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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.
Analysis of lumped parameter models for blood flow simulations and their relation with 1D models
, 2004
"... Abstract. This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward onedimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to dis ..."
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Abstract. This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward onedimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to discretize continuous 1D models at first order in space. We derive the complete set of equations useful for the blood flow networks, new schemes for electric circuit analogy, the stability criteria that guarantee the convergence, and the energy estimates of the limit 1D equations.
Parameter identification for a onedimensional blood flow model
 in &quot;Proceedings of CEMRACS 2004 on Mathematics and Applications to Biology and Medicine&quot;, E. CANCÈS, J.F. GERBEAU (editors). , ESAIM Proceedings
"... Abstract. The purpose of this work is to use a variational method to identify some of the parameters of onedimensional models for blood flow in arteries. These parameters can be fit to approach as much as possible some data coming from experimental measurements or from numerical simulations perform ..."
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Abstract. The purpose of this work is to use a variational method to identify some of the parameters of onedimensional models for blood flow in arteries. These parameters can be fit to approach as much as possible some data coming from experimental measurements or from numerical simulations performed using more complex models. A nonlinear least squares approach to parameter estimation was taken, based on the optimization of a cost function. The resolution of such an optimization problem generally requires the efficient and accurate computation of the gradient of the cost function with respect to the parameters. This gradient is computed analytically when the onedimensional hyperbolic model is discretized with a second order TaylorGalerkin scheme. An adjoint approach was used. Some preliminary numerical tests are shown. In these simulations, we mainly focused on determining a parameter that is linked to the mechanical properties of the arterial walls, the compliance. The synthetic data we used to estimate the parameter were obtained from a numerical computation performed with a more accurate model: a threedimensional fluidstructure interaction model. The first results seem to be promising. In particular, it is worth noticing that the estimated compliance which gives the best fit is quite different from the values that are commonly used in practice.
Veneziani A. ”Geometrical Multiscale Models for the Cardiovascular System
 ABIOMED Lecture Notes
, 2005
"... The human cardiovascular system is formed by a closed network with a high level of interdependency. The distribution of blood flow inside the various vascular grounds affects the local flow patterns in each ..."
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The human cardiovascular system is formed by a closed network with a high level of interdependency. The distribution of blood flow inside the various vascular grounds affects the local flow patterns in each
Sketchbased Dynamic Illustration of Fluid Systems
"... Figure 1: Illustrations created using our system to describe the surgical repair procedure of tricuspid atresia (TA). The user interactively edits the illustration and the system continuously presents the corresponding blood flow computed by simplified fluid simulation. This paper presents a lightwe ..."
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Figure 1: Illustrations created using our system to describe the surgical repair procedure of tricuspid atresia (TA). The user interactively edits the illustration and the system continuously presents the corresponding blood flow computed by simplified fluid simulation. This paper presents a lightweight sketching system that enables interactive illustration of complex fluid systems. Users can sketch on a 2.5dimensional (2.5D) canvas to design the shapes and connections of a fluid circuit. These input sketches are automatically analyzed and abstracted into a hydraulic graph, and a new hybrid fluid model is used in the background to enhance the illustrations. The system provides rich simple operations for users to edit the fluid system incrementally, and the new internal flow patterns can be simulated in real time. Our system is used to illustrate various fluid systems in medicine, biology, and engineering. We asked professional medical doctors to try our system and obtained positive feedback from them.
Minimal haemodynamic modelling of the circulation
, 2008
"... The knowledge of the flowstructures in the human arteries is limited. The medical staff have the wish to have a better side to this phenomenon. In a specific mathematical research of the flow through the carotid bifurcation there is attention for this problem. To make it possible to do this research ..."
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The knowledge of the flowstructures in the human arteries is limited. The medical staff have the wish to have a better side to this phenomenon. In a specific mathematical research of the flow through the carotid bifurcation there is attention for this problem. To make it possible to do this research a mathematical model of the whole cardiovascular system (CVS) is needed. Models found in literature simulate specific areas of the CVS while others are either overly complex, difficult to solve, and/or unstable. This thesis develops a minimal model with the primary goal of having the possibility to reflect accurately a small part of the cardiovascular system. The focus is just on the simplicity of the overall structure, with a reasonable reflection of the heartfunction. A novel mixedformulation approach to simulating blood flow in lumped parameters CVS models is outlined that adds minimal complexity, but significantly improves physiological accuracy. The minimal model is shown to match a Wiggers ’ diagram and was also verified to simulate different heartdiseases. The model offers a tool that can be used in conjunction with experimental research to improve understanding of the blood flow. i ii
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"... approximate method for solving incompressible NavierStokes problem with flow rate conditions ∗ ..."
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approximate method for solving incompressible NavierStokes problem with flow rate conditions ∗
REVIEW Open Access Review of ZeroD and 1D Models of Blood Flow in the Cardiovascular System
"... Background: Zerodimensional (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. ZeroD models provide a concise way to evaluate the haemodynamic int ..."
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Background: Zerodimensional (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. ZeroD models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst oneD (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 Results: The 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