organized fabric, and appears tobe well made from a structural point of view. The wall isﬁnely layered (
10 layers), and the different layers(
15
2
20
m
m in thickness) have ﬁbre orientations of allazimuthal angles (Canham et al., 1991a, 1999). Aneurysmwall thicknesses are reported between 16 and 212
m
m(MacDonald et al., 2000), which is of the same order as thehealthy adventitia (Smith et al., 1981).The development of aneurysms is accompanied by anincrease in collagen production, which is mainly attributedto an enhanced activity of ﬁbroblasts (Halloran andBaxter, 1995; Whittaker et al., 1988). When a woundappears in the body, ﬁbroblasts migrate towards thatwound, proliferate, and synthesize a new collagen-richmatrix (Eastwood et al., 1998). Mature ﬁbroblasts perceiveapplied loads through the surrounding collagen matrix,and an arterial injury causes an increased cellular activityin ﬁbroblasts in terms of replication, recruitment andcollagen synthesis (Barnes, 1985; Bishop and Lindahl,1999; Sluijter et al., 2004). More speciﬁcally, increasedmechanical loading is known to increase the proliferationand collagen production of ﬁbroblasts (Aumailley et al.,1982; Bishop and Lindahl, 1999; Butt et al., 1995). It hasbeen observed that ﬁbroblasts in the aneurysmal wall tendto be elongated, and they often show a parallel alignment(Kamphorst et al., 1991; Mimata et al., 1997; Scanariniet al., 1978).Hemodynamics also play an important role in thedevelopment of aneurysms. There are three hemodynamicforces to consider: shear stress, dynamic (blood) pressure,and static (blood) pressure. The shear stresses areconsiderably higher at the apex of artery bifurcations. Thisis caused by the sudden deﬂection of the central stream atthe apex, which results in increased shear stresses and animpulse at the apex region of the bifurcation (Foutrakiset al., 1999). Shear stresses and impulse may well determinethe initial development of aneurysms, e.g., in terms of endothelial cell protein activity and resulting elastindegradation. But the main driving force leading to growthand rupture of an aneurysm appears to be the static bloodpressure (Steiger, 1990).Hence, the development of saccular cerebral aneurysmsseems to occur roughly as follows: increased shear stressesat an arterial bifurcation stimulate endothelial cells toenzymatic activities, which cause a degradation of theinternal elastic lamina (Miskolczi et al., 1998). This in turninduces degenerative changes in the medial layer (Kimet al., 1992), including apoptosis (genetically programmedcell death) of smooth muscle cells (Kondo et al., 1998;Pentimalli et al., 2004). As this degradation processadvances, more and more of the load imposed by theblood pressure has to be carried by the adventitial layer. Atthe same time, a balloon-like dilation starts to form. In themature aneurysm, the media has virtually disappeared, andthe aneurysm wall consists of remnants of the adventitiaand newly produced collagen tissue.
3. A model for saccular cerebral aneurysm growth
In the proposed model it is assumed that the only load-bearing constituent in the aneurysm wall is collagen, andthat the aneurysm wall is a development of the adventitia.Remnants of media, internal elastic lamina, and intima areconsidered not to give any contribution to the mechanicalbehaviour of the aneurysm wall. Thus, from a mechanicalpoint of view, the structural organization and properties of the collagen completely determine the mechanical responseof the aneurysm wall. The remodelling and turnover of collagen is assumed to be accomplished by ﬁbroblasts,which are continuously spread throughout the collagennetwork.The aneurysm wall is assumed to consist of
n
discreteand distinct layers of collagen (plys that form thelaminate). Within a layer (ply) with index
i
, the collagenﬁbres and the ﬁbroblasts are perfectly aligned in a direction
f
i
, deﬁned with respect to a 2D reference conﬁguration, seeFig. 1. There is a continuously ongoing process of production and degradation of collagen within each layer,and this is accomplished by the ﬁbroblasts. Since theﬁbroblasts are oriented in the same direction
f
i
as thecollagen ﬁbres, newly produced collagen is deposited inthat direction as well, and the orientation of the ﬁbres inthe different layers is, therefore, preserved during thegrowth process.The collagen production is assumed to depend on thestretching and proliferation of ﬁbroblasts. The stretchingof individual ﬁbroblasts affects the collagen production
ARTICLE IN PRESS
φ
i
i
-1
ii
+1
Fig. 1. The aneurysm wall is assumed to consist of
n
layers (plys). Layer
i
with ﬁbres oriented in a direction
f
i
is displayed.
M. Kroon, G.A. Holzapfel / Journal of Theoretical Biology 247 (2007) 775–787
777
rate per cell, and the proliferation of ﬁbroblasts governs thetotal number of collagen producing cells. Both effects aretaken to depend on the global straining of the material.Furthermore, the processes of proliferation and increase inproduction rate of ﬁbroblasts are assumed to occur veryrapidly compared to the overall process of collagenturnover. Thus, the collagen mass production rate, say
_
m
i
ð
t
Þ
, per unit reference volume in layer
i
at time
t
isproposed to be
_
m
i
ð
t
Þ¼
b
0
C
a
i
, (1)where
b
0
is the normal production rate per unit volume inthe reference conﬁguration (cf. Baek et al., 2006). It shouldbe noted, that this reference conﬁguration is not necessarilya stress-free or unloaded conﬁguration. The factor
b
0
pertains to the healthy adventitia, and may be interpretedas the density of ﬁbroblasts multiplied by the collagenproduction rate per ﬁbroblast in the healthy adventitia.The scalar
C
i
is deﬁned as
C
i
¼
C
:
A
ð
f
i
Þ
, where
C
is theright Cauchy–Green tensor,
A
ð
f
i
Þ¼
M
M
a structuretensor, and
M
¼ð
cos
f
i
sin
f
i
Þ
T
(cf. Holzapfel, 2000;Holzapfel et al., 2000). Thus,
C
i
is the projection of
C
inthe direction
f
i
of the ﬁbres, and the inﬂuence of this scalaron the collagen production rate is modulated by theexponent
a
.From time
t
!1
and on, in each layer new collagenﬁbres are continuously deposited. By assuming thatcollagen deposition occurs at a speciﬁc time, say
t
dp
, therelated deformation gradient is
F
ð
t
dp
Þ
(note that thedeposition may occur at any time between
1
and
t
).Subsequently, the current deformation gradient at time
t
isdecomposed according to
F
ð
t
Þ¼
F
loc
ð
t
;
t
dp
Þ
F
ð
t
dp
Þ
, where
F
loc
ð
t
;
t
dp
Þ¼
F
ð
t
Þ
F
1
ð
t
dp
Þ
is the current local deformationgradient to which collagen is exposed, see Fig. 2(cf. Humphrey and Rajagopal, 2002). Thus, the localityof
F
loc
ð
t
;
t
dp
Þ
has both a space aspect and a time aspect.Furthermore, the collagen ﬁbres are deposited by theﬁbroblasts in a pre-stretched condition, deﬁned by the pre-stretch
l
pre
. The resulting deformation in the individualﬁbres, say
C
fib
, can thus be expressed as
C
fib
¼
l
2pre
C
loc
:
A
ð
f
i
Þ
, (2)where
C
loc
¼
F
Tloc
F
loc
.As a start, the strain energy
c
fib
per unit mass stored inthe ﬁbres is characterized by a simple polynomial account-ing for the highly nonlinear response of the wave-likecollagen. Thus,
c
fib
¼
m
ð
C
fib
1
Þ
3
;
C
fib
X
1, (3)where
m
4
0 is a positive material parameter associated withthe stiffness of collagen ﬁbres. Note that Eq. (3) is validwhen the ﬁbres are in tension
ð
C
fib
X
1
Þ
, whereas the ﬁbresare assumed to have zero stiffness in compression
ð
C
fib
o
1
Þ
.The total strain energy
C
i
per unit volume for layer
i
is thenintegrated according to
C
i
ð
t
Þ¼
Z
t
1
g
ð
t
;
t
dp
Þ
_
m
i
ð
t
dp
Þ
c
fib
ð
t
;
t
dp
Þ
d
t
dp
, (4)where the
life cycle function
, denoted by
g
, has beenintroduced. This function accounts for the turnover of thecollagen ﬁbres, and, in principle, it can be used to includeeffects of collagen maturing and degradation, which affectthe stiffness of the ﬁbres (cf. Humphrey and Rajagopal,2002). However, in this study a simple pulse function isadopted according to
g
ð
t
;
t
dp
Þ¼
H
ð
t
t
dp
Þ
H
ð
t
t
dp
t
lf
Þ
, (5)where
H
ð
t
Þ
is the Heaviside step function, and
t
lf
is the life-time of the collagen ﬁbres. This formulation implies thatcollagen ﬁbres are assumed to have the same mechanicalproperties from the moment they are deposited to the endof their life time. The pulse function adopted here alsoimplies that maturation and degradation of collagen ﬁbresare taken to occur instantly at
t
¼
t
dp
and
t
¼
t
dp
þ
t
lf
,respectively.
4. Numerical formulation
4.1. FE membrane formulation
Saccular cerebral aneurysms are normally extremelythin-walled,sphere-like structures and appear to have low(negligible) bending stiffness. Furthermore, the media andthe internal elastic lamina tend to taper off quite abruptlyat the neck of the aneurysm. Therefore, the saccularcerebral aneurysm is here modelled as an axisymmetricmembrane, which is hinged along the neck and exposed toan inﬂation pressure
p
. The membrane formulation usedhere is based on a work by Fried (1982) (also utilized byKyriacou and Humphrey, 1996), and a brief review of hisformulation is provided below.Consider the axisymmetric membrane, as illustrated inFig. 3. The surface proﬁle can be parameterized using
ARTICLE IN PRESS
Time –
∞
Time
t
F
(
t
)
F
(
t
dp
)
F
loc
(
t
)Time
t
dp
Fig. 2. Schema of deformation. Original conﬁguration at
t
!1
; fromthere on new collagen is continuously deposited in each layer. Conﬁgura-tion at
t
¼
t
dp
is the one where collagen deposition occurs, with the relateddeformation gradient
F
ð
t
dp
Þ
. The current deformation gradient
F
ð
t
Þ
isdecomposed according to
F
loc
ð
t
;
t
dp
Þ
F
ð
t
dp
Þ
, where
F
loc
ð
t
;
t
dp
Þ
is the currentlocal deformation gradient to which collagen is exposed.
M. Kroon, G.A. Holzapfel / Journal of Theoretical Biology 247 (2007) 775–787
778
coordinates
S
and
s
in the reference and current conﬁg-urations, respectively. Coordinates
R
ð
S
Þ
,
Z
ð
S
Þ
,
r
ð
s
Þ
, and
z
ð
s
Þ
denote cylindrical coordinates in the reference andcurrent conﬁgurations, respectively. The membrane ishinged at
R
¼
R
0
. Principal directions 1 and 2 coincidewith the direction of
s
and the circumferential direction,respectively, as indicated in Fig. 3. The ﬁbre angles
f
i
aredeﬁned with respect to these principal directions, and
f
i
¼
0 corresponds to the principal direction 1. The principalstretches in the plane of the membrane can then beexpressed as
l
1
¼
d
s
d
S
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
d
r
d
S
2
þ
d
z
d
S
2
s
;
l
2
¼
2
p
r
2
p
R
¼
rR
. (6)The potential energy
P
of the membrane inﬂated by agiven constant pressure
p
is
P
¼
Z
O
0
C
d
V
þ
p
Z
q
O
s
d
au
¼
p
Z
R
0
0
2
R
X
ni
¼
1
T
i
C
i
þ
pr
2
d
z
d
S
!
d
S
,
ð
7
Þ
where
C
is the total strain energy per unit volume of themembrane,
O
0
is the reference region of the membrane,with the inﬁnitesimal volume element d
V
deﬁned in thatregion, and
q
O
s
q
O
is the current boundary surface onwhich the pressure boundary condition acts. The secondterm in Eq. (7) denotes the energy contribution due to thepressure
p
. In that term d
a
¼
d
a
n
is a vector element of aninﬁnitesimal small area deﬁned in the current conﬁgura-tion, where
n
is the direction of the (pointwise) outwardunit vector, which is perpendicular to the pressure loadedsurface
q
O
s
of the membrane region, and d
a
is aninﬁnitesimal surface element in the current conﬁguration(see Fig. 3). The displacement vector is denoted by
u
. Notethat the pressure load is deformation dependent (Holzap-fel, 2000). Making use of the symmetry conditions, thevolume integrals can be recast into a one-dimensionalform, as indicated in Eq. (7)
2
, where
T
i
denotes thereference thickness of the membrane layer, and
n
denotesthe total number of membrane layers.Eq.
ð
7
Þ
2
is solved by using the ﬁnite element method withquadratic line elements. The membrane is discretized in the
S
direction by nodes distributed with a constant distance
h
in the reference conﬁguration. The current geometry for anelement is approximated according to
r
ð
x
Þ¼
12
x
ð
x
1
Þ
r
1
þð
1
x
2
Þ
r
2
þ
12
x
ð
x
þ
1
Þ
r
3
, (8)where
x
2½
1
;
1
, and
r
1
,
r
2
and
r
3
are the nodal values of
r
.In a similar way
R
and
z
are approximated. According toEq.
ð
7
Þ
2
, the potential energy
P
e
for an element can thus beexpressed as
P
e
¼
p
Z
1
1
2
R
X
ni
¼
1
T
i
C
i
þ
pr
2
z
0
1
h
!
h
d
x
, (9)where
z
0
denotes the differentiation with respect to
x
, andsince
S
can be written as, e.g.,
S
¼
S
0
þ
h
x
, it follows thatd
S
¼
h
d
x
. The integration in Eq. (9) is approximated usingthe Gauss integration, i.e.
P
e
p
h
X
2
j
¼
1
2
R
ð
x
j
Þ
X
ni
¼
1
T
i
C
i
ð
x
j
Þþ
pr
2
ð
x
j
Þ
z
0
ð
x
j
Þ
1
h
" #
, (10)where
x
1
,
x
2
denote the coordinates for the Gauss points,and the relating weights are 1. The approximated totalpotential energy is achieved by a summation over allelements. Nodal values for
r
and
z
are stored in a vector
q
¼ð
r
1
;
. . .
;
r
n
n
z
1
;
. . .
;
z
n
n
Þ
T
, where
n
n
is the number of nodes in the model. To ﬁnd the equilibrium state thepotential energy is then minimized with respect to
q
using aNewton–Raphson scheme. Boundary conditions are im-posed, requiring that
r
¼
R
0
,
z
¼
0 for
R
¼
R
0
, and
r
¼
0for
R
¼
0, see Fig. 3.
4.2. Strain-energy function
Since the quantities in Eq. (7) evolve with time, themodel has to be discretized with respect to time as well.The strain-energy function in Eq. (4) is approximated bythe summation
C
i
;
k
¼
X
k l
¼
k
n
g
_
m
i
ð
t
l
Þ
c
fib
ð
t
k
;
t
l
Þ
D
t
l
, (11)where
D
t
l
¼
t
l
t
l
1
denotes the time increment. Thisintegration scheme yields the value of the strain-energyfunction in layer
i
at time step
k
. Due to the life cyclefunction (5), the integration is conﬁned to the closedinterval
½
t
t
lf
;
t
, and
n
g
is the number of discretizationpoints within this domain.Initial conditions are required for the strain-energyfunction. In a healthy artery, the load is mainly carriedby the media, but as the media is degraded in a developinganeurysm, the load from the blood pressure is transferredto the adventitia. The initial conditions used herecorrespond to an instant transfer of the load from themedia to the adventitia. Thus, in the half-closed timeinterval
t
2ð1
;
0
, the membrane existed in the reference
ARTICLE IN PRESS
R, r, S Z, zs ,
12
R
0
Fundus (
S
= 0)Neck (
S
=
R
0
)Midpoint (
S
= 0.5
R
0
)d
a = n
da
p
Fig. 3. Proﬁle of the membrane, deﬁnitions of coordinates and principaldirections, and loading and boundary conditions.
M. Kroon, G.A. Holzapfel / Journal of Theoretical Biology 247 (2007) 775–787
779

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