A thick-walled fluid-solid-growth model of abdominal aortic aneurysm evolution: application to a patient-specific geometry.

We propose a novel thick-walled fluid-solid-growth (FSG) computational framework for modeling vascular disease evolution. The arterial wall is modeled as a thick-walled nonlinearly elastic cylindrical tube consisting of two layers corresponding to the media-intima and adventitia, where each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component. Blood is modeled as a Newtonian fluid with constant density and viscosity; no slip and no-flux conditions are applied at the arterial wall. Disease progression is simulated by growth and remodeling (G&R) of the load bearing constituents of the wall. Adaptions of the natural reference configurations and mass densities of constituents are driven by deviations of mechanical stimuli from homeostatic levels. We apply the novel framework to model abdominal aortic aneurysm (AAA) evolution. Elastin degradation is initially prescribed to create a perturbation to the geometry which results in a local decrease in wall shear stress (WSS). Subsequent degradation of elastin is driven by low WSS and an aneurysm evolves as the elastin degrades and the collagen adapts. The influence of transmural G&R of constituents on the aneurysm development is analyzed. We observe that elastin and collagen strains evolve to be transmurally heterogeneous and this may facilitate the development of tortuosity. This multiphysics framework provides the basis for exploring the influence of transmural metabolic activity on the progression of vascular disease.

A chemo-mechano-biological modeling framework for cartilage evolving in health, disease, injury, and treatment.

BACKGROUND AND OBJECTIVE: Osteoarthritis (OA) is a pervasive and debilitating disease, wherein degeneration of cartilage features prominently. Despite extensive research, we do not yet understand the cause or progression of OA. Studies show biochemical, mechanical, and biological factors affect cartilage health. Mechanical loads influence synthesis of biochemical constituents which build and/or break down cartilage, and which in turn affect mechanical loads. OA-associated biochemical profiles activate cellular activity that disrupts homeostasis. To understand the complex interplay among mechanical stimuli, biochemical signaling, and cartilage function requires integrating vast research on experimental mechanics and mechanobiology-a task approachable only with computational models. At present, mechanical models of cartilage generally lack chemo-biological effects, and biochemical models lack coupled mechanics, let alone interactions over time. METHODS: We establish a first-of-its kind virtual cartilage: a modeling framework that considers time-dependent, chemo-mechano-biologically induced turnover of key constituents resulting from biochemical, mechanical, and/or biological activity. We include the "minimally essential" yet complex chemical and mechanobiological mechanisms. Our 3-D framework integrates a constitutive model for the mechanics of cartilage with a novel model of homeostatic adaptation by chondrocytes to pathological mechanical stimuli, and a new application of anisotropic growth (loss) to simulate degradation clinically observed as cartilage thinning. RESULTS: Using a single set of representative parameters, our simulations of immobilizing and overloading successfully captured loss of cartilage quantified experimentally. Simulations of immobilizing, overloading, and injuring cartilage predicted dose-dependent recovery of cartilage when treated with suramin, a proposed therapeutic for OA. The modeling framework prompted us to add growth factors to the suramin treatment, which predicted even better recovery. CONCLUSIONS: Our flexible framework is a first step toward computational investigations of how cartilage and chondrocytes mechanically and biochemically evolve in degeneration of OA and respond to pharmacological therapies. Our framework will enable future studies to link physical activity and resulting mechanical stimuli to progression of OA and loss of cartilage function, facilitating new fundamental understanding of the complex progression of OA and elucidating new perspectives on causes, treatments, and possible preventions.

A constrained mixture-micturition-growth (CMMG) model of the urinary bladder: Application to partial bladder outlet obstruction (BOO).

We present a constrained mixture-micturition-growth (CMMG) model for the bladder. It simulates bladder mechanics, voiding function (micturition) and tissue adaptations in response to altered biomechanical conditions. The CMMG model is calibrated with both in vivo and in vitro data from healthy male rat urinary bladders (cystometry, bioimaging of wall structure, mechanical testing) and applied to simulate the growth and remodeling (G&R) response to partial bladder outlet obstruction (BOO). The bladder wall is represented as a multi-layered, anisotropic, nonlinear constrained mixture. A short time scale micturition component of the CMMG model accounts for the active and passive mechanics of voiding. Over a second, longer time scale, G&R algorithms for the evolution of both cellular and extracellular constituents act to maintain/restore bladder (homeostatic) functionality. The CMMG model is applied to a spherical membrane model of the BOO bladder utilizing temporal data from an experimental male rodent model to parameterize and then verify the model. Consistent with the experimental studies of BOO, the model predicts: an initial loss of voiding capacity followed by hypertrophy of SMC to restore voiding function; bladder enlargement; collagen remodeling to maintain its role as a protective sheath; and increased voiding duration with lower average flow rate. This CMMG model enables a mechanistic approach for investigating the bladder's structure-function relationship and its adaption in pathological conditions. While the approach is illustrated with a conceptual spherical bladder model, it provides the basis for application of the CMMG model to anatomical geometries. Such a mechanistic approach has promise as an in silico tool for the rational development of new surgical and pharmacological treatments for bladder diseases such as BOO.

Modeling intracranial aneurysm stability and growth: an integrative mechanobiological framework for clinical cases.

We present a novel patient-specific fluid-solid-growth framework to model the mechanobiological state of clinically detected intracranial aneurysms (IAs) and their evolution. The artery and IA sac are modeled as thick-walled, non-linear elastic fiber-reinforced composites. We represent the undulation distribution of collagen fibers: the adventitia of the healthy artery is modeled as a protective sheath whereas the aneurysm sac is modeled to bear load within physiological range of pressures. Initially, we assume the detected IA is stable and then consider two flow-related mechanisms to drive enlargement: (1) low wall shear stress; (2) dysfunctional endothelium which is associated with regions of high oscillatory flow. Localized collagen degradation and remodelling gives rise to formation of secondary blebs on the aneurysm dome. Restabilization of blebs is achieved by remodelling of the homeostatic collagen fiber stretch distribution. This integrative mechanobiological modelling workflow provides a step towards a personalized risk-assessment and treatment of clinically detected IAs.

Shear stress rosettes capture the complex flow physics in diseased arteries.

Wall shear stress (WSS) is an important parameter in arterial mechanobiology. Various flow metrics, such as time averaged WSS (TAWSS), oscillatory shear index (OSI), and transWSS, have been used to characterize and relate possible WSS variations in arterial diseases like aneurysms and atherosclerosis. We use a graphical representation of WSS using shear rosettes to map temporal changes in the flow dynamics during a cardiac cycle at any spatial location on the vessel surface. The presence of secondary flows and flow reversals can be interpreted directly from the shape of the shear rosette. The mean WSS is given by the rosette centroid, the OSI by the splay around the rosette origin, and the transWSS by its width. We define a new metric, anisotropy ratio (AR), based on the ratio of the length to width of the shear rosette, to capture flow bi-directionality. We characterized the flow physics in controls and patient specific geometries of the ascending aorta (AA) and internal carotid artery (ICA) that have fundamentally different flow dynamics due to differences in the Reynolds and Womersley numbers. The differences in the flow dynamics are well reflected in the shapes of the WSS rosettes and the corresponding flow metrics.

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.

Optimization schemes for endovascular repair with parallel technique based on hemodynamic analyses.

Endovascular repair with parallel stent-grafts (SG) is a challenging technique that reconstructs the luminal flow pathways by implanting parallel-placed SGs into the vessel. After treatment, occlusion and shifting of the parallel SGs are sometimes reported, which could be fatal and difficult to be re-operated. These issues are highly related to the local hemodynamic conditions in the stented region. In this study, a patient case treated by the octopus endograft technique (a head-SG with three limb-SGs) and experienced limb-SG occlusion is studied. 3-D models are established based on computed tomography (CT) angiography datasets pretreatment and posttreatment as well as during follow-ups. Hemodynamic quantities such as pressure drop, wall shear stress-related parameters, and flow division in limb-SGs and visceral arteries are quantitatively investigated. Optimizations on the length of the head-SG and diameter of the limb-SGs are analyzed based on various scenarios. The results indicate that when reconstructing the flow pathways via octopus stenting, it is important to ensure the flow distribution as physiologically required with this new morphology. Position (or length) of the head-SG and diameter of the limb-SGs play an important role in controlling flow division, and high time average wall shear stress (TAWSS) around the head-SG acts as a main factor for graft immigration. This study, by proposing optimization suggestions with hemodynamic analyses for a specific case, implicates that pretreatment SG scenarios may assist in wise selection and placement of the device and thus may improve long-term effectiveness of this kind of challenging endovascular repair techniques.

The unexplained success of stentplasty vasospasm treatment : Insights using Mechanistic Mathematical Modeling.

BACKGROUND: Cerebral vasospasm (CVS) following subarachnoid hemorrhage occurs in up to 70% of patients. Recently, stents have been used to successfully treat CVS. This implies that the force required to expand spastic vessels and resolve vasospasm is lower than previously thought. OBJECTIVE: We develop a mechanistic model of the spastic arterial wall to provide insight into CVS and predict the forces required to treat it. MATERIAL AND METHODS: The arterial wall is modelled as a cylindrical membrane using a constrained mixture theory that accounts for the mechanical roles of elastin, collagen and vascular smooth muscle cells (VSMC). We model the pressure diameter curve prior to CVS and predict how it changes following CVS. We propose a stretch-based damage criterion for VSMC and evaluate if several commercially available stents are able to resolve vasospasm. RESULTS: The model predicts that dilatation of VSMCs beyond a threshold of mechanical failure is sufficient to resolve CVS without damage to the underlying extracellular matrix. Consistent with recent clinical observations, our model predicts that existing stents have the potential to provide sufficient outward force to successfully treat CVS and that success will be dependent on an appropriate match between stent and vessel. CONCLUSION: Mathematical models of CVS can provide insights into biological mechanisms and explore treatment approaches. Improved understanding of the underlying mechanistic processes governing CVS and its mechanical treatment may assist in the development of dedicated stents.

Correction to: The unexplained success of stentplasty vasospasm treatment : Insights using Mechanistic Mathematical Modeling.

Correction to: Clin Neuroradiol 2019 https://doi.org/10.1007/s00062-019-00776-2 The original version of this article unfortunately contained a mistake. The Acknowledgements were missing. The correct information is given ….

A biomechanical model for fibril recruitment: Evaluation in tendons and arteries.

Simulations of soft tissue mechanobiological behaviour are increasingly important for clinical prediction of aneurysm, tendinopathy and other disorders. Mechanical behaviour at low stretches is governed by fibril straightening, transitioning into load-bearing at recruitment stretch, resulting in a tissue stiffening effect. Previous investigations have suggested theoretical relationships between stress-stretch measurements and recruitment probability density function (PDF) but not derived these rigorously nor evaluated these experimentally. Other work has proposed image-based methods for measurement of recruitment but made use of arbitrary fibril critical straightness parameters. The aim of this work was to provide a sound theoretical basis for estimating recruitment PDF from stress-stretch measurements and to evaluate this relationship using image-based methods, clearly motivating the choice of fibril critical straightness parameter in rat tail tendon and porcine artery. Rigorous derivation showed that the recruitment PDF may be estimated from the second stretch derivative of the first Piola-Kirchoff tissue stress. Image-based fibril recruitment identified the fibril straightness parameter that maximised Pearson correlation coefficients (PCC) with estimated PDFs. Using these critical straightness parameters the new method for estimating recruitment PDF showed a PCC with image-based measures of 0.915 and 0.933 for tendons and arteries respectively. This method may be used for accurate estimation of fibril recruitment PDF in mechanobiological simulation where fibril-level mechanical parameters are important for predicting cell behaviour.

Layer-dependent role of collagen recruitment during loading of the rat bladder wall.

In this work, we re-evaluated long-standing conjectures as to the source of the exceptionally large compliance of the bladder wall. Whereas these conjectures were based on indirect measures of loading mechanisms, in this work we take advantage of advances in bioimaging to directly assess collagen fibers and wall architecture during biaxial loading. A custom biaxial mechanical testing system compatible with multiphoton microscopy was used to directly measure the layer-dependent collagen fiber recruitment in bladder tissue from 9 male Fischer rats (4 adult and 5 aged). As for other soft tissues, the bladder loading curve was exponential in shape and could be divided into toe, transition and high stress regimes. The relationship between collagen recruitment and loading curves was evaluated in the context of the inner (lamina propria) and outer (detrusor smooth muscle) layers. The large extensibility of the bladder was found to be possible due to folds in the wall (rugae) that provide a mechanism for low resistance flattening without any discernible recruitment of collagen fibers throughout the toe regime. For more extensible bladders, as the loading extended into the transition regime, a gradual coordinated recruitment of collagen fibers between the lamina propria layer and detrusor smooth muscle layer was found. A second important finding was that wall extensibility could be lost by premature recruitment of collagen in the outer wall that cut short the toe region. This change was correlated with age. This work provides, for the first time, a mechanistic understanding of the role of collagen recruitment in determining bladder extensibility and capacitance.

Growth Description for Vessel Wall Adaptation: A Thick-Walled Mixture Model of Abdominal Aortic Aneurysm Evolution.

(1) Background: Vascular tissue seems to adapt towards stable homeostatic mechanical conditions, however, failure of reaching homeostasis may result in pathologies. Current vascular tissue adaptation models use many ad hoc assumptions, the implications of which are far from being fully understood; (2) Methods: The present study investigates the plausibility of different growth kinematics in modeling Abdominal Aortic Aneurysm (AAA) evolution in time. A structurally motivated constitutive description for the vessel wall is coupled to multi-constituent tissue growth descriptions; Constituent deposition preserved either the constituent's density or its volume, and Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), in-Thickness Volume Growth (TVG) and No Volume Growth (NVG) describe the kinematics of the growing vessel wall. The sensitivity of key modeling parameters is explored, and predictions are assessed for their plausibility; (3) Results: AAA development based on TVG and NVG kinematics provided not only quantitatively, but also qualitatively different results compared to IVG and PVG kinematics. Specifically, for IVG and PVG kinematics, increasing collagen mass production accelerated AAA expansion which seems counterintuitive. In addition, TVG and NVG kinematics showed less sensitivity to the initial constituent volume fractions, than predictions based on IVG and PVG; (4) Conclusions: The choice of tissue growth kinematics is of crucial importance when modeling AAA growth. Much more interdisciplinary experimental work is required to develop and validate vascular tissue adaption models, before such models can be of any practical use.

Quantitative multiphoton microscopy of murine urinary bladder morphology during in situ uniaxial loading.

Urodynamic tests are the gold standard for the diagnosis of bladder dysfunction, and the mechanical compliance of the bladder is an important parameter in these tests. The bladder wall has a layered structure, differentially affected by pathology, so knowledge of the contribution and role of these layers and their constituents to overall bladder compliance will enhance interpretation of these clinical tests. In this study we document the functional morphology of the detrusor and lamina propria of the murine bladder wall using a custom in-situ tensile loading system under multiphoton microscopy (MPM) observation in unloaded state and under incremental uniaxial stretch. Features in the stress-stretch curves of bladder samples were then directly related to corresponding MPM images. Collagen organisation across wall depth was quantified using image analysis techniques. The hypothesis that the lamina propria deformed at low strain by unfolding of the rugae and rearranging collagen fibrils was confirmed. A novel 'pocket' feature in the detrusor was observed along with extensive rearrangement of fibrils in two families at different depths, providing higher stiffness at high stretches in the detrusor. The very different deformations of detrusor and lamina propria were accommodated by the highly coiled structure of collagen in the lamina propria. Imaging and mechanical studies presented here allow gross mechanical response to be attributed to specific components of the bladder wall and further, may be used to investigate the impact of microstructural changes due to pathology or aging, and how they impair tissue functionality. STATEMENT OF SIGNIFICANCE: This article reports the first in-situ multiphoton microscopy observations of microstructural deformation under uniaxial tensile loading of ex vivo bladder. We describe collagen rearrangement through the tissue thickness and relate this directly to the stress-stretch behaviour. We confirm for the first time the unfolding of rugae and realignment of fibrils in the lamina propria during extension and the rapid stiffening as two fibril families in the detrusor are engaged. This technique provides new insight into microstructure function and will enhance understanding of the impact of changes due to pathology or aging.

Hemodynamic parameters that may predict false-lumen growth in type-B aortic dissection after endovascular repair: A preliminary study on long-term multiple follow-ups.

Thoracic endovascular aortic repair (TEVAR) is commonly applied in type-B aortic dissection. For patients with dissection affects descending aorta and extends downward to involve abdominal aorta and possibly iliac arteries, false lumen (FL) expansion might occur post-TEVAR. Predictions of dissection development may assist in medical decision on re-intervention or surgery. In this study, two patients are selected with similar morphological features at initial presentation but with different long-term FL development post-TEVAR (stable and enlarged FL). Patient-specific models are established for each of the follow-ups. Flow boundaries and computational validations are obtained from Doppler ultrasound velocimetry. By analyzing the hemodynamic parameters, the false-to-true luminal pressure difference (PDiff) and particle relative residence time (RRT) are found related to FL remodeling. It is found that (i) the position of the first FL flow entry is the watershed of negative-and-positive PDiff and, in long-term follow-ups, and the position of largest PDiff is consistent with that of the greatest increase of FL width; (ii) high RRT occurs at the FL proximal tip and similar magnitude of RRT is found in both stable and enlarged cases; (iii) comparing to the RRT at 7days post-TEVAR, an increase of RRT afterwards in short-term is found in the stable case while a slight decrease of this parameter is found in the enlarged case, indicating that the variation of RRT in short-term post-TEVAR might be potential to predict long-term FL remodeling.

A novel chemo-mechano-biological model of arterial tissue growth and remodelling.

Arterial growth and remodelling (G&R) is mediated by vascular cells in response to their chemical and mechanical environment. To date, mechanical and biochemical stimuli tend to be modelled separately, however this ignores their complex interplay. Here, we present a novel mathematical model of arterial chemo-mechano-biology. We illustrate its application to the development of an inflammatory aneurysm in the descending human aorta. The arterial wall is modelled as a bilayer cylindrical non-linear elastic membrane, which is internally pressurised and axially stretched. The medial degradation that accompanies aneurysm development is driven by an inflammatory response. Collagen remodelling is simulated by adaption of the natural reference configuration of constituents; growth is simulated by changes in normalised mass-densities. We account for the distribution of attachment stretches that collagen fibres are configured to the matrix and, innovatively, allow this distribution to remodel. This enables the changing functional role of the adventitia to be simulated. Fibroblast-mediated collagen growth is represented using a biochemical pathway model: a system of coupled non-linear ODEs governs the evolution of fibroblast properties and levels of key biomolecules under the regulation of Transforming Growth Factor (TGF)-ß, a key promoter of matrix deposition. Given physiologically realistic targets, different modes of aneurysm development can be captured, while the predicted evolution of biochemical variables is qualitatively consistent with trends observed experimentally. Interestingly, we observe that increasing the levels of collagen-promoting TGF-ß results in arrest of aneurysm growth, which seems to be consistent with experimental evidence. We conclude that this novel Chemo-Mechano-Biological (CMB) mathematical model has the potential to provide new mechanobiological insight into vascular disease progression and therapy.

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.

Multiscale modeling of intracranial aneurysms: cell signaling, hemodynamics, and remodeling.

The genesis, growth, and rupture of intracranial aneurysms (IAs) involve physics at the molecular, cellular, blood vessel, and organ levels that occur over time scales ranging from seconds to years. Comprehensive mathematical modeling of IAs, therefore, requires the description and integration of events across length and time scales that span many orders of magnitude. In this letter, we outline a strategy for mulstiscale modeling of IAs that involves the construction of individual models at each relevant scale and their subsequent combination into an integrative model that captures the overall complexity of IA development. An example of the approach is provided using three models operating at different length and time scales: 1) shear stress induced nitric oxide production; 2) smooth muscle cell apoptosis; and 3) fluid-structure-growth modeling. A computational framework for combining them is presented. We conclude with a discussion of the advantages and challenges of the approach.

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.

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.