T.Christian Gasser June 28, 2017

Aorta T.Christian Gasser KTH Solid Mechanics, School of Engineering Sciences KTH Royal Institute of Technology, Stockholm, Sweden

June 28, 2017

Abstract The aorta is a dynamic structure that is able to maintain conditions for optimal mechanical operation through the continuous turnover of its internal structure. The aorta’s properties are critical to the entire cardiovascular system, and the study of its biomechanics may, amongst others, help to understand the role of tissue stress and strain in aortic aging and pathology, help to optimize medical devices, and improve therapeutic and diagnostic methods that are currently used in clinics. The present chapter reviews aortic wall histology and morphology in relation to its key mechanical properties. Specifically, the biomechanical role of cells (endothelial cells, smooth muscle cells, fibroblasts, etc.) as well as the extracellular matrix components (elastin, collagen, proteoglycans, water, etc.) will be discussed. Then this information is related to reported constitutive descriptions for aortic tissues. The focus is on histo-mechanical approaches and modeling frames, related to hyperelasticity as well as a superposition of fiber contributions according to a general theory of fibrous connective tissue. Concluding remarks relate to open problems in aorta biomechanics like uncertainty and variability of input information. Remarks are also made on the admissible degree of complexity in aortic simulations in the context of such uncertainties.

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Contents 1 Introduction

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2 Histology and morphology of the aorta 2.1 Extra cellular matrix (ECM) . . . . . . 2.1.1 Collagen structure . . . . . . . . 2.1.2 Elastin structure . . . . . . . . . 2.2 Cells . . . . . . . . . . . . . . . . . . . .

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3 Mechanical properties and experimental observations 4 Constitutive descriptions 4.1 Modeling frameworks . . . . . . . . . . 4.2 Purely phenomenological descriptions 4.3 Histo-mechanical descriptions . . . . . 4.4 Damage and failure descriptions . . . . 4.5 Growth and Remodeling descriptions .

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5 Conclusion

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6 Acknowledgement

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1

Introduction

The aorta is the first arterial segment of the systemic blood circulation, directly connected to the heart. The aorta is the largest artery in the human body with a diameter of 3 cm at its origin (ascending aorta), 2.5 cm in the descending portion (thoracic aorta), and 1.82 cm in the abdomen (abdominal aorta). In addition to the conduit function, the aorta also accomplishes a buffering function, i.e. its (visco)-elastic compliance plays a pivotal role in regulating left ventricular performance, myocardial perfusion and arterial function in the entire cardiovascular system [Nichols et al., 2011]. Consequently, relation between aortic stiffness and left ventricular hypertrophy, as well as cardiovascular morbidity/mortality in general has been well documented [Laurent et al., 2006]. In addition to this circulatory implications, the aorta itself is particularly prone to mechanics-triggered injuries, with formation of aneurysms being the most commonly observed aortic pathology. Most interestingly, histopathological changes of the aortic wall show striking spatial correlation with biomechanical stress. For example, dilations of the ascending aorta occurs preferentially at the outer curvature of the vessel, i.e. at the site of maximum axial stress [Gomez et al., 2009], and blood flow alterations from above-knee amputations causes a five-fold higher prevalence for abdominal aortic aneurysm development [Vollmar et al., 1989]. The aorta is also highly vulnerable to aging, and age-related increase in diameter and stiffness [L¨anne et al., 1992] are much more pronounced in the aorta than other vessels. There has, therefore, been enormous motivation to assess the mechanical properties of the aorta so as to understand cardiovascular physiology and the mechanisms of cardiovascular disease. Computational biomechanics may assist reasonably in such investigations. The main challenge, however, is to relate the complex architecture of the aortic wall to engineering concepts in order to draw robust conclusions towards aortic wall pathology development and progress.

2

Histology and morphology of the aorta

A sound histological understanding is imperative for the mechanical characterization of soft biological tissue. Similar to other vessels, extracellular matrix (ECM) components (elastin, collagen, proteoglycans (PGs), fibronectin, fibrilin, etc.) ensure the aortic wall’s structural integrity, whereas cells (endothelial cells, smooth muscle cells (SMC), fibroblasts, myofibroblasts, etc.) maintain its metabolism. Specifically, the proteins elastin and collagen almost entirely define the aorta’s passive mechanical properties, while SMCs are responsible for its active properties, and, together with fibroblasts, also for the production of ECM components [McDonald, 2011]. The aorta should always be regarded as dynamic structure which, within certain physiologic ranges, is able to adapt to functional needs. The aorta’s geometrical, histological and mechanical properties change from the ascending aorta towards the abdominal aorta, (see Figure 2), likely to maintain conditions for optimal mechanical operation [Rachev et al., 2013]. In

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addition aortic properties and histology change with age [L¨anne et al., 1992]. Such changes are associated with alterations in chemical composition (decreased elastin content, increased collagen content, increased levels of metalloproteinase2 (MMP2)), as well as in wall histology (thinner, splitted and fraying of Medial Lamellar Units (MLUs), increased glycation of elastin, increased crosslinking of collagen), see [Bailey et al., 1998, Nichols et al., 2011, Astrand et al., 2011] and references therein. The aorta is composed of intimal, medial and adventitial layers, see Figure 1. The intima is the innermost layer and comprises primarily of a single layer of endothelial cells lining the arterial wall. Endothelial cells rest on a thin basal membrane on top of a subendothelial layer, whose thickness changes with topography, age and disease. The media is the middle layer of the artery (see Figure 1) and consists of a complex three-dimensional network of SMCs, elastin and collagen fibers and fibrils. These structural components are preferentially aligned along the circumferential vessel direction [O’Connell et al., 2008, Gasser et al., 2012] and organized in repeating MLUs of 13-15 µm thickness [Clark and Glagov, 1985, Dingemans et al., 2000, O’Connell et al., 2008]. The thickness of MLUs is independent of the radial location in the wall and the number of MLUs increases with increasing vessel diameter, such that the tension carried by a single MLU in the normal wall remains constant at about 2±0.4 N/m [Clark and Glagov, 1985]. The media’s layered structure is gradually lost towards the periphery, and a discrete laminated architecture is hardly present in the abdominal aorta. The adventitia is the outermost layer of the artery (see Figure 1) and consists mainly of fibroblasts and fibrocytes and ECM of thick bundles of collagen fibrils. The adventitia is surrounded by loose connective tissue that anchors the aorta to its surrounding. Medial and adventitial thicknesses depend strongly on the physiological function of the blood vessel and its topographical site, see Figure 2.

2.1

Extra cellular matrix (ECM)

The 3D organization of ECM components is vital towards accomplishing proper physiological aortic functions. The ECM rather than being merely a system of scaffolding for the surrounding cells, is a mechanical structure that controls the micro-mechanical and macro-mechanical environments to which vascular tissue is exposed, i.e. it quantifies the amount of stress and/or strain that is transmitted to the individual cells of vascular tissue and influences cell metabolism [Carey, 1991]. Specifically, elastin and collagen are dominating ECM vessel wall components and their organization is schematically illustrated in Figure 1. 2.1.1

Collagen structure

Collagen fibrils of diameters ranging from fifty to a few hundreds of nanometers are the basic building blocks of fibrous collagenous tissues [Fratzl, 2008], and their organization into suprafibrilar structures has a large impact on the tissue’s macroscopic mechanical properties.

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Collagen fibers are assembled by interlinked collagen fibrils of different undulations Collagen fibrils are assembled by interlinked collagen triple helix molecules

Elastin fibers are assembled by an amorphous core of cross-linked elastin protein and an outer mantle of undulated and cross-linked fibrillin

Proteoglycan bridges Collagen fibril

Elastin core Microfibril

200 nm

Cross-link

2 µm 100 nm The adventitia is formed by thick bundles of collagen fibers, fibroblasts and elastin fibers. Fibers in the adventitia are highly dispersed in orientations

Elastin sheet

Elastin fiber

The media is formed by Medial Lamellar Units (MLU) within which elastin fibers, collagen fibers and SMC fibers are predominanltly aligned along the circumferential direction

Thick radial strut

Fibroblast cell Smooth Muscle Cell (SMC) Collagen fibrils and fibers Elastic lamina externa Elastin fiber Elastic lamina interna

3 µm

Endothelial cell

5 mm

Elastin is organized in sheets, in rope-like elastin fibers, and as thick radial struts

Figure 1: Histological idealization of the normal aorta. It is composed of three layers: intima (I), media (M), adventitia (A). The intima is the innermost layer consisting of a single layer of endothelial cells, a thin basal membrane and a subendothelial layer. Smooth Muscle Cells (SMCs), elastin and collagen are key mechanical constituents in the media and are arranged in a number (up to 60) of 13-15 µm thick Medial Lamellar Units (MLUs). (In the image only three MLUs are shown). In the adventitia the primary constituents are collagen fibers and fibroblasts. Collagen fibers with a thickness in the range of micrometers are assembled by collagen fibrils (50 to 300 nm thick) of different undulations. Load transition between collagen fibrils is maintained by Proteoglycan (PG) bridges. Elastin fibers with a thickness of hundreds of nanometers are formed by an amorphous core of highly cross-linked elastin protein that is encapsulated by 5nm thick microfibrils. Elastin fibers are organized in thin concentric elastic sheets, in a rope-like interlamellar elastin fibers, and as thick radial struts.

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18.0 2.0 Medial layer thickness

12.0 Outer vessel diameter

1.0

6.0

Adventitial layer thickness 0.0

Asc. Thor.

Arch

3.0 Desc. Thor.

Abdominal

Desc. Thor.

Abdominal

Outer Vessle Diameter (mm)

Vessle Layer Thickness (mm)

(a)

(b) Relative area density (%)

80

Adventitial collagen

60

40 Medial elastin 20

Medial collagen Adventitial elastin

0

Asc. Thor.

Arch

Figure 2: Variation of geometrical properties and histological composition of the porcine aorta [Sokolis, 2007]. (a) Change of outer aortic diameter and thickness of medial and adventitial layers. Intimal layer thickness covers less than 1% of the total wall thickness. (b) Relative area density of elastin and collagen in medial and adventitial layers. The relative area density represents the area covered by a constituent in histological stains. Asc.Thor. - Ascending thoracic aorta; Arch. - Aortic arch; Desc.Thor. - Descending thoracic aorta; Abdominal - Abdominal aorta. Within the MLU, i.e. between the elastic lamellae of the media, collagen fibrils or bundles of fibrils (24±15 fibers per bundle) run in parallel closely enveloping the SMCs [O’Connell et al., 2008]. Consequently, collagen fibers

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are not woven together but aligned in parallel (like in a tendon or ligament), probably to better cope with mechanical load [O’Connell et al., 2008]. Collagen gives stiffness, strength and toughness to the vascular wall. Earlier observations indicated that the collagen-rich abdominal aorta was stiffer than the collagen-poor thoracic aorta [Bergel, 1961, Langewouters et al., 1984] and later regional variations of aortic properties were specifically documented, see [Sokolis, 2007]. Apart from the amount of collagen in the wall, its spatial orientation [Fratzl, 2008] (including the spread in orientations [Gasser et al., 2006]) also strongly influences macroscopic mechanical properties. At physiological load only 6% to 7% of collagen fibers are engaged [Armentano et al., 1991, Greenwald et al., 1997]. Collagen fibrils seem to be interlinked by Proteoglycan (PG) bridges [Scott, 2003, 2008] to provide interfibrillar load transition. Specifically, small PGs, such as decorin, bind noncovalently but specifically to collagen fibrils and crosslink adjacent collagen fibrils at about 60 nm intervals [Scott, 2003]. Reversible deformability of the PG bridges is crucial to serve as shape-maintaining modules [Scott, 2003] and, fast and slow deformation mechanisms have been identified. The fast (elastic) deformation is supported by the sudden extension of about 10% of the L-iduronate (an elastic sugar) at a critical load of about 200 pN [Haverkamp et al., 2005]. The slow (viscous) deformation is based on a sliding filament mechanism of the twofold helix of the glycan [Scott, 2003], and may explain the large portion of macroscopic visco-elasticity of collagen. PG-based cross-linking is supported by numerous experimental studies showing that PGs play a direct role in inter-fibril load sharing [Liao and Vesely, 2007, Robinson et al., 2005, Scott, 2003, Sasaki and Odajima, 1996], and has been verified through theoretical investigations [Fessel and Snedeker, 2011, Redaelli et al., 2003, Vesentini et al., 2005]. However, the biomechanical role of PGs is still somewhat uncertain, and some data indicates minimal, if any, PG contribution to the tensile properties of the tissue [Fessel and Snedeker, 2011, Rigozzi et al., 2009, 2010]. Collagen is in a continuous state of deposition and degradation at a normal half-life time of 60-70 days [Nissen et al., 1978]. Physiological maintenance of the collagen structure relies on a delicate (coupled) balance between degradation (mainly through MMPs) and synthesis by cells like SMCs, fibroblasts and myofibroblasts, [Nichols et al., 2011]. 2.1.2

Elastin structure

Elastin functions in partnership with collagen. Elastin mainly determines the mechanical properties of arterial tissue at low strain levels [Roach and Burton, 1957] and recoils the tissue during each pulse cycle. In the vessel wall, elastin is predominantly seen in the media, see Figure 2(b). Specifically, it is organized as thin concentric elastin sheets with a fibrous surface which encapsulate the MLU (71%; about 1 to 2.2 µm thick), in a rope-like interlamellar elastin fibers (27%; about 100nm to 500nm thick), and as thick radial struts (2%; about 1.5µm thick) [O’Connell et al., 2008, Dingemans et al., 2000, Berry and Greenwald, 7

1976]. The elastin lamellae are perforated and gusseted by elastin fibers. Microscopy studies also indicate that elastin is made up of repeating self-similar structures at many length-scales [Tamburro et al., 1995]. Elastin fibers are about 100nm thick and composed of two significant components; 90% of which is an amorphous core of highly cross-linked elastin protein and the remaining 10% is a fibrillar mantle of about 5 nm thick microfibrils [R. Ross and Bornstein, 1969, Cleary, 1987]. Elastin is synthesized and secreted by vascular SMCs and fibroblasts, a process that normally stops soon after puberty once the body reaches maturity. Although the dense lysyl cross linking makes elastin fibrils extremely insoluble and stable (half-life times of tens of years [Alberts et al., 1994]), elastin may be degraded by selective MMPs, collectively known as elastases. Elastases cause disruption of elastin fiber integrity and subsequently diminishes mechanical tissue properties. Elastin is a critical autocrine factor that maintains vascular homeostasis through a combination of biomechanical support and biologic signaling, see [B¨ack et al., 2013] and references therein. While elastin degradation is related to several diseases (atherosclerosis, Marfan syndrome, Cutis laxa, etc. ), it is also important for many physiological processes such as growth, wound healing, pregnancy and tissue remodeling [Werb et al., 1982]. Consequently, the proteolytic degradation of elastin may have important consequences for normal elastogenesis and repair processes [Vrhovski and Weiss, 1998]. Repair of protease-damaged elastin can occur but does not appear to produce elastin of the same quality as when originally laid down during development, i.e. primary vascular growth [Soskel and B.Sandberg, 1987]. Elastin and rubber show some mechanical similarities. For example, both are entropic elastic and go through a glassy transition. However, elastin’s hydrophobic interactions are a determining factor in its elasticity, such that elastin is only elastic when swollen in water, see [Vrhovski and Weiss, 1998] and references therein.

2.2

Cells

Cells like endothelial cells, SMCs, fibroblasts, etc. in the aorta sense and respond to mechanical loads, allowing the aorta to undergo many changes during normal development, ageing, in response to disease or implanted devices, etc. Endothelial cells are constantly exposed to wall shear stress (WSS) and provide an anti-thrombogenic and low-resistance lining between the blood and aortic tissue. WSS experienced by the endothelial cells is associated with changes in gene expression patterns through positive and negative WSS responsive elements in their promoter regions [Chen et al., 2003]. In the aneurysmatic aorta the frequently seen intra-luminal thrombus layer [Hans et al., 2005] shields endothelial cells from blood flow and, if not die out, they likely loose their blood flow regulatory role. Elongated vascular SMCs are layered between elastic lamellae, i.e within the MLU. They are aligned with the circumferential direction and at a radial tilt of about 20 degrees [O’Connell et al., 2008, Fujiwara and Uehara, 1992]. In their 8

differentiated/contractile phenotype vascular SMCs serve as contractile unit in the vessel wall and actively influence aortic diameter [Milewicz et al., 2010]. In addition, in their dedifferentiated/synthetic phenotype vascular SMCs respond to cyclic stretch, as well as to mediators that are convected through the aortic wall with the transmural interstitial flow [Milewicz et al., 2010]. The effects of cyclic stretching of vascular SMCs on collagen production were described almost 40 years ago [Leung et al., 1976], where increased strain up-regulates MMPs in vascular SMCs [Grote et al., 2003], such that increased mechanical strain may lead to enhanced ECM degradation. In addition to these direct vascular SMC response, endothelium-derived vasoactive factors, released in response to WSS for example, act on the aortic SMCs, see [B¨ack et al., 2013] and references therein. Fibroblast cells are tightly anchored to collagen fibers and, together with SMCs (at dedifferentiated/synthetic phenotype) and MMPs, control the delicate balance between synthesis and degradation of collagen. Mature fibroblasts respond to mechanical strain or stress and adjust their expression and synthesis of collagen molecules [Grote et al., 2003, Bishop and Lindahl, 1999], perhaps similar to cardiac fibroblasts, where members of the Mitogen-activated protein (MAP) kinase family upregulate procollagen gene expression in response to cyclic mechanical load [Papakrivopoulou et al., 2004].

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Mechanical properties and experimental observations

The pressure-diameter property (compliance) of the aorta is of critical importance to the entire cardiovascular system and determines the non-linearity of its pressure-flow relationship [Fung, 1990]. The aorta contributes almost the entire capacity of the cardiovascular system, i.e. defines its Windkessel properties, see [Westerhof et al., 2009] and references therein. The aortic volume compliance (C = ∆V /∆p) is constant over a wide range of pressures, and the thoracic aorta alone contributes 85% to the aortic compliance [Guo and Kassab, 2003]. The aortic wall can be regarded as a mixture of solid components (elastin, collagen, SMCs, PGs, etc) and water, much of which is not particularly mobile but bound to PGs, elastin, and the like. While the radial convection of fluid (which is established due to pressure gradient between the arterial circulation and the interstitial pressure in the adventitia) is essential for aorta physiology [B¨ ack et al., 2013], for many mechanical problems the aortic wall can be regarded as an incompressible homogenized solid; in-vivo the volumetric strain is three orders of magnitude smaller than the circumferential strain [Carew et al., 1968]. The normal (non-calcified) aorta is highly deformable and exhibits a nonlinear [Roy, 1880–82] stress versus strain response with a typical stiffening at around the physiological strain level. The aorta’s circumferential stiffness is highest at the level of the diaphragm [Tanaka and Fung, 1974, Guo and Kassab, 2003], increases with age and might also be higher in males than in females

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[Sonesson et al., 1993]. The thick and collagen-rich adventitial layer (see Figure 2) of the abdominal aorta leads to the observed increase of stiffness of the abdominal aorta when compared to the thoracic aorta. Vascular tissue characterization after selective digestion of elastin or collagen [Roach and Burton, 1957, Dobrin and Canfield, 1984, Marsh et al., 2004, Gundiah et al., 2013] contributed considerably to our current understanding of vessel wall mechanics. Experimental data suggest that the vessel wall stiffens (at physiological deformations) in response to the recruitment of the embedded wavy collagen fibrils [Roach and Burton, 1957, Wolinsky and Glagov, 1964, Samila and Carter, 1981], a mechanism that in turn explains the non-linear elasticity of the aorta. Such a collagen fiber engagement mechanism also explains the anisotropic properties of the aorta [Patel et al., 1969, Vande Geest et al., 2006b] with a higher stiffness along the circumferential direction; direction along which more collagen fibers are oriented [O’Connell et al., 2008, Gasser et al., 2012]. Analyzing in-vivo pressure diameter properties of the abdominal aorta revealed that the load-bearing fraction of collagen between diastole and systole oscillates between 10% and 30%, respectively [Astrand et al., 2011]. The constant turnover of wall constituents at the aorta’s in-vivo configuration explains residual stresses in its load-free configuration. Residual stresses in the arteries have been known for at least half a century [Bergel, 1961] and their biomechanical consequences are well discussed in the literature [Choung and Fung, 1986, Fung, 1991, Rachev and Greenwald, 2003, Vaishnav and Vossoughi, 1987]. Specifically, it is reported that both, circumferential [Choung and Fung, 1986, Vaishnav and Vossoughi, 1987] and longitudinal [Vossoughi, 1992] strips change their curvature when excised from a load-free artery, i.e. residual stresses in the vascular wall are clearly multi-dimensional. Neglecting residual stresses in the load-free configuration can be a severe limitation [de Putter et al., 2007, Lu et al., 2007] and for purely passive simulations typically leads to considerable stress gradients across the wall thickness. Such a wall stress field is non physiological, i.e. in contradiction to the uniform stress hypothesis [Fung, 1991]. In humans, the aorta is axially pre-stretched at 5% [Astrand et al., 2011], a property that, however, changes with age [Horny et al., 2013] and between the different aortic segments [Guo and Kassab, 2003]. Experimental data indicated that at in-vivo pre-stretch the axial force acting on the vessel remains independent from the inflation pressure [Weizs¨acker et al., 1983], i.e. inside the body pulsatile axial strain is prevented. In-vitro testing of arterial tissue typically displays pronounced stress softening under the first few cycles of loading until the tissue is preconditioned and a stable cyclic response is observed. Even when preconditioned, the aortic wall shows typical strain rate dependency like creep, relaxation and dissipation under cyclic loading. Most interestingly, the dissipation during cyclic loading over a frequency range of five orders of magnitude does not change by a factor of more than two [Tanaka and Fung, 1974]. Exposing vascular tissue to supra-physiological mechanical stresses rearranges the tissues microstructure through irreversible deformations; damage-related effects [Emery et al., 1997a, Oktay et al., 1991] and plasticity-related effects [Oktay et al., 1991, Salunke 10

and Topoleski, 1997] have been documented. Probably there is some correlation between the mechanisms of pre-conditioning and inelastic phenomena when exceeding the physiological loading. Due to its high clinical relevance, the aneurysmatic infrarenal aorta has been studied extensively. The biomechanical properties of the infrarenal aneurysm’s wall differ considerably from the properties of the normal aorta. Specifically, the stress strain law of the aneurysmatic wall is more nonlinear, the wall deforms less [Vande Geest et al., 2006b] and has a lower strength when compared to the normal aortic wall strength (descending thoracic: 1.95 ± 0.6 MPa [Adham et al., 1996]; the midthoracic: 1.470 ± 0.91 MPa [Mohan and Melvin, 1982]; abdominal aorta: 1.21 ± 0.33 MPa [Vorp et al., 1996]; 1.710 ± 0.14 MPa [Vorp et al., 2003]). Table 1 presents a review of reported thickness and strength data of the infrarenal aortic aneurysm wall. The data indicates a remarkable negative correlation (Pearson’s correlation coefficient = -0.71) between thickness and strength, which has also been reported elsewhere, see for example [Reeps et al., 2013, Forsell et al., 2014]. SMC at the differentiated/contractile phenotype regulate the aortic diameter and stiffness. Specifically, contractile cell fibers span the vascular SMC cell body to generate force against the ECM [Alberts et al., 1994]. The level of vascular SMC activation or basal tone changes in response to biomechanical stimuli such as flow [Davies, 1995] or pressure [Fridez et al., 2001], hormonal stimuli, neural stimuli, and drugs. Vascular SMC in its normal tone state appears to be sensitive to both circumferential and longitudinal stretching of the vessel wall, see [Zulliger et al., 2004b] and references therein. The normal vessel seems to follow the concept of homeostasis, i.e. it adapts to changes in the mechanical environment such that target mechanical properties are kept relatively constant. Specifically, target values for WSS [Guzman et al., 1997, Castier et al., 2005], circumferential wall stress [Wolinsky, 1971, Matsumoto and Hayashi, 1996] and axial stretch [Gleason et al., 2007] have been shown to be regained after alterations. Normal vessel remodeling is diminished in the aneurysmatic aorta and the wall in larger aneurysm seems to lose its ability to respond to mechanical stress [Martufi et al., 2016, Nchimi et al., 2014]. In addition, during aneurysm development wall thickness remains relatively constant, such that wall stress increases proportionally with the size of the aneurysm [Gasser et al., 2014], i.e. is not kept at its homeostatic level.

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Constitutive descriptions

Constitutive modeling of vascular tissue is an active field of research and numerous descriptions with application to the aorta have been reported. While purely phenomenological approaches can successfully fit experimental data, such models show limited robustness for predictions beyond the strain range within which model parameters have been identified, and they cannot allocate stress or strain to the different histological constituents in the vascular wall. Structural constitutive descriptions overcome such limitations and integrate histological

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and mechanical information of the arterial wall, which in turn is not only more robust but also helps to understand load carrying mechanisms in the vessel wall. Clearly, modeling assumptions need to fit the objective of the particular simulation [Sargent, 2011], and for most applications the aortic wall can be regarded as a single phase incompressible solid. Even the highly porous intraluminal thrombus [Adolph et al., 1997, Gasser et al., 2010] can be modeled as a homogenized solid [Polzer et al., 2012] for a wall stress-based aneurysm rupture risk assessment, for example. However, the suitability of all applied modeling assumptions needs to be carefully validated, and more complex frames like poro-elasticity [Simon and Gaballa, 1988] for example, can be motivated.

4.1

Modeling frameworks

Hyperelasticity for incompressible solids is a popular modeling framework to derive the Cauchy stress σ = 2F

∂ψ(C) T F − pI ∂C

(1)

of aortic tissue from the strain energy density ψ(C) in reference volume. Here, tissue deformation is determined by the right Cauchy-Green strain C = F TF, which is a function of the deformation gradient F. The hydrostatic pressure p acts on the identity I and serves as a Lagrange parameter to enforce incompressibility. All constitutive information (mechanical and histological) is captured by a particular form of the strain energy function ψ, and Table 2 lists some models for aortic tissues. The stress in the aortic wall can also be expressed according to the general theory of fibrous connective tissue [Lanir, 1983]. Such descriptions are also denoted as angular integration models and allow integrating fibers of dispersed ordination and undulation. Specifically, an orientation density function ρ(ϕ, θ) reflects the orientation of fibers in the reference configuration with respect to elevation ϕ and azimuthal θ angles. The superposition of such fibers determines the tissue’s Cauchy stress 2 σ= π

∫π/2 ∫π/2 ρ(ϕ, θ)σ(λ)dev(m ⊗ m) cos ϕdϕdθ − pI ,

(2)

ϕ=0 θ=0

where m = FM denotes the push forward of the referential fiber direction M. Here, σ(λ) expresses the Cauchy stress in a fiber as a function of the fiber stretch λ, i.e. reflects a constitutive model on the fiber level, and dev(•) = (•)−tr(•)I/3 denotes the spatial deviator operator. Model parameters, i.e. constitutive parameters appearing in ψ of eq.(1) as well as in ρ and σ of eq.(2) are identified from experimental data by leastsquare optimization, for example. The uniform stress hypothesis [Fung, 1991] and other plausible assumptions [Weizs¨acker et al., 1983] have been used as constraints in parameter estimation [Takamizawa and Hayashi, 1987, Rachev et al., 12

1996, Taber and Humphrey, 2001, St˚ alhand and Klarbring, 2005] to improve the reliability of the estimated model parameters.

4.2

Purely phenomenological descriptions

Due to some similarities between vascular tissue and rubber (like entropic elasticity), models that originally have been proposed for rubber are also frequently used for the aorta, see Table 2(a). In contrast, models that have especially been developed for vascular tissue include exponential terms in the strain energy ψ [Demiray, 1972, Fung et al., 1979] and even consider anisotropy [Vaishnav et al., 1972, Chuong and Fung, 1983, Takamizawa and Hayashi, 1987, Horgan and Saccomandi, 2003]. For such models, the strain energy is typically expressed as a function of the components of the Green-Lagrange strain E = (C − I)/2 within the local cylindrical coordinate system, i.e. ψ(Eij ) for i, j = r, ϑ, z with r, ϑ, z denoting the radial, circumferential and axial vessel directions, respectively. Such models have also been enriched by techniques to estimate the residual strain field from the uniform stress hypothesis [Polzer et al., 2013], or to back-calculate the zero-load configuration from inflated geometries [de Putter et al., 2007, Riveros et al., 2013].

4.3

Histo-mechanical descriptions

Motivated by the results from selective digestion of elastin and collagen [Roach and Burton, 1957, Dobrin and Canfield, 1984], constitutive models started to regard the vessel wall as a mixture (composition) of collagen, elastin and SMCs [Oka and Azuma, 1970, Holzapfel et al., 2000, Humphrey and Rajagopal, 2002, Sokolis et al., 2006]. Nowadays, most studies assume that elastin and collagen are major independent determinants (each of which is reflected by an additive strain energy contribution, see examples in Table 2) of the mechanical properties of the aortic wall at low and high stresses, respectively. In addition to composition-based models, the actual load carrying of collagen fibers attracted much attention. Almost 40 years ago a tissue model based on aligned wavy collagen fibers that engage during loading [Decraemer et al., 1980] has been reported, and later very similar ideas have been adopted to the vessel wall [Wuyts et al., 1995, Zulliger et al., 2004a, Martufi and Gasser, 2011], see Table 3). Alternatively, the strong stiffening effect of the collagen fibers during stretching was phenomenologically integrated in the strain energy function by an exponential expression [Holzapfel et al., 2000] (HGO model) or a 6thorder polynomial term [Basciano and Kleinstreuer, 2009]. Collagen and elastin fiber orientations in the vessel wall are dispersed [Schriefl et al., 2012, Polzer et al., 2015, Thunes et al., 2016]. Specifically, the collagen in the adventitia of normal [Finlay et al., 1995] and aneurysmatic [Gasser et al., 2012] aortas is highly dispersed, and Table 4 summarizes models to express such dispersions. Consequently, the basic assumption of two (or more) families of parallel aligned collagen fibers seems unrealistic [Polzer et al., 2015] and the GOH model proposed an efficient way of integrating the dispersion of fiber orientations in the 13

strain energy function [Gasser et al., 2006]. The GOH model considers all collagen fibers within the i-th (dispersed) family of fibers being strained homogenously at Ei = Hi : C − 1 with Hi denoting a general structural tensor. This assumption seems appropriate for the aorta at physiological loading, where homogenously stressed collagen is continuously deposited. However, for other load cases (prior to tissue failure, during automobile accidents, etc.) collagen fibers are likely stressed inhomogenously, and the more general framework according to eq.(2) is motivated. Apart from the above described homogenization approaches, Representative Volume Element (RVE) models have been reported to integrate histological and mechanical properties of the MLU composites [Thunes et al., 2016]. On top of passive stresses, SMC contraction acts along the vessel’s circumference and influences vessel wall stress. Constitutive models accounted either directly [Zulliger et al., 2004b], or indirectly through the calcium concentration [Sharifimajd and St˚ alhand, 2014], for (active) SMC stress. Interestingly, considering active stress from SMCs seems, in addition to residual strain in the load-free configuration, reducing the transmural stress gradient at physiological loading [Rachev and Hayashi, 1999].

4.4

Damage and failure descriptions

Exposing biological soft tissue to supraphysiological mechanical stresses rearranges vascular tissue’s microstructure through irreversible deformations. Specifically, damage-related effects [Emery et al., 1997a,b, Oktay et al., 1991] and plasticity-related effects [Oktay et al., 1991, Salunke and Topoleski, 1997] have been proposed, which in turn triggered the development of models that account for damage [Hokanson and Yazdani, 1997, Balzani et al., 2006, Volokh and Vorp, 2008, Calvo et al., 2009, Marini et al., 2011, Noble et al., 2016], plasticity [Tanaka and Yamada, 1990, Gasser and Holzapfel, 2002] and fracture [Ionescu et al., 2006, Gasser and Holzapfel, 2006, 2007, Ferrara and Pandolfi, 2008, Forsell and Gasser, 2011]. Most commonly a macroscopic (single scale) framework was followed, which, however, fails to describe the experimentally reported [Quinn and Winkelstein, 2008, Kn¨orzer et al., 1986] localized irreversible deformation of individual collagen fibers. In contrast, irreversible collagen fiber deformation (related to PG-based deformation mechanisms) has been integrated in eq.(2) to model aneurysm wall failure [Gasser, 2011].

4.5

Growth and Remodeling descriptions

The aorta responds to mechanical stimuli, which is a necessity to locally alter its mechanical properties towards approaching conditions for optimal mechanical operation [Rachev et al., 2013]. Modeling vascular adaptation (growth and remodeling) is an active field of research [Humphrey and Rajagopal, 2002], much of which is related to aneurysm development and progression [Volokh and Vorp, 2008, Watton and Hill, 2009, Kroon and Holzapfel, 2009, Zeinali-Davarani and Baek, 2012, Wilson et al., 2012, Martufi and Gasser, 2012]. Most of these models 14

are characterized by a high degree of phenomenology, and all of them are poorly validated. Consequently, significant further development is required to provide robust predictions that could augment clinical decisions, for example.

5

Conclusion

The aorta’s biomechanical properties are not only related to aortic pathologies but also critical to the biomechanics of the entire cardiovascular system. The aortic wall is a dynamic structure able to maintain conditions for optimal mechanical operation by continuous turnover of its internal structure. Numerous constitutive description for aortic tissues at different levels of details have been proposed. Specifically, histo-mechanical approaches are able to examine local micro-architectural features, which in turn constrains internal load carrying mechanisms, and provide more robust predictions when compared to purely phenomenological descriptions. In contrast, such histo-mechanical models can be quite complex, involving many structural and mechanical parameters, which need to be carefully and consistently identified. Naturally, every model involves making modeling assumptions and reflects the real object or process always only up to a certain degree of completeness (”Everything should be made as simple as possible, but no simpler”; quote attributed to A. Einstein, see http://quoteinvestigator.com/2011/05/13/einsteinsimple/#more-2363), and a model should be validated to the degree needed for the model’s intended purpose or application [Sargent, 2011]. Consequently, a good model will only include modeling details that improve the intended simulation objective, and, even if in contradiction to our current knowledge about the actual biomechanical problem, disregard all the other information. For example, the required degree of complexity of the aortic wall model (isotropic versus anisotropic modeling; single phase versus multiphase modeling; constant wall thickness versus variable wall thickness; etc.) used to provide clinical diagnostic information can only be evaluated in relation to the model’s implication on the clinical outcome - a complex model does not necessarily give better diagnostic information. Despite encouraging progress in vascular tissue biomechanics, the variability of biomechanical predictions due to uncertainty of input information remains a challenging limitation. Specifically, for clinical applications key input information like detailed geometrical features and the local biomechanical properties of aortic wall tissue often remains unknown. Owing to this lack of input information, homogeneous mean population input information is often used, and the extent to which this simplification influences the value of model predictions remains to be explored for each individual application. In view of these key limitations, many constitutive models reported in the literature seem overdone. Probabilistic approaches seem to be a promising way to deal with input uncertainty of aorta biomechanics, and such approaches have been reported recently [Biehler et al., 2014, Polzer et al., 2015]. Even under well-defined laboratory conditions, the in-vitro experimental

15

characterization of aortic wall properties show huge variability and the estimated tissue properties usually vary at least by one order of magnitude. While the detailed causes of this variability remain largely unknown, several influential parameters have already been identified. For example it seems that Chronic Obstructive Pulmonary Disease (COPD) [Forsell et al., 2012], bicuspid aortic valve anatomy [C.Y. Shim, 2011, Forsell et al., 2014], Marfan syndrome [Marque et al., 2001], diabetes mellitus [Maier et al., 2012], as well as the administration of drugs (like beta blocker and Angiotensin-Converting-Enzyme (ACE) inhibitors [Maier et al., 2012]) alter biomechanical vessel wall properties. Further exploring such influential factors may help to improve the reliability of aorta biomechanics.

6

Acknowledgement

The author would like to thank Andrii Grytsan and Prashanth Srinivasa from KTH Royal Institute of Technology, Stockholm for the valuable feed-back on the manuscript.

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30

Reference

Sample specification

[O’Leary et al., 2015]

[Vande Geest et al., 2006a]

N=31; N=38; N=28 N=83 N=26 N=25; N=65; N=76

[Reeps et al., 2013] [Raghavan et al., 2006] [Xiong et al., 2008]

N=163 N=374/48 N=14

1.57 1.48 1.5 to 1.9

[Forsell et al., 2012] [Thubrikar et al., 2001]

N=16 Anterior: N=29

2.06 2.73

Lateral: N=9

2.52

Posterior: N=9

2.09

Intact AAA: N=26 Ruptured AAA: N=13 Intact: N=278/56 Ruptured: N= 141/21 Long.: N=45 Circ.: N=19 ILT layer thick. > 4mm: N=7 ILT layer thick. < 4mm: N=7

2.5 3.6 1.5 1.7 -

[O’Leary et al., 2014] [Vande Geest et al., 2006d] [Vande Geest et al., 2006b] [Monteiro et al., 2013]

[DiMartino et al., 2006] [Raghavan et al., 2011] [Raghavan et al., 1996] [Vorp et al., 2001]

fibrous partly calcified

AAA diam. < 55mm AAA diam. > 55mm

Thickness [mm] 1.2 1.5 1.18 1.32 1.53 1.58 -

Strength [MPa] 1.2 0.87 0.81 0.77 1.03 Female: 0.68 Male: 0.88 1.42 1.26 Long.: 0.93 Circ.: 1.15 0.57 Long.: 0.38 Circ.: 0.52 Long.: 0.51 Circ.: 0.73 Long.: 0.47 Circ.: 0.45 0.82 0.54 0.98 0.95 0.86 1.02 1.38 2.16

Table 1: Abdominal Aortic Aneurysm (AAA) wall thickness and wall strength measured from in-vitro tensile testing. The number of test samples is denoted by N (used for thickness/ strength measurements). Circ. - circumferential; Long. - longitudinal; ILT - intra-luminal thrombus.

31

32

i

2 ∑

(λ4i − 1)

3 ∑

ψ = c0 (I1 − 3) +

[Gasser et al., 2006] i=1

N ∑

c1 i (I4 i − 1)i

c1 [exp(c2 Ei2 ) − 1] ; Ei = Hi : C − 1

i=1

2 ∑

(1 − c3 )(I1 − 3)2 ]

c1 {exp[Q] − 1} ,

[Noble et al., 2016]

[Celi and Berti, 2012]

[Rodr´ıguez et al., 2008, 2009, Vande Geest et al., 2006b]

N=2: [Zeinali-Davarani et al., 2013, Zulliger et al., 2004a, St˚ alhand et al., 2004] N=4: [Baek et al., 2009, Ferruzzi et al., 2010]

[Basciano and Kleinstreuer, 2009, Vande Geest et al., 2006b]

[Vande Geest et al., 2006b] [Riveros et al., 2013]

[Zulliger et al., 2004a]

Parameter identification

[Gasser et al., 2008]

Parameter identification N = 2: [Raghavan and Vorp, 2000, Raghavan et al., 1996, Reeps et al., 2013, Vande Geest et al., 2006c] N = 5: [Polzer et al., 2013] [Polzer et al., 2013, Riveros et al., 2013]

Table 2: Hyperelastic strain energy functions to model aortic tissue stress according to eq.(1). I1 = trC denotes the first invariant of the right Cauchy-Green strain C. I4 i = C : (a0 i ⊗ a0 i ) denotes the fourth invariant of C and the structural tensor a0 i ⊗ a0 i , where the unit direction vector a0 i defines the fiber orientation of the i−th family of parallel-aligned (collagen) fibers. For collagen fibers that are dispersed in orientation, a0 i denotes their mean orientation and κ is a measure of their dispersion. Eij ; i, j = r, ϑ, z denotes the components of the Green-Lagrange strain E = (C − I)/2 with respect to the local cylindrical coordinate system of radial r, circumferential ϑ and axial z vessel directions. Model parameters to be identified from experimental data are denoted by ci , a0 i and κ.

General structural tensor: Hi = κI + (1 − 3κ)(a0 i ⊗ a0 i )

i=1

i=1

2 ∑

c1 i {exp[c2 i (I4 i − 1)2 ] − 1}

c0 i (I1 − 3)i +

ψ=

[Celi and Berti, 2012]

3 ∑

Q = c2 [c3 (I4 i −

ψ = c0 (I1 − 3) +

[Rodr´ıguez et al., 2008]

i=1

N ∑

c1 (I4 i − 1)6

i=1 c4 )2 +

ψ = c0 (I1 − 3) +

ψ = c0 (I1 − 3) +

Strain energy function ψ = c0 {exp[Q] − 1} , 2 2 2 Q = c1 Eϑϑ + c2 Ezz + c3 Err + c4 Eϑϑ Ezz + c5 Ezz Err 2 2 2 +c{6 Err Eϑϑ + c7 Eϑz + c8 Erz + c9 Erϑ } 2 2 ψ = c0 exp[c1 Eϑϑ ] + exp[c2 Ezz ] + exp[c3 Eϑϑ Ezz ] − 3 ψ = c0 {exp[c1 (I1 − 3)] − 1} + c2 {exp[c3 (I4 − 1)] − 1}

ψ = c0

[Holzapfel and Gasser, 2001]

[Basciano and Kleinstreuer, 2009]

[Choi and Vito, 1990] [Riveros et al., 2013]

(b) Anisotropic Model frame [Chuong and Fung, 1983]

[Ogden, 1972]

ψ = c0 exp[c1 (I1 − 3) − 1]

i=1

[Demiray, 1981]

ci (I1 − 3)i

ψ=

N ∑

Strain energy function

[Yeoh, 1993]

(a) Isotropic Model frame

33

−∞

ε

(ε − x)ρ(x)dx

(

)

σ(λ) =

{

0 c1 (λ − c0 )

, ,

λ ≤ c0 c0 < λ ≤ ∞

1 c0 λ λ − c2 +c , c2 < λ ≤ ∞. 2 CDF∆ (x): Cumulative Density Function of the triangular probability distribution { 0 , λ≤1 σ(λ) = 2c λ(1 + c1 sin θ)(λ2 − 1) exp[c2 (λ2 − 1)] , 1 < λ ≤ ∞ { 0 0 , λ ≤ c0 σ(λ) = c1 λ2 (λ/c0 − 1) , c0 < λ ≤ ∞

 

with Cauchy-Lorentz probability distribution: ρ(x) =

c1 2 {2π[c /4 + (x − c2 )2 ]} { c (x−c 1/c )c1 −1 1 3 2 for x ≤ c3 c2 [1+(x−c3 /c2 )c2 ]2 with log-logistic probability distribution: ρ(x) = 0 for x > c3  0 , λ ≤ c1    ∫ λ 1  2c0 λ (λ−c1 )32 , c < λ ≤ c2 +c 1 2 3(c2 −c1 ) ] [ σ(λ) = c0 λ CDF∆ (x)dx = (λ−c2 )3 c2 +c1 c2 +c1  , < λ ≤ c2 −∞  c0 λ λ − 2 3(c2 −c1 )2 − 2 2

σ(ε) = c0

∫

[Miyazaki and Hayashi, 1999, Zulliger and Stergiopulos, 2007]

[Gasser, 2011]

[Gasser et al., 2012]

[Martufi and Gasser, 2011, Gasser et al., 2012, Polzer et al., 2015]

[Zulliger et al., 2004a] (carotid)

[Wuyts et al., 1995]

Parameter identification

Table 3: Constitutive assumptions to model the (gradual) engagement of collagen fibrils. Fiber stress σ is expressed with respect to the applied fiber stretch λ or the fiber strain ε = λ − 1. Model parameters to be identified from experimental data are denoted by ci , and θ denotes the azimuthal fiber orientation angle, i.e. its orientation with respect to the vessel’s circumferential direction.

[Thunes et al., 2016]

[Gasser, 2011]

[Gasser et al., 2012]

[Martufi and Gasser, 2011]

[Decraemer et al., 1980]

Model frame

34

[Polzer et al., 2015] [Gasser et al., 2012]

ρ(ϕ) = c exp[c0 cos(2ϕ)] ρ(ϕ, θ) = c exp[c0 (cos ϕ cos θ)2 + c1 (cos ϕ sin θ)2 ]

Planar von Mises

Bingham [Bingham, 1974]

Table 4: Modeling the orientation of collagen fibers in the aortic wall. The orientation of a fiber in space is describe by azimuthal θ and elevation ϕ angles. Model∫parameters to be identified from experimental data are denoted by ci and c denotes a normalization parameter to ensure that ω ρdω = 1, i.e. when integrating over the all directions.

[Pichamuthu et al., 2013, Thunes et al., 2016]

[Schriefl et al., 2012]

ρ(ϕ) = c exp{c0 [cos(2ϕ) + 1]}

Transverse isotropy [Gasser et al., 2006]

Direct experimental histogram data

Parameter identification

Collagen orientation density model