Abstract. In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled.In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A, Q), area-velocity (A, u), pressure-velocity (p, u) and pressure-mass flux(p, Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A, u) variables the extension of the single vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves.The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second order Adams-Bashforth time integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)).Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.

In this paper we numerically investigate a one‐dimensional model of blood flow in human arteries using both a discontinuous Galerkin and a Taylor–Galerkin formulation. The derivation of the model and the numerical schemes are detailed and applied to two model numerical experiments. We first study the effect of an intervention, such the implantation of a vascular prosthesis (e.g. a stent), which leads to an abrupt variation of the mechanical characteristics of an artery. We then discuss the simulation of the propagation of pressure and velocity waveforms in the human arterial tree using a simplified model consisting of the 55 main arteries. Copyright © 2003 John Wiley & Sons, Ltd.

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Abstract. In this paper we outline the hyperbolic system of governing equations describing onedimensional blood flow in arterial networks. This system is numerically discretised using a discontinuous Galerkin formulation with a spectral/hp element spatial approximation. We apply the numerical model to arterial networks in the placenta. Starting with a single placenta we investigate the velocity waveform in the umbilical artery and its relationship with the distal bifurcation geometry and the terminal resistance. We then present results for the waveform patterns and the volume fluxes throughout a simplified model of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. The effects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role of terminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by the computational model and demonstrate the applicability of the methods to the simulation of flows in arterial networks.Mathematics Subject Classification. 92C35, 76Z05.

Pressure and blood flow waveforms in the human body can be recorded using techniques such as sphygmomanometry and Doppler ultrasound. The presence of abnormal waveforms in arteries may indicate a pathological state. In this paper arterial wave patterns are modelled using a simplified one dimensional model for blood flow. The hyperbolic system of governing equations is discretised using the discontinuous Galerkin method. Results are presented for patterns of blood flow and pressure waves throughout a simplified arterial system consisting of the main 55 arteries. The method is also employed to determine unusual wave patterns in monochorionic twin pregnancies where an anastomosis between the twins is present across the placenta equator.