Paul N. Watton

Modelling the mechanical response of elastin for arterial tissue.

We compare two constitutive models proposed to model the elastinous constituents of an artery. Holzapfel and Weizsäcker [1998. Biomechanical behavior of the arterial wall and its numerical characterization. Comput. Biol. Med. 28, 377-392] attribute a neo-Hookean response, i.e. Psi=c(I(1)-3)), to the elastin whilst Zulliger et al. [2004a. A strain energy function for arteries accounting for wall composition and structure. J. Biomech. 37, 989-1000] propose Psi=c(I(1)-3)(3/2). We analyse these constitutive models for two specific cases: (i) uniaxial extension of an elastinous sheet; (ii) inflation of a cylindrical elastinous membrane. For case (i) we illustrate the functional relationships between: (a) the Cauchy stress (CS) and the Green-Lagrange (GL) strain; (b) the tangent modulus (gradient of the CS-GL strain curve) and linearised strain. The predicted mechanical responses are compared with recent uniaxial extension tests on elastin [Gundiah, N., Ratcliffe, M.B., Pruitt, L.A., 2007. Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests. J. Biomech. 40, 586-594; Lillie, M.A., Gosline, J.M., 2007a. Limits to the durability of arterial elastic tissue. Biomaterials 28, 2021-2031; 2007b. Mechanical properties of elastin along the thoracic aorta in the pig. J. Biomech. 40, 2214-2221]. The neo-Hookean model accurately predicts the mechanical response of a single elastin fibre. However, it is unable to accurately capture the mechanical response of arterial elastin, e.g. the initial toe region of arterial elastin (if it exists) or the gradual increase in modulus of arterial elastin that occurs as it is stretched. The alternative constitutive model (n=32) yields a nonlinear mechanical response that departs from recent uniaxial test data mentioned above, for the same stretch range. For case (ii) we illustrate the pressure-circumferential stretch relationships and the gradients of the pressure-circumferential stretch curves: significant qualitative differences are observed. For the neo-Hookean model, the gradient decreases rapidly to zero, however, for n=32, the gradient decreases more gradually to a constant value. We conclude that whilst the neo-Hookean model has limitations, it appears to capture more accurately the mechanical response of elastin.

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

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.

Modelling the growth and stabilization of cerebral aneurysms

Experimental and theoretical guidance is needed to understand how the collagen fabric evolves during the development of aneurysms. In this paper, we model the development of an aneurysm as a cylindrical/spherical membrane subject to 1D enlargement; these conceptual models reflect the development of fusiform and saccular cerebral aneurysms. The mechanical response is attributed to the elastin and collagen. We introduce variables which define the elastin and collagen fibre concentration; these evolve to simulate growth/atrophy of the constituents. A hypothetical aneurysm model is analysed: collagen stretch is constant, elastin degrades and collagen fibre concentration can adapt to maintain mechanical equilibrium. An analytic expression for the rate of evolution of the fibre concentration is derived. The functional form is dependent on (i) the current collagen fibre concentration, (ii) the deviations in the collagen fibre stretch from the attachment stretch, (iii) the rate of change of fibre stretch, (iv) the rate of loss of elastin and (v) the ratio of load borne by elastinous and collagenous constituents. Finally, numerical examples of aneurysm development are considered. Suitable candidates for the fibre concentration evolution equations are identified that yield stabilization of the aneurysm even when there is complete loss of elastin. This theoretical analysis provides the basis for the development of physiologically realistic models of aneurysm development.

Modelling evolution and the evolving mechanical environment of saccular cerebral aneurysms

A fluid–solid-growth (FSG) model of saccular cerebral aneurysm evolution is developed. It utilises a realistic two-layered structural model of the internal carotid artery and explicitly accounts for the degradation of the elastinous constituents and growth and remodelling (G&R) of the collagen fabric. Aneurysm inception is prescribed: a localised degradation of elastin results in a perturbation in the arterial geometry; the collagen fabric adapts, and the artery achieves a new homeostatic configuration. The perturbation to the geometry creates an altered haemodynamic environment. Subsequent degradation of elastin is explicitly linked to low wall shear stress (WSS) in a confined region of the arterial domain. A sidewall saccular aneurysm develops, the collagen fabric adapts and the aneurysm stabilises in size. A quasi-static analysis is performed to determine the geometry at diastolic pressure. This enables the cyclic stretching of the tissue to be quantified, and we propose a novel index to quantify the degree of biaxial stretching of the tissue. Whilst growth is linked to low WSS from a steady (systolic) flow analysis, a pulsatile flow analysis is performed to compare steady and pulsatile flow parameters during evolution. This model illustrates the evolving mechanical environment for an idealised saccular cerebral aneurysm developing on a cylindrical parent artery and provides the guidance to more sophisticated FSG models of aneurysm evolution which link G&R to the local mechanical stimuli of vascular cells.

Impact of transmural heterogeneities on arterial adaptation

Recent experimental and computational studies have shown that transmurally heterogeneous material properties through the arterial wall are critical to understanding the heterogeneous expressions of constituent degrading molecules. Given that expression of such molecules is thought to be intimately linked to local magnitudes of stress, modelling the transmural stress distribution is critical to understanding arterial adaption during disease. The aim of this study was to develop an arterial growth and remodelling framework that can incorporate both transmurally heterogeneous constituent distributions and residual stresses, into a 3-D finite element model. As an illustrative example, we model the development of a fusiform aneurysm and investigate the effects of elastinous and collagenous heterogeneities on the stress distribution during evolution. It is observed that the adaptive processes of growth and remodelling exhibit transmural variations. For physiological heterogeneous constituent distributions, a stress peak appears in the media towards the intima, and a stress plateau occurs towards the adventitia. These features can be primarily attributed to the underlying heterogeneity of elastinous constituents. During arterial adaption, the collagen strain is regulated to remain in its homoeostatic level; consequently, the partial stress of collagen has less influence on the total stress than the elastin. However, following significant elastin degradation, collagen plays the dominant role for the transmural stress profile and a marked stress peak occurs towards the adventitia. We conclude that to improve our understanding of the arterial adaption and the aetiology of arterial disease, there is a need to: quantify transmural constituent distributions during histopathological examinations, understand and model the role of the evolving transmural stress distribution.

Rest versus exercise hemodynamics for middle cerebral artery aneurysms: a computational study

BACKGROUND AND PURPOSE: Exercise is an accepted method of improving cardiovascular health; however, the impact of increases in blood flow and heart rate on a cerebral aneurysms is unknown. This study was performed to simulate the changes in hemodynamic conditions within an intracranial aneurysm when a patient exercises. MATERIALS AND METHODS: Rotational 3D digital subtraction angiograms were used to reconstruct patient-specific geometries of 3 aneurysms located at the bifurcation of the middle cerebral artery. CFD was used to solve for transient flow fields during simulated rest and exercise conditions. Inlet conditions were set by using published transcranial Doppler sonography data for the middle cerebral artery. Velocity fields were analyzed and postprocessed to provide physiologically relevant metrics. RESULTS: Overall flow patterns were not significantly altered during exercise. Across subjects, during the exercise simulation, time-averaged WSS increased by a mean of 20% (range, 4%–34%), the RRT of a particle in the near-wall flow decreased by a mean of 28% (range, 13%–40%), and time-averaged pressure on the aneurysm wall did not change significantly. In 2 of the aneurysms, there was a 3-fold order-of-magnitude spatial difference in RRT between the aneurysm and surrounding vasculature. CONCLUSIONS: WSS did not increase significantly during simulated moderate aerobic exercise. While the reduction in RRT during exercise was small in comparison with spatial differences, there may be potential benefits associated with decreased RRT (ie, improved replenishment of nutrients to cells within the aneurysmal tissue).

Transitional flow in aneurysms and the computation of haemodynamic parameters

Haemodynamic forces appear to play an influential role in the evolution of aneurysms. This has led to numerous studies, usually based on computational fluid dynamics. Their focus is predominantly on the wall shear stress (WSS) and associated derived parameters, attempting to find correlations between particular patterns of haemodynamic indices and regions subjected to disease formation and progression. The indices are generally determined by integration of flow properties over a single cardiac cycle. In this study, we illustrate that in some cases the transitional flow in aneurysms can lead to significantly different WSS distributions in consecutive cardiac cycles. Accurate determination of time-averaged haemodynamic indices may thus require simulation of a large number of cycles, which contrasts with the common approach to determine parameters using data from a single cycle. To demonstrate the role of transitional flow, two exemplary cases are considered: flow in an abdominal aortic aneurysm and in an intracranial aneurysm. The key differences that are observed between these cases are explained in terms of the integral timescale of the transitional flows in comparison with the cardiac cycle duration: for relatively small geometries, transients will decay before the next cardiac cycle. In larger geometries, transients are still present when the systolic phase produces new instabilities. These residual fluctuations serve as random initial conditions and thus seed different flow patterns in each cycle. To judge whether statistics are converged, the derived indices from at least two successive cardiac cycles should be compared.

Investigating the Influence of Haemodynamic Stimuli on Intracranial Aneurysm Inception

We propose a novel method to reconstruct the hypothetical geometry of the healthy vasculature prior to intracranial aneurysm (IA) formation: a Frenet frame is calculated along the skeletonization of the arterial geometry; upstream and downstream boundaries of the aneurysmal segment are expressed in terms of the local Frenet frame basis vectors; the hypothetical healthy geometry is then reconstructed by propagating a closed curve along the skeleton using the local Frenet frames so that the upstream boundary is smoothly morphed into the downstream boundary. This methodology takes into account the tortuosity of the arterial vasculature and requires minimal user subjectivity. The method is applied to 22 clinical cases depicting IAs. Computational fluid dynamic simulations of the vasculature without IA are performed and the haemodynamic stimuli in the location of IA formation are examined. We observe that locally elevated wall shear stress (WSS) and gradient oscillatory number (GON) are highly correlated (20/22 for WSS and 19/22 for GON) with regions susceptible to sidewall IA formation whilst haemodynamic indices associated with the oscillation of the WSS vectors have much lower correlations.

Influence of differing material properties in media and adventitia on arterial adaptation — application to aneurysm formation and rupture

Experimental and computational studies suggest a substantial variation in the mechanical responses and collagen fibre orientations of the two structurally important layers of the arterial wall. Some observe the adventitia to be an order of magnitude stiffer than the media whilst others claim the opposite. Furthermore, studies show that molecular metabolisms may differ substantially in each layer. Following a literature review that juxtaposes the differing layer-specific results we create a range of different hypothetical arteries: (1) with different elastic responses, (2) different fibre orientations, and (3) different metabolic activities during adaptation. We use a finite element model to investigate the effects of those on: (1) the stress response in homeostasis; (2) the time course of arterial adaptation; and (3) an acute increase in luminal pressure due to a stressful event and its influence on the likelihood of aneurysm rupture. Interestingly, for all hypothetical cases considered, we observe that the adventitia acts to protect the wall against rupture by keeping stresses in the media and adventitia below experimentally observed ultimate strength values. Significantly, this conclusion holds true in pathological conditions.

Modelling evolution of saccular cerebral aneurysms

A computational model for the evolution of a saccular cerebral aneurysm of the internal carotid artery is developed. The artery is modelled as a two-layered non-linear elastic cylindrical membrane using a physiologically realistic constitutive model. It is subject to a constant systolic pressure and a physiological axial prestretch. The development of the aneurysm is assumed to be a consequence of the growth and remodelling of its material constituents. The elastinous constituents are prescribed to degrade in a localized circular patch. Microstructural recruitment and fibre concentration variables for the collagen fibres enable the growth and remodelling of collagen fabric to be simulated. The model predicts realistic saccular aneurysm geometries that stabilize in size. It provides the foundations for the development of more sophisticated models of cerebral aneurysm evolution.