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Research Papers

Coupling the Hemodynamic Environment to the Evolution of Cerebral Aneurysms: Computational Framework and Numerical Examples

[+] Author and Article Information
Paul N. Watton1

Department of Engineering Science and Institute of Biomedical Engineering, University of Oxford, Oxford OX1 3PJ, UKpaul.watton@eng.ox.ac.uk

Nikolaus B. Raberger

 ETH Zurich, 8092 Zurich, Switzerlandnikolaus.raberger@alumni.ethz.ch

Gerhard A. Holzapfel

 Graz University of Technology, Institute of Biomechanics, Centre of Biomedical Engineering, 8010 Graz, Austria; Department of Solid Mechanics Royal Institute of Technology, School of Engineering Sciences, Osquars Backe 1, 100 44 Stockholm, Swedenholzapfel@tugraz.at

Yiannis Ventikos

Department of Engineering Science and Institute of Biomedical Engineering, University of Oxford, Oxford OX1 3PJ, UKyiannis.ventikos@eng.ox.ac.uk

1

Corresponding author.

J Biomech Eng 131(10), 101003 (Sep 01, 2009) (14 pages) doi:10.1115/1.3192141 History: Received November 20, 2008; Revised June 10, 2009; Published September 01, 2009

The physiological mechanisms that give rise to the inception and development of a cerebral aneurysm are accepted to involve the interplay between the local mechanical forces acting on the arterial wall and the biological processes occurring at the cellular level. In fact, the wall shear stresses (WSSs) that act on the endothelial cells are thought to play a pivotal role. A computational framework is proposed to explore the link between the evolution of a cerebral aneurysm and the influence of hemodynamic stimuli that act on the endothelial cells. An aneurysm evolution model, which utilizes a realistic microstructural model of the arterial wall, is combined with detailed 3D hemodynamic solutions. The evolution of the blood flow within the developing aneurysm determines the distributions of the WSS and the spatial WSS gradient (WSSG) that act on the endothelial cell layer of the tissue. Two illustrative examples are considered: Degradation of the elastinous constituents is driven by deviations of WSS or the WSSG from normotensive values. This model provides the basis to further explore the etiology of aneurysmal disease.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 17

Axial profiles of the evolution of the (a) radius, (b) elastin concentration mE, (c) WSS, and (d) WSSG for t=5, 6, 7, and 8 years. Degradation of elastin is explicitly linked to the level of WSSG.

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Figure 1

Overview of the artery model in the unloaded reference configuration Ω0 and the initial loaded configuration Ωt

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Figure 2

Attachment of collagen fibers in altered configurations: (1) Ω0 is the unloaded reference configuration of the tissue in which the fibers have a characteristic waviness and the stretch λ of the elastin is 1; (2) Ω0R is the initial recruitment configuration of the collagen fibers. The fibers have straightened out (λC=1) and begin to bear load; (3) Ω0t is the initial loaded configuration such that the stretch in the collagen fibers is the equilibrium value, i.e., at systolic pressure the stretch of the fibers is λatC. Thereby, the tissue is in homeostasis. Although the fibers are in a continual state of degradation and deposition, new fibers will attach with identical levels of stretch to those that decay. Thus, no changes occur to the mechanical properties of the tissue; (4) suppose the tissue is stretched further and, for the purposes of this example, held at fixed length. The stretch of the tissue is then λ=λCλ0R, where λC>λatC. The collagen fabric is no longer in material equilibrium: Subsequent fiber deposition and degradation will result in a change in the configuration of the collagenous tissue. New fibers attach to the tissue so that at the new systolic configuration their stretch is λatC. The old fibers decay, and the distribution of collagen fibers changes. The naturally occurring turnover of the fibers will proceed to restore stretch in all the fibers to equilibrium levels; (5) all of the old fibers have decayed and have been replaced with new fibers of stretch λatC. The artery reaches a new equilibrium configuration Ωt; (6) ΩR is the new recruitment configuration of the collagen fibers following remodeling of the tissue; (7) if the tissue is contracted back to the reference configuration Ω0, the crimp of the collagen will have increased. Equivalently, the factor the tissue must be stretched for the collagen to be recruited has increased. Hence, the effects of deposition and degradation in altered configurations can be captured by remodeling the recruitment stretch. Note that the time for the tissue to remodel from state (4) to state (5) is dependent on the turnover rate of the collagen fibers, i.e., the half-life. Equivalently, the rate at which the recruitment stretch remodels is dependent on the turnover rate of the fibers.

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Figure 3

Pressure-diameter relationship for the model of the internal carotid artery. Relative radius change is 10% between systole and diastole (diastolic radius is 1.82 mm) and the onset of load bearing of the collagen begins in the initial diastolic configuration. Note that at systolic pressure the material parameters are determined such that the elastinous constituents bear 80% of the load (input data for the model are provided in Table 1).

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Figure 15

Evolution of the Green–Lagrange strains (a) E11 and (b) E22 in the elastin (left) and the Green–Lagrange strain in the collagen in the media (c) EM+C and the adventitia (d) EA+C, for t=5, 6, 7, and 8 years. Even though the aneurysm is subject to large deformation the collagen strains increase only slightly due to remodeling of the configuration at which they are recruited to load bearing.

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Figure 16

Collagen fiber concentration mC at t=5, 6, 7, and 8 years

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Figure 4

Vessel geometry with developed aneurysm. The central section illustrated in the figure is the deformable region where the proposed wall model is used. Note that to achieve fully developed flow in the region where the aneurysm develops, extensions are attached to the computational domain.

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Figure 5

Computational mesh for the fluid domain

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Figure 6

The elastinous constituents are prescribed to degrade in a circular patch in the center of a domain in the unloaded reference configuration Ω0. The degradation within the inner circular patch of radius RiE is prescribed. The degradation in the outer circular annulus, with outer radius RoE, is linearly interpolated using radial splines between the values on the inner annulus and the values on the outer annulus where no degradation occurs. Throughout the remaining domain no degradation of the elastinous constituents occurs.

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Figure 7

Degradation of elastin FD versus wall shear stress τ utilizing the values τcrit=2 Pa and τX=0.5 Pa; see Eq. 26

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Figure 8

Degradation of elastin FD versus wall shear stress gradient utilizing the values |∇τ|crit=0.5 Pa m−1 and |∇τ|X=2 Pa m−1; see Eq. 27

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Figure 9

A small localized outpouching of the artery is generated (for the geometry compare with Fig. 6). According to Eq. 19 the elastin concentration mE is prescribed to degrade in a circular small patch in the center of the domain. The collagen remodels until the artery achieves a new homeostasis.

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Figure 10

Spatial distribution of Green–Lagrange strains E11 and E22 of the elastin at t=4 and t=5. Note that there is negligible change in the magnitudes and thus the geometry as the aneurysm stabilizes.

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Figure 11

Green–Lagrange strains of the collagen fibers EM+C in the media at t=4 and t=5 ((a) and (c)). At t=4 the collagen strains are slightly elevated; however, at t=5, they have restored to homeostatic levels throughout the domain. Average fiber concentration mC in the medial and adventitial layers at t=4 and t=5 ((b) and (d)). The concentration increases to compensate for the loss of elastin and the artery achieves a new homeostasis.

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Figure 12

The small outpouching of the artery alters the hemodynamics and thus the spatial distribution of the magnitudes of the (a) WSS and (b) the WSSGs. The distributions are illustrated at t=5.

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Figure 13

Distributions of wall shear stress τ ((a)–(c)) and elastin concentration mE in the media and adventitia ((d)–(f)) at t=6, 7, and 8 years

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Figure 14

Axial profiles of the evolution of the (a) radius, (b) elastin concentration mE, (c) WSS, and (d) WSSG for t=5, 6, 7, and 8 years. Degradation of elastin is linked to the level of WSS.

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