Modelling the influence of endothelial heterogeneity on the progression of arterial disease: application to abdominal aortic aneurysm evolution.

We sophisticate a fluid-solid growth computational framework for modelling aneurysm evolution. A realistic structural model of the arterial wall is integrated into a patient-specific geometry of the vasculature. This enables physiologically representative distributions of haemodynamic stimuli, obtained from a rigid-wall computational fluid dynamics analysis, to be linked to growth and remodelling algorithms. Additionally, a quasistatic structural analysis quantifies the cyclic deformation of the arterial wall so that collagen growth and remodelling can be explicitly linked to the cyclic deformation of vascular cells. To simulate aneurysm evolution, degradation of elastin is driven by reductions in wall shear stress (WSS) below homeostatic thresholds. Given that the endothelium exhibits spatial and temporal heterogeneity, we propose a novel approach to define the homeostatic WSS thresholds: We allow them to be spatially and temporally heterogeneous. We illustrate the application of this novel fluid-solid growth framework to model abdominal aortic aneurysm (AAA) evolution and to examine how the influence of the definition of the WSS homeostatic threshold influences AAA progression. We conclude that improved understanding and modelling of the endothelial heterogeneity is important for modelling aneurysm evolution and, more generally, other vascular diseases where haemodynamic stimuli play an important role.

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.

Effect of bending rigidity in a dynamic model of a polyurethane prosthetic mitral valve.

We investigate the behaviour of a dynamic fluid-structure interaction model of a chorded polyurethane mitral valve prosthesis, focusing on the effects on valve dynamics of including descriptions of the bending stiffnesses of the valve leaflets and artificial chordae tendineae. Each of the chordae is attached at one end to the valve annulus and at the other to one of two chordal attachment points. These attachment points correspond to the positions where the chords of the real prosthesis would attach to the left-ventricular wall, although in the present study, these attachment points are kept fixed in space to facilitate comparison between our simulations and earlier results obtained from an experimental test rig. In our simulations, a time-dependent pressure difference derived from experimental measurements drives flow through the model valve during diastole and provides a realistic pressure load during systole. In previous modelling studies of this valve prosthesis, the valve presents an unrealistically large orifice at beginning of diastole and does not close completely at the end of diastole. We show that including a description of the chordal bending stiffness enables the model valve to close properly at the end of the diastolic phase of the cardiac cycle. Valve over-opening is eliminated only by incorporating a description of the bending stiffnesses of the valve leaflets into the model. Thus, bending stiffness plays a significant role in the dynamic behaviour of the polyurethane mitral valve prosthesis.

Multi-scale interaction of particulate flow and the artery wall.

We discuss, from the perspective of basic science, the physical and biological processes which underlie atherosclerotic (plaque) initiation at the vascular endothelium, identifying the widely separated spatial and temporal scales which participate. We draw on current, related models of vessel wall evolution, paying particular attention to the role of particulate flow (blood is not a continuum fluid), and proceed to propose, then validate all the key components in a multiply-coupled, multi-scale modeling strategy (in qualitative terms only, note). Eventually, this strategy should lead to a quantitative, patient-specific understanding of the coupling between particulate flow and the endothelial state.

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.

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).

Impact of transmural heterogeneities on arterial adaptation: application to aneurysm formation.

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.

Evolving mechanical properties of a model of abdominal aortic aneurysm.

The novel three-dimensional (3D) mathematical model for the development of abdominal aortic aneurysm (AAA) of Watton et al. Biomech Model Mechanobiol 3(2): 98-113, (2004) describes how changes in the micro-structure of the arterial wall lead to the development of AAA, during which collagen remodels to compensate for loss of elastin. In this paper, we examine the influence of several of the model's material and remodelling parameters on growth rates of the AAA and compare with clinical data. Furthermore, we calculate the dynamic properties of the AAA at different stages in its development and examine the evolution of clinically measurable mechanical properties. The model predicts that the maximum diameter of the aneurysm increases exponentially and that the ratio of systolic to diastolic diameter decreases from 1.13 to 1.02 as the aneurysm develops; these predictions are consistent with physiological observations of Vardulaki et al. Br J Surg 85:1674-1680 (1998) and Lanne et al. Eur J Vasc Surg 6:178-184 (1992), respectively. We conclude that mathematical models of aneurysm growth have the potential to be useful, noninvasive diagnostic tools and thus merit further development.

Computational modelling for cerebral aneurysms: risk evaluation and interventional planning.

Advanced computational techniques offer a new array of capabilities in the healthcare provision for cerebral aneurysms. In this paper information is provided on specific simulation methodologies that address some of the unanswered questions about intracranial aneurysm and their treatment. These include the evaluation of rupture risk, the thrombogenic characteristics of specific lesions and the efficacy assessment of particular interventional techniques and devices (e.g. endovascular coil embolisation and flow diversion using stents). The issues connected with ease-of-use and interactivity of computed simulations is discussed, and it is concluded, that the potential of these techniques to optimise planning of complex and multifaceted interventions is very significant, in spite of the fact that most of the methodologies described are still being developed and perfected.

Modelling the interaction of haemodynamics and the artery wall: current status and future prospects.

Clinical research has historically focused on the two main strategies of in vivo and in vitro experimentation. The concept of applying scientific theory to direct clinical applications is relatively recent. In this paper we focus on the interaction of wall shear stress with the endothelium and discuss how 'state of the art' computer modelling techniques can provide valuable data to aid understanding. Such data may be used to inform experiment and further, may help identify the key features of this complex system. Current emphasis is on coupling haemodynamics with models of biological phenomena to test hypotheses or predict the likely outcome of a disease or an intervention. New technologies to enable the integration of models of different types, levels of complexity and scales, are being developed. As will be discussed, the ultimate goal is the translation of this technology to the clinical arena.

Dynamic modelling of prosthetic chorded mitral valves using the immersed boundary method.

Current artificial heart valves either have limited lifespan or require the recipient to be on permanent anticoagulation therapy. In this paper, effort is made to assess a newly developed bileaflet valve prosthesis made of synthetic flexible leaflet materials, whose geometry and material properties are based on those of the native mitral valve, with a view to providing superior options for mitral valve replacement. Computational analysis is employed to evaluate the geometric and material design of the valve, by investigation of its mechanical behaviour and unsteady flow characteristics. The immersed boundary (IB) method is used for the dynamic modelling of the large deformation of the valve leaflets and the fluid-structure interactions. The IB simulation is first validated for the aortic prosthesis subjected to a hydrostatic loading. The predicted displacement fields by IB are compared with those obtained using ANSYS, as well as with experimental measurements. Good quantitative agreement is obtained. Moreover, known failure regions of aortic prostheses are identified. The dynamic behaviour of the valve designs is then simulated under four physiological pulsatile flows. Experimental pressure gradients for opening and closure of the valves are in good agreement with IB predictions for all flow rates for both aortic and mitral designs. Importantly, the simulations predicted improved physiological haemodynamics for the novel mitral design. Limitation of the current IB model is also discussed. We conclude that the IB model can be developed to be an extremely effective dynamic simulation tool to aid prosthesis design.

A mathematical model for the growth of the abdominal aortic aneurysm.

We present the first mathematical model to account for the evolution of the abdominal aortic aneurysm. The artery is modelled as a two-layered, cylindrical membrane using nonlinear elasticity and 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 remodelling of its material constituents. Microstructural 'recruitment' and fibre density variables for the collagen are introduced into the strain energy density functions. This enables the remodelling of collagen to be addressed as the aneurysm enlarges. An axisymmetric aneurysm, with axisymmetric degradation of elastin and linear differential equations for the remodelling of the fibre variables, is simulated numerically. Using physiologically determined parameters to model the abdominal aorta and realistic remodelling rates for its constituents, the predicted dilations of the aneurysm are consistent with those observed in vivo. An asymmetric aneurysm with spinal contact is also modelled, and the stress distributions are consistent with previous studies.

Modelling chorded prosthetic mitral valves using the immersed boundary method.

The Immersed Boundary (IB) Method is an efficient method of modelling fluid structure interactions. However, it has two main limitations: ease of use and ability to model static loading. In this paper, the method is developed, so that it can efficiently and easily model any multileaflet elastic structure. The structure may include chordae, which attach to the leaflets and continue through the leaflet surfaces. In addition, an external surface pressure may be applied to the leaflets, thus enabling the deformations that arise under steady loads to be solved. This method is validated for a model of the native mitral valve under systolic loading and for a prosthetic aortic valve under static loading. It is then applied to a new chorded prosthetic mitral valve, housed in a cylindrical tube, subject to a physiological periodic fluid flow. Results are compared with those obtained by using the commercial package ANSYS as well as with experimental measurements. Qualitative agreements are obtained. There are some discrepancies due to the current IB method being unable to model bending and shear behaviour. In particular, the fibre structures of the new prosthetic valve model developed using the IB method may be prone to crimping. Further development of the IB method is necessary to include bending effects. This will improve the accuracy of both the dynamic and static analysis.