Histo-Mechanical Modeling of the Wall of Abdominal Aorta Aneurysms

Histo-Mechanical Modeling of the Wall of Abdominal Aorta Aneurysms Giampaolo Martufi* and T. Christian Gasser** 

Solid Mechanics Department, The Royal Institute of Technology, Stockholm, Sweden, e-mail: *[email protected], ** [email protected] Abstract: Proteolytic degradation of elastin and collagen in the aortic wall may result in an Abdominal Aortic Aneurysms (AAAs), which rupture can be prevented by elective (surgical or minimal invasive) AAA repair. The biomechanical rupture risk assessment provides mechanical parameters like Peak Wall Stress (PWS) and Peak Wall Rupture Risk (PWRR) that support a patient-specific indication for AAA repair. Computing these parameters involve a non-linear biomechanical analysis, which to some extend depends on the constitutive description for the AAA wall. The late stage of AAA disease involves degradation of elastin and compensatory increased collagen in the AAA wall, such that its passive mechanical response can be modeled as fibrous collagenous tissue. Consequently, the present work is based on the assumption that the tissue's macroscopic mechanical properties being entirely governed by interlinked collagen fibrils. According to this approach, the spatial orientation and undulation of collagen fibrils are the most influential micro-histological parameters and macroscopic properties are derived through two integrations, i.e. once over the undulation and twice over the spatial orientation of fibers. Finally, the model accounts for collagen turn-over, where a stretch-based stimulus determines the production as well as the spatial orientation of collagen fibers. A micro-fiber approach was used to derive continuum mechanical properties of the AAA wall, where t-designs supported a fast integration over the spatial orientations of collagen fibers. The basic model response was studied with respect to a stepchange in model parameters on top of the homeostatic solution. The results from creep and relaxation tests indicated that the pre-stretch of collagen fibrils, that are deposited as part of collagen turnover, determine if vascular tissue expands or shrinks in time. Keywords: collagen turn-over, remodeling, growth, rupture, Finite Element, micro-plane  

1. INTRODUCTION Rupture of AAAs account for a large number of deaths particularly in men and elective repair (surgical or endovascular) is indicated if the risk of aneurysm rupture exceeds the interventional risks. Based on clinical studies a maximum AAA diameter of 55 mm (or more) is accepted as AAA repair indication (The United Kingdom Small Aneurysm Trial). However, the maximum diameter fails to describe AAA rupture risk patient-specifically (Brow and Powell 1999), and hence, results from these studies do not directly apply to individual patients. Consequently, the diameter criterion has clear limitations and other AAA rupture risk indicators have been suggested in the literature (Fillinger et al. 2007). One of these alternative approaches is the biomechanical rupture risk assessment, according to which the aneurysm wall will rupture as soon as the local mechanical stress overcomes the strength of the wall. Consequently, Peak Wall Stress (PWS; the maximum stress in the wall) and the Peak Wall Rupture Risk (PWRR; the maximum stress/strength ratio in the wall) are key biomechanical rupture risk indices. PWS and PWRR integrate a large number of well-known risk factors for AAA rupture. Several independent studies demonstrated that PWS and PWRR are more reliable rupture risk indicators than the maximum diameter, i.e. they

discriminate better between ruptured and non-ruptured AAAs (Fillinger et al. 2002; Venkatasubramaniam et al. 2004; Heng et al. 2008; Gasser et al. 2010; Maier et al. 2010). The computation of PWS and PWRR requires reliable constitutive descriptions of the AAA wall. Constitutive modeling of vascular tissue is an active field of research and numerous descriptions have been reported. However, the phenomenological approaches (Vaishnav et al. 1972; Fung et al. 1979; Choung et al. 1983; Takamizawa et al. 1987; Humphrey 1995) that have been successfully used to fit experimental data cannot allocate stress to the different histological constituents in the vascular wall. Structural constitutive descriptions (Lanir 1983; Wuyts et al. 1995, Holzapfel et al. 2000; Zullinger et al 2004; Gasser et al. 2006; Gasser et al. 2011a; Martufi et al. 2011) overcome this limitation and integrate histological and mechanical information of the arterial wall. In addition to the mathematical description of mechanical properties an efficient and implicit numerical implementation (Gasser et al. 2002; Gasser et al. 2006; Federico and Gasser 2010; Gasser et al. 2011a, b) of constitutive formulations is beneficial for analyzing clinically relevant problems. Already 60 years ago (Roach at al. 1957) suggested that collagen had a main impact on the mechanical properties of arterial tissue at high strain levels. Since that time a direct correlation between the collagen content and the stiffness and

strength has become generally accepted. 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 (Sokolis 2007). In addition to the volume fraction of collagen, its spatial arrangement, including the spread in orientations significantly affects the macroscopic mechanical properties (Gasser et al. 2006). The present work proposes a novel histo-mechanical constitutive model for aneurysmatic tissue, where collagen fibers are assembled by proteoglycan cross-linked collagen fibrils (CFPG-complex) and reinforce an otherwise isotropic matrix material (elastin). Multiplicative kinematics account for the straightening and stretching of collagen fibrils, and an orientation density function captures the spatial organization of collagen fibers in the tissue. Likewise, collagen fibers are dynamically formed by a continuous stretch-mediated process (Humphrey 1999), deposited in the current configuration and removed by a constant degradation rate. Finally the micro-fiber approach is used for the Finite Element implementation of the constitutive law. 2. THE STRUCTURE OF THE AAA WALL The extracellular matrix (ECM) provides an essential supporting scaffold for the structural and functional properties of vessel walls. The ECM mainly contains elastin, collagen, and proteoglycans (PGs) (Carey 1991) and their three-dimensional organization is vital to accomplish proper physiological functions. The ECM, therefore, rather than being merely a system of scaffolding for the surrounding cells, is an active mechanical structure that controls the micro-mechanical and macro-mechanical environments to which vascular tissue is exposed. Specifically, a proper understanding of the mechanical properties of the ECM is critically important to estimate and quantify the amount of stress and/or strain transmitted from the macroscopic to the cellular levels of vascular tissue. Collagen is one of the most dominant structural proteins in the ECM and critically involved in the gradual remodeling and weakening of the aneurysmal wall (Choke et al. 2005). Specifically, collagen fibrils (with diameter ranging from 50 to a few hundred of nanometers) are the basic building blocks of fibrous collagenous tissues (Fratzl 2008) and their organization into suprafibrilar structures determines the tissue's macroscopic mechanical properties. Consequently, biomechanical (MacSweeney et al 1992; He et al 1996, Gasser et al. 2006) and clinical studies (Inahara 1979) invariably show that the mechanics of the arterial wall essentially relies on fibrillar collagens in media and adventitia. The late stage of AAA disease is characterized by irreversible pathological remodeling of the aortic wall connective tissue, which, amongst many others, involve degradation of the elastin and compensatory increased collagen synthesis and content, see Choke et al. 2005 and references therein. Consequently, the passive mechanical response of a larger AAA can be modeled as fibrous collagenous tissue with

negligible contribution from the degraded and fragmented elastin. Information of collagen formation in the AAA wall permits a qualitative biomechanical understanding. However, the challenge is to relate it to engineering concepts and constitutive models, i.e. mathematical descriptions of biomechanical properties. The macroscopic properties of the AAA wall depend to a large extent on the 3D structural arrangement of collagen that is supposed to develop according to collagen turnover in the wall. The orientation of a collagen fiber in the threedimensional space is uniquely defined by its azimuthal angle ȣ and its elevation angleȰ. 3. CONSTITUTIVE MODELING For soft biological tissues numerous constitutive models have been reported, where some of them, histo-mechanical constitutive models say, aim at integrating collagen fiber density and orientation. Specifically, the pioneering work by Lanir (Lanir 1983) assumed macroscopic mechanical properties being governed by the arrangement of fibrous tissue components like collagen. According to his approach, the spatial orientation and undulation of collagen are the most influential micro-histological parameters that together with the fibers' constitution determine the macroscopic mechanical properties. Following a structural constitutive approach the macroscopic mechanical tissue properties are derived through two numerical integrations, i.e. (i) over the undulation and (ii) over the fibers' spatial orientation. This requires extensive computations and makes the application of such an approach somewhat limited. However, either the use of phenomenological collagen fiber models or specific assumptions regarding the constitution and undulation of collagen can avoid the numerical integration over the undulation of collagen. Consequently, the biomechanics of an entire AAAs can be analyzed by such a two-scale approach within reasonable computational times. 2.1 The CFPG-complex AAA wall tissue was regarded as a fibrous collagenous composite, where fibers of collagen reinforced an otherwise isotropic matrix material. Each collagen fiber is assembled by a bundle of collagen fibrils mutually interconnected by proteoglycan (PG) bridges that provide interfibrillar load transition, see Fig.1. Here, small proteoglycans like decorin bind noncovalently but specifically to the collagen fibril and cross-link adjacent collagen fibrils at about 60 nm intervals. Reversible deformability of the PG bridges is crucial to serve as shape-maintaining modules 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 sliding filament mechanisms of the twofold helix of the glycan (Scott et al. 2003). Alternatively, the close packing and cross-linking of collagen molecules in fibrils defines a virtually inextensible fiber, such that the strain within collagen fibrils is always much smaller than the

macroscopic strain in collagenous tissues, which also points towards the existence of gliding processes occurring at the interfibrilar and/or the interfiber levels (Gupta et al. 2010). Stretching a collagen fiber involves continuous recruitment of collagen fibrils, which if straightened, start carrying load. Here, a straightening stretch ɉୱ୲ defines the stretch beyond which the collagen fibril is stretched elastically and elastic energy is stored in the CFPG-complex, i.e. in the collagen fibril itself and in the PG-rich matrix between the fibrils. Finally, the spatial orientation of collagen fibers is captured by the orientation density functionɏሺȰǡ ȣ).

2.3 The micro-fiber approach Following Lanir's pioneering work (Lanir 1983) we assumed that the macroscopic Cauchy stress is defined by a superposition of individual collagen fiber contributions, i.e ஠ ଶ

࣌ൌ

ʹ ඵ ߩሺȰǡ ȣሻ ߪ†‡˜ሺ‫ܕ ٔ ܕ‬ሻ …‘• Ȱ †Ȱ†ȣ ൅ ‫݌‬۷ሺʹሻ Ɏ ଴

where ‫ ܕ‬ൌ ۴‫ۻ‬Τȁ۴‫ۻ‬ȁdenotes the spatial orientation vector of the collagen fiber and dev(●) denotes the spatial deviator operator. In Eq. 2 the constitution of the collagen fiber is incorporated through its Cauchy stress ߪሺߣሻ and the term ‫݌‬۷ denotes the hydrostatic stress with the Lagrange parameter ‫݌‬ that is defined by the problem's boundary conditions, i.e. cannot be computed from the tissue's constitutive law. The integral in Eq. 2 was numerically approximated by spherical designs (Hardin and Sloane 1996; Federico and ௟౟౤౪ ሺȈሻ௟ where Gasser 2010), i.e. ‫׬‬ఠሺȈሻ †߱ ൎ ሺͶߨΤ݈୧୬୲ ሻ σ௟ୀଵ ݈୧୬୲ denoted the total number of integration points. 2.4 Collagen turn-over model

Fig. 1. The hierarchical structure of a collagen fiber built up of statistically-distributed collagen fibrils interlinked by proteoglycan (PG) bridges. The constitution of collagen fibrils is defined by a virtually linear response stress-strain law and a triangular probability density function (PDF) relates the stretch that is required to engage collagen fibrils (engagement stretch) when exposing the collagen fiber to a stretchߣ. Here, the first and last fibrils within a collagen fiber straighten at the fiber stretches of ߣ୫୧୬ and ɉ୫ୟ୶ respectively. As detailed elsewhere (Martufi et al. 2011) these assumptions yield the piece-wise analytical expressions for the collagen fiber’s Cauchy stress ͲǡͲ ൏ ߣ ൑ ɉ୫୧୬ ‫ۓ‬ ۖ ʹ ɉሺɉ െ ɉ ሻଷ ǡɉ ത ۖ ୫୧୬ ୫୧୬ ൏ ߣ ൑ ɉ ͵ȟɉଶ ሺͳሻ
ɐሺɉሻ ൌ ଷ ‫ ۔‬ɉ ቈɉ െ ʹሺɉ െ ɉ୫ୟ୶ ሻ െ ɉത቉ɉത ൏ ߣ ൑ ɉ  ୫ୟ୶ ͵ȟɉଶ ۖ ۖ ‫ە‬ɉ൫ɉ െ ɉത൯ǡɉ୫ୟ୶ ൏ ߣ ൏ λ where ȟɉ ൌ ɉ୫ୟ୶ െ ɉ୫୧୬ ƒ†ɉത ൌ ሺɉ୫ୟ୶ ൅ ɉ୫୧୬ ሻΤʹ and  defines the stiffness of the CFPG-complex. Finally, an affine deformation between the continuum and the collagen fiber is considered, which completely describes the multi-scale kinematics of the fibrous tissue.

Collagen turnover in the AAA wall is thought to be accomplished by fibroblast cells, which are spread throughout the collagen network. Fibroblasts sense the state of strain and continuously produce and degrade collagen fibrils. Specifically, the ratio ߣΤߣ୬ୣ୵ ୮୦ between the present ߣ ୬ୣ୵ and physiological (homeostatic)ߣ୮୦ stretches is thought to stimulate the production of collagen fibrils. To maintain a macroscopic stress during turnover of collagen, the collagen fibrils have to be deposited at a certain prestretch. The applied structural concept implicitly defines the pre-stretch of collagen fibrils through setting the limits of the triangular distribution, i.e. by defining the undulation limits ୬ୣ୵ ߣ୬ୣ୵ ୫୧୬  and ߣ୫ୟ୶ of the newly deposited fibrils. Moreover, we assumed that collagen is resolved isotropically and has a halflife time of 60 days (Humphrey 1999). Finally, for biological reasons, the collagen production is limited in the model. 4. RESULTS Model predictions have been investigated in response to a step-change in model parameters. To this end a single cubic tissue element was exposed to a mechanical environment that is in-vivo experienced by the AAA wall. Specifically, the model parameters were set such that homeostasis was reached at a stress of 100 kPa and a stretch of 1.06. To this end an initially isotropic collagen distribution (ߩሺȰǡ ȣሻ ൌ …‘•–) was used and the collagen fibril deposition stretch was ୬ୣ୵ set toߣ୬ୣ୵ ୫୧୬ ൌ ͲǤͻ͹ƒ†ߣ୫ୟ୶ ൌ ͳǤʹ͵. Then collagen turnover was simulated until homeostasis was reached, which in turn defined the initial condition for alternating the pre-stretch of collagen fibrils. The impact of the alternation was studied (i) by a creep test, i.e. recording the displacements at fixed stress (Fig. 2) and (ii) by a relaxation test, i.e. recording the stress at fixed displacement (Fig. 3).

The computed results indicated a remarkable impact of the pre-stretch of collagen fibrils that were deposited as part of the collagen turnover in the AAA wall. Specifically, depositing collagen fibrils at a certain stretch defined homeostatic conditions and the mechanical state variable remained fixed in time. AAAs not always enlarge, sometimes they are stable or even shrink over time (Brady et al. 2004). This response of AAAs is captured by our model. While, depositing collagen fibrils at high undulation triggered continuous growth, placing the newly-formed collagen fibrils at lower undulation caused the tissue to shrink.

1 Onmeiwn = 0.97 Onmeawx = 1.23 Onmeiwn = 0.98 Onmeawx = 1.24

0.8

displacement (mm)

Onmeiwn = 0.96 Onmeawx = 1.22 0.6

0.4

0.2

0

-0.2

0

20

40

60

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Fig. 2. Creep test response of the Abdominal Aortic Aneurysm (AAA) wall with respect to a step-change in collagen pre-stretch at. Increasing the undulation of collagen fibrils, i.e. decreasing their pre-stretch at deposition, caused the tissue to creep (creep test) and to relax (relaxation). The opposite response is observed when decreasing the undulation of collagen. Note that the creep test protocol closer reflects the in-vivo conditions of the AAA wall. At creep test conditions and for the case of increased collagen undulation, the time to reach a new homeostatic condition is so large that the tissue virtually keeps on growing.

First Piola Kirchhoff stress (kPa)

250

200

0.98

Onmeawx

= 1.24

Onmeiwn = 0.96 Onmeawx = 1.22

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The proposed model has a strong biological motivation and integrates the fibril and fiber levels of collagen with the tissue’s macroscopic properties. Apart from modeling the tissue’s passive response, the model is helpful to understand the impact of collagen turnover on the macroscopic properties of the tissue and to predict saline feature of aneurysm growth and remodeling. Likewise, the constitutive concept renders a highly efficient and robust multi-scale approach that allows simulating in-vivo growth of patientspecific vascular geometry. Finally, although the model could successfully capture some features of AAA development, a rigorous validation against experimental data would be crucial to evaluate its descriptive and predictive capabilities.

REFERENCES

Onmeiwn = 0.97 Onmeawx = 1.23 Onmeiwn =

The numerical integration of Eq. 2 required sufficiently accurate spherical t-designs, and a 21-design (involving 240 integration points) was used in the present study. However, the concentrated (anisotropic) collagen orientation distribution that develops as a response of the uniaxial tension protocol might require even higher-order integration schemas.

100

120

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Fig. 3. Relaxation test response of the Abdominal Aortic Aneurysm (AAA) with respect to a step-change in collagen pre-stretch. 5. DISCUSSIONS AND CONCLUSIONS The present work enriched our previously reported multiscale constitutive model (Martufi et al. 2011) by considering collagen turnover. Specifically, the model’s collagen fibril level provided an interface to integrate the vascular wall biology and to study the impact of collagen turnover on the macroscopic properties of AAAs.

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