ORIGINAL PAPER

P. N. Watton Æ N. A. Hill Æ M. Heil

A mathematical model for the growth of the abdominalaortic aneurysm

Received: 12 December 2003 / Accepted: 14 July 2004 / Published online: 25 September 2004� Springer-Verlag Berlin Heidelberg 2004

Abstract We present the first mathematical model toaccount for the evolution of the abdominal aortic aneu-rysm. The artery is modelled as a two-layered, cylindricalmembrane using nonlinear elasticity and a physiologicallyrealistic constitutive model. It is subject to a constantsystolic pressure and a physiological axial prestretch. Thedevelopment of the aneurysm is assumed to be a conse-quence of the remodelling of its material constituents.Microstructural ‘recruitment’ and fibre density variablesfor the collagen are introduced into the strain energydensity functions. This enables the remodelling of colla-gen to be addressed as the aneurysm enlarges. An axi-symmetric aneurysm, with axisymmetric degradation ofelastin and linear differential equations for the remodel-ling of the fibre variables, is simulated numerically. Usingphysiologically determined parameters to model theabdominal aorta and realistic remodelling rates for itsconstituents, the predicted dilations of the aneurysm areconsistent with those observed in vivo. An asymmetricaneurysm with spinal contact is also modelled, and thestress distributions are consistent with previous studies.

1 Introduction

This paper is concerned with the modelling of theabdominal aortic aneurysm (AAA), which is character-

ised by a bulge in the abdominal aorta. Development ofthe AAA is associated with a weakening and dilation ofthe arterial wall and the possibility of rupture; 80–90%of ruptured aneurysms will result in death (Wilminket al. 1999). Surgery to remove the aneurysm is an op-tion, but it is a high-risk procedure with a 5% mortalityrate (Raghavan and Vorp 2000). The aims of mathe-matical modelling of aneurysms are to lead to a greaterunderstanding of the pathogenesis of the disease andimproved criteria for the prediction of rupture.

The evolution of an aneurysm is assumed to be aconsequence of the remodelling of the artery’s materialconstituents. The microstructural characteristics of theartery need to be incorporated into the model so thattissue remodelling can be addressed. The abdominalaorta is a large artery of elastic type (Holzapfel et al.2000; Humphrey 1995). It consists of three layers, theintima, media and adventitia. The intima is the inner-most layer, the adventitia the outermost. The media issandwiched between these two layers and is separatedfrom them by the internal elastic lamellae, which are thinelastic sheets composed of elastin. A substantial part ofthe arterial wall volume consists of an intricate networkof macromolecules consitituting the extra-ceullularmatrix (e.c.m.). This matrix is composed of a variety ofversatile proteins that are secreted locally, by fibroblastcells, and assembled into an organized meshwork inclose association with the surface of the cell that createdthem. The e.c.m. determines the tissue’s structuralproperties. For an artery it consists primarily of elastin,collagen, smooth muscle and ground substance; ahydrophilic gel within which the collagen and elastintissues are embedded. Elastin and collagen are the mainload bearers. Collagen is the stiffer and most nonlinearof the two materials, but at low strains elastin bearsmost of the load. This is because at physiological strainsthe collagen is tortuous in nature, and is crimped in theunstrained artery (Shadwick 1999; Raghavan et al.1999). The elastin gives rise to the artery’s isotropic andrubber-like behaviour for small deformations whereasthe collagen gives rise to the anisotropy and high

P. N. Watton (&)Department of Cardiac Surgery,University of Glasgow, Royal Infirmary,10 Alexandra Parade, Glasgow, G31 2ER, UKE-mail: P.Watton@clinmed.gla.ac.ukTel.: +44-141-2227863

N. A. HillDepartment of Mathematics,University of Glasgow, University Gardens,Glasgow, G12 8QW, UK

M. HeilDepartment of Mathematics,University of Manchester, Oxford Road,Manchester, M13 9PL, UK

Biomechan Model Mechanobiol (2004) 3: 98–113DOI 10.1007/s10237-004-0052-9

nonlinearity at larger deformations. For modellingpurposes, the mechanical contribution from the smoothmuscle is ignored (Holzapfel et al. 2000). Dilation of ananeurysm is accompanied by a loss of elastin and aweakening of the arterial wall (He and Roach 1993). Asthe wall dilates, strains in the elastin and collagen in-crease. Elastin is a stable protein with a long half-life(Alberts et. al. 1994), whereas collagen is in a continualstate of deposition and degradation (Humphrey 1999)with a considerably faster turnover. The highly nonlin-ear tensile nature of the collagen implies that collagenremodelling is necessary to account for the large dila-tions observed in aneurysms. Without remodelling, evenif all the elastin degraded, the strains in the collagen, andthus the dilation of the wall, would not have to increasesignificantly for the total load to be borne.

The remodelling of arteries in response to changes intheir external environment is a recognised phenomenonand has been considered by, for example, Fung et al.(1993), Rachev and Meister (1998) and Rodriguez et al.(1994). However, their analyses were phenomenologi-cally based, the remodelling driven, for example, bychanges in the thickness, opening angle or compliance ofthe arterial wall, rather than explicitly associated withadaptations of its microstructural architecture. Morerecently, Humphrey (1999) has proposed a microstruc-turally based model which addresses the remodelling ofcollagen in altered configurations and considers thesimple case of remodelling at extended and contractedlengths for a uniaxial model in which the fibres attachwith zero strain. This idea is employed in this work;however, here it is assumed that the collagen fibreattaches and configures itself such that its strain atsystole is a constant, ea>0. This remodelling process maybe advantageous for the artery because it adapts so thatthe load is primarily borne by the elastin during thecardiac cycle, and the collagen adapts such that it actsonly to prevent large strains occurring in the arterial wall.

The abdominal aorta is modelled as a cylindrical tubesubject to an axial pre-stretch and a constant internalsystolic pressure. The distal and proximal ends of theabdominal aorta are fixed to simulate vascular tetheringby the renal and iliac arteries. The constitutive equationsproposed by Holzapfel et al. (2000) to model the arterialwall are developed to include microstructural features ofthe structure of the collagen, namely fibre recruitmentand density variables; these enable the remodelling ofcollagen to be simulated. The development of aneurysmis modelled as the deformation that arises as the materialconstituents of the artery remodel whilst it is subject to aconstant pressure. Where possible we have basedparameter values on those from the literature, however,some may be coarsely estimated/assumed and furtherexperimental research is required to accurately deter-mine values.

The systolic blood pressure is 16 KPa whilst the peakwall shear stress is 2.5 Pa (Cheng et al. 2002), thusmechanically the shear stress has a negligible effect onthe deformation. However, the permeability of the

intima is dependent on the magnitude of the shear stress(Lever 1995). As the aneurysm develops, the dynamicsof the fluid flow will change. Consequently, the magni-tudes of the shear stress exerted on the intima, and thepermeability of the intima will change. The permeabilityof the intima may play an important role in the bio-chemical processes that lead to the destruction of theelastin, thus the fluid mechanics may be an importantaspect of aneurysm development. However, there is littledata on the temporal and spatial degradation of elastin,let alone the physiological processes linking this to theuptake of substances from the bloodstream. Conse-quently, we have decided to ignore the fluid dynamicaleffects and simply prescribe the degradation of elastin.

As the aneurysm develops in size, the naturallyoccurring process of fibre turnover acts to restore thestrain in the fibres. This is desirable mechanical behaviouras it shifts the load bearing back to the more compliantelastin. In our model, we introduce recruitment variables,which define the factors by which the tissue must bestretched with respect to the undeformed configuration,for the collagen to begin to bear load. This allows us tosimulate the gross mechanical effects of fibre turnover inaltered configurations. As the material constituents re-model, the new equilibrium displacement field for thearterial wall needs to be calculated. The wall is treated as amembrane and so a variational equation over a two-dimensional domain governs the ensuing deformationand evolution of the aneurysm. A finite element method isemployed to solve the resulting equations. We obtainrealistic rates of dilation for physiologically realisticgeometric,material and remodelling parameters.Dilationrates are seen to increase in hypertensive conditions. Tofurther test the consistency of our model we compare thepredicted stress distributions with those predicted byprevious mathematical models of developed aneurysms.

2 The recruitment variable

An integral part of this work is defining recruitmentvariables, which define the factors by which the tissuemust be stretched, for the collagen fibres to bear load.This enables key aspects of the naturally occurringphysiological process of collagen fibre deposition anddegradation in the altered configuration to be modelled.Consider a specimen of homogeneous arterial tissue,with the collagen fibres arranged parallel along its length(Fig. 1a). If the tissue is stretched parallel to the axis of

Fig. 1 a Collagen fibres are recruited, i.e. begin to bear load, whenunstrained tissue is stretched by a factor r. b The uni-axial model.In the unstrained state, the collagen is crimped and the length ofelastin is L. A constant force is applied to both ends. In the initialequilibrium state the strain in the collagen is ea

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the collagen fibres, then beyond a critical stretch, thefibres will be recruited, and begin to bear load. Thefactor the tissue must be stretched for this to occur isdefined by the recruitment variable r. In fact, at themicroscopic level, the fibres are distributed with a rangeof waviness (Armeniades et al. 1973) and here r relates tothose fibres that are recruited first. Once recruitmentbegins, the collagen bears a load and the subsequentmechanical behaviour of the whole population of fibrescan be modelled (Armentano et al. 1991).

The strain in the collagen is defined with respect tothe configuration in which it is recruited, whilst thestrain in the elastin is always defined with respect to theinitial reference configuration. Consider Fig. 1b, whichdepicts a parallel arrangement of elastin and collagen.The length L denotes the length of the unstrained tissueand x its extension. In unstrained tissue, the strain in theelastin is taken to be zero, and thus the strain in theelastin is measured with respect to the undeformedconfiguration. The Green’s strain in the elastin is

eE ¼ððLþ xÞ=LÞ2 � 1

2: ð1Þ

Collagen fibres are recruited when the unstrained tissueis stretched by a factor r. Hence, the strain in the col-lagen, eC, is defined with respect to onset of collagenrecruitment, i.e.

eC ¼½ðrLþ yÞ=rL�2 � 1

2; y ¼ Lþ x� rL: ð2Þ

Combining Eqs. 1 and 2 gives

eC ¼eE þ ð1� r2Þ=2

r2: ð3Þ

The significance of this relationship, i.e. Eq. 3, is that rcan remodel to maintain the strain in the collagen tosome equilibrium value, whilst the strain in the elastinincreases as the aneurysm dilates.

To model the deformations that occur as the elastindegrades and collagen remodels, it is necessary to definean equilibrium level of strain for the fibres, which thecollagen attempts to maintain by remodelling. However,the choice of this equilibrium strain is not straightfor-ward. In the physiological range of strains, collagencontributes partly to the load bearing of the arterialwall. As strains in the arterial wall increase, collagenbecomes the main load bearer and, due to its highlynonlinear mechanical behaviour, quickly preventsstrains becoming excessive. Consequently, to accountfor the large dilations observed in aneurysms, remodel-ling of collagen must be occurring. Thus, it is hypothe-sized: collagen remodels to maintain an equilibrium levelof strain in its fibres.

This is consistent with the fact that collagen fibres arein a continual state of degradation and deposition(Humphrey 1999), and that the fibres attach in a state ofstrain (Alberts et al. 1994). In this work, the equilibriumstrain, ea, is defined to be the strain in the collagen at

systole. The justification for this definition of ea is basedon the following reasoning.

– The fibroblasts work on the collagen they havesecreted, crawling over it and tugging on it—helpingto compact it into sheets and draw it out into cables,i.e. they act to attach the collagen fibres to the e.c.m ina state of strain (Alberts et al. 1994, p 984). Therefore,it may be hypothesized that there is a maximumattachment strain that the fibroblasts can achieve.

– We assume that the time taken for the fibroblasts toconfigure the collagen fibres is much longer than theduration of a pulse, so that the maximum fibreattachment strain will occur at the systolic (peak)strain of the cardiac cycle. This implies the peak strainin the fibres is equal to the peak attachment strain.

– Fibres that achieve maximum attachment strains arethe first to be recruited as the tissue is strained. Thusthe definition of a maximum attachment strain isconsistent with the definition of r, which relates tothose fibres that are recruited first. Other fibres withlower levels of attachment strain are recruited at alater stage, which is accounted for by the proposedstress functions to model the gross mechanicalbehaviour of the population of collagen fibres.

Note, not all fibres attach with the maximumattachment strain. There would be a distribution ofattachment strains achieved; this is consistent with thevariability in waviness of a population of collagen fibres.In this work, we assume that the recruitment distribu-tion function is unchanged throughout the remodelling.This is unlikely, as changes in the distribution of thepopulation of collagen fibres may occur, and could bean important aspect of the remodelling. However, thiswould lead to added complexity, and at this stage, moreinsight may be gained with a simpler model—so here weassume that the mechanical behaviour of a populationof collagen fibres is fixed throughout the remodelling.However, such features could be included in laterdevelopments.

If collagen is recruited when the tissue has beenstretched by a factor r, and if a systolic stretch isachieved when tissue has been stretched by a factor ks,then from Eq. 3 the attachment strain is

ea ¼k2s � r2

2r2: ð4Þ

Other definitions may be proposed for this parameter.Further experimental work is needed to clarify the nat-ure of the attachment of collagen fibres in altered con-figurations as the aneurysm develops, and test thevalidity of these assumptions.

We now consider a conceptual one-dimensionalmodel to gain insight into the mechanical aspects of theremodelling. Consider Fig. 1b, which illustrates a par-allel arrangement of elastin and collagen, subject to aconstant force F0. The force balance equation is

AEðtÞrEðeEÞ þ ACðtÞrCðeCÞ � F0 ¼ 0; ð5Þ

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where AE (t) and AC(t) represent cross-sectional areas,and rE(eE) and rC(eC) the Piola–Kirchhoff stresses forthe elastin (E) and collagen (C), respectively. Initially,the system (t=0) is assumed to be in equilibrium, andthus the strain in the collagen is ea. If the elastin de-grades then, since a fixed axial force is acting, the systemwill stretch and the strain in the collagen will increase sothat eC>ea. If the strain in the collagen deviates from theequilibrium strain, it is hypothesized that a remodellingprocess acts to restore the strain to ea. If it is assumedthat the population distribution of fibres remains thesame, then for the collagen to restore its strain to ea, twoprocesses can occur:

– The recruitment variable r can increase in value until anew equilibrium configuration is achieved.

– The collagen can thicken. This will act to contract theuniaxial system and restore the strain in the collagen.This is achieved by remodelling the cross-sectionalarea, AC(t), of collagen. An increase in AC(t) is con-sistent with the fact that an increase in the collagencontent is observed in aneurysms (He and Roach1993). This may be seen as a desirable remodellingresponse to the development of the aneurysm for itlimits the rate, and extent of the dilation. In this workwe attribute the thickening of the collagen to an in-crease in the density of collagen fibres whilst thedimensions of individual fibres remain the same.However, it may be that it is the number of fibres thatremain constant and thicker fibres are laid down.Experimental work is needed to clarify this matter sothat more suitable remodelling mechanisms can beproposed.

The remodelling of the recruitment variable is subtleand it is helpful to consider its mechanical effects, in aconceptual one-dimensional model before analysing themore complex geometry of an aneurysm. When elastindegrades, the system will stretch, an increased load willbe borne by the collagen, and the compliance will de-crease. If the collagen fibres are in a continual state ofdeposition and degradation then this turnover will causethe tissue to stretch yet further until a new equilibrium isrestored. This will shift some of the load-bearing back tothe elastin and reduce the load borne by the collagen,and the strain of the fibres, thus increasing the compli-ance. Figure 2 considers the effects of fibre depositionand degradation, on the scale of the collagen fibres, for atissue that is stretched and held at fixed length whilstremodelling occurs, and illustrates that this physiologi-cal process can be simulated by remodelling therecruitment variable r.

3 Mathematical description of the deformation

The strain field through the thickness of the normalarterial wall is approximately uniform about the physi-ological range of strains (Fung and Choung 1986). If it isassumed that the mechanism by which collagen fibres

attach to the artery, is independent of the current con-figuration of the artery, then this process may naturallymaintain a uniform strain field (in the collagen)throughout the thickness of the arterial wall as theaneurysm develops. The deformations at a constantsystolic pressure are considered. Thus for the purposesof studying the deformations that arise as the aneurysmdevelops it is sufficient to consider only the strains in themidplane, and this is the approach we adopt; we havemodelled the arterial wall as a membrane, using a geo-metric nonlinear membrane theory (see e.g. Wempner1973; Heil 1996).

Fig. 2 Attachment of collagen fibres in altered configurations.1 The reference state is the original undeformed configuration ofthe tissue. In the undeformed state, the fibres have a characteristicwaviness. 2 The initial physiological state is such that the strain inthe collagen fibres is the equilibrium value. At systole, the strain inthe fibres is ea. The tissue is currently in equilibrium. Although thefibres are in a continual state of degradation and deposition, newfibres will attach with identical levels of strain to those that decayand thus no changes occur in the mechanical properties of thetissue. 3 Suppose the tissue is stretched further and, for thepurposes of this example, held at fixed length. New fibres attach tothe tissue so that at the new systolic configuration their strain is ea.The old fibres decay, and the distribution of collagen fibreschanges. The naturally occurring turnover of the fibres, willproceed to restore strain in all the fibres to equilibrium levels.4 All of the old fibres have decayed and have been replaced by newfibres of strain ea. The artery reaches a new equilibriumconfiguration. 5 If the tissue is contracted back to the referenceconfiguration, the crimp of the collagen will have increased.Equivalently, the factor the tissue must be stretched for thecollagen to be recruited has increased. Hence the effects ofdeposition and degradation in altered configurations can becaptured by remodelling the recruitment variable r—which relatesto the waviness of collagen in the undeformed configuration

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In general, a body-fitted coordinate system is used todescribe the membrane with Lagrangian coordinates xa

(a=1, 2) parallel to the midplane and the x3 coordinateperpendicular to it. The midplane is positioned at x3=0;the upper and lower surfaces of the membrane havecoordinates x3=±h/2, where h is the thickness of themembrane. Lower case letters refer to the undeformedreference state, upper case to the deformed state. TheEinstein summation convention is used throughout. Theposition of a material point on the midplane ofthe membrane is given by q (x1, x2, x3=0)=q0 (x1, x2).The covariant base vectors, which are tangent to thecoordinate lines, xa, on the midplane, are given by

aa ¼@q0

@xa¼ q0;a; ð6Þ

where the comma denotes partial differentiation withrespect to xa . The unit normal to the two midplane basevectors is chosen as the third midplane base vector, i.e.

a3 ¼a1 ^ a2

a1 ^ a2j j : ð7Þ

The covariant midplane metric tensors in the unde-formed configuration are aij=ai Æ aj. During the defor-mation a material point on the membrane’s midplane,which was at a position = q

0 (xa), in the undeformedreference configuration, is displaced to a new position

Q0ðxaÞ ¼ q0ðxaÞ þ vðxaÞ; ð8Þ

where v(xa) is the midplane displacement field. Thetangent vectors to the deformed midplane are thus

Aa ¼ Q0;a ¼ aa þ v;a: ð9Þ

The normal to the midplane base vectors, A3, is chosenas the third midplane base vector, with the magnitudedefined such that the incompressibility condition is sat-isfied on the midplane, i.e.

A3 ¼ðA1 ^ A2Þ a1 ^ a2j j

A1 ^ A2j j : ð10Þ

The covariant midplane metric tensors in the deformedconfiguration are Aij=Ai Æ Aj (i, j=1, 2, 3). If it is as-sumed that off-midplane strain field is equal to that of themidplane then the Green’s strain tensor can be expressedin terms of the difference of the midplane metrics,

eij ¼Aij � aij

2: ð11Þ

We now turn to the specific geometry of our model ofthe aneurysm. The abdominal aorta is modelled as a thincylinder of undeformed radius R, length L, and thick-ness h. The thicknesses of the media and adventitia aredenoted hM and hA respectively. It is subject to a phys-iological axial pre-stretch, kz

0, and a constant systolicpressure p0 (= 16 kPa) which causes a circumferentialstretch of kh

0. The initial in vivo configuration is thus acylindrical tube of length kz

0L, radius kh0R, and thick-

ness h/kz0kh

0. The ends of the initial deformed configu-ration are fixed throughout the deformation to simulatevascular tethering by the renal and iliac arteries. Thecylindrical polar Lagrangian coordinates, xi (i=1,2,3),denote arc lengths in the axial, azimuthal and radialdirections, respectively. The vector to a material pointon the midplane before the deformation is thus

q0 ¼ R sinx2

R

� �;R cos

x2

R

� �; x1

� �

for x1 2 ½0; L�; x2 2 ½0; 2pRÞ: ð12Þ

After the deformation, the material point on the mid-plane is displaced to a new position,

Q0 ¼ ½ðRþ v3Þ sin hþ v2 cos h; ðRþ v3Þ cos h� v2 sin h; x1 þ v1�; ð13Þ

where h=x2/R, and v1, v2, v3 denote the displacements inthe axial, azimuthal and radial directions, respectively.The strain field for the midplane is calculated usingEqs. 6, 7, 8, 9, 10 and 11.

4 Governing equations

The principle of virtual displacements,

dPstrain � dPload ¼ 0; ð14Þ

governs the deformation, where dPstrain is the variationof the strain energy stored in the deformed aneurysmwall and dPload is the work done by the pressure,p, acting on the tube surface during a virtual displace-ment, i.e.Zv

dw dv�IS

pðA3 � dQ0ÞdS ¼ 0; ð15Þ

where dQ0=d(q0+v)=dviai, dS=|A1 � A2| dx1 dx2, and

A3 ¼ ðA1 ^ A2Þ= A1 ^ A2j j: Here, we make the approxi-mation that the virtual displacement of the upper andlower surfaces is equal to that of the midplane and thatthe area elements of the upper and lower surfaces areequal to that of the midplane. The assumption that thestrain field in the aneurysm tissue is uniform through thethickness of the arterial wall, i.e. the strain field of theoff-midplane is equal to that of the midplane, impliesthat the strain energy density functions (SEDFs), w, areindependent of x3 . This enables an immediate integra-tion yielding

Z2pR

0

ZL

0

½dðhMwMþhAwAÞ�pðA1^A2Þ �dviai�dx1 dx2¼ 0:

ð16Þ

Appropriate functional forms for the SEDFs for themedia, wM, and adventitia, wA, are required.

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5 Strain energy density functions for aneurysmal tissue

The aneurysm develops as the material constituents re-model; thus it is important to use a constitutive modelthat accounts for the microstructure of the artery. Hol-zapfel et al. (2000) have recently proposed a generalconstitutive model that explicitly accounts for the indi-vidual contributions of the elastin and the collagen in thearterial wall. It is assumed that the mechanical responseof the artery is due to the media and the adventitia,i.e. the intima is not considered to contribute mechani-cally. Each layer is modelled as a fibre reinforced com-posite. The structure of the collagen was determinedexperimentally. It was found that the fibres are arrangedin double helical structures of pitches ±cM in the mediaand of pitches ±cA in the adventitia. The SEDF is splitinto two parts, one for isotropic properties and the otherfor anisotropic properties. It is supposed that themechanical response of the artery at low strains is gov-erned by the non-collagenous constituents (elastin) thatrespond isotropically. The mechanical response at highstrains is governed chiefly by the collagenous constitu-ents that are anisotropic. A classical neo-Hookeanmodel is used to model the isotropic response in eachlayer. An exponential functional form was used to de-scribe the energy stored in the collagen fibres, so that thestrong stiffening effect that occurs at high pressurescould be captured. Holzapfel et al. (2000) proposeSEDFs for the media, wM, and adventitia, wA, given by

wM ¼ cMðe11 þ e22 þ e33Þþ

Xe2Mp�0;p¼�

kM exp axe2Mp

h i� 1

� �; ð17Þ

wA ¼ cAðe11 þ e22 þ e33Þ þX

e2Ap�0;p¼�

kA exp axe2Ap

h i� 1

� �;

ð18Þ

where

eJp ¼ e11 sin2 cJp þ e22 cos

2 cJp þ 2e12 sin cJp cos cJp ; ð19Þ

is the Green’s strain resolved in the direction of acollagen fibre which has an orientation of cJp to theazimuthal axis. The subscript J takes values M and A,referring to the media and adventitia respectively, andp denotes the pitch ±cJ. For a more completedescription of this constitutive model see Holzapfelet al. (2000). Note, Holzapfel et al. (2000) express (17)and (18) in terms of the invariants of the deformationand we have written them explicitly in terms of theGreen’s strain. The constants cM and cA are associatedwith the non-collagenous matrix of the material, andax, kM and kA with the collagenous part. Previousstudies (Holzapfel 2000) have found that the neo-Hookean elastic properties of the adventitia are anorder of magnitude lower than that for the media

(6–14 times depending on location) and thus they setcA=cM/10. Holzapfel determined specific parametersfor the carotid artery of a rat and hence the actualvalues of the material parameters are not of relevancehere. Note, the fibres are assumed to have no com-pressive resistance, i.e. the contribution to the strainenergy is zero if the fibres are in compression.

These SEDFs can be adapted to include microstruc-tural features for the collagen and elastin. To modelaneurysm development we need to prescribe a degra-dation of elastin. However, little is known other than theinitial and final concentrations of elastin (MacSweeneyet al. 1992) and thus appropriate assumptions need to bemade. The fact that the dilation of the aorta is localizedto a specific region would suggest that elastin is beinglost principally in a confined region. Here, the concen-tration of elastin in the arterial wall cE(x

1, x2, t) is takento be a double Gaussian exponential

cEðx1; x2; tÞ ¼ 1� 1� ðcminÞt=T� �

� exp �m1L1 � 2x1

L1

� �2

� m2L2 � 2x2

L2

� �2" #

; ð20Þ

where L1 and L2 denote the Lagrangian lengths of themembrane in the axial and circumferential directions, m1

(>0) and m2 (>0) are parameters that control the de-gree of localization of the degradation, and cmin is theminimum concentration of elastin at time t =T. Thisgives a point of minimum concentration of elastin, andthe elastin concentration tends towards normal valuestowards the distal and proximal ends of the abdominalaorta. The timescale of aneurysm development may varyfrom person to person. Given that average growth rate is0.4 cm/year (Humphrey 1995), and the decision whetherto operate on an aneurysm occurs when the diameterreaches 5–6 cm, then the timescale of development is ofthe order of 10 years. Estimates for the concentration ofelastin in the developed aneurysm suggest that between63 and 92% of the elastin is lost (He and Roach 1993).We choose T=10 years and cmin= 0.1. The spatialdegradation is unknown and further experimental workis needed to clarify the distributions before more accu-rate mathematical models can be proposed. For anaxisymmetric degradation Eq. 20 reduces to

cEðx1; tÞ ¼ 1� 1� ðcminÞt=T� �

exp �m1L1 � 2x1

L1

� �2" #

:

ð21Þ

The SEDF is then the product of the concentration ofelastin within the tissue and the neo-Hookean SEDF forthe elastin, i.e.

welastin ¼ kEcEðx1; x2; tÞðe11 þ e22 þ e33Þ; ð22Þ

where kE is a material parameter to be determined thatrepresents the mechanical behaviour of elastin for thehealthy abdominal aorta.

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Although the geometry is now more complex,recruitment variables for the collagen fibres can beintroduced throughout the tissue. The Green’s strains inthe collagen, eCJp ; are related to the Green’s strains of the

elastin resolved in the directions of the collagen fibres,eJp ; via the recruitment variables rJpðx1; x2; tÞ:

eCJp ¼eJp þ 1� r2Jp

� �.2

� �r2Jp

; ð23Þ

where rJp defines the factor by which the tissue must bestretched, in the direction of a fibre, with respect to theundeformed configuration, for the collagen fibre to be-gin to bear a load, and eJp is given by (19). Note again,the subscript J takes values M and A, referring to themedia and adventitia respectively, and p denotes thepitch ±cJ. The initial values for recruitment variables,r0Jp ; are chosen so that the fibres in the media andadventitia have strains equal to ea at systole, i.e. collagenfibres in the adventitia have strains identical to those inthe media at the systolic pressure at t=0, i.e. fromEq. 23

r2Jp ¼1þ 2eJpðkh; kzÞ

1þ 2ea; ð24Þ

where the strain in the elastin resolved in the direction ofthe collagen fibres, eJp ; is calculated with Eq. 19. A valuefor ea now needs to be proposed; experiments to deter-mine its value precisely would be desirable. The mediahas collagen fibres orientated primarily in the azimuthaldirection and is roughly twice the thickness of theadventitia (Table 1). Consequently, the media is thepredominant load bearer during radial inflation. Usingthe pressure–radius graph for a human abdominal aortagiven in Lanne et al. (1992) and subtracting the ‘‘esti-mated’’ contribution from elastin gives an estimate for

the circumferential strain, khrec, at which the medial

collagen begins to be recruited. This has been achievedquite crudely and more precise experiments would bedesirable. Now Eq. 23 implies,

r2Mp¼

1þ 2eMp

1þ 2eCMp

: ð25Þ

At the onset of recruitment the strain in the medialcollagen fibres is zero, i.e. eCMp

¼ 0: The Green’s strainsare

e11 ¼k0z� �2�1

2; e22 ¼

krech

� �2�12

; e12 ¼ 0; ð26Þ

and so the value of the recruitment variable for themedial collagen at t ¼ 0 r0Mp

� �is given by Eq. (25) as

r0Mp¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik0z� �2

sin2 cM þ krech

� �2cos cM

q; ð27Þ

where eMpk0z ; k

rech

� �is calculated using Eq. 19. The col-

lagen fibre attachment strain, ea, is defined to be thestrain in the fibres at the systolic pressure, (kh=kh

0,kz=kz

0), and thus from Eq. 23,

ea ¼eMp

k0h; k0z

� �þ 1� r0Mp

� �2� �2

ðr0MpÞ2

: ð28Þ

The initial values for recruitment variable for theadventitia, r0Ap

; are chosen so that the fibres in theadventitia have strains equal to ea at systole, i.e. collagenfibres in the adventitia have strains identical to those inthe media at the systolic pressure at t=0. Thus Eq. (24)implies that

r0Ap¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ 2eAp

ðk0h; k0z ÞÞ=ð1þ 2eaÞq

: ð29Þ

where the strain in the elastin resolved in the direction ofthe adventitial collagen fibres, eAp

ðk0h; k0z Þ; is calculatedwith Eq. (19).

6 Strain energy density function for heterogeneousaneurysmal tissue

A normalized density variable is introduced, which de-fines the density of the collagen fibres with respect to theundeformed configuration of the tissue. The recruitmentvariables and normalized fibre density variables areincorporated into the strain energy representation forthe collagen in Holpzapfel et al.’s (2000) model. Note weassume that the neo-Hookean response in the media hasan elastinous and a non-elastinous part, kg, due to theground substance. Furthermore, we assume thatthe contribution from the ground substance is equal inthe media and adventitia. Note that in Holzapfel et al.’s(2000) model, since there is no degradation of elastin,only one material parameter is required for the

Table 1 Material constants and physiological data used for mod-elling the human abdominal aorta

Wall thicknessMedia (hM) 1.33 mmAdventitia (hA) 0.67 mmCollagen fibre anglesMedia (cM) 8.4�Adventitia (cA) 41.9�Systolic pressure (p0) 120 mmHgUndeformed radius (R) 6.6 mmAxial pre-stretch (kz

0) 1.5Circumferential stretchAt onset of recruitment (kh

rec) 15.2/13.2At systole (kh

0) 17.2/13.2At 200 mmHg (kh

200) 18.5/13.2Material parametersExponential constant for collagen (ax) 40Elastin (kE) 97.6 kPaGround substance (kg) 9.76 kPaCollagen media (kM) 3.52 kPaCollagen adventitia (kA) 0.88 kPa

This data is estimated and determined from the pressure radiuscurve for the abdominal aorta in Lanne et al. (1992)

104

neo-Hookean contribution in the media. The SEDFs,wJ, are thus

wM ¼ðkg þ cEðx1; x2; tÞkEÞðe11 þ e22 þ e33ÞþX

eCMp�0;p¼� rMp

ðx1; x2ÞnMpðx1; x2ÞkM

� exp ax eCMpeMp

; rMp

� �� �2� �� 1

� �ð30Þ

for x3 2 ½ð�hÞ=2; ð�hþ 2hMÞ=2�; and

wA ¼ kgðe11 þ e22 þ e33ÞþX

eCAp�0;p¼� rAp

ðx1; x2ÞnApðx1; x2ÞkA

� exp ax eCApeAp

; rAp

� �� �2� �� 1

� �ð31Þ

for x3 2 ððh� 2hAÞ=2; h=2�: Recall that hM, hA, h (=hM+ hA) denote the thicknesses of the media, adventitiaand arterial wall respectively, and cE(x

1, x2, t) representsthe concentration of elastin in media. eCJp x1; x2; t

� �de-

notes the Green’s strain, and rJp x1; x2; t� �

andnJp x1; x2; t� �

; the recruitment and density variables of thecollagen fibres, of pitch p in layer J. At t = 0, cE(x

1, x2,0) = nM (x1, x2, 0) = nA (x1, x2, 0) = 1. FollowingHolzapfel et al. (2000) we set kg to be an order ofmagnitude lower than the contribution from the elastin,i.e. kg = kE/10. Due to a lack of available data on thehuman abdominal aorta, we have based some of theremaining parameters on those determined by Holzapfelet al. (2000) for the carotid rat artery, namely the fibreangles in the media and adventitia, and the relativemagnitude of kM, kA; here we choose kA = kM/4. This isquestionable; experimental work is needed to correctlydetermine these values. The undetermined constants, kE,kM and ax, are set so that the strain energy functionsmodel the mechanical behaviour of the abdominal aortausing physiological data (Lanne et al. 1992). Note, thecontributions to the strain energy from the collagen fi-bres arise only when the fibre strains are positive, eC

Jp > 0;

i.e. fibres do not contribute to the strain energy when incompression. This SEDF enables remodelling in alteredconfigurations to be addressed. In all, four recruitmentvariables and four fibre density variables are required. Inthe absence of physiological data, linear differentialequations are proposed for the remodelling of therecruitment and density variables:

@rJp

@t¼ a eCJp � ea

� �;

@nJp

@t¼ b eCJp � ea

� �; ð32Þ

where the constants a, b > 0. The remodelling param-eter a was determined to correspond to a prescribedturnover rate of the collagen fibres, as follows. Theabdominal aorta is modelled using a physiologicallydetermined set of material parameters and is initiallysubject to physiological axial and systolic circumferen-tial stretches. The circumferential stretch is theninstantaneously increased. The strains in the collagenfibres increase to eSJ (>ea), and the artery is held in this

altered configuration whilst the collagen remodels torestore its strain ea. Two remodelling mechanisms areconsidered for restoring strain:

1. A half-life model (Humphrey 1999) which models themechanical effects of fibres being deposited in thealtered configuration with strain ea and the decay ofthe strained fibres (with strain eSJ).

2. Remodelling via the recruitment variable. Thekinetics and mechanical effects of the two remodel-ling mechanisms are compared to give a value for aassociated with the turnover rate of the collagenfibres.

It is important to remember that the strain in theelastin is defined with respect to the undeformed con-figuration, whereas the strain in the collagen is definedwith respect to an altered configuration, via therecruitment variable. The recruitment variable remodelsto maintain the strain in the collagen to ea. This is tosimulate the effect of fibres being degraded and depos-ited in altered configurations. It is not regarded as theeffect of disattachment and reattachment of fibres, whichmay occur if the tissue is stretched over much shortertime periods. The fibre density, nJ, is defined with respectto the undeformed configuration of the tissue, and givesan interpretation of the relative density of the fibres asthe tissue remodels. If nJ stays constant whilst therecruitment variable remodels then one can picture thisas the number of fibres in the arterial wall remainingconstant. This allows for a structural interpretation ofthe remodelled tissue and may be used to predict chan-ges in mass content of collagen. This may be useful, sincethe mass content of collagen could be measured.Although this may be less easy to determine than thethickness of the tissue, it may yield a better under-standing of the structural changes in the tissue. Morecomplex models may be proposed in the future asknowledge of the changing structure of the collagen inthe developing aneurysms improves. Ideally the degra-dation and deposition of the entire population of col-lagen fibres would be modelled; however, such anapproach would be computationally expensive in a full3D model and it would seem preferable to use variableswhich are gross mechanical descriptors.

7 Numerical solution

Equation 16 can be written as

Z2pR0

0

ZL

0

ðui � piÞdvi þ uiadvi;a

h idx1 dx2 ¼ 0; ð33Þ

where ui ¼ @ðhMwM þ hAwAÞ=@vi; uia ¼ @ðhMwM þhAwAÞ=@vi

;a and pi ¼ pðA1 ^ A2Þ � ai for i = 1, 2, 3 anda=1, 2. This equation is solved by applying a dis-placement-based finite element technique. The displace-ments are approximated by an ansatz

105

viðxaÞ ¼XN

g¼1V igwgðxaÞ ð34Þ

for N basis functions, wg; piecewise bilinear shapefunctions are employed to approximate the displace-ment field. Similarly, the remodelling variables rJp ; nJpare defined at nodal points of the domain, with thevariables Rg

Jp and N gJp ; and interpolated using the shape

functions:

rJpðxaÞ ¼XN

g¼1Rg

JpwgðxaÞ; nJpðxaÞ ¼XN

g¼1N g

JpwgðxaÞ ð35Þ

Now

dviðxaÞ ¼XN

g¼1dV igwgðxaÞ; dvi

;aðxaÞ ¼XN

g¼1dV igwg;aðx

aÞ

ð36Þ

and so insertion of Eq. (35) into Eq. (32) yields

fig ¼XN

g¼1

Z2pR0

0

ZL

0

½ðui � piÞwg þ uiawg;a�dV igdx1dx2 ¼ 0;

ð37Þ

where ui ¼ uiðcEðx1; x2; tÞ; rJpðRgJp ;wgÞ; nJpðN

gJp ;wgÞ;

viðV ig;wgÞ; vi;aðV ig;wg;aÞÞ; with a similar expression for

/i,a. Equation 37 must hold for all variations dVig. IfnBC of the boundary conditions are specified then thisequation yields 3N-nBC equations for the 3N-nBC un-knowns. It is solved using a Newton–Raphson scheme,using the displacement variables as the independentvariables. The initial values of the nodal values of therecruitment are given by Eqs. 27 and 29. The nodalvalues of the fibre density are set equal to 1 throughoutthe domain. At each time step the remodelling variablesare updated with a forward Euler time-integrationscheme applied to Eqs. 32, i.e.

RgJ ðtnþ1Þ ¼ Rg

J ðtnÞ þ aðeCgJp ðtnÞ � eaÞdt;

NgJ ðtnþ1Þ ¼ N g

J ðtnÞ þ aðeCgJp ðtnÞ � eaÞdt;

ð38Þ

where the nodal value of the strain in the collagen eCgJp is

taken to be the average of the strains in the surroundingelements. The degradation of the elastin is the drivingforce for the remodelling, it causes the strains in theaneurysm to increase and then the recruitment anddensity variables remodel to restore the strain in thecollagen (with respect to the configuration it is recruitedin). A timestep of 0.04 years was employed; decreasingthe timestep further made negligible difference to theresults and merely increased computational time. Theasymmetric model (see below) employed a domain sizeof 200·50 elements, and the axisymmetric model, 200line elements. Increasing the number of elements furtherhas a negligible effect on the results (Watton).

Partial integration of Eq. 33 yields ui � uia;a ¼ 0: Theinitial displacement field for the artery subject to apressure p axial stretch kz and radial stretch kh withundeformed radius R is v1=(kz�1)x1, v2=0,v3=(kh�1)R. This displacement field can be shown toimply that /i- /ia,a=0 (i=1, 2, 3 and a=1, 2) and /1=0,/2=0. Thus /3=0, which implies that

p ¼ 1

Rkz

� ðhM þ hAÞkg|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

ð1Þ

þ hMcEkE

�1� 1

k2z k4h

�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð2Þ

�

þX

J¼M;A;eCJp�0

2nJ hJ kJ axeCJp exp

�axðeCJpÞ

2

�cos2 cJp

r2Jp

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ð3Þ

!: ð39Þ

The first term (1) is a contribution from the non-elas-tinous neo-Hookean constituents in the media andadventitia, the second term (2) arises due to the contri-bution from the elastin in the media, and the thirdcontribution (3) is due to the collagen in the media andthe adventitia. For an artery subject to a physiologicalaxial pre-stretch (kz=kz

0), the onset of collagenrecruitment is estimated to begin at kh=kh

rec. This en-ables values for rM, rA, ea to be determined using theanalysis detailed earlier. For a healthy artery nJ (x

1, x2,0)=1, and thus in Eq. 39 there are three unknowns to bedetermined, namely kE, kM, ax (given that kA=kM/4). Itis assumed that at the initial systolic pressure (kz=kz

0,kh=kh

0, p=p0) the neo-Hookean constituents bear 80%of the load. This enables kE to be determined using Eq.39, i.e.

kE ¼ Rkz0P0

E:Cp0h10þ hM

� �1� 1

k0z� �2

k0h� �4

! !

ð40Þ

where PE:C0=0.8 (and h=hM+hA, kg=kE/10). At a

pressure of 200 mmHg the circumferential stretch, kh200,

is determined from Lanne et al. (1992). The proportionof load borne by the elastin, PE:C

200, and collagen(1�PE:C

200) are found at this increased pressure usingEq. 39. Consequently, there are two equations for theproportion of load borne by the collagen, at the systolicpressure and a pressure of 200 mmHg, which allows theconstants ax and kM to be determined to achievethe appropriate mechanical nonlinear behaviour for thecollagen. Table 1 details the material parameters used inthe model, and Fig. 3, the resulting pressure–diameterrelationship.

8 Example of an axisymmetric aneurysm

We will now consider the development of an aneurysmthat arises due to an axisymmetric degradation of

106

elastin. The degradation of elastin is given by Eq. 21with m1=20 (see Fig. 4a). Appropriate remodeling rateparameters need to be specified for Eqs. 32. Given avalue for the remodelling parameter a that correspondsto a physiologically realistic turnover rate of the collagenfibres, the remodelling constant b is then determined sothat the aneurysm dilation is physically realistic, i.e. by afactor of 2–3 over a period of 10 years. Humphrey(1999) suggested collagen turnover rates, i.e. the time for50% of the fibres to decay and be replaced, of3–90 days, although acknowledging that the timescalesmay be longer. It was found that correspondings valuesof a resulted in a very nonlinear remodelling profile, withlittle dilation initially followed by rapid dilation in thefinal few years (Watton 2002). Although aneurysms cansuddenly expand, the average rate of dilation is 0.4cm/year (Humphrey 1995) and dilation rates are in therange of 0.3 to 1.1 cm/year (Humphrey 2002). Thus, itwas considered physiologically more consistent tochoose a=12 (130<b< 150), which corresponds to a(longer) half-life of 180 days. This yields an increasingrate of dilation but in a less nonlinear manner (Fig. 5a).It would be desirable if turnover rates of collagen fibresin aneurysms were more precisely known to both im-prove and validate our modelling.

An unusual feature of aneurysm growth is high-lighted in the axisymmetric model (Fig. 5c). Large axialdeformations occur in addition to the visually apparent,radial deformations. This is a consequence of the factthat, under in vivo normotensive conditions, theabdominal aorta is subject to a large axial stretch. As theelastin degrades, the central region of the aneurysm di-lates axially, while the distal and proximal end regions ofthe aorta retract. As the end regions of the aneurysmretract, the collagen remodels about the new state, whichpromotes further retraction. Conversely, as the centralregion dilates axially, the collagen remodels about thisnew configuration, allowing further axial dilation. Thiseffect is highlighted by the difference in the profiles of theelastin degradation as defined in Lagrangian coordinates(Fig. 4a) and in a spatially fixed Eulerian frame(Fig. 4b). Physiologically, this would have the effect ofreducing the in vivo axial stretch of the artery as theconstituents remodel about the new basal state.

The remodelling parameters are determined by firstselecting a value of a that corresponds to a realisticturnover rate for the collagen fibres, and then choosingb such that dilation is physiologically realistic over atimescale of 10 years. The free variable is the remodel-ling of the collagen density. This must be examined tosee if, once a and b have been prescribed, the remodel-ling of the density of fibres is physiologically consistent.Not surprisingly, the density remodels to the greatestextent in the central region of the aneurysm, whichundergoes the greatest deformation. The adventitialcollagen density remodels more than medial colla-gen—this may be attributed to the adventitial collagenbeing more axially orientated so it is subject to greaterstrains due to the high axial deformation in the centralregion of the aneurysm. The remodelling of the collagendensity nJ in the medial and adventitial layers at firstseems excessive, with maximum increases of 25 times inthe media (Fig. 6a) and 30 times in the adventitia(Fig. 6b). However, nJ defines the collagen density withrespect to the undeformed configuration of the tissue.The remodelled in vivo aneurysmal tissue may retractonly slightly due to the loss of elastin and remodelling ofconstituents about the new basal state. In fact, Fukuiet al. (2002) report mean axial and circumferential invivo stretches of aneurysmal arterial tissue of 5 and 12%respectively. The in vivo physiological pre-stretch of thehealthy abdominal aorta is 1.5, therefore the tissue musthave remodelled about the deformed configuration.Thus, it may be more appropriate to define the collagen

Fig. 3 The pressure–diameter relationship for the model of theabdominal aorta. Here the relationship is shown for an in vivophysiological axial stretch of kz=1.5

Fig. 4 a Concentration of elastin as a function of the Lagrangiancoordinate x1 for aneurysm in Fig. 5c. b Concentration profile asplotted against a spatially fixed Eulerian axial coordinate. t denotestime in years. The Eulerian profile occupies a wider region than theLagrangian profile due to the axial distortion as the aneurysmdevelops

107

density with respect to the current deformed configura-tion, ~nJ ðx1; tÞ ¼ nJ ðx1; tÞ= A1 ^ A2j j; where A1 and A2 arethe deformed midplane basis vectors. This gives in-creases in density of a factor of 2–3 (Fig. 6c and d).

The axisymmetric model may be employed to exam-ine the effect that changes in the pressure have on therate of remodelling. To simulate the onset of hyperten-sion the pressure is increased stepwise after 2 years byfactors of 1.1, 1.2 and 1.3. As expected, greater pressuresgive greater dilation rates (Fig. 7). Importantly, for apressure change of up to 30%, the predicted dilations

are still physiologically consistent, i.e. 0.3–1.1 cm peryear (Humphrey 2002). These results have an obviousclinical application in that patients with aneurysmsshould have therapy to prevent hypertensive conditions.

9 Example of an asymmetric aneurysm

Asymmetric aneurysms can be developed either byassuming an asymmetric degradation of elastin or byassuming an axisymmetric degradation of elastin andpostulating that the spine limits posterior expansion.The physiologically determined remodelling-rateparameters for the axisymmetric model (a=12, b=140)are used in the subsequent analysis. Here the elastin isdegraded axisymmetrically, and a penalty pressure ap-proach is used to model spinal contact. The spine ismodelled as a stiff spring-backed plate. Where theaneurysm wall penetrates the plate, a penalty pressureacts normal to the aneurysm. The effective pressure

Fig. 5 a Plots of the maximum diameter of the axisymmetricaneurysm model vs. time showing that a physiologically realisticdilation is obtained for a=12 (corresponding to a half-life of6 months) and 130<b<150. This range of parameters givesphysiologically realistic dilation, i.e. the aneurysm diameterincreases by a factor of 2–3 over 10 years. b The initial undeformedgeometry at t=0. c The axisymmetric solution at 10 years usinga=12, b=140 (m1=20, m2=0). The Lagrangian mesh superim-posed on the midplane of the aneurysm is attached to the materialpoints and illustrates the deformation from the initial state

Fig. 6 Remodelling of thea medial nM and b adventitialnA, collagen density vs. Eulerianaxial coordinate z for theaxisymmetric model shown inFig. 4b, t denotes time in years.The predicted results seem high.However, if one takes intoaccount the distortion of thetissue in the alteredconfiguration, more realisticresults are obtained:remodelling of c medial nM andd adventitial nA collagen densitywith respect to the deformedconfiguration. Note that therapid increase in dilation of thecentral region in later yearsresults in a drop in density withrespect to the deformedconfiguration in this region

108

acting on the membrane, p(x1, x2, t) is given by thedifference of internal physiological pressure and thepenalty pressure. For further details, see Watton (2002).The contact with the spine results in a preferentialanterior bulging in the aneurysm (Fig. 8a).

Axial and azimuthal Cauchy stresses, normalizedwith respect to the initial stresses in the healthyabdominal aorta at systolic pressure, are considered onthe anterior and posterior wall. A rupture criterionbased on the stresses on the tissue is not known—indeed,the use of such a criterion would be useful only if thestresses in an aneurysm in a living person could beaccurately measured. Thus a criterion based on thediameter serves more usefully. However, mathematicalmodels can easily define the stresses in the aneurysm,which may be useful in determining the shapes ofaneurysm that are more likely to rupture. The Cauchystresses relate to the stress that the tissue ‘feels’; relativechanges in the magnitude of stresses with respect to theirnormotensive levels may be a suitable indicator forlikelihood of rupture. The purpose of our model is toshow that a model for an aneurysm can be developed byconsidering it to be a consequence of changes to thetissues composition. To further test the validity of ourmodel, the stresses it predicts will be compared withothers predicted in the literature.

The stresses, averaged across the thickness of thearterial wall, will be considered on the anterior surfacesand posterior surfaces, along deformed coordinate lines,where x2=0 and x2=pR. Due to the symmetry of theaneurysm along these deformed coordinate lines, there

are no shear strains. Thus, the average Cauchy stressthrough the thickness of the wall is given by

t11av ¼ ð2e11 þ 1ÞS11av ; t22av ¼ ð2e22 þ 1ÞS22

av ; ð41Þ

where tav11 and tav

22 denote the average Cauchy stressesin the direction of the deformed x1 and x2 coordinates.The quantities Sav

11 and Sav22 are the average second

Piola–Kirchhoff stresses. The media is twice the thick-ness of the adventitia and thus the average stress isweighted accordingly,

S11av ¼

@ðð2wM þ wAÞ=3Þ@e11

; S22av ¼

@ðð2wM þ wAÞ=3Þ@e22

:

ð42Þ

The stresses can be normalized with respect to axial andazimuthal stresses at the systolic pressure in the healthyabdominal aorta,

t11 ¼ t11avt11av0

; t22 ¼ t22avt22av0

; ð43Þ

where t11av0 and t22av0 denote the average axial and azi-muthal Cauchy stresses in the abdominal aorta at systoleat t = 0. Plots of the normalized Cauchy stresses on theanterior and posterior wall are shown in Fig. 9. Stressesincrease in the region of maximum dilation. The azi-muthal stresses increase more than the axial stresses, andstresses are greater on the anterior wall than the pos-terior wall. At the proximal and distal ends of thisaneurysm, the axial stresses have dropped almost to zeroand there is a small reduction in the azimuthal stresses.The stress analysis agrees with the previous analysis(Elger et al. 1996) in that the azimuthal stresses increasemore than the axial stresses. Also, the axial stress de-creases towards the distal and proximal ends of theaneurysm. However, the model predicts increases in theCauchy stresses that are particularly greater than inprevious models, i.e. 10–18 times, whereas Elger et al.(1996) calculate increases by a factor of 2–3. This is aconsequence of the axial strains in the central region ofthe aneurysm increasing to such an extent that, in thedeformed configuration, a substantial thinning of themembrane has occurred (Fig. 8b). Thus, the Cauchystresses, which define stresses with respect to deformedarea elements, are very high.

In this work, collagen remodelling has been achievedby changing the collagen fibre density. However, depo-sition of collagen may arise by adding more tissue,

Fig. 7 The rate of dilation of the axisymmetric aneurysm increasesin hypertensive conditions and is physiologically consistent. Theremodelling parameters are fixed (a=12, b =140) and the samedegradation of elastin (Fig. 5a) is used for each case

Fig. 8 Profile of an aneurysmusing an axisymmetricdegradation of elastin (m1=5,m2=0.25) and introducingcontact at y=15 mm. Spinalcontact has resulted inpreferential anterior bulging.b The deformed thickness of theaneurysm is very thin

109

rather than increasing the density. Mathematically, theequilibrium deformation field is dependent on the strainenergy stored inside the arterial wall. The deformationfield for a thickening of the wall would be equivalent tothat of an increase in density, provided that this repre-sented the same number of fibres per unit area of themembrane, and the wall remained thin enough forthe membrane approximation to be made. The density isthe preferred variable to consider for remodelling pur-poses, since this relates to the mass of the collagen in thewall, whereas there is ambiguity in the microstructuralinterpretation of a remodelled thickness. Nevertheless,to correctly estimate stresses, it may be more appropri-ate if they are defined with respect to a physiologicallyrealistic aneurysm wall thickness. Elger et al. (1996)considered the wall thickness of the aneurysm to beuniform in the deformed configuration. The stress dis-tributions in this model can be examined by assumingthat the wall thickness in the deformed configuration isconstant.

The Cauchy stresses are defined with respect to thedeformed arterial wall thickness hdef, i.e.hdef ¼ h= A1 ^ A2j j; where h=hM+hA is the undeformedthickness, and A1 ^ A2j j is the determinant of the mid-plane metric tensor. Now consider the forces acting on amembrane element: if the thickness of the element in-creases by a factor f, then the stress must decrease by afactor f for the force to remain the same. The (virtual)factor by which the deformed aneurysm membrane mustthicken to achieve a uniform systolic thickness isf ¼ A1 ^ A2j jt¼T= A1 ^ A2j jt¼0; the initial systolic thick-ness for the healthy axisymmetric model of the abdom-inal aorta is hsys ¼ h= A1 ^ A2j jt¼0: Thus the stresses(acting in the deformed configuration) defined with re-spect to an arterial wall of uniform systolic thickness,taaunif; are given by

taaunif ¼

taa A1 ^ A2j jt¼0A1 ^ A2j jt¼T

for a ¼ 1; 2: ð44Þ

Plots of the stress distributions on the posterior andanterior walls, for an aneurysm wall of uniform systolicthickness, are shown in Fig. 10. The relative magnitudesand distributions of the stresses are now in agreementwith Elger et al.’s (1996) findings. Furthermore, thedistribution of the stress compares favourably with thestress analysis of an asymmetric aneurysm model used byRaghavan and Vorp (2000). They proposed an isotropichomogeneous SEDF for aneurysmal tissue and used anidealized asymmetric geometry for the aneurysm, withno attempt to model spinal contact. The von Misesstresses were found to be lower towards the proximaland distal ends, had peaks in the necks of the aneurysm,and had a local maximum stress that was located at thecentre of the posterior wall. We obtained similar resultswith the exception that there is a local maximum azi-muthal stress on the anterior wall and not the posteriorwall (Fig. 10). However, this difference may be attrib-uted to two differences in the modelling. Firstly, ourmodel includes spinal contact, which would act to sup-port the posterior wall and thus will reduce the stresses inthis region. Secondly, the geometry of the asymmetricaneurysm differs. Raghavan and Vorp’s (2000) aneu-rysm’s geometry is more asymmetric, the anterior wallbulges substantially whilst there is negligible bulging ofthe posterior wall. We did find though, that if anasymmetric aneurysm is developed using an asymmetricdegradation of elastin, with no spinal contact, thenmaximum azimuthal and axial stresses occurred on thecenter of the posterior wall (Watton 2002). An interest-ing and possibly important result is obtained when oneexamines the distribution of the axial stress. On theanterior wall it has a localized minimum close to thedistal and proximal ends of the aneurysm. This mayindicate a propensity for the aneurysm to buckle.

The model allows for the strain fields in the elastinand collagen to be examined independently. The strainsin the collagen fibres are defined with respect to thedeformed configuration by means of the recruitment

Fig. 9 Normalized Cauchystresses in the asymmetricaneurysm with spinal contact at10 years: a posterior, b anterior.Azimuthal stresses increase by agreater factor than the axialstresses. The magnitudes arehigh due to the substantialthinning of the membrane in themodel

Fig. 10 Cauchy stresses(t=10 years) for asymmetricaneurysm with constant wallthickness in deformedconfiguration a posterior,b anterior. This yieldsdistributions and magnitudes ingeneral qualitative agreementwith previous studies

110

variables, whilst the strains in the elastin are defined withrespect to the initial undeformed configuration of theartery. This may be an important feature if the eventualrupture of the aneurysm is attributed to the failure of aparticular material. Figure 11a shows the distribution ofthe strains in the elastin and Fig. 11b the strains in thecollagen fibres, on the anterior of the aneurysm. Similarresults are found for the posterior wall although, asexpected, the strains on this wall increase slightly less.The azimuthal strains in the elastin are highest in thecentral region of the aneurysm and close to normoten-sive values towards the distal and proximal ends of theabdominal aorta. The axial strains in the elastin peak inthe central region of maximum dilation, and fall to zerotowards the proximal and distal ends of the aorta.Collagen fibre strains are greatest in the central region ofmaximum dilation and have remodelled to normotensivevalues towards the distal and proximal ends of theabdominal aorta. The adventitial fibre strains increase toa greater extent than the medial fibre strains because theadventitial fibres are more axially orientated and thegreatest deformation occurs in the axial direction.Overall strains in both the elastin and collagen increaseto the greatest extent on the anterior wall at the point ofmaximum dilation. The small increase in collagen strainsis a consequence of the remodelling of the recruitmentvariables and the fibres strains being defined with respectto an altered configuration. The much larger increase inthe elastin strains are a result of the artery being subjectto an initial axial pre-stretch which causes the centralregion of the artery to dilate and the distal and proximalends to retract as the elastin degrades.

10 Conclusions and discussion

The model developed here is the first mathematicalmodel to consider the evolution of the AAA. It uses arealistic structural model for the healthy abdominalaorta and implements the remodelling of the constitu-ents using structural remodelling equations. However,this model does not explain why aneurysms are mostlikely to occur in the abdominal aorta and not elsewherein the body. Furthermore, the central assumption thatdrives the remodelling, the degradation of elastin, isassumed and prescribed. No attempt is made to model

the cause of the loss of elastin, nor to explain why thisprimarily occurs in the abdominal aorta. These areimportant matters, which require increased physiologi-cal knowledge; new mathematical models may guideresearch. However, there are more immediate criticismsof the model that could be addressed in subsequentwork.

During development of the aneurysm, the Green’sstrains in the elastin become very large. The isotropicneo-Hookean SEDF used to model the elastin’smechanical behaviour may not be valid at such largestrains; an alternative SEDF may be required. However,it is likely that some creeping of the elastinous tissueoccurs. To model such a process would require that theelastin strains are defined with respect to a deformedconfiguration, in which case the isotropic neo-HookeanSEDF (defined with respect to the strains in an alteredconfiguration) may be valid. Also, the ground substancemay creep. This is probably not significant in our cal-culations because it bears only a small proportion of thetotal load, yet it is probably able to resist a compressiveload and could lead to in vivo aneurysms not retractingfully to the undeformed state.

The spatial degradation of elastin may have beenunrealistic. The Lagrangian function, which is used tomodel the degradation, deforms with the materialcoordinates. Thus, due to the distortion of the tissue inthe central region of the aneurysm, the relative concen-tration of elastin would be of the order of10%=ððA1 ^ A2Þjt¼T� 1%: This is approximately anorder lower than the lowest experimental estimates, i.e.8% (He and Roach 1993). The substantial axial distor-tion of the aneurysm is a consequence of degrading theelastin in the central region whilst assuming it does notdegrade towards the distal and proximal ends. Theconcentration difference causes the axial dilation of thecentral region and retraction of the ends. If the elastinwere degraded uniformly along the length of the artery,then such high axial distortions would not occur. Thismay account for the unrealistic thinning of the mem-brane that is predicted by the model. Further experi-mental work is necessary to accurately determine spatialdistributions in developed aneurysms to improve themathematical model.

It has been assumed that the collagen fibre anglesdeform with the tissue as the aneurysm deforms; this is

Fig. 11 Green’s strains onanterior wall of aneurysm att=10 years for a elastin bcollagen fibres. Note that thestrains in the elastin increasesubstantially since they aredefined with respect to theundeformed configuration ofthe tissue, whereas collagenfibre strains are much lower dueto the remodelling

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questionable. The structural arrangement of fibre anglesin the healthy abdominal aorta is known, but thestructure in an aneurysm is not. It is likely that the fibresdistribute themselves so that the tissue behaves in anoptimum manner. If a physiological/mechanical crite-rion for fibre alignment were known, then the fibre an-gles could be treated as free variables and allowed toremodel. Indeed if the fibre angles in diseased tissue wereknown, remodelling hypotheses for fibre alignmentcould be tested with this model.

The remodelling equations for the recruitment andfibre density variables have been given simple linearforms and the remodelling parameters, a and b, havebeen assumed to take the same values in the media andadventitia. It is questionable as to whether a linearfunction realistically represents the remodelling behav-iour and whether the remodelling rate parameters for themedia should be equal to those for the adventitia. Forexample, if turnover rates and increased rates of depo-sition are affected by cellular activity, then, since elastindegradation occurs in media, the remodelling ratesshould be higher in the media. In vivo, aneurysms arereported to have an appreciably thicker adventitia thanmedia (Humphrey 2002). A suitably sensitive, nonlinearremodelling function could allow for greater increases inthe adventitia and less in the media, or the value of theremodelling parameter b could be increased for theadventitia and decreased for the media. However, suchmodelling is phenomenological and does not yield aninsight into the underlying physical mechanisms causingthe increased deposition in the adventitia. An alternativeapproach, that may be more appropriate, would be touse a remodelling equation for the thickness of thearterial wall that acts to maintain an equilibrium stress.However, this would not yield information aboutchanges in the structural composition. Although thismodel does not accurately predict thicknesses of thearterial wall it does predict how much mass of collagenthere will be in the developed aneurysm wall, whichcould be measured experimentally. Further experimentalinvestigation is needed to assist in determining the mostappropriate form the remodelling functions should take.

The abdominal aorta has been modelled as a perfectlycylindrical tube. In fact, there is a small degree oftapering and a small amount of tortuosity; an axialsection of abdominal aorta of length 11.0 cm has anactual straight length of 11.5 cm (Raghavan and Vorp2000). It was seen in this work that the axial stress couldfall to a minimum of zero, close to the neck of theaneurysm. The reduction in axial stress (that occurs asthe elastin is lost and the collagen remodels) coupledwith an initial asymmetric geometry may predict thehighly tortuous aneurysms often seen in vivo.

The remodelling hypothesis assumes that the fibresattach and configure to achieve a peak attachmentstrain. It was assumed that the peak strains in the fibresoccur at systolic pressure. Thus, fibre attachment can bemodelled by considering the steady deformations that

occur as the constituents remodel at systolic pressure.However, for a more complex geometry, the fibres maynot achieve maximum strains at systolic pressure. Forexample, towards the distal and proximal ends of theaneurysm the strains may increase as the pressure isreduced. The remodelling of fibres should be dependenton the peak strains of the fibres during the cardiac cycle.The remodelling assumption should thus be generalizedso that no remodelling occurs if the peak strain in thefibre is equal to eA during the cardiac cycle, i.e.

drJp

dt¼ a eCJp

���max�ea

� �ð45Þ

where eCJp

���max

is the maximum strain that occurs in the

collagen during the cardiac cycle (diastolic to systolic).This would increase computational cost but would berelatively easy to include. Physically this would act toreduce the rate of remodelling towards the distal andproximal ends of the aorta.

The model does not include other physical featuresthat would act to bear the load and limit the rate ofdilation, such as the calcification of the wall or the in-traluminal thrombus (Di Martino et al. 1998; Wanget al. 2001). Moreover, the effective transmural pressurewould reduce as the aneurysm comes into contact withthe surrounding tissue which would reduce the rate ofdilation. Modelling of such features may be incorpo-rated in future developments of the model.

All of the criticisms discussed so far could be ad-dressed. However, one criticism is integral to the model.Increased deposition of collagen has been achieved byremodelling the fibre density within the tissue. Themodel does not predict by how much the wall willthicken. Consequently, the precise stress distributions inthe aneurysm wall are unknown and some assumptionregarding the remodelled thickness of the aneurysm isrequired.

Despite its shortcomings, our model yields resultsthat are consistent with those observed in vivo andpredicted by other mathematical analyses of developedaneurysms. Where possible we have based parametervalues on those from the literature; however some of theestimates are coarse and further experimental research isrequired to accurately determine values. We believe thismodel may provide a theoretical basis for furtherexperimentation and validation in the field.

Acknowledgements P. N. Watton gratefully acknowledges theaward of a Research Studentship funded by the UK Medical Re-search Council. The authors are indebted to the Consultant Vas-cular Surgeons, Mr S. Dodds (Good Hope Hospital, SuttonColdfield, UK) and Mr D.A.J. Scott (St. James’s University Hos-pital, Leeds, UK) for many helpful discussions about the clinicalaspects and physiology of abdominal aortic aneurysms. We alsoacknowledge the Harwell Software Library (http://www.hsl.ac.uk )for granting UK academics the free use of its Fortran subroutinesin non-commercial applications. MA38 was employed to solve thelinear system that arises in the Newton iteration, which is requiredto update the deformation at successive timesteps.

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