Numerical modelling of fracture in human arteries

Computer Methods in Biomechanics and Biomedical Engineering Vol. 11, No. 5, October 2008, 553–567

Numerical modelling of fracture in human arteries A. Ferrara and A. Pandolfi* Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, -20133 Milano, Italy ( Received 11 July 2007; final version received 30 September 2007 )

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We present 3D finite element models of atherosclerotic arteries, used to investigate the influence of the geometry and tissue properties on the plaque rupture caused by overexpansion. We adopted a geometry reconstructed from a contiguous set of in vitro magnetic resonance images of a damaged artery. The artery wall is divided in three layers (adventitia, media and intima) and is discretized into tetrahedral finite elements. The artery material is described with a hyperelastic two-fiber anisotropic model proposed by Holzapfel et al. 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elasticity 61(1):1 – 48, while the plaque is assumed to be transversely isotropic. Cracks induced by mechanical actions are represented through cohesive surfaces, and are allowed to develop along solid elements boundaries only. Fractures are explicitly introduced in the discretized model at the locations where the tensile strength of the material is reached. Keywords: cohesive model; plaque fracture; two-fiber reinforced soft tissue; finite element method

1.

Introduction

Atherosclerosis is a vascular disease, characterised by accumulation of lipids, collagen, muscle fibers, macrophages, calcium and necrotic tissue into the internal layer of the artery walls. The disease may evolve to form atherosclerotic plaque, which can cause partial occlusion of the lumen (stenosis) and reduction of the blood flow. Rupture of atherosclerotic plaque is a common cause of acute myocardial infarction and unstable angina. The typical atherosclerotic plaque is made of plaque core and underlying arterial wall, including media and adventitia. In advanced lesions, the plaque core may contain a lipid pool and calcifications. In the latter case, the lipid pool is separated from the lumen by a fibrous cap of connective tissue. If no lipid pool is present, the cap denotes the luminal part of the plaque. The most crucial evolution of atherosclerosis is thrombosis, which may occur via two mechanisms. The first is characterised by endothelial erosion; in this case a thrombus separates from the surface of the plaque. The second is characterised by disruption of the cap of a lipid-rich plaque (intra-plaque thrombosis). The risk of plaque rupture is related to the vulnerability of individual plaques (intrinsic disease) and to rupture triggers (extrinsic dynamic forces imposed on the plaque). Plaque ruptures manifest at the locations where the stresses resulting from biomechanical and haemodynamic forces exceed the intrinsic strength of the material, in general at the shoulders of atherosclerotic plaque

*Corresponding author. Email: [email protected] ISSN 1025-5842 print/ISSN 1476-8259 online q 2008 Taylor & Francis DOI: 10.1080/10255840701771743 http://www.informaworld.com

(Lendon et al. 1991; Loree et al. 1992). In vitro tests showed that human atherosclerotic materials generally fracture under stresses exceeding 300 kPa; therefore, the peak stress is often used as a predictor of the location of plaque rupture. In presence of local inhomogeneities, though, it has been observed that rupture may not occur at the highest stress locations (Cheng et al. 1993). The mechanism of plaque rupture is controversial. Various factors have been proposed as the initial cause of plaque rupture, including: hemodynamic shear stresses (Gertz and Roberts 1990), turbulent flow (Loree et al. 1991), transient collapse of the stenotic lesions (Binns and Ku 1989), mechanical shear stresses (Vito et al. 1990), rupture of the vasa vasorum (vessels of the blood vessel wall; Barger et al. 1991) and the concentration of tensile stresses in the wall of the lesion. Other mechanisms have been reported, such as plaque erosion – abrasion of the endothelium with no plaque rupture or less frequently, calcified nodules – lesions with fibrous cap disruption and thrombi – protruding into the artery lumen (Virmani et al. 2000). In recent years, a number of noninvasive imaging methods have been developed and used to study vascular diseases, such as atherosclerosis, and to drive therapies. A typical image acquisition method is the magnetic resonance imaging (MRI). MRI characterises the lesions in detail in terms of size, shape and even plaque tissue components (lipid core, fibrosis, calcifications and thrombosis deposits; Frank 2001; Fayad and Fuster


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2001; Larose et al. 2005). MRI can differentiate tissue content within atheroma on the basis of the signal intensities and morphological appearance of the plaque. Besides its intrinsic importance as a diagnostic tool, the interest on MRI techniques is growing from the side of computational mechanics. The numerical modelling of healthy and diseased human arteries can be used to predict the behaviour of the tissues under mechanical action. In recent years, several finite element analyses of the biomechanical behaviour of arteries were presented in the literature, see e.g. (Auricchio et al. 2001; Migliavacca et al. 2002; Petrini et al. 2004; Lally et al. 2005). Different material models have been used to describe the mechanical behaviour of healthy arteries. The complexity of the proposed models varies from approach to approach. As remarkable examples, the twofiber reinforced hyperelastic model by Holzapfel et al. (2000) and its recent extension to fiber dispersion (Gasser et al. 2006), have been developed expressly for the artery walls; subsequently they have extended to include inelastic behaviours (Gasser and Holzapfel 2007). The two-fiber model has been used to simulate balloon angioplasty intervention (Holzapfel et al. 2002) and, in combination with X-FEM (extended finite element, see Moe¨s et al. (1999)), balloon induced fissuring and intralayer dissection processes (Gasser and Holzapfel 2006, 2007). Fracture models based on cohesive theories have been proved to be extremely efficient to model failure in brittle materials (Ruiz et al. 2000, 2001) and in transversely isotropic polymeric materials (Yu et al. 2002). A few applications can be found in biological materials (Gasser and Holzapfel 2006, 2007). The inelastic behaviour of the artery wall can be described by alternative models based on damage theories (Gasser and Holzapfel 2002; Balzani et al. 2006) or on plastic theories (Gasser et al. 2006). More sophisticated approaches combine rupture of plaque with tissue growth, to model in-stent restenosis (Zohdi et al. 2004; Kuhl et al. 2007). Possibly due to the intrinsic difficulties in the characterisation of the material properties, the mechanical behaviour of atherosclerotic arteries still represents a challenging research topic. This work aims to define a 3D numerical model of fissuring atherosclerotic plaques by using cohesive theories of fracture. Our work is intended as a first step towards a patient specific numerical tool, to be used for the simulation of mechanical induced plaque ruptures, as the ones resulting from balloon angioplasty. As qualifying feature, the model here proposed is based on a realistic geometry obtained from a manual segmentation of MRI images, and on the use of anisotropic cohesive elements for modelling the fracture surfaces.

2. Materials and methods 2.1 Artery wall Arterial walls are composite structures containing elastin, collagen, cells (endothelial cell, smooth muscle cell, fibroblast) and ground matrix. The isotropic ground matrix contains several sets of collagen fibers that straighten as the pressure increases, preventing the overstretching of the artery. At microscopic level the arterial wall appears as a layered structure, composed of three concentric zones, intima, media, and adventitia, separated by elastic membranes. The reinforcing fibers are parallel to the middle plane of each layer. Histological evidence (Shekhonin et al. 1985) shows a large dispersion of collagen fibers in the intima and adventitia. Contrariwise, in the media the orientation of the fibers is better defined: most of the fibers are inclined of a small angle with respect to the circumferential direction and present a very little dispersion. The intima is made of a single layer of endothelial cells embedded in extracellular matrix and an underlying thin basal lamina. The intima is thin, but with aging and atherosclerosis it becomes thicker and stiffer. The media is the middle and thickest layer of the artery, made of collagen arranged in repetitive lamellar units, forming concentric medial layers. The adventitia is the outermost layer of the artery and is surrounded continuously by loose connective tissue, which often provides additional structural support.

2.1.1 Experimental evidence Experimental tests are fundamental to understand the arterial properties and to collect the parameters for the constitutive equations. A variety of methods have been used to determine the mechanical properties of blood vessels, including classical tensile tests, inflation tests, torsion tests, and pressure –diameter tests of tubular specimens. Pressure – diameter tests are preferable to tensile tests, because the tubular shape of blood vessels is preserved. In other tests the vessel wall is dissected – causing unpredictable alteration of the vessel mechanical properties – and each layer is tested separately. Under physiological conditions, arteries are regarded as nearly incompressible solids. Healthy arteries behave as highly deformable composite structures and exhibit a nonlinear stress – strain response with a typical stiffening at the physiological strain level. The anisotropic properties of the composite structures have been evidenced in several experiments conducted in canine and porcine arteries. Anisotropy is related to the function and the location of the artery. In particular, the arterial wall exhibits anisotropic behaviour with different elastic constants in the radial, circumferential and axial directions (Patel et al. 1973).


Computer Methods in Biomechanics and Biomedical Engineering Additionally, the material properties are quite different in the three layers; for example, experimental tests presented in Xie et al. (1995) show that the elastic properties of the media and adventitia may be significantly different. Regrettably, scarce experimental data on the deformation of human artery are available. Uniaxial tensile stress – strain tests on human coronary arterial tissue were published by Yamada (1970). Inflation experiments on intact human and porcine arteries, leading to relationship between lumen pressure and circumferential strain, were presented in separate works by Carmines et al. (1992) and van Andel et al. (2003). Schulze-Bauer et al. (2003) investigated the performance of various anisotropic material models in the simulation of the expansion of an aged iliac human artery. An important phenomenon typical of arteries is the presence of a residual stress. When an arterial ring is cut, it opens assuming the shape of an open sector under a spring effect. In general, the cut open sector is not stress free, since the natural opening angles of the separate layers are different. Residual stresses influence the behaviour under loading and the stress and strain distributions through the arterial wall. Additional effects that should be accounted for in the description of the arterial behaviour are the growth (Driessen et al. 2004; Gleason and Humphrey 2005; Rodriguez et al. 2007) and remodelling (Pasterkamp et al. 1998; Smits et al. 1999; Schoenhagen et al. 2000) of the tissue. The features of the material model are nonlinear elasticity, characterised by a stiffening of the load – displacement curve due to the activation of reinforcing constituents, near-incompressibility, and marked anisotropy associated to a two-fiber reinforcement. For sake of simplicity, in our numerical applications the material is assumed to be stress-free and no tissue growth is considered.

2.1.2 Artery material model According to the histological evidence, (see Section 2.1.1), in the media layer it is possible to identify two main sets of fibers inclined of a constant angle g. The resulting material structure is thus orthotropic. As an assumption, the orthotropic structure of the material is extended to the intima and the media, considering a different angle g for each layer. The material model used here is the one proposed by Holzapfel et al. (2000). The collagenous fibers are characterised by two unit vectors a and g, giving their direction in the reference configuration, see Figure 1. The hyperlastic model is defined by a strain –energy function C, C ¼ CðF; A; GÞ;

A ¼ aâ; G ¼ g^g;

ð1Þ

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Figure 1. Schematic representation of the fibers orientation assumed for each arterial layers. From outside to inside: adventitia, media and intima.

dependent on the deformation gradient F and on A and G, structural tensors that account for the orientation of the reinforcing fibers. To separate the dilatational and the distortional parts of the deformation, the deformation gradient F is decomposed into the product F ¼ ðJ 1=3 IÞF, 1/3 where J I represents the volumetric deformation and F the isochoric deformation gradient. The volume ratio J is given by det F . 1. The multiplicative decomposition of the deformation gradient corresponds to the decomposition of the strain energy function in additive form, i.e. iso ðF; A; GÞ. Assuming full decoupling C ¼ Cvol ðJÞ þ C iso in turn between isotropic and anisotropic behaviour, C i decomposes additionally into an isotropic potential C for the non-collagenous ground matrix and the a for the two transversely isotropic potentials C embedded families of collagen fibres as: i ðFÞ a ðF; þC A; GÞ: C ¼ Cvol ðJÞ þ C

ð2Þ

According to Spencer (1982), material frame A and indifference requires that the dependence on F, ¼ G to be expressed through three true invariants of C F T F ¼ J 22=3 C and six pseudo-invariants related to the fibers. A simplified form of the strain – energy function, that discards the four higher order invariants, reads: a ðI4 ; I6 Þ; i ðI1 ; I2 Þ þ C C ¼ Cvol ðJÞ þ C

ð3Þ

where I1 ¼ trC;

1 2 2 ; I2 ¼ ðtrCÞ 2 trðCÞ 2

ð4Þ

are the first and second invariant of the modified right and Cauchy – Green tensor C, I4 ¼ a·Ca

I6 ¼ g·Cg;

ð5Þ

556


are the two pseudo-invariants measuring the square of the stretch in the direction of the fibers. The expression of the volumetric contribution assumed here is, according to Holzapfel et al. (2000) and Holzapfel and Gasser (2001): Cvol ðJÞ ¼

K log2 J; 2

ð6Þ

where K is the bulk modulus of the material. The isotropic ground substance matrix is described through a neo-Hookean model: i ðI1 Þ ¼ m ðI1 2 3Þ; C 2

ð7Þ


where m is the shear modulus of the matrix. The reinforcing behaviour of each fiber family is represented through an exponential function, which rules out compressive loads (I4 . 1 and I6 . 1): 4 ðI4 Þ þ C 6 ðI6 Þ; a ðI4 ; I6 Þ ¼ C C

ð8Þ

with n ¼ Kn C 2k

(

exp½kðIn 2 1Þ2 2 1

0

2.2.1

Anisotropic resistance

When the two sets of reinforcing fibers in the directions a and g, forming the angle 2g, are equivalent in stiffness and resistance, the material exhibits an orthotropic structure. The three principal material directions in the reference configuration are given by the two bisectors of the angles formed by the fibers G1 (bisector of the minimum angle) and G2 (bisector of the maximum angle) and the normal G3 to the plane of the fibers (see Figure 2(a)):

for In . 1 for In # 1

ð9Þ

n ¼ 4; 6; where k is a dimensionless constants and K4 ¼ K6 are the stiffness moduli related to the collagen fiber sets. The specific form of the proposed constitutive Equation (8) requires the definition of four material parameters, i.e. K, m, k, and K4, whose interpretations can be partially based on the underlying histological structure. A transversely isotropic model, in the following applications used to describe the plaque behaviour, is directly derived for the previous orthotropic material by setting the stiffness of the second set of fibers to zero, i.e. K6 ¼ 0.

2.2

of cracks in unidirectional composites by Yu et al. (2002), by means of a simple extension of the insertion criterion for isotropic material described in Pandolfi and Ortiz (1998). Here, we use the cohesive models proposed in Ortiz and Pandolfi (1999) extended to account for the presence of a double fiber reinforcement in the bulk. While the anisotropy of the bulk has been included in the material description through structural pseudo-invariants, the characterisation of anisotropy along the interfaces requires the identification of the planes of local material symmetry.

Interface and fracture surfaces

In the framework of finite element discretization, crack nucleation and propagation are conveniently modelled through the cohesive approach (Dugdale 1960; Barenblatt 1962). Cohesive models have been used recently for the analysis of fracture in biological tissues by Gasser and Holzapfel (2003) and in simulation of delamination by Wells et al. (2002), in the context of the extended finite element method (X-FEM). By way of contrast, here we adopt the approach proposed by Ortiz and Pandolfi (1999), combined with the automatic fragmentation procedure described in Pandolfi and Ortiz (2002). Large deformation anisotropic cohesive models have been used for the analysis of dynamic propagation

G1 ¼

aþg g2a G1 £ G2 ; G2 ¼ ; G3 ¼ : ð10Þ ja þ gj jg 2 aj jG 1 £ G 2 j

The GI define a set of orthonormal basis vectors; we introduce the cartesian coordinates ZI. While the anisotropy of the stiffness is intrinsically built in the adopted hyperelastic behaviour, the variation of the tensile resistance with the material direction requires to be characterised. We extend the anisotropic resistance model used in Yu et al. (2002), and introduce an ellipsoidal resistance surface, see also (Boone et al. 1987). The equation of the resistance surface referred to the principal axes of anisotropy is given by: Z 21 Z2 Z2 þ 22 þ 23 2 1 ¼ 0: 2 sc1 sc2 sc3

ð11Þ

The resistance surface is visualised in Figure 2(b). We assume that each principal material direction is characterised by a different tensile resistance, i.e. sc1 $ sc2 $ sc3, and that the corresponding critical energy release rates scale proportionally as Gc1 $ Gc2 $ Gc3. The cohesive resistance sc(N) and the critical energy release rate Gc(N) in a specific direction N are then derived as follows. The reference configuration is characterised by the orthonormal basis vectors EI and coordinates XI. In general, the principal anisotropy basis vectors GI are rotated by a proper rotation tensor R A, of components RAIJ ¼ cosðE I ; G J Þ ¼ E I ·G J . Thus, the following relations hold between the coordinates of the two

Computer Methods in Biomechanics and Biomedical Engineering

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Figure 3. Linearly decreasing cohesive laws considered in the present application. (a) Anisotropic cohesive laws expressed in terms of an effective opening displacement D and traction T. The area enclosed by the loading envelopes starting from sci measures the critical energy release rate Gci. Cohesive strengths and critical energy release rates scale proportionally. The characteristic opening displacement Dc does not change with the anisotropy direction. (b) Irreversible behaviour, showing unloading to the origin.

Likewise, we assume that the critical energy release rate Gc(N) is related to the three principal values: Gc ðNÞ ¼

Figure 2. Definition of the principal anisotropy axes Gi in a two-fiber reinforced material: the two sets of fibers are characterised by equal stiffness and resistance. The minimum resistance is in the direction normal to the fiber plane. (a) Two principal anisotropy axes in the fiber plane are given by the two bisectors of the angles formed by the fibers. The third (weaker) axis is orthogonal to the fiber plane, i.e. G3. (b) Associated resistance surface in the reference configuration.

21=2

Z 21 =s2c1 þ Z 22 =s2c2 þ Z 23 =s2c3 2 1 ¼ 0 M1Z 1 þ M2Z 2 þ M3Z 3 ¼ 0

ZI ¼

RAJI X J ;

XI ¼

RAIJ Z J :

Z J ¼ sM J ¼ sRAIJ N I :

:

ð12Þ

A vector D ¼ sN ¼ D1 E 1 þ D2 E 2 þ D3 E 3 , with jNj ¼ 1, is described in the principal anisotropy system through the components ZJ ¼ sMJ: DI ¼ sN I ¼ sRAIJ M J ;

ð15Þ

;

leading to an anisotropic cohesive law characterised by a direction dependent scaling along the traction axis (see Figure 3(a)). Under a finite element discretization of a solid body, the direction N of interest identifies the normal to a interelemental surface in the reference configuration. The principal directions of anisotropy in the fracture plane are given by the axes of the ellipsis resulting from the intersection of the resistance ellipsoid with the plane of normal N. The ellipsis equation may be obtained by solving the system (

systems:

M 21 M 22 M 23 þ þ G2c1 G2c2 G2c3

ð13Þ

The resistance of the material in the direction N is the solution of Equation (11) combined with Equation (13)2 with respect to s: 2 21=2 M M2 M2 sc ðNÞ ¼ 21 þ 22 þ 23 : ð14Þ sc1 sc2 sc3

The resistance ellipsis on the fracture plane is more conveniently written by performing a further basis change. Let R N be the rotation matrix from the principal anisotropy system to the oriented surface (see Appendix 1) and Z I ¼ RNIJ Z~ J be the corresponding coordinate change. The in-plane anisotropy ellipsis equation is thus obtained by the solution of the system: 8 > ~2 ~2 ~2 ~ ~ ~ ~ > < a11 Z1 þ a22 Z2 þ a33 Z3 þ 2a12 Z1 Z2 þ 2a23 Z2 Z3 ~ ~ þ2a31 Z3 Z1 2 1 ¼ 0 ð16Þ : > > ~ : Z3 ¼ 0 which leads to the equation of an ellipsis with the origin of the reference system in its centre:

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A. Ferrara and A. Pandolfi 2

2

a11 Z~ 1 þ a22 Z~ 2 þ 2a12 Z~ 1 Z~ 2 ¼ 0:

ð17Þ

The canonic form of the anisotropy ellipse is finally recovered by introducing the in-plane rotation matrix R U (see Appendix 1) about the ellipsis centre, with corresponding coordinate change Z~ I ¼ RU IJ U J : U 21 U 22 þ 2 1 ¼ 0: ðscM 1 Þ2 ðscM 2 Þ2

ð18Þ

The coefficients scM1 and scM2 in (18) define the maximum and minimum resistance on the fracture plane in the reference configuration. The orientations M1 and M2 of the principal axes of resistance on the fracture surface are then recovered in terms of reference coordinates XI as


M1 ¼ RARN RU I1;

M2 ¼ RARN RU I2;

ð19Þ

where I1 ¼ {1,0,0} and I2 ¼ {0,1,0}. Once the finite element discretization of the solid is performed, each inter-element surface is equipped with its anisotropy parameters M1, M2, scM1, and scM2. With reference to alternative approaches to explicit modelling of fracture based on the concept of partition of unit (X-FEM; (Gasser and Holzapfel 2003, 2005)), the present numerical approach has an intrinsic drawback. The failure criterion can be satisfied at the finite element interfaces, predefined by the discretization. Such surfaces may not coincide with the surfaces where the maximum stress is reached and hence, the model may predict a stiffer structural response. A partial remedy to this issue is the adoption of a fine discretization.

Alternative formulations of transversely isotropic cohesive laws, suitable for X-FEM formulation, where the cohesive surfaces are not constrained to follow the solid element boundaries, can be found in Gasser and Holzapfel (2003, 2005). In the reference configuration, a cohesive interface S0 is defined by two coincident surfaces S^ 0 , which separate the body into two parts B^ , see Figure 4(a). At each point 0 P of the undeformed cohesive surface S0, we identify two principal directions of anisotropy M1 and M2, given in Equation (19), consistent with the presence of two sets of equivalent fibers (see Figure 4(a)). With the normal N ¼ M3 to the undeformed surface, M1 and M2 define a set of orthonormal reference basis vectors. Under a deformation mapping w applied to the ^ continuum, with w ^ on S^ 0 , the two cohesive surfaces S may separate (see Figure 4(b)). To univocally identify the deformed cohesive interface S, one possible choice is to refer to the mean value w: 1 w ¼ ðw þ þ w 2 Þ: 2

Assuming that the cohesive surface S remains smooth, a unique unit normal n pointing from w 2 to w þ is well-defined everywhere. The normal n is computed from the spatial (deformed) principal anisotropy directions m1 and m2. Under the deformation mapping w, the two vectors M1 and M2 in Equation (19) transform into m1 and m2, respectively: m1 ¼

2.2.2

Anisotropic cohesive law

In cohesive theories, the opening displacements D across a cohesive surface play the role of a deformation measure, while the tractions T furnish the workconjugate stress measure. Supported by experimental evidence, the cohesive behaviour, for both isotropic and anisotropic materials, is different for opening (Mode I separation) and sliding (Mode II and III separation). Additionally, an anisotropic cohesive model must be characterised by a law with material properties (both resistance and toughness) dependent on the direction of sliding on the cohesive surface. An accurate theory of isotropic cohesive surfaces in solids was developed in Ortiz and Pandolfi (1999). In the present application, we adopt an anisotropic extension of the model by Ortiz and Pandolfi. The model has been developed extensively in a parallel work (Ferrara and Pandolfi); here, we recall only the main features that apply to orthotropic materials. We assume that the behaviour of cohesive surfaces is local, and confine our attention to one point on the cohesive surface.

ð20Þ

n¼

7 S0 wM 7 S0 wM 1 2 ; m2 ¼ ; j7 S0 wM j7 S0 wM 1j 2j m1 £ m2 ; jm 1 £ m 2 j

ð21Þ

Figure 4. (a) Undeformed configuration S0 a cohesive surface traversing a body. Definition of the reference anisotropy basis vectors N, M1 and M2. (b) Deformed configuration S. Definition of the current anisotropy basis vectors N, m1 and m2, cohesive tractions T and opening displacements D (at the point P).

Computer Methods in Biomechanics and Biomedical Engineering where the components of the surface deformation gradient 7 S0 w are the covariant derivatives of the components of w. For a general set of fibers, the deformed anisotropy directions do not preserve reciprocal orthogonality and the three unit vectors m1, m2 and n define general basis vectors. In the case here considering equivalent sets of fibers, the orthonormality of the three vectors upon deformation is preserved. The description of the more general case will be presented in Ferrara and Pandolfi. In order to derive the cohesive law, we follow (Ortiz and Pandolfi 1999) and postulate the existence of a free energy density F per unit undeformed area, that acts as a potential for the displacement jump, so that the cohesive tractions are computed as:


T¼

›F : ›D

ð22Þ

The simplified form of free energy here considered exclude dependence on thermal processes and on dilatation and distortion of the mean cohesive surface. The material frame indifference requirement imposes that F is dependent on the individual components of the opening displacement on the deformed anisotropy system: F ¼ FðD1 ; D2 ; Dn ; qÞ;

ð23Þ

where D1 ¼ D·m 1 ;

D2 ¼ D·m 2 ; Dn ¼ D·n;

D ¼ D1 m 1 þ D2 m 2 þ Dn n;

ð24Þ

and the scalar variable q is introduced to account for irreversibility. The cohesive law (22) becomes T¼

›F ›F ›F m1 þ m2 þ n ›D1 ›D 2 ›D n

¼ T 1 m 1 þ T 2 m 2 þ T n n:

energy density F depends on D only through the EOD F ¼ FðD; qÞ;

T¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi b 2 D21 þ a 2 D22 þ D2n :

T¼

ð28Þ

T 2 b D1 m 1 þ a 2 D2 m 2 þ Dn n : D

ð29Þ

A simple calculation (see Appendix 2) gives the identity T¼

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 22 T 21 þ a 22 T 22 þ T 2n :

ð30Þ

Relation (30) shows that b defines the ratio between the maximum shear and the normal critical tractions, while a defines the ratio between the minimum and maximum inplane shear resistance of the cohesive surface. With reference to Equation (18), the material parameters b and a are obtained as:

b2 ¼

scM 1 ; sc ðNÞ

a2 ¼

scM2 : scM1

ð31Þ

Note that the cohesive relations account only for positive normal opening displacements; contact and friction are considered as independent phenomena to be modelled outside the cohesive law, for example as discussed in Pandolfi et al. (2002). The cohesive law is rendered irreversible by assuming unloading to the origin (see Figure 3(b)). Since irreversibility manifests itself upon unloading, the appropriate choice of the internal variable q is the maximum attained effective opening displacement. The corresponding kinetic equation is: (

D_

_ $0 if D ¼ q and D

0

otherwise

;

ð32Þ

where the first case corresponds to loading and the second to unloading. For purposes of display of results, we define a monotonically growing damage parameter D:

ð26Þ

The parameter b assigns different weights to the sliding and normal opening displacements, while the parameter a distinguishes between the shear behaviour in the two directions m1 and m2. We assume that the free

›F ðD; qÞ: ›D

Under these conditions, the cohesive law (25) reduces to a scalar expression

q_ ¼

D¼

ð27Þ

so that it delivers an effective cohesive traction:

ð25Þ

A further simplification, which holds only for positive normal opening displacements, follows by the introduction of an effective scalar opening displacement (EOD); Camacho and Ortiz (1996); Ortiz and Pandolfi (1999).

559

D¼

FðqÞ ; Gc1

_ . 0: D

ð33Þ

The limits of variability of D are 0 and 1; they correspond to an intact and a fully decohered cohesive surface, respectively.

560


2.2.3 Insertion criterion In order to decide about the insertion of a cohesive surface, at the end of each loading step, on each interelement surface, the effective traction T is computed and compared with the normal fracture resistance sc(N) of the surface: T¼

ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 22 T 21 þ a 22 T 22 þ T 2n # sc ðNÞ:

ð34Þ

summarised in Table 1. The bulk modulus K adopted for all the layers was estimated from a Young’s modulus of 100 kPa (Loree et al. 1992; Cheng et al. 1993) and a Poisson’s ratio of 0.499. Finally, for the lipid pool, we assumed a neo-Hookean behaviour and used average elastic data from Loree et al. (1994). The cohesive strengths used in the numerical analyses are listed in Table 2. The labels sc1, sc2 and sc3 refer to the


If the criterion is violated, the topology of the mesh is updated with the insertion of a new surface that explicitly reproduce the tissue separation.

2.3 Finite element model A 3D model of a stenotic artery, reconstructed by manual segmentation of in vitro high MRI, has been used in our finite element simulations. We started from a set of contiguous MR images published in the work of Yang et al. (2003). The images reproduced four transversal cross sections of a human arterial specimen, at 1.2 mm distance one from the other. According to the chromatism of the images, on each cross-section we traced four borders: (i) the outer border, (ii) the inner border, (iii) the interface between diseased intima and media and (iv) the interface between media and adventitia. Subsequently, we reconstructed the interface separating two arterial layers between two adjacent sections as ruled surface. The final solid model was generated by an automatic solid modeler starting from the boundary surfaces. The geometry reproduces an atherosclerotic lesion with a 40% stenosis, characterised by an eccentric fibrous plaque. In our analyses, the anatomy of the model is taken as baseline geometry, and referred to as Model 1. We analysed two alternative geometries, representative of two possible anatomic diversities: (i) the stenosis severity is increased up to 80%, Model 2; (ii) an extracellular lipid pool is included in the plaque core, Model 3. The tree geometries are illustrated in Figure 5. The models are discretized in 10-node tetrahedral elements, preserving the separation between the layers. Figure 6 shows the finite element mesh of the Model 3. The mesh consists of 8130 nodes and 4705 elements (1673 for the adventitia, 1134 for the media, 1547 for the plaque core, and 351 for the lipid pool). No interface elements are present in the original meshes. Most of the material properties used in the calculations have been recovered from the literature. According to Holzapfel et al. (2000), the mean value of fiber angle g is assumed to be 498 for the adventitia, 78 for the media and 08 for both healthy and diseased intima. The material constants for the two-fiber reinforced materials have been calibrated by fitting experimental data presented by Holzapfel et al. (2004), and are

Figure 5. 3D views of the three geometrical models of stenotic human arteries. (a) Baseline geometry of atherosclerotic plaque with 40% of stenosis, Model 1. (b) Atherosclerotic plaque with 80% of stenosis, Model 2. (c) Diseased artery consisting of the adventitia, media, and the deposit composed of the plaque and a lipid pool, with 40% of stenosis, Model 3. Available in colour online.



Figure 6. Finite element discretization for Model 3: (a) assembled mesh; (b) individual components: adventitia (A), media (M), diseased intima (I þ P), and lipid pool (Lp). Available in colour online.

circumferential, longitudinal and radial strengths, respectively. The values of sc1 and sc2 are chosen from the experimental data reported in Holzapfel et al. (2004). For the radial strength, we select the average value of 62 kPa reported by MacLean et al. (1999). In consideration of the paucity of experimental data, in the simulations we used the same fracture energy Gc for all the layers. The value of Gc ¼ 1.4 kJ m22, taken from Purslow (1983), corresponds to a tearing test, and was estimated as the ratio between the total work done to break the tissue and the nominal area of the crack. For sake of comparison, in Table 2 the alternative value Gc ¼ 0.159 kJ m22 provided by Carson and Roach (1990) is reported. The low value of Carson’s fracture energy corresponds to a dissection test along the fibers of the

Table 1. Material constants of the arterial layers (adventitia, media and healthy and diseased intima). Material constants

K (kPa)

m (kPa)

K4 (kPa)

k

g (deg)

Adventitia Media Diseased intima Lipid pool

1667 1667 1667 1.15

3.97 10.77 4.59 0.12

37.71 4.83 25.22 –

35.74 4.71 10.14 –

49 7 0 –

Experimental data from Holzapfel et al. (2004).

561

medial layer. For the interfaces between the artery layers we assumed the cohesive properties of the media layer. The finite element model is constrained to exclude rigid body motions. The two basis are constrained in the axial direction. The bottom nodes are fixed, while the top nodes may undergo an imposed displacement to provide the in situ axial prestretch observed in the experiments, i.e. l ¼ 1.2 for healthy arteries (Schulze-Bauer et al. 2003). The confinement offered by the surrounding tissues is simulated by applying linear elastic constraints all around the model. The stiffness of the elastic constraints is equivalent to an isotropic material with Young’s modulus E ¼ 1 kPa (Veress et al. 2002). The finite element code in finite deformations was explicitly developed for the analysis of nonlinear elastic biological tissues showing anisotropic behaviour. The numerical procedure is able to recover the unloaded configuration, i.e. the geometry of the vessel without blood pressure. It is evident that MR images refer to the deformed geometry of the artery, thus a realistic simulation should account correctly for the initial configuration. For sake of simplicity, in the simulations of fracture propagation presented here, we disregard the recovery of the unstressed configuration, as well as the presence of the circumferential residual stresses observed in the experiments (Schulze-Bauer et al. 2003).

3.

Numerical results

We perform quasi-static analyses by applying a growing uniform pressure on the internal wall of the vessel, using small load increments, from 0 to 260 mmHg, i.e. twice the maximum average physiological blood pressure. In the following we present the results by means of contour levels of the von Mises stress, defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsI 2 sII Þ2 þ ðsII 2 sIII Þ2 þ ðsIII 2 sI Þ2 ; sM ¼ 2 ð35Þ where sI, sII and sIII are the eigenvalues of the Cauchy stress tensor. As reference for the subsequent numerical results, Figure 7 shows the contour levels of the von Mises stress at 100 mmHg blood pressure (physiological value) for the Model 1. The stress is rather uniform, and the maximum stress is < 173 kPa. Similar results are obtained for the Model 3, while in the Model 2 the stresses are about an order of magnitude smaller. We conducted a first set of analyses to evaluate the influence of the geometry on the peak stress and on the location of the first fracture event. For all the models, we assumed high cohesive strength for the diseased intima (Table 2, Diseased intima 1). The luminal pressure

562

A. Ferrara and A. Pandolfi Table 2.

Cohesive strength and fracture energy used in the numerical simulations.

Material constants

sc1 (kPa)a

sc2 (kPa)a

sc3 (kPa)b

Gc (kJ m22)c

Gc (kJ m22)d

Adventitia Media Diseased intima 1 Diseased intima 2

1031.6 202.0 776.8 254.8

951.8 188.8 277.5 468.6

62 62 62 62

1.4 1.4 1.4 1.4

0.16 0.16 0.16 0.16

a

Holzapfel and Sommer (2004). MacLean et al. (1999). c Purslow (1983). d Carson and Roach (1990).


b

is increased up to the formation of a first crack in the arterial wall. In the three models, the first crack develops at different luminal pressures and at different values of the peak stress (see Table 3). Figure 8 compares the contour levels of the von Mises stress for the three geometries at the first fracture event (see Table 3). Figure 9(a) shows the final crack patterns for the Model 1 at the end of the numerical analysis, i.e. 260 mmHg blood pressure. Fractures originate from the shoulders of the diseased plaque, where the maximum stress is registered. Then, cracks propagate in the radial direction across the intima, exposing the media to the lumen. Lateral side cracks are spread around along longitudinal channels. At highest pressure values, cracks may occur also in the healthy layers of arterial wall. As a consequence of the crack, the section of the artery wall narrows locally. The whole model shows a reduced structural stiffness, and concentrations of strains are observed in correspondence of the thinned walls. Similar crack patterns are obtained with Model 3, while in Model 2 an extended rupture is observed only at 380 mmHg blood pressure (see Figure 9(b)). To evaluate the influence of the cohesive strength on the fracture propagation, we performed an additional analysis using the Model 1. We modified the values of the cohesive circumferential and longitudinal strengths for the intima (Table 2, Diseased intima 2). In particular, while in the original analysis the circumferential strength was higher than the longitudinal one, in the second analysis the circumferential strength was assumed to be lower than the longitudinal one. The crack patterns testify different fracture mechanisms in the two cases (see Figure 10): cracks develop diagonally when the circumferential fibers are stronger, and develop in the longitudinal direction when the circumferential fibers are weaker. The evolution of the internal shape of the artery with increasing blood pressure is visualised in Figure 11. The history of the corresponding deformations along the two main directions of the elliptic lumen are plotted in Figure 12. Note that at low blood pressures the artery deforms in both main directions preserving the original elliptic shape, while, at higher pressures, the section

assumes a more circular shape. Upon fracture, the mechanical integrity of the vessel is broken and the circular shape is lost, as shown by the oscillations of the deformation measures in Figure 12. The only material constants arbitrarily chosen in our analysis are the bulk modulus for all the layers. To analyse the influence of such material constants on the initiation of the fracture, we performed additional parametric analyses, but no relevant changes were observed in the peak stress nor in the crack patterns. Also, the reduction of the cohesive energy from the high value 1.4 kJ m22 to the low value 0.16 kJ m22 does not affect remarkably the pattern of the cracks.

4.

Discussion

We set up a numerical model of atherosclerotic artery, where most of the mechanical features observed experimentally are considered. In particular, we account for the longitudinal stretch and apply deformable boundary conditions to simulate the presence of surrounding tissues. The finite element model includes fracture and fragmentation procedures and uses cohesive models to describe the fracture surfaces. We performed a set of finite element analyses to evaluate the stress distribution in different geometries and to predict the locations of plaque rupture and the mechanisms of fracture propagation in diseased tissues. According to Richardson et al. (1989), stress concentrations usually occur near the junction of the atherosclerotic plaque and healthy arterial wall, or are located close to the soft lipid pool, which is unable to bear significant stresses. Stress concentrations are Table 3. Effects of the geometrical variable (stenosis and presence of lipid pool) on numerical simulations.

Model 1 Model 2 Model 3

P (mmHg)

P (kPa)

smax (kPa)

smax/P

114 263 111

15.2 35.1 14.8

223 215 303

14.7 6.1 20.4

Numerical results in terms of the luminal pressure P and peak stress smax.


563


Figure 7. Reference geometry with a 40% stenosis and no lipid pool (Model 1). von Mises stress map at physiological pressure 100 mmHg. The maximum stress is < 173 kPa.

Figure 9. Evolution of crack patterns and von Mises stress map in MPa, at 260 mmHg blood pressure for Model 1.

Figure 8. von Mises stress contour levels in MPa, at first fracture event for the three geometrical models. (a) Model 1, luminal pressure 114 mmHg, peak stress 223 kPa. (b) Model 2, luminal pressure 263 mmHg, peak stress 215 kPa. (c) Model 3, luminal pressure 111 mmHg, peak stress 303 kPa.

Figure 10. von Mises stress contour levels in MPa and crack patterns for different cohesive strength of the intima at 170 mmHg blood pressure. (a) sc1 ¼ 776.8 kPa, sc2 ¼ 277.5 kPa. (b) sc1 ¼ 254.8 kPa, sc2 ¼ 468.6 kPa.


564

Figure 11.


Deformed meshes during the overpressure expansion. (a) 100 mmHg, (b) 120 mmHg, (c) 180 mmHg, (d) 260 mmHg.

Figure 12. Time history of the deformation of the artery lumen along the main ellipsis axes A– A and B – B.

critically dependent on the geometry of the fibrous cap, on the extension of lipid pool, and on the material properties of the lesion components. Contrariwise, stress concentrations do not depend remarkably on the percentage of occlusion (Loree et al. 1991). Stress analysis on the three geometries show a good agreement with the experimental observations; our finite element simulations revealed that the stress reaches higher values at the shoulders of the lesion, i.e. in the zones where the diseased intima merges with the healthy tissue. Additionally, the maximum values of the stress may differ considerably in the three geometries. In our study, we also investigated the influence of the material constants on the stress distribution and on the location of the peak stress. In particular, we analyzed the effects of the variation of the material properties of the diseased intima and of the lipid pool. No appreciable variations on the stress distributions were observed.


Computer Methods in Biomechanics and Biomedical Engineering Thus, according to our finite element results, we may conclude that the distribution of the stress depends strongly on the geometry and is less influenced by the material properties of the plaque components. In vitro tests documented in the literature (Lendon et al. 1991; Cheng et al. 1993; Huang et al. 2001) report that human atherosclerotic walls generally fracture under stresses exceeding 300 kPa. For this reason, the peak stress is often used as a predictor of the location of rupture of atherosclerotic plaque (Pasterkamp and Falk 2000). Our numerical analyses confirmed the experimental observations (see Figure 9 and Table 3). All the ruptures obtained from the numerical tests originated from the internal intima, at the point where the maximum circumferential stress was registered, and developed in a longitudinal direction. To understand the reasons of such behaviour, we arbitrary modified the strength of the diseased intima assigning a higher resistance to the circumferential direction than to the longitudinal direction. The resulting cracks developed in a diagonal direction, always starting from the internal wall. We observed that eccentric lumens of atherosclerotic vessels may change their shape from elliptical to circular under growing blood pressure. An additional observation is that plaques with high level of stenosis (combined with little eccentricity of the lumen) do not reach high stresses and are less prone to rupture than thinner vessels. Cheng et al. (1993) reported the occurrence of plaques rupture in regions of secondary stress concentration. Possible explanations of such behaviours are that the atherosclerotic materials are heterogeneous, and the strength throughout the lesion is not uniform. Strength may also vary in time in dependence on the biologically dynamic environment. These issues have not been investigated in the present study, but our finite element model may be easily adapted to account for the presence of calcifications or inhomogeneities. A parametric study of the effects of inhomogeneities is under development. The results obtained during this research represent a first step towards the development of a patient-specific computer assisted tool that may help surgeons in the prediction of the mechanical evolution of artery lesions. The numerical instrument already possesses the ability to evaluate quantitatively the stress field under strong mechanical actions, as the ones induced by balloon angioplasty or by a periodic high pressure loading. Future extensions of the material model include rate dependency and inelasticity in the behaviour of the bulk, as well as inclusion of growth which may manifest under cyclic or repeated loads. In particular, progressive damage of an incipient crack due to fatigue can be accounted for by simple modification of the cohesive law developed herein, on the wake of existing models (Nguyen et al. 2001; Pandolfi and Ortiz 2003).

565

The proposed approach is potentially powerful, but certainly it calls for the support of experimental investigations, able to provide the necessary data (geometrical, material constant for elasticity and fracture) all together. The weakness of the results here presented are in fact that the mechanical and geometrical data are collected from different sources available in the literature, and therefore they must be considered only in qualitative terms.

Acknowledgements This research has been partially supported by the Italian MIURCofin2005 programme:‘Interfacial resistance and failure in materials and structural systems’.

References Auricchio F, Di Loreto M, Sacco E. 2001. Finite-element analysis of a stenotic revascularization through a stent insertion. Comput Methods Biomech Biomed Eng 4:249–264. Balzani D, Schro¨der J, Gross D. 2006. Simulation of discontinuous damage incorporating residual stresses in circumferentially overstretched atherosclerotic arteries. Acta Biomater 2(6): 609–618. Barenblatt GI. 1962. The mathematical theory of equilbrium of cracks in brittle fractures. Adv Appl Mech 7:55– 129. Barger AC, Beeuwkes R, Lainey LL, Silverman KJ. 1991. Hypothesis: vasa vasorum and neovascularization of human coronary arteries – a possible role in the pathophysiology of atherosclerosis. N Engl J Med 88:8154–8158. Binns RL, Ku DN. 1989. Effect of stenosis on wall motion: a possible mechanism of stroke and transient ischemic attack. Artherosclerosis 9:842–847. Boone TJ, Wawrzynek PA, Ingraffea AR. 1987. Finite element modelling of fracture propagation in orthotropic materials. Eng Fract Mech 26:185–201. Camacho GT, Ortiz M. 1996. Computational modelling of impact damage in brittle materials. Int J Solids Struct 33:2899– 2938. Carmines DV, McElhaney JH, Stack R. 1992. A piece-wise non-linear elastic stress expression of human and pig coronary arteries tested in vitro. J Biomech 24:899–906. Carson MW, Roach MR. 1990. The strength of the aorta media and its role in the propagation of aortic dissection. J Biomech 23(6): 579–588. Cheng GC, Loree HM, Kamm RD, Fishbein MC, Lee RT. 1993. Distribution of circumferential stress in ruptured and stable atherosclerotic lesions. A structural analysis with histopathological correlation. Circulation 87:1179–1187. Driessen NJB, Wilson W, Bouten CVC, Baaijens FPT. 2004. A computational model for collagen fibre remodelling in the arterial wall. J Theor Biol 226(1). Dugdale DS. 1960. Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–104. Fayad ZA, Fuster V. 2001. Clinical imaging of the high-risk or vulnerable atherosclerotic plaque. Circ Res 89:305–316. Ferrara A, Pandolfi A. Three-dimensional anisotropic cohesive models for the analyis of dissection and delamination in highly deformable tissues. In preparation. Frank H. 2001. Characterization of atherosclerotic plaque by magnetic resonance imaging. Am Heart J 141(2 Suppl):S45–S48. Gasser TC, Holzapfel GA. 2002. A rate-independent elastoplastic constitutive model for (biological) fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation. Comput Mech 29:340– 360.


566


Gasser TC, Holzapfel GA. 2003. Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Comput Methods Appl Mech Eng 192: 5059–5098. Gasser TC, Holzapfel GA. 2005. Modeling 3D crack propagation in unreinfoced concrete using pufem. Comput Methods Appl Mech Eng 194:2859 –2896. Gasser TC, Holzapfel GA. 2006. Modeling the propagation of arterial dissection. Eur J Mech A/Solids 25:617–633. Gasser TC, Holzapfel GA. 2007. Finite element modeling of balloon angioplasty by considering overstrtech of remnant-nondiseased tissues in lesions. Comput Mech 40:47–60. Gasser TC, Holzapfel GA. 2007. Modeling plaque fissuring and dissedction during balloon angioplasty intervention. Ann Biomed Eng 35:711–723. Gasser TC, Ogden RW, Holzapfel GA. 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3:15–35. Gertz SD, Roberts WC. 1990. Hemodynamic shear force in rupture of coronary arterial atherosclerotic plaques. Am J Cardiol 66: 1368–1372. Gleason RL, Humphrey JD. 2005. Effects of a sustained extension on arterial growth and remodeling: a theoretical study. J Biomech 38(6):1255–1261. Holzapfel GA, Gasser TC. 2001. A viscoelastic model for fiberreinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Methods Appl Mech Eng 190:4379 –4403. Holzapfel GA, Sommer G, Regitnig P. 2004. Anisotropic mechanical properties of tissue components in human atherosclerotic plaques. J Biomech Eng 126:657–665. Holzapfel GA, Gasser TC, Ogden RW. 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elasticity 61(1):1–48. Holzapfel GA, Schulze-Bauer CAJ, Stadler M. 2000. Mechanics of angioplasty: wall, balloon and stent. In: Casey J, Bao G, editors. Mechanics in biology. AMD-Vol. 242/BED-Vol. 46. New York (NY): The American Society of Mechanical Engineers (ASME). p. 141 –156. Holzapfel GA, Stadler M, Schulze-Bauer CAJ. 2002. A layer-specific 3D model for the simulation of balloon angioplasty using MR imaging and mechanical testing. Ann Biomed Eng 30(6):753–767. Huang H, Virmani R, Younis H, Burke AP, Kamm RD, Lee RT. 2001. The impact of calcification on the biomechanical stability of atherosclerotic plaques. Circulation 103:1051–1056. Kuhl E, Maas R, Himpel G, Menzel A. 2007. Computational modeling of arterial wall growth. Attempts towards patient-specific simulations based on computer tomography. Biomech Model Mechanobiol 6(5):321– 331. Lally C, Dolanb F, Prendergast PJ. 2005. Cardiovascular stent design and vessel stresses: a finite element analysis. J Biomech 38: 1574–1581. Larose E, Yeghiazarians Y, Libby P, Yucel E, Aikawa M, Kacher DF, Aikawa E, Kinlay S, Schoen FJ, Selwyn AP, Ganz P. 2005. Characterization of human atherosclerotic plaques by intravascular magnetic resonance imaging. Circulation 112:2324–2331. Lendon CL, Davies MJ, Born GVR, Richardson PD. 1991. Atherosclerotic plaque caps are locally weakened when macrophages density is increased. Atherosclerosis 87:87–90. Loree HM, Kamm RD, Atkinson CM, Lee RT. 1991. Turbulent pressure fluctuations on surface of model vascular stenoses. Am J Physiol 261:H644 –H650. Loree HM, Kamm RD, Stringfellow RG, Lee RT. 1992. Effects of fibrous cap thickness on peak circumferencial stress in model atherosclerotic vessels. Circ Res 71:850–858. Loree HM, Tobias BJ, Gibson LJ, Kamm RD, Small DM, Lee RT. 1994. Mechanical properties of model atherosclerotic lesion lipid pools. Arterioscler Thromb 14:230–234. MacLean NF, Dudek NL, Roach MR. 1999. The role of radial elastic properties in the development of aortic dissections. J Vasc Surg 29:703–710.

Migliavacca F, Petrini L, Colombo M, Auricchio F, Pietrabissa R. 2002. Mechanical behavior of coronary stents investigated through the finite element method. J Biomech 35:803– 811. Moe¨s N, Dolbow J, Belytschko T. 1999. A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131 –501. Nguyen O, Repetto EA, Ortiz M, Radovitzky R. 2001. A cohesive model of fatigue crack growth. Int J Fract 110:351–369. Ortiz M, Pandolfi A. 1999. Finite-deformation irreversible cohesive elements for three-dimensional crack propagation analysis. Int J Numer Methods Eng 44:1267–1282. Pandolfi A, Ortiz M. 1998. Solids modeling aspects of threedimensional fragmentation. Eng Comput 14:287–308. Pandolfi A, Ortiz M. 2002. An efficient adaptive procedure for threedimensional fragmentation simulations. Eng Comput 28:148– 159. Pandolfi A, Ortiz M. 2003. A cohesive model for fatigue crack. In: International conference on fatigue crack paths, FCP 2003. p. 1–7, Parma, 18–20 September. Pandolfi A, Kane C, Marsden JE, Ortiz M. 2002. Time-discretized variational formulation of non-smooth frictional contact. Int J Numer Methods Eng 53(8):1801– 1829. Pasterkamp G, Falk E. 2000. Atherosclerotic plaque rupture: an overview. J Clin Basic Cardiol 3:81–86. Pasterkamp G, Schoneveld AH, van der Wal AC, Haudenschild CC, Clarijs RJG, Becker AE, Hillen B, Borst C. 1998. The relation of arterial geometry with luminal narrowing and plaque vulnerability: the remodeling paradox. J Am Coll Cardiol 32:655– 662. Patel DJ, Janicki JS, Vaishnav RN, Youg JT. 1973. Dynamic anisotropic viscoelastic properties of the aorta in living dogs. Circ Res 32:93–107. Petrini L, Migliavacca F, Aricchio F, Dubini G. 2004. Numerical investigation of the intravascular coronary stent flexibility. J Biomech 37:495–501. Purslow P. 1983. Positional variations in fracture toughness, stiffness and strength of descending thoracic pig aorta. J Biomech 16: 947 –955. Richardson PD, Davis MJ, Born GVR. 1989. Influence of plaque configuration and stress distribution on fissuring of coronary atherosclerosisplaque. Lancet 2:941–944. Rodriguez J, Goicolea JM, Gabaldon F. 2007. A volumetric model for growth of arterial walls with arbitrary geometry and loads. J Biomech 40(5). Ruiz G, Ortiz M, Pandolfi A. 2000. Three-dimensional finite element simulation of the dynamic brazilian tests on concrete cylinders. Int J Numer Methods Eng 48(7):963–994. Ruiz G, Pandolfi A, Ortiz M. 2001. Three-dimensional cohesive modeling of dynamic mixed-mode fracture. Int J Numer Methods Eng 52(1-2):97–120. Schoenhagen P, Ziada KM, Kapadia SR, Crowe TD, Nissen SE, Tuzcu EM. 2000. Extent and direction of arterial remodeling in stable and unstable coronary syndromes. Circulation 101: 598 –603. Schulze-Bauer CAJ, Mo¨rth C, Holzapfel GA. 2003. Passive biaxial mechanical response of aged human iliac arteries. J Biomech Eng 125:395– 406. Shekhonin BV, Domogatsky SP, Muzykantov VR, Idelson GL, Rukosuev VS. 1985. Distribution of type I, III, IV and V collagen in normal and atherosclerotic human arterial wall: immunomorphological characteristic. Coll Relat Res 5:355–368. Smits PC, Pasterkamp G, de Jaegere PPT, de Feyter PJ, Borst C. 1999. Angioscopic complex lesions are predominantly compensatory enlarged. Circ Res 41:458– 464. Spencer AJM. 1982. Deformation of fiber-reinforced materials. Oxford (England): Oxford University Press. Veress AI, Weiss JA, Gullberg GT, Vince DG, Rabbitt RD. 2002. Strain measurement in coronary arteries using intravascular ultrasound and deformable images. J Biomech Eng 124:734–741. Virmani R, Kolodgie FD, Burke AP, Farb A, Schwartz SM. 2000. Lessons from sudden coronary death: a comprehensive morphological classification scheme for atherosclerotic lesions. Arterioscler Thromb Vasc Biol 20:1262–1275.


Computer Methods in Biomechanics and Biomedical Engineering Vito RP, Whang MC, Giddens DP, Zarins CK, Glagov S. 1990. Stress analysis of the diseased arterial cross-section. ASME Adv Bioeng 17. Wells GN, de Borst R, Sluys LJ. 2002. A consistent gemetrically nonlinear approach for delamination. Int J Numer Methods Eng 54: 1333–1355. Xie J, Zhou J, Fung YC. 1995. Bending of blood vessel wall: stress– strain laws of the intima-media and adventitia layers. J Biomech Eng 117:136–145. Yamada H. 1970. Strength of biological materials. Baltimore (MD): Williams and Wilkins. Yang F, Holzapfel G, Schulze-Bauer C, Stollberger R, Thedens D, Bolinger L, Stolpen A, Sonka M. 2003. Segmentation of wall and plaque in in vitro vascular MR images. Int J Cardiovasc Imaging 19:419–428. Yu C, Pandolfi A, Ortiz M, Coker D, Rosakis AJ. 2002. Threedimensional modeling of intersonic crack-growth in asymmetrically loaded unidirectional composite plates. Int J Solids Struct 39:6135–6157. Zohdi TI, Holzapfel GA, Berger SA. 2004. A phenomenological model for atherosclerotic plaque growth and rupture. J Theor Biol 227: 437– 443. van Andel CJ, Pistecky PV, Borst C. 2003. Mechanical properties of porcine and human arteries: implications for coronary anastomotic connectors. Ann Thorac Surg 76:58–65.

567

The axis of the anisotropy ellipsis on the cohesive surface ~ 1 of the angle u given by: are inclined with respect to the axis G tan 2u ¼

2a12 ; 2 a222

a211

ð38Þ

where the coefficients a12,a11 and a22 appear in Equation (17). The in-plane rotation matrix R U that recovers the principal axis of the ellipsis is thus defined as: 3 2 cosu sinu 0 7 6 7 ð39Þ RU ¼ 6 4 2sinu cosu 0 5: 0 0 1

Appendix 2 From (25), we take the squares of the components of T: T 2n ¼

T2 2 Dn ; D2

T 21 ¼

T2 4 2 T2 b D1 ; T 22 ¼ 2 b 4 a 4 D22 ; 2 D D

and write them alternatively as: T 2n D2 ¼ T 2 D2n ; b 22 T 21 D2 ¼ T 2 b 2 D21 ;

Appendix 1 ~ I }; I ¼ 1; 2; 3 be a basis vector with G ~ 3 coincident with Let {G the normal N. The rotation matrix R N is given by: ~ 3 ¼ N; G

~ ~2¼G ~ 1 ¼ G1 £ G3 ; G ~3£G ~ 1; G ~ 3j jG 1 £ G

2

~1 G 1 ·G 6 ~ G ·G RN ¼ 6 4 2 1 ~1 G 3 ·G

~2 G 1 ·G ~2 G 2 ·G ~2 G 3 ·G

~33 G 1 ·G ~37 7: G 2 ·G 5 ~3 G 3 ·G

ð36Þ

b 22 a 22 T 22 D2 ¼ T 2 b 2 a 2 D22 : By summing the left hand sides of the above equations, we write 2 T n þ b 22 T 21 þ b 22 a 22 T 22 D2 ¼ T 2 D2n þ b 2 D21 þ b 2 a 2 D22 :

ð37Þ

and by using Equation (26), we obtain Equation (30).