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of
19
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Onedimensional modeling of a vascular network in spacetime variables
 Journal of Engineering Mathematics
"... Abstract. In this paper a onedimensional model of a vascular network based on spacetime variables is investigated. Although the onedimensional system has been more widely studied using a spacefrequency decomposition, the spacetime formulation oers a more direct physical interpretation of the d ..."
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Cited by 59 (9 self)
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Abstract. In this paper a onedimensional model of a vascular network based on spacetime variables is investigated. Although the onedimensional system has been more widely studied using a spacefrequency decomposition, the spacetime formulation oers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the spacetime representation of the linear and nonlinear onedimensional system can be theoretically and numerically modelled. In deriving the governing equations from rst principles, the assumptions involved in constructing the system in terms of areamass
ux (A;Q), areavelocity (A;u), pressurevelocity (p; u) and pressuremass
ux(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.
Lumped parameter outflow models for 1D blood flow simulations: Effect on pulse waves and parameter estimation
 Commun. Comput. Phys
"... Abstract. Several lumped parameter, or zerodimensional (0D), models of the microcirculation are coupled in the time domain to the nonlinear, onedimensional (1D) equations of blood flow in large arteries. A linear analysis of the coupled system, together with in vivo observations, shows that: ( ..."
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Cited by 19 (4 self)
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Abstract. Several lumped parameter, or zerodimensional (0D), models of the microcirculation are coupled in the time domain to the nonlinear, onedimensional (1D) equations of blood flow in large arteries. A linear analysis of the coupled system, together with in vivo observations, shows that: (i) an inflow resistance that matches the characteristic impedance of the terminal arteries is required to avoid nonphysiological wave reflections; (ii) periodic mean pressures and flow distributions in large arteries depend on arterial and peripheral resistances, but not on the compliances and inertias of the system, which only affect instantaneous pressure and flow waveforms; (iii) peripheral inertias have a minor effect on pulse waveforms under normal conditions; and (iv) the time constant of the diastolic pressure decay is the same in any 1D model artery, if viscous dissipation can be neglected in these arteries, and it depends on all the peripheral compliances and resistances of the system. Following this analysis, we propose an algorithm to accurately estimate peripheral resistances and compliances from in vivo data. This algorithm is verified against numerical data simulated using a 1D model network of the 55 largest human arteries, in which the parameters of the peripheral windkessel outflow models are known a priori. Pressure and flow waveforms in the aorta and the first generation of bifurcations are reproduced with relative rootmeansquare errors smaller than 3%.
Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network
 Journal of Computational Physics
"... Reduced models of human arterial networks are an efficient approach to analyse quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in ..."
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Cited by 10 (2 self)
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Reduced models of human arterial networks are an efficient approach to analyse quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour can not be made unless the effect of such data uncertainty is quantified. In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a highorder stochastic collocation method based on the generalized polynomial
TIME DOMAIN COMPUTATIONAL MODELLING OF 1D ARTERIAL NETWORKS IN MONOCHORIONIC PLACENTAS
"... 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 ..."
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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 arterioarterial anastomosis and an arteriovenous 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. 1.
S.J.: Analysing the pattern of pulse waves in arterial networks: a timedomain study
 J. Eng. Math
, 2009
"... Abstract. The aim of this work is to study the mechanisms that determine the shape of arterial pulse waves in normal conditions using a time domain analysis of the onedimensional (1D) equations of blood flow in compliant vessels. Based on the reservoirwave hypothesis, we first propose an algorith ..."
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Cited by 8 (4 self)
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Abstract. The aim of this work is to study the mechanisms that determine the shape of arterial pulse waves in normal conditions using a time domain analysis of the onedimensional (1D) equations of blood flow in compliant vessels. Based on the reservoirwave hypothesis, we first propose an algorithm to calculate the reservoir component of the pressure waveform at an arbitrary location in the arterial network from pressure measurements only. This algorithm is applied to analyse the shape of the pulse waves simulated using a nonlinear 1D model of the 55 largest systemic arteries in the human. This study demonstrates that the reservoir pressure component makes a higher contribution to the total pressure waveforms than the wave components. The wave components are tightly related to the outflow from the left ventricle in early systole. Later in the cardiac cycle, the wave components are the result of reflections at the junctions and terminal branches of the network. We also present a novel postprocessing algorithm to study the wave component of the pressure waveforms and the crosssectional velocity waveforms. This algorithm describes the waves generated at an arbitrary location in a linear 1D model network by a single wavefront starting at the root. Although the number of reflected waves increases approximately as 3m, with m being the number of reflection sites encountered, the magnitude of each reflected wave tends to decrease exponentially. As a result, wave activity is almost nonexistent during the last period of diastole, as is typically observed in vivo. This algorithm also provides valuable information on identifying the parameters and pathways that have a larger effect on the simulated pulse waveforms.
Topics in Ultrascale Scientific Computing with Application in Biomedial Modeling
, 2009
"... In this Thesis we focus on simulations of blood flow in threedimensional patientspecific arterial networks. We employ highorder spectral/hpelement spatial discretization and concentrate on computational efficiency in solving multimillion degrees of freedom (DOF) flow problems on petaflop compu ..."
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Cited by 2 (0 self)
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In this Thesis we focus on simulations of blood flow in threedimensional patientspecific arterial networks. We employ highorder spectral/hpelement spatial discretization and concentrate on computational efficiency in solving multimillion degrees of freedom (DOF) flow problems on petaflop computers. We develop new twolevel domain decomposition method and multilevel communicating interface for ultraparallel flow simulations. Specifically, at the coarse level the computational domain is subdivided into several big patches. Within each patch a spectral element discretization (fine level) is employed. New interface conditions for the NavierStokes equations are developed. The proposed numerical approach has been tested in arterial flow simulations with up to 147 arteries. Solution of 2.87B DOF problem was computed on 18,576 processors in less than one second at each time step. A scalable and fast parallel lowenergy bases preconditioner (LEBP) in conjunction with coarsespace linear vertex solver is developed. We provide details on optimization, parallel performance and implementation of the coarsespace solver and show scalability of LEBP on thousands processors of the IBM BlueGene/L and the Cray XT3. An embarrassingly parallel but extremely efficient accelerator for iterative solver has been proposed. The new
Verification and comparison of four numerical schemes for a 1D viscoelastic
, 2013
"... blood flow model ..."
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13. Algorithms and arteries: Multidomain spectral/hp
"... methods for vascular flow modelling ..."
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OneDimensional Spectral/hp Element Simulation of Wave Propagation in Human Arterial Networks by
, 2003
"... This thesis is concerned with the modelling of blood
ow and wave propagation in arterial systems using computational
uid dynamics. The arterial networks are simpli ed to onedimensional systems and the governing equations are solved using a high order spectral/hp element method with a discontinuou ..."
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This thesis is concerned with the modelling of blood
ow and wave propagation in arterial systems using computational
uid dynamics. The arterial networks are simpli ed to onedimensional systems and the governing equations are solved using a high order spectral/hp element method with a discontinuous Galerkin discretisation. Two dierent human systemic arterial networks are considered. The rst is a composite network that has been studied by other investigators. A number of different parameters such as the in
ow and out
ow boundary conditions and cardiac period are varied and the eects on the pressure and velocity waveforms discussed. The second network is entirely new and is constructed from invivo geometry data obtained using magnetic resonance imaging of a single human subject. The numerically computed waveforms are compared with clinical measurements for the same human subject obtained using Doppler ultrasound and applanation tonometry. The eects of arterial taper and illmatched bifurcations are assessed. The eect of geometrical alterations to the network are investigated and the impact on the predicted waveforms of articial devices, such as bypass grafts and stents, presented. Wave propagation is also investigated in placental arterial networks focusing on monochorionic placentas where twin fetuses share a single placenta in which the blood vessels of the two fetoplacental circulations are often joined. The geometry and connectivity of the placental arterial networks are obtained by producing a plastic cast of the recently delivered placentas. The onedimensional code is used to investigate the dynamic interaction between the two fetoplacental circulations. The volume
uxes are calculated and the predicted waveforms are compared with invivo measurements. The conclusions discuss the usefulness and further applications of the numerical modelling and make suggestions for the incorporation of additional parameters.