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SUMMARYThere is an important interaction between the pumping performance of the ventricle, arterial haemodynamics and coronary blood flow. While previous non-linear 1D models have focused only on one of these components, the model presented in this study includes coronary and systemic arterial circulations, as well as ventricular pressure and an aortic valve that opens and closes 'independently' and based on local haemodynamics. The systemic circulation is modelled as a branching network of elastic tapering vessels. The terminal element applied at the extremities of the network is a single tapering vessel which is shown to adequately represent the input characteristics of the downstream vasculature. The coronary model consists of left and right coronary arteries which both branch into two 'equivalent' vessels that account for the lumped characteristics of subendocardial and subepicardial flows. As contracting heart muscle causes significant compression of the subendocardial vessels, a time-varying external pressure proportional to ventricular pressure is applied to the distal part of the equivalent subendocardial vessel. The aortic valve is modelled using a variable reflection coefficient with respect to backward-running aortic waves, and a variable transmission coefficient with respect to forward-running ventricular waves. A realistic ventricular pressure is the input to the system; however, an afterload-corrected ventricular pressure is calculated and results in pressure gradients between the ventricle and aorta that are similar to those observed in vivo. The 1D equations of fluid flow are solved using the locally conservative Galerkin method, which provides explicit element-wise conservation, and can naturally incorporate vessel branching. Each component of the model is verified using a number of tests to ensure accuracy and reveal the underlying processes that give rise to complex pressure and flow waveforms. The complete model is then implemented, and simulations are performed with input parameters representing 'at rest' and exercise states for a normal adult. The resulting waveforms contain all of the important features seen in vivo, and standard measures of haemodynamic state are found to be normal. In addition, one or several characteristics of some common diseases are imposed on the model and are found to produce haemodynamic changes that agree with experimental observations in the published literature. Using a patient-specific carotid bifurcation geometry, 1D velocity waveforms are also compared with waveforms obtained from a three-dimensional model. The 1D and 3D results show good agreement.

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SUMMARYHaemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of threedimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

SUMMARYIn this paper, an artiÿcial compressibility scheme using the ÿnite element method is introduced. The multi-purpose CBS scheme is implemented in its fully explicit form to solve incompressible uid dynamics problems. It is important to note that the scheme developed here includes split and velocity correction. The proposed method takes advantage of good features from both velocity correction and standard artiÿcial compressibility schemes. Unlike many other artiÿcial compressibility schemes, the proposed one works on a variety of grids and gives results for a wide range of Reynold's numbers. The paper presents some bench mark two-and three-dimensional steady and unsteady incompressible ow solutions obtained from the proposed scheme.