Noradrenaline has opposing effects on the hydraulic conductance of arterial intima and media.

The uptake of circulating macromolecules by the arterial intima is thought to be a key step in atherogenesis. Such transport is dominantly advective, so elucidating the mechanisms of water transport is important. The relation between vasoactive agents and water transport in the arterial wall is incompletely understood. Here we applied our recently-developed combination of computational and experimental methods to investigate the effects of noradrenaline (NA) on hydraulic conductance of the wall (Lp), medial extracellular matrix volume fraction (ϕECM) and medial permeability (K11) in the rat abdominal aorta. Experimentally, we found that physiological NA concentrations were sufficient to induce SMC contraction and produced significant decreases in Lp and increases in ϕECM. Simulation results based on 3D confocal images of the extracellular volume showed a corresponding increase in K11, attributed to the opening of the ECM. Conversion of permeabilities to layer-specific resistances revealed that although the total wall resistance increased, medial resistance decreased, suggesting an increase in intimal resistance upon application of NA.

An adaptable parallel algorithm for the direct numerical simulation of incompressible turbulent flows using a Fourier spectral/hp element method and MPI virtual topologies.

A hybrid parallelisation technique for distributed memory systems is investigated for a coupled Fourier-spectral/hp element discretisation of domains characterised by geometric homogeneity in one or more directions. The performance of the approach is mathematically modelled in terms of operation count and communication costs for identifying the most efficient parameter choices. The model is calibrated to target a specific hardware platform after which it is shown to accurately predict the performance in the hybrid regime. The method is applied to modelling turbulent flow using the incompressible Navier-Stokes equations in an axisymmetric pipe and square channel. The hybrid method extends the practical limitations of the discretisation, allowing greater parallelism and reduced wall times. Performance is shown to continue to scale when both parallelisation strategies are used.

Intimal and medial contributions to the hydraulic resistance of the arterial wall at different pressures: a combined computational and experimental study.

The hydraulic resistances of the intima and media determine water flux and the advection of macromolecules into and across the arterial wall. Despite several experimental and computational studies, these transport processes and their dependence on transmural pressure remain incompletely understood. Here, we use a combination of experimental and computational methods to ascertain how the hydraulic permeability of the rat abdominal aorta depends on these two layers and how it is affected by structural rearrangement of the media under pressure. Ex vivo experiments determined the conductance of the whole wall, the thickness of the media and the geometry of medial smooth muscle cells (SMCs) and extracellular matrix (ECM). Numerical methods were used to compute water flux through the media. Intimal values were obtained by subtraction. A mechanism was identified that modulates pressure-induced changes in medial transport properties: compaction of the ECM leading to spatial reorganization of SMCs. This is summarized in an empirical constitutive law for permeability and volumetric strain. It led to the physiologically interesting observation that, as a consequence of the changes in medial microstructure, the relative contributions of the intima and media to the hydraulic resistance of the wall depend on the applied pressure; medial resistance dominated at pressures above approximately 93 mmHg in this vessel.

Techniques for automated local activation time annotation and conduction velocity estimation in cardiac mapping.

Measurements of cardiac conduction velocity provide valuable functional and structural insight into the initiation and perpetuation of cardiac arrhythmias, in both a clinical and laboratory context. The interpretation of activation wavefronts and their propagation can identify mechanistic properties of a broad range of electrophysiological pathologies. However, the sparsity, distribution and uncertainty of recorded data make accurate conduction velocity calculation difficult. A wide range of mathematical approaches have been proposed for addressing this challenge, often targeted towards specific data modalities, species or recording environments. Many of these algorithms require identification of activation times from electrogram recordings which themselves may have complex morphology or low signal-to-noise ratio. This paper surveys algorithms designed for identifying local activation times and computing conduction direction and speed. Their suitability for use in different recording contexts and applications is assessed.

A combined numerical and experimental framework for determining permeability properties of the arterial media.

The medial layer of the arterial wall may play an important role in the regulation of water and solute transport across the wall. In particular, a high medial resistance to transport could cause accumulation of lipid-carrying molecules in the inner wall. In this study, the water transport properties of medial tissue were characterised in a numerical model, utilising experimentally obtained data for the medial microstructure and the relative permeability of different constituents. For the model, a new solver for flow in porous materials, based on a high-order splitting scheme, was implemented in the spectral/hp element library nektar++ and validated. The data were obtained by immersing excised aortic bifurcations in a solution of fluorescent protein tracer and subsequently imaging them with a confocal microscope. Cuboidal regions of interest were selected in which the microstructure and relative permeability of different structures were transformed to a computational mesh. Impermeable objects were treated fictitiously in the numerical scheme. On this cube, a pressure drop was applied in the three coordinate directions and the principal components of the permeability tensor were determined. The reconstructed images demonstrated the arrangement of elastic lamellae and interspersed smooth muscle cells in rat aortic media; the distribution and alignment of the smooth muscle cells varied spatially within the extracellular matrix. The numerical simulations highlighted that the heterogeneity of the medial structure is important in determining local water transport properties of the tissue, resulting in regional and directional variation of the permeability tensor. A major factor in this variation is the alignment and density of smooth muscle cells in the media, particularly adjacent to the adventitial layer.

From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method.

We investigate the relative performance of a second-order Adams-Bashforth scheme and second-order and fourth-order Runge-Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd.

Blood flow in the rabbit aortic arch and descending thoracic aorta.

The distribution of atherosclerotic lesions within the rabbit vasculature, particularly within the descending thoracic aorta, has been mapped in numerous studies. The patchy nature of such lesions has been attributed to local variation in the pattern of blood flow. However, there have been few attempts to model and characterize the flow. In this study, a high-order continuous Galerkin finite-element method was used to simulate blood flow within a realistic representation of the rabbit aortic arch and descending thoracic aorta. The geometry, which was obtained from computed tomography of a resin corrosion cast, included all vessels originating from the aortic arch (followed to at least their second generation) and five pairs of intercostal arteries originating from the proximal descending thoracic aorta. The simulations showed that small geometrical undulations associated with the ductus arteriosus scar cause significant deviations in wall shear stress (WSS). This finding highlights the importance of geometrical accuracy when analysing WSS or related metrics. It was also observed that two Dean-type vortices form in the aortic arch and propagate down the descending thoracic aorta (along with an associated skewed axial velocity profile). This leads to the occurrence of axial streaks in WSS, similar in nature to the axial streaks of lipid deposition found in the descending aorta of cholesterol-fed rabbits. Finally, it was observed that WSS patterns within the vicinity of intercostal branch ostia depend not only on local flow features caused by the branches themselves, but also on larger-scale flow features within the descending aorta, which vary between branches at different locations. This result implies that disease and WSS patterns in the vicinity of intercostal ostia are best compared on a branch-by-branch basis.

Effect of reverse flow on the pattern of wall shear stress near arterial branches.

Atherosclerotic lesions have a patchy distribution within arteries that suggests a controlling influence of haemodynamic stresses on their development. The distribution near aortic branches varies with age and species, perhaps reflecting differences in these stresses. Our previous work, which assumed steady flow, revealed a dependence of wall shear stress (WSS) patterns on Reynolds number and side-branch flow rate. Here, we examine effects of pulsatile flow. Flow and WSS patterns were computed by applying high-order unstructured spectral/hp element methods to the Newtonian incompressible Navier-Stokes equations in a geometrically simplified model of an aorto-intercostal junction. The effect of pulsatile but non-reversing side-branch flow was small; the aortic WSS pattern resembled that obtained under steady flow conditions, with high WSS upstream and downstream of the branch. When flow in the side branch or in the aortic near-wall region reversed during part of the cycle, significantly different instantaneous patterns were generated, with low WSS appearing upstream and downstream. Time-averaged WSS was similar to the steady flow case, reflecting the short duration of these events, but patterns of the oscillatory shear index for reversing aortic near-wall flow were profoundly altered. Effects of reverse flow may help explain the different distributions of lesions.

Endothelial nitric oxide production and transport in flow chambers: The importance of convection.

A computational model of Nitric Oxide (NO) production and transport within a parallel-plate flow chamber coated with endothelial cells is presented. The relationship between NO concentration and Wall Shear Stress (WSS) at the endothelium is investigated in detail. An increase in WSS is associated with two phenomena: enhanced NO production by the endothelial cells, and an increase in the velocity at which NO is convected out of the chamber. These two phenomena have opposite effects on endothelial NO concentration. In physiologically realistic cases, the balance between them is found to vary as WSS is raised, resulting in a complex non-monotonic dependence of endothelial NO concentration on WSS. Also, it is found that a NO concentration boundary layer develops within the chamber, leading to substantial spatial variations in NO concentration along the length of the device. Finally, the implications of a negative feedback mechanism (that affects NO production) are presented. The results emphasize the role of convection on NO transport within flow chambers, which has been overlooked or misinterpreted in most previous theoretical studies. It is hoped that the conclusions of this study can be used to aid accurate interpretation of related experimental data.

The effect of the endothelial glycocalyx layer on concentration polarisation of low density lipoprotein in arteries.

It has been postulated that a flow-dependent (and hence spatially varying) low density lipoprotein (LDL) concentration polarisation layer forms on the luminal surface of the vascular endothelium. Such a layer has the potential to cause heterogeneity in the distribution of atherosclerotic lesions by spatially modulating the rate of LDL transport into the arterial wall. Theoretical analysis suggests that a transmural water flux which is spatially heterogeneous at the cellular scale can act to enhance LDL concentration polarisation in a shear dependent fashion. However, such an effect is only observed if a relevant Peclet number (i.e. the ratio of LDL convection to LDL diffusion) is of order unity or greater. Based on the diffusivity of LDL in blood plasma, such a Peclet number is found to be far less than unity, implying that the aforementioned enhancement and shear dependence will not occur. However, this conclusion ignores the existence of the endothelial glycocalyx layer (EGL), which may inhibit the diffusion of LDL near the luminal surface of the endothelium, and hence raise any Peclet number associated with the transport of LDL. The present study numerically investigates the effect of the EGL, as well as a heterogeneous transmural water flux, on arterial LDL concentration polarisation. Particular attention is paid to measures of LDL concentration polarisation thought relevant to the rate of transendothelial LDL transport. It is demonstrated that an EGL is unlikely to cause any additional shear dependence of such measures directly, irrespective of whether or not LDL can penetrate into the EGL. However, it is found that such measures depend significantly on the nature of the interaction between LDL and the EGL (parameterized by the height of the EGL, the depth to which LDL penetrates into the EGL, and the diffusivity of LDL in the EGL). Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.

Mixing through stirring of steady flow in small amplitude helical tubes.

In this paper we numerically simulate flow in a helical tube for physiological conditions using a co-ordinate mapping of the Navier-Stokes equations. Helical geometries have been proposed for use as bypass grafts, arterial stents and as an idealized model for the out-of-plane curvature of arteries. Small amplitude helical tubes are also currently being investigated for possible application as A-V shunts, where preliminary in vivo tests suggest a possibly lower risk of thrombotic occlusion. In-plane mixing induced by the geometry is hypothesized to be an important mechanism. In this work, we focus mainly on a Reynolds number of 250 and investigate both the flow structure and the in-plane mixing in helical geometries with fixed pitch of 6 tube diameters (D), and centerline helical radius ranging from 0.1 D to 0.5 D. High-order particle tracking, and an information entropy measure is used to analyze the in-plane mixing. A combination of translational and rotational reference frames are shown to explain the apparent discrepancy between flow field and particle trajectories, whereby particle paths display a pattern characteristic of a double vortex, though the flow field reveals only a single dominant vortex. A radius of 0.25 D is found to provide the best trade-off between mixing and pressure loss, with little increase in mixing above R=0.25 D, whereas pressure continues to increase linearly.

Effect of Reynolds number and flow division on patterns of haemodynamic wall shear stress near branch points in the descending thoracic aorta.

Atherosclerotic lesions are non-uniformly distributed at arterial bends and branch sites, suggesting an important role for haemodynamic factors, particularly wall shear stress (WSS), in their development. The pattern of lesions at aortic branch sites depends on age and species. Using computational flow simulations in an idealized model of an intercostal artery emerging perpendicularly from the thoracic aorta, we studied the effects of Reynolds number and flow division under steady conditions. Patterns of flow and WSS were strikingly dependent on these haemodynamic parameters. With increasing Reynolds number, WSS, normalized by the fully developed aortic value, was lowered at the sides of the ostium and increased upstream and downstream of it. Increasing flow into the side branch exacerbated these patterns and gave rise to a reversing flow region downstream of the ostium. Incorporation of more realistic geometric features had only minor effects and patterns of mean WSS under pulsatile conditions were similar to the steady flow results. Aspects of the observed WSS patterns correlate with, and may explain, some but not all of the lesion patterns in human, rabbit and mouse aortas.

Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows.

Blood flow in the circle of Willis (CoW) is modelled using the 1-D equations of pressure and flow wave propagation in compliant vessels. The model starts at the left ventricle and includes the largest arteries that supply the CoW. Based on published physiological data, it is able to capture the main features of pulse wave propagation along the aorta, at the brachiocephalic bifurcation and throughout the cerebral arteries. The collateral ability of the complete CoW and its most frequent anatomical variations is studied in normal conditions and after occlusion of a carotid or vertebral artery (VA). Our results suggest that the system does not require collateral pathways through the communicating arteries to adequately perfuse the brain of normal subjects. The communicating arteries become important in cases of missing or occluded vessels, the anterior communicating artery (ACoA) being a more critical collateral pathway than the posterior communicating arteries (PCoAs) if an internal carotid artery (ICA) is occluded. Occlusions of the VAs proved to be far less critical than occlusions of the ICAs. The worst scenario in terms of reduction in the mean cerebral outflows is a CoW without the first segment of an anterior cerebral artery combined with an occlusion of the contralateral ICA. Furthermore, in patients without any severe occlusion of a carotid or VA, the direction of flow measured at the communicating arteries corresponds to the side of the CoW with an absent or occluded artery. Finally, we study the effect of partial occlusions of the communicating arteries on the cerebral flows, which again confirms that the ACoA is a more important collateral pathway than the PCoAs if an ICA is occluded.

Can the modified Allen's test always detect sufficient collateral flow in the hand? A computational study.

Blood flow in the largest arteries of the arm up to the digital arteries is numerically modelled using the one-dimensional equations of pressure and flow wave propagation in compliant vessels. The model can be applied to different anatomies of arterial networks and can simulate compression of arteries, these allowing us to simulate the modified Allen's test (MAT) and to assess its suitability for the detection of sufficient collateral flow in the hand if radial blood supply is interrupted. The test measures blood flow in the superficial palmar arch before and during compression of the radial artery. The absence of reversal flow in the palmar arch with the compression indicates insufficient collateral flow and is referred to as a positive MAT. This study shows that small calibres of the superficial palmar arch and insufficient compression of the radial artery can lead to false-positive results. Measurement of the drop in digital systolic pressures with compression of the radial artery has proved to be a more sensitive test to predict the presence of sufficient ulnar collateral flow in networks with small calibres of the superficial palmar arch. However, this study also shows that digital pressure measurements can fail in detecting enough collateral flow if the radial artery is insufficiently compressed.

Local and global geometric influence on steady flow in distal anastomoses of peripheral bypass grafts.

We consider the effect of geometrical configuration on the steady flow field of representative geometries from an in vivo anatomical data set of end-to-side distal anastomoses constructed as part of a peripheral bypass graft. Using a geometrical classification technique, we select the anastomoses of three representative patients according to the angle between the graft and proximal host vessels (GPA) and the planarity of the anastomotic configuration. The geometries considered include two surgically tunneled grafts with shallow GPAs which are relatively planar but have different lumen characteristics, one case exhibiting a local restriction at the perianastomotic graft and proximal host whilst the other case has a relatively uniform cross section. The third case is nonplanar and characterized by a wide GPA resulting from the graft being constructed superficially from an in situ vein. In all three models the same peripheral resistance was imposed at the computational outflows of the distal and proximal host vessels and this condition, combined with the effect of the anastomotic geometry, has been observed to reasonably reproduce the in vivo flow split. By analyzing the flow fields we demonstrate how the local and global geometric characteristics influences the distribution of wall shear stress and the steady transport of fluid particles. Specifically, in vessels that have a global geometric characteristic we observe that the wall shear stress depends on large scale geometrical factors, e.g., the curvature and planarity of blood vessels. In contrast, the wall shear stress distribution and local mixing is significantly influenced by morphology and location of restrictions, particular when there is a shallow GPA. A combination of local and global effects are also possible as demonstrated in our third study of an anastomosis with a larger GPA. These relatively simple observations highlight the need to distinguish between local and global geometric influences for a given reconstruction. We further present the geometrical evolution of the anastomoses over a series of follow-up studies and observe how the lumen progresses towards the faster bulk flow of the velocity in the original geometry. This mechanism is consistent with the luminal changes in recirculation regions that experience low wall shear stress. In the shallow GPA anastomoses the proximal part of the native host vessel occludes or stenoses earlier than in the case with wide GPA. A potential contribution to this behavior is suggested by the stronger mixing that characterizes anastomoses with large GPA.

Automated classification of peripheral distal by-pass geometries reconstructed from medical data.

Abnormal haemodynamic conditions are implicated in the development of anastomotic myointimal hyperplasia (MIH). However, these conditions are difficult to determine in vivo, prompting research using ex vivo idealised models. To relate the understanding gained in idealised geometries to anatomically correct conditions we have investigated a reproducible approach to classify in vivo distal graft anastomoses and their inter-patient variability. In vivo distal anastomotic geometries were acquired by magnetic resonance (MR) angiography from 13 patients who had undergone infrageniculate autologous venous by-pass surgery. On average, the images were acquired 2 weeks post-operatively. Five patients also underwent repeat examinations 2 to 7 weeks later. For each geometry, the surface of the arterial lumen is represented by the zero level set of an implicit function constructed from radial basis functions that minimise curvature. The three-dimensional binary image created from the interpolated surface is processed using a skeletonisation algorithm to obtain the centreline of each branch in the geometry. This allows for the measurement of the branching angles between straight line approximations of the centrelines of each vessel, averaging them over a characteristic length of each anastomosis. The main finding in the application of the proposed classification methodology to this set of patients is that the spectrum of anastomoses can be reduced to a small subset of cases characterised by two angles: the angle between the graft and the plane of the host artery and the angle between the graft and the proximal branch of the artery.

The influence of out-of-plane geometry on pulsatile flow within a distal end-to-side anastomosis.

We present an experimental and computational investigation of time-varying flow in an idealized fully occluded 45 degrees distal end-to-side anastomosis. Two geometric configurations are assessed, one where the centerlines of host and bypass vessels lie within a plane, and one where the bypass vessel is deformed out of the plane of symmetry, respectively, termed planar and non-planar. Flow experiments were conducted by magnetic resonance imaging in rigid wall models and computations were performed using a high order spectral/hp algorithm. Results indicate a significant change in the spatial distribution of wall shear stress and a reduction of the time-averaged peak wall shear stress magnitude by 10% in the non-planar model as compared to the planar configuration. In the planar geometry the stagnation point follows a straight-line path along the host artery bed with a path length of 0.8 diameters. By contrast in the non-planar case the stagnation point oscillates about a center that is located off the symmetry plane intersection with the host artery bed wall, and follows a parabolic path with a 0.7 diameter longitudinal and 0.5 diameter transverse excursion. A definition of the oscillatory shear index (OSI) is introduced that varies between 0 and 0.5 and that accounts for a continuous range of wall shear stress vector angles. In both models, regions of elevated oscillatory shear were spatially associated with regions of separated or oscillating stagnation point flow. The mean oscillatory shear magnitude (considering sites where OSI>0.1) in the non-planar geometry was reduced by 22% as compared to the planar configuration. These changes in the dynamic behavior of the stagnation point and the oscillatory shear distribution introduced by out-of-plane graft curvature may influence the localization of vessel wall sites exposed to physiologically unfavorable flow conditions.

Unsteady near wall residence times and shear exposure in model distal arterial bypass grafts.

Building on previous studies of unsteady flow within model distal bypass grafts we analyse the near wall residence times and shear exposure in a 45 degrees anastomosis under symmetrical and symmetry breaking geometric configurations. We define residence time as the minimum time for a particle to exit a spherical region and shear exposure as a temporal integral of the Huber-Henky-von-Mises criterion along a particle path over a fixed time interval. Decomposing the pulsatile cycle into four equal intervals we find that the interval of peak residence time in the host vessel is from mid-deceleration to peak diastole and peak diastole to mid-acceleration. The asymmetric model is shown to have a significantly lower residence time during these intervals. Considering the shear exposure prior to the residence time evaluation we determine that a higher average shear exposure exists in the asymmetric model associated with the upstream geometry modification. Analysis of the regions of high residence time and shear exposure suggests that the "toe" region and the interface between the "heel" and bulk flow are more significant than the bed and heel region. Although the asymmetric model considered in this study reduces residence times in the host artery, the product of the measure of shear exposure and residence time is not found to be preferable. If shear exposure were to be considered as an important factor in particle activation, the findings imply that for junction optimisation, greater consideration needs to be given both to the local junction asymmetry and upstream influence on the shear history.

Combined MR imaging and numerical simulation of flow in realistic arterial bypass graft models.

We report methods for (a) transforming a three-dimensional geometry acquired by magnetic resonance angiography (MRA) in vivo, or by imaging a model cast, into a computational surface representation, (b) use of this to construct a three dimensional numerical grid for computational fluid dynamic (CFD) studies, and (c) use of the surface representation to produce a stereo-lithographic replica of the real detailed geometry, at a scale convenient for detailed magnetic resonance imaging (MRI) flow studies. This is applied to assess the local flow field in realistic geometry arterial bypass grafts. Results from a parallel numerical simulation and MRI measurement of flow in an aorto-coronary bypass graft with various inlet flow conditions demonstrate the strong influence of the graft inlet waveform on the perianastomotic flow field. A sinusoidal and a multi harmonic coronary flow waveform both with a mean Reynolds number (Re) of 100 and a Womersley parameter of 2.7 were applied at the graft inlet. A weak axial flow separation region just distal to the toe was found in sinusoidal flow near end deceleration (Re = 25). At the same location and approximately the same point in the cycle (Re = 30) but in coronary flow, the axial flow separation was stronger and more spatially pronounced. No axial flow separation occurred in steady flow for Re = 100. Numerical predictions indicate a region in the vicinity of the suture line (where there is a local narrowing of the graft) with a wall shear magnitude in excess of five times that associated with fully developed flow at the graft inlet.