Andrew Comerford

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.

Structured Tree Impedance Outflow Boundary Conditions for 3D Lung Simulations

In this paper, we develop structured tree outflow boundary conditions for modeling the airflow in patient specific human lungs. The utilized structured tree is used to represent the nonimageable vessels beyond the 3D domain. The coupling of the two different scales (1D and 3D) employs a Dirichlet–Neumann approach. The simulations are performed under a variety of conditions such as light breathing and constant flow ventilation (which is characterized by very rapid acceleration and deceleration). All results show that the peripheral vessels significantly impact the pressure, however, the flow is relatively unaffected, reinforcing the fact that the majority of the lung impedance is due to the lower generations rather than the peripheral vessels. Furthermore, simulations of a hypothetical diseased lung (restricted flow in the superior left lobe) under mechanical ventilation show that the mean pressure at the outlets of the 3D domain is about 28% higher. This hypothetical model illustrates potential causes of volutrauma in the human lung and furthermore demonstrates how different clinical scenarios can be studied without the need to assume the unknown flow distribution into the downstream region.

Endothelial Nitric Oxide Synthase and Calcium Production in Arterial Geometries: An Integrated Fluid Mechanics/Cell Model

It is well known that atherosclerosis occurs at very specific locations throughout the human vasculature, such as arterial bifurcations and bends, all of which are subjected to low wall shear stress. A key player in the pathology of atherosclerosis is the endothelium, controlling the passage of material to and from the artery wall. Endothelial dysfunction refers to the condition where the normal regulation of processes by the endothelium is diminished. In this paper, the blood flow and transport of the low diffusion coefficient species adenosine triphosphate (ATP) are investigated in a variety of arterial geometries: a bifurcation with varying inner angle, and an artery bend. A mathematical model of endothelial calcium and endothelial nitric oxide synthase cellular dynamics is used to investigate spatial variations in the physiology of the endothelium. This model allows assessment of regions of the artery wall deficient in nitric oxide (NO). The models here aim to determine whether 3D flow fields are important in determining ATP concentration and endothelial function. For ATP transport, the effects of a coronary and carotid wave form on mass transport is investigated for low Womersley number. For the carotid, the Womersley number is then increased to determine whether this is an important factor. The results show that regions of low wall shear stress correspond with regions of impaired endothetial nitric oxide synthase signaling, therefore reduced availability of NO. However, experimental work is required to determine if this level is significant. The results also suggest that bifurcation angle is an important factor and acute angle bifurcations are more susceptible to disease than large angle bifurcations. It has been evidenced that complex 3D flow fields play an important role in determining signaling within endothelial cells. Furthermore, the distribution of ATP in blood is highly dependent on secondary flow features. The models here use ATP concentration simulated under steady conditions. This has been evidenced to reproduce essential features of time-averaged ATP concentration over a cardiac cycle for small Womersley numbers. However, when the Womersley number is increased, some differences are observed. Transient variations are overall insignificant, suggesting that spatial variation is more important than temporal. It has been determined that acute angle bifurcations are potentially more susceptible to atherogenesis and steady-state ATP transport reproduces essential features of time-averaged pulsatile transport for small Womersley number. Larger Womersley numbers appear to be an important factor in time-dependent mass transfer.

A novel formulation for Neumann inflow boundary conditions in biomechanics

Neumann boundary conditions prescribing the total momentum flux at inflow boundaries of biomechanical problems are proposed in this study. This approach enables the simultaneous application of velocity/flow rate and pressure curves at inflow boundaries. As the basic numerical method, a residual-based variational multiscale (or stabilized) finite element method is presented. The focus of the numerical examples in this work is on respiratory flows with complete flow reversals. However, the proposed formulation is just as well suited for cardiovascular flow problems with partial retrograde flow. Instabilities, which were reported for such problems in the literature, are resolved by the present approach without requiring the additional consideration of a Lagrange multiplier technique. The suitability of the approach is demonstrated for two respiratory flow examples, a rather simple tube and complex tracheobronchial airways (up to the fourth generation, segmented from end-expiratory CT images). For the latter example, the boundary conditions are generated from mechanical ventilation data obtained from an intensive care unit patient suffering from acute lung injury. For the tube, analytical pressure profiles can be replicated, and for the tracheobronchial airways, a correct distribution of the prescribed total momentum flux at the inflow boundary into velocity and pressure part is observed.

Coupled and reduced dimensional modeling of respiratory mechanics during spontaneous breathing

In this paper, we develop a total lung model based on a tree of 0D airway and acinar models for studying respiratory mechanics during spontaneous breathing. This model utilizes both computer tomography-based geometries and artificially generated lobe-filling airway trees to model the entire conducting region of the lung. Beyond the conducting airways, we develop an acinar model, which takes into account the alveolar tissue resistance, compliance, and the intrapleural pressure. With this methodology, we compare four different 0D models of airway mechanics and determine the best model based on a comparison with a 3D-0D coupled model of the conducting airways; this methodology is possible because the majority of airway resistance is confined to the lower generations, that is, the trachea and the first few bronchial generations. As an example application of the model, we simulate the flow and pressure dynamics under spontaneous breathing conditions, that is, at flow conditions driven purely by pleural space pressure. The results show good agreement, both qualitatively and quantitatively, with reported physiological values. One of the key advantages of this model is the ability to provide insight into lung ventilation in the peripheral regions. This is often crucial because this is where information, specifically for studying diseases and gas exchange, is needed. Thus, the model can be used as a tool for better understanding local peripheral lung mechanics without excluding the upper portions of the lung. This tool will be also useful for in vitro investigations of lung mechanics in both health and disease.

Computer Model of Nucleotide Transport in a Realistic Porcine Aortic Trifurcation

Adenosine triphosphate (ATP) is a ubiquitous blood borne agonist which is responsible for the regulation of vascular tone via purinogenic signalling pathways. The present study models the transport of ATP in a realistic porcine aortic trifurcation, which includes multiple branches and bifurcations. The focus of the present study is understanding how pulsatile flow effects mass transfer, observing both mean and transient variations. Unlike in the many idealized models which model transport of low diffusion coefficient species, the realistic geometry leads to very different mass transfer characteristics. These include spiral patterns in the distribution of low concentration fluid. Furthermore, the mean ATP distribution was found to be elevated compared with the steady state; this is attributed to the effects of convective mixing. The results strongly implicate that under certain conditions mass transport in pulsatile flow exhibits different hydrolysis characteristics at the endothelium compared with steady state. Transient variations throughout the the cardiac cycle were found to be small. This small transient response is primarily due to low ATP diffusivity.

A stable approach for coupling multidimensional cardiovascular and pulmonary networks based on a novel pressure‐flow rate or pressure‐only Neumann boundary condition formulation

In many biomedical flow problems, reversed flows along with standard treatment of Neumann boundary conditions can cause instabilities. We have developed a method that resolves these instabilities in a consistent way while maintaining correct pressure and flow rate values. We also are able to remove the necessary prescription of both pressure and velocities/flow rates to problems where only pressure is known. In addition, the method is extended to coupled 3D/reduced-D fluid and fluid-structure interaction models. Numerical examples mainly focus on using Neumann boundary condition in cardiovascular and pulmonary systems, particularly, coupled with 3D-1D and 3D-0D models. Inflow pressure, traction, and impedance boundary conditions are first tested on idealized tubes for various Womersley numbers. Both pressure and flow rate are shown to match the analytical solutions for these examples. Our method is then tested on a coupled 1D-3D-1D artery example, demonstrating the power and simplicity of extending this method toward fluid-structure interaction. Finally, the proposed method is investigated for a coupled 3D-0D patient-specific full lung model during spontaneous breathing. All coupled 3D/reduced-D results show a perfect matching of pressure and flow rate between 3D and corresponding reduced-D boundaries. The methods are straight-forward to implement in contrast to using Lagrange multipliers as previously proposed in other studies.

Effects of Arterial Bifurcation Geometry on Nucleotide Concentration at the Endothelium

A mathematical and computational analysis of nucleotide concentration for a two-dimensional artery bifurcation has been developed. This region of the vasculature is known to be exposed to spatially varying wall shear stress (WSS), hence the variation of adenosine nucleotides is of interest. A previously derived similarity solution for mass transport in blood boundary layers for arbitrary wall shear stress function has been used. For the analytical model, the geometric condition has been incorporated into the system using the theory for flow past a wedge. As the bifurcation angle varies different characteristics are exhibited, such as maxima in ADP concentration and the existence of a low wall shear stress region around the stagnation point, for large angles. In the limiting case of two-dimensional stagnation point flow the concentration was constant throughout the domain length. The computational simulations provided a more detailed understanding into how nucleotides vary. Similarities existed between the two solution methods in the vicinity of the stagnation point, but deviated as the fully developed condition prevailed. Additionally, the effect of pulsatile flow has been included, leading to considerable gradients in wall shear stress, both temporal and spatial. However, the resulting nucleotide concentration is determined by the time-averaged wall shear stress. The effects of flow-induced ATP release have also been included, leading to significant changes in ATP concentration. Under rapid release, the concentration at the surface increased relative to the bulk concentration.

Concentration of blood-borne agonists at the endothelium

The delivery of endothelial ligands and macromolecules, such as lipoproteins, from the circulation to the arterial wall is of central importance in modulating endothelial cell function and physiology and, consequently, in the onset of vascular disease. Given the strong spatial correlation between areas of disturbed blood flow and occurrence of atherosclerotic plaque, a detailed understanding of the effects of different fluid flow characteristics on the delivery of factors in the bloodstream to the endothelium is an essential step towards understanding the observed localization of vascular disease to certain focal sites within the vasculature. In this paper, a model of biochemical mass transport in a two-dimensional flow chamber with spatially varying wall shear stress is presented. The advantage of the relatively simple arterial geometry is that all the essential features of blood flow in vivo are captured, but the underlying effects on mass transport, and, hence, on endothelial cell function, are not masked by complex three-dimensional flow. Indeed, it is demonstrated that a previously derived similarity solution, in terms of the shear stress on the endothelial wall, is an asymptotically close approximation to the exact solution to the advection–diffusion equation. The mathematical analysis is then used to identify fundamental links between the wall shear stress and the distribution of chemical species along the endothelium. The physiological implications of these links are discussed, in particular the location of maximum chemical concentration on the endothelium, which is of great significance to the regulation of intracellular signalling and, consequently, endothelial cell function.

The effects of curvature and constriction on airflow and energy loss in pathological tracheas

This paper considers factors that play a significant role in determining inspiratory pressure and energy losses in the human trachea. Previous characterisations of pathological geometry changes have focussed on relating airway constriction and subsequent pressure loss, however many pathologies that affect the trachea cause deviation, increased curvature, constriction or a combination of these. This study investigates the effects of these measures on tracheal flow mechanics, using the compressive goitre (a thyroid gland enlargement) as an example. Computational fluid dynamics simulations were performed in airways affected by goitres (with differing geometric consequences) and a normal geometry for comparison. Realistic airways, derived from medical images, were used because idealised geometries often oversimplify the complex anatomy of the larynx and its effects on the flow. Two mechanisms, distinct from stenosis, were found to strongly affect airflow energy dissipation in the pathological tracheas. The jet emanating from the glottis displayed different impingement and breakdown patterns in pathological geometries and increased loss was associated with curvature.