Spencer J. Sherwin

Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476–3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10−6) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.

Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.

A triangular spectral element method; applications to the incompressible Navier-Stokes equations

Abstract Encouraged by the success of spectral elements methods in computational fluid dynamics and p-type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers. We introduce here a complete formulation using a modal basis which has been implemented in a new code NeκTαr . The new basis has the following properties: Jacobi polynomials of mixed weights; semi-orthogonality; hierarchical structure; generalized tensor ( warped ) product; variable order; and a new apex co-ordinate system allowing automated integration with Gaussian quadrature. We have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations. We use this notation to formulate the linear advection and Helmholtz equations in an efficient manner and show that we recover the following properties: well conditioned matrices; asymptotic operation count of O ( N 3 ); scaling O ( N 2 of the spectral radius of the weak convective operator; and exponential convergence using polynomials up to ∼ 40. Having constructed and numerically analysed these equations we are then able to solve the incompressible Navier-Stokes equations using a high-order splitting scheme. We demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the Stokes and Navier-Stokes problems.

Does low and oscillatory wall shear stress correlate spatially with early atherosclerosis? A systematic review

Low and oscillatory wall shear stress is widely assumed to play a key role in the initiation and development of atherosclerosis. Indeed, some studies have relied on the low shear theory when developing diagnostic and treatment strategies for cardiovascular disease. We wished to ascertain if this consensus is justified by published data. We performed a systematic review of papers that compare the localization of atherosclerotic lesions with the distribution of haemodynamic indicators calculated using computational fluid dynamics. The review showed that although many articles claim their results conform to the theory, it has been interpreted in different ways: a range of metrics has been used to characterize the distribution of disease, and they have been compared with a range of haemodynamic factors. Several studies, including all of those making systematic point-by-point comparisons of shear and disease, failed to find the expected relation. The various pre- and post-processing techniques used by different groups have reduced the range of shears over which correlations were sought, and in some cases are mutually incompatible. Finally, only a subset of the known patterns of disease has been investigated. The evidence for the low/oscillatory shear theory is less robust than commonly assumed. Longitudinal studies starting from the healthy state, or the collection of average flow metrics derived from large numbers of healthy vessels, both in conjunction with point-by-point comparisons using appropriate statistical techniques, will be necessary to improve our understanding of the relation between blood flow and atherogenesis.

The Influence of Out-of-Plane Geometry on the Flow Within a Distal End-to-Side Anastomosis

This paper describes a computational and experimental investigation of flow in a proto-type model geometry of a fully occluded 45 deg distal end-to-side anastomosis. Previous investigations have considered a similar configuration where the centerlines of the bypass and host vessels lie within a plane, thereby producing a plane of symmetry within the flow. We have extended these investigations by deforming the bypass vessel out of the plane of symmetry, thereby breaking the symmetry of the flow and producing a nonplanar geometry. Experimental data were obtained using magnetic resonance imaging of flow within perspex models and computational data were obtained from simulations using a high-order spectral/hp element method. We found that the nonplanar three-dimensional flow notably alters the distribution of wall shear stress at the bed of the anastomosis, reducing the peak wall shear stress peak by approximately 10 percent when compared with the planar model. Furthermore, an increase in the absolute flux of velocity into the occluded region, proximal to the anastomosis, of 80 percent was observed in the nonplanar geometry when compared with the planar geometry.

Nektar++: An open-source spectral/hp element framework

Nektar++ is an open-source software framework designed to support the development of high-performance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains. Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland No. of lines in distributed program, including test data, etc.: 1052456 No. of bytes in distributed program, including test data, etc.: 42851367 External routines: Boost, PFTW, MPI, BLAS, LAPACK and METIS (www.cs.umn.edu) Nature of problem: The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations. Running time: The tests provided take a few minutes to run. Runtime in general depends on mesh size and total integration time.

Convective instability and transient growth in flow over a backward-facing step

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

Flow past a square-section cylinder with a wavy stagnation face

Numerical investigations have been performed for the flow past square-section cylinders with a spanwise geometric deformation leading to a stagnation face with a sinusoidal waviness. The computations were performed using a spectral/ hp element solver over a range of Reynolds numbers from 10 to 150. Starting from fully developed shedding past a straight cylinder at a Reynolds number of 100, a sufficiently high waviness is impulsively introduced resulting in the stabilization of the near wake to a time-independent state. It is shown that the spanwise waviness sets up a cross-flow within the growing boundary layer on the leading-edge surface thereby generating streamwise and vertical components of vorticity. These additional components of vorticity appear in regions close to the inflection points of the wavy stagnation face where the spanwise vorticity is weakened. This redistribution of vorticity leads to the breakdown of the unsteady and staggered Karman vortex wake into a steady and symmetric near-wake structure. The steady nature of the near wake is associated with a reduction in total drag of about 16% at a Reynolds number of 100 compared with the straight, non-wavy cylinder. Further increases in the amplitude of the waviness lead to the emergence of hairpin vortices from the near-wake region. This wake topology has similarities to the wake of a sphere at low Reynolds numbers. The physical structure of the wake due to the variation of the amplitude of the waviness is identified with five distinct regimes. Furthermore, the introduction of a waviness at a wavelength close to the mode A wavelength and the primary wavelength of the straight square-section cylinder leads to the suppression of the Karman street at a minimal waviness amplitude.

From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations

The spectral/hp element method can be considered as bridging the gap between the - traditionally low-order - finite element method on one side and spectral methods on the other side. Consequently, a major challenge which arises in implementing the spectral/hp element methods is to design algorithms that perform efficiently for both low- and high-order spectral/hp discretisations, as well as discretisations in the intermediate regime. In this paper, we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders. Furthermore, based upon this efficient implementation, we analyse which spectral/hp discretisation (which specific combination of mesh-size h and polynomial order P) minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy. We investigate this question for a set of both smooth and non-smooth problems.

Disturbed Flow Promotes Endothelial Senescence via a p53-Dependent Pathway

Objective—Although atherosclerosis is associated with systemic risk factors such as age, high cholesterol, and obesity, plaque formation occurs predominately at branches and bends that are exposed to disturbed patterns of blood flow. The molecular mechanisms that link disturbed flow–generated mechanical forces with arterial injury are uncertain. To illuminate them, we investigated the effects of flow on endothelial cell (EC) senescence. Approach and Results—LDLR−/− (low-density lipoprotein receptor−/−) mice were exposed to a high-fat diet for 2 to 12 weeks (or to a normal chow diet as a control) before the assessment of cellular senescence in aortic ECs. En face staining revealed that senescence-associated &bgr;-galactosidase activity and p53 expression were elevated in ECs at sites of disturbed flow in response to a high-fat diet. By contrast, ECs exposed to undisturbed flow did not express senescence-associated &bgr;-galactosidase or p53. Studies of aortae from healthy pigs (aged 6 months) also revealed enhanced senescence-associated &bgr;-galactosidase staining at sites of disturbed flow. These data suggest that senescent ECs accumulate at disturbed flow sites during atherogenesis. We used in vitro flow systems to examine whether a causal relationship exists between flow and EC senescence. Exposure of cultured ECs to flow (using either an orbital shaker or a syringe-pump flow bioreactor) revealed that disturbed flow promoted EC senescence compared with static conditions, whereas undisturbed flow reduced senescence. Gene silencing studies demonstrated that disturbed flow induced EC senescence via a p53-p21 signaling pathway. Disturbed flow–induced senescent ECs exhibited reduced migration compared with nonsenescent ECs in a scratch wound closure assay, and thus may be defective for arterial repair. However, pharmacological activation of sirtuin 1 (using resveratrol or SRT1720) protected ECs from disturbed flow–induced senescence. Conclusions—Disturbed flow promotes endothelial senescence via a p53-p21–dependent pathway which can be inhibited by activation of sirtuin 1. These observations support the principle that pharmacological activation of sirtuin 1 may promote cardiovascular health by suppressing EC senescence at atheroprone sites.