Rodney Hose

Review of Zero-D and 1-D Models of Blood Flow in the Cardiovascular System

BackgroundZero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models.Method and ResultsThe purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/. They are freely available to the research and education communities.ConclusionEach published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

Influence of inlet boundary conditions on the local haemodynamics of intracranial aneurysms

Haemodynamics is believed to play an important role in the initiation, growth and rupture of intracranial aneurysms. In this context, computational haemodynamics has been extensively used in an effort to establish correlations between flow variables and clinical outcome. It is common practice in the application of Dirichlet boundary conditions at domain inlets to specify transient velocities as either a flat (plug) profile or a spatially developed profile based on Womersleys analytical solution. This paper provides comparative haemodynamics measures for three typical cerebral aneurysms. Three dimentional rotational angiography images of aneurysms at three common locations, viz. basilar artery tip, internal carotid artery and middle cerebral artery were obtained. The computational tools being developed in the European project @neurIST were used to reconstruct the fluid domains and solve the unsteady Navier–Stokes equations, using in turn Womersley and plug-flow inlet velocity profiles. The effects of these assumptions were analysed and compared in terms of relevant haemodynamic variables within the aneurismal sac. For the aneurysm at the basilar tip geometries with different extensions of the afferent vasculature were considered to study the plausibility of a fully-developed axial flow at the inlet boundaries. The study shows that assumptions made on the velocity profile while specifying inlet boundary conditions have little influence on the local haemodynamics in the aneurysm, provided that a sufficient extension of the afferent vasculature is considered and that geometry is the primary determinant of the flow field within the aneurismal sac. For real geometries the Womersley profile is at best an unnecessary over-complication, and may even be worse than the plug profile in some anatomical locations (e.g. basilar confluence).

Development and validation of models for the investigation of blood clotting in idealized stenoses and cerebral aneurysms

An in vitro model of blood clotting is presented using hypercoaguable milk as an analog for blood. Milk clot formation was studied for periods of 2, 5, 10, 20, and 30 min within an idealized stenosis geometry. Clot formation was recorded using photography, clot casting, and clot mass calculation. The distribution of clot within the fluid was seen to be in good agreement with a previous study that used a residence time model to predict areas of clot formation in thrombin solution. A numerical model was formulated within computational fluid dynamics package CFX that allowed local activation of blood clotting to be simulated. This model was applied to the analysis of an idealized cerebral aneurysm geometry. An idealized coil geometry was included within the aneurysm and clotting fluid concentration and fluid residence time were modeled using transport equations within CFX. The viscosity of the fluid was defined as a function of both residence time and clotting fluid concentration. The model was seen to produce features consistent with observations of thrombosis within cerebral aneurysms, while avoiding the unrealistic build up of clot in near-wall regions that is associated with a pure residence time model.

Identification of artery wall stiffness: in vitro validation and in vivo results of a data assimilation procedure applied to a 3D fluid-structure interaction model

We consider the problem of estimating the stiffness of an artery wall using a data assimilation method applied to a 3D fluid-structure interaction (FSI) model. Recalling previous works, we briefly present the FSI model, the data assimilation procedure and the segmentation algorithm. We present then two examples of the procedure using real data. First, we estimate the stiffness distribution of a silicon rubber tube from image data. Second, we present the estimation of aortic wall stiffness from real clinical data.

Heart rate reduction with ivabradine promotes shear stress-dependent anti-inflammatory mechanisms in arteries

Blood flow generates wall shear stress (WSS) which alters endothelial cell (EC) function. Low WSS promotes vascular inflammation and atherosclerosis whereas high uniform WSS is protective. Ivabradine decreases heart rate leading to altered haemodynamics. Besides its cardio-protective effects, ivabradine protects arteries from inflammation and atherosclerosis via unknown mechanisms. We hypothesised that ivabradine protects arteries by increasing WSS to reduce vascular inflammation. Hypercholesterolaemic mice were treated with ivabradine for seven weeks in drinking water or remained untreated as a control. En face immunostaining demonstrated that treatment with ivabradine reduced the expression of pro-inflammatory VCAM-1 (p<0.01) and enhanced the expression of anti-inflammatory eNOS (p<0.01) at the inner curvature of the aorta. We concluded that ivabradine alters EC physiology indirectly via modulation of flow because treatment with ivabradine had no effect in ligated carotid arteries in vivo, and did not influence the basal or TNFα-induced expression of inflammatory (VCAM-1, MCP-1) or protective (eNOS, HMOX1, KLF2, KLF4) genes in cultured EC. We therefore considered whether ivabradine can alter WSS which is a regulator of EC inflammatory activation. Computational fluid dynamics demonstrated that ivabradine treatment reduced heart rate by 20 % and enhanced WSS in the aorta. In conclusion, ivabradine treatment altered haemodynamics in the murine aorta by increasing the magnitude of shear stress. This was accompanied by induction of eNOS and suppression of VCAM-1, whereas ivabradine did not alter EC that could not respond to flow. Thus ivabradine protects arteries by altering local mechanical conditions to trigger an anti-inflammatory response.

Stable periodic vortex shedding studied using computational fluid dynamics, laser sheet flow visualization, and MR imaging

Recirculating and detached flow patterns close to the carotid bifurcation are thought to play an important role in the development of carotid stenoses by promoting atherosclerosis. The aim of this study was to investigate a flow regime with strong transient characteristics, including vortex shedding and transport to develop methodologies appropriate to the analysis of carotid stenoses. The existence of a regular periodic vortex street behind a cylindrical flow obstruction was predicted and analysed in detail by Theodore van Karman in the early 20th century. This model was chosen in our study for both ease of phantom construction and of theoretical modelling using finite element computational fluid dynamics (CFD). The results of the theoretical calculations have been compared with two methods of flow visualization-laser sheet imaging and real-time echo planar magnitude MR imaging. Flow was investigated over a range of Reynolds number from 40 through 400 through which vortex shedding is predicted. Good overall agreement was obtained between the theoretical (16 mm-CFD) and experimental (16+/-2 mm-Laser, 17+/-2 mm-MRI) estimates of the Karman Vortex street wavelength for a Reynolds number of 200.

209 The Development and Validation of an in silico Model of Coronary Arterial Physiology for Diagnostic Use

Background Coronary revascularisation guided by physiological assessment with fractional flow reserve (FFR) improves outcomes and expenditure. However, due to practical limitations, FFR is only used in <10% of percutaneous coronary interventions (PCI) and almost none of the 250,000 diagnostic coronary angiograms performed every year in the UK. We aimed to develop an in silico model of coronary physiology which computes FFR using computational fluid dynamics (CFD) with data from coronary angiography (CAG), which also computes real-time coronary flow. Methods We studied 40 patients with stable coronary artery disease undergoing PCI. The in silico workflow reconstructed coronary geometry from CAG images into 3-D geometries using image registration and segmentation protocols. The 3-D domain was coupled to an algebraically encoded zero-D model representing coronary microvascular resistance. CFD simulated pressure and flow. Computed pressures were validated against patient data. Flow was computed from those pressures and validated in a benchtop flow circuit constructed from patient-specific 3-D printed coronary arterial models, a blood analogue, and patient-specific coronary flow patterns using a programmable pump. Pressure and flow were measured with clinical pressure and flow sensitive wires. To reduce computation time, a novel pseudo-transient CFD method was developed and investigated. Results Using generic boundary conditions, the initial model computed ‘virtual’ FFR (vFFR) with ±0.06 error and diagnosed physiological lesion significance (FFR <0.80) with 97% accuracy. Ultra-precise, transient CFD took 24 h to compute, whereas the novel ‘pseudo-transient’ method reduced computation to <4 mins on a standard PC with no significant loss of accuracy. Using case-specific boundary conditions (based on invasive measurements) error was reduced to ±0.004 and more useful, artery-specific, patient averaged values yielded ±0.01 error. Coronary flow was computed in all cases and, with the 3-D printed arteries, we demonstrated that the in silico model computed reliable coronary flow with 5.3% error in 3 min. Conclusions Our in silico model computes reliable intra-coronary physiology, including pressure and flow, in timescales that are practical for clinical use in the catheter laboratory. Virtual modelling of FFR and hyperaemic stenosis resistance (HSR) is now feasible, which may facilitate increased uptake and impact of physiologically-guided revascularisation planning for every patient undergoing CAG, rather than a small minority undergoing selective PCI. Improved accuracy will follow from better characterisation of the individual’s microvascular resistance. coronary stenosis. Abstract 209 Figure 1 A patient case (a) with pressure and flow simulation results (b and c) across a severe right

ONE-DIMENSIONAL SIMULATION OF HAEMODYNAMICS IN AORTIC COARCTATION

Aortic coarctation is a narrowing of part of the aorta due to the congenital defect. It causes abnormally elevated after-load to the left heart and thus impaired blood supply to the peripheral organs. Currently diagnosis and treatment of aortic coarctation are still difficult issues due to the complex haemodynamic changes involved [Keshavarz-Motamed, 2011]. The EU FP7 euHeart project is developing numerical tools to assist clinicians in the diagnostic evaluation of aortic coarctation based on MRI images of patients. The pressure gradient across the aorta in a patient-specific anatomy is computed by 3D computational fluid dynamics, but the result is critically dependent on the balance of the measured ascending aortic flow that is distributed to the supra-aortic vessels and the descending aorta. The balance is achieved using Windkessel models at the four outlets. The process of tuning the Windkessels in combination with the 3D model to match the (noisy) clinical flow measurements is computationally expensive and is sensitive to the initialisation. We propose the use of an intermediate 1D/0D model, based on the geometry of the patient aorta, to pre-tune the system.