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30
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.
SelfConsistent Effective Equations Modeling Blood Flow in MediumtoLarge Compliant Arteries
 SIAM J. Multiscale Analysis and Simulation
, 2005
"... Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time dependent pressure head difference. The NavierStokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a ..."
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Cited by 21 (10 self)
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Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time dependent pressure head difference. The NavierStokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a curved, linearly elastic membrane to model the wall. Employing the asymptotic techniques typically used in thin domains, we derive a set of effective equations that hold in mediumtolarge compliant vessels for laminar flow regimes. The main novelty is the derivation of the effective equations that do not assume any ad hoc closure, typically assumed in the derivation of onedimensional models. Using ideas from homogenization theory for porous media flows, we obtain a closed system of effective equations that are of Biot type with memory. Memory accounts for the wavelike phenomena in the problem. Although the equations are twodimensional, their simple structure enables a design of a numerical algorithm that has complexity of a onedimensional solver. Our numerical simulations show that our model captures twodimensional effects that cannot be captured using standard onedimensional methods. Key words. Blood flow, compliant arteries, fluidstructure interaction, effective equations. AMS subject classifications. 35Q30, 74K15, 76D27 1. Introduction. In
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%.
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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Cited by 14 (0 self)
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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.
On the coupling of systems of hyperbolic conservation laws with ordinary differential equations
 Nonlinearity
"... Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided ..."
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Cited by 10 (2 self)
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Motivated by applications to the piston problem, to a manhole model, to blood flow and to supply chain dynamics, this paper deals with a system of conservation laws coupled with a system of ordinary differential equations. The former is defined on a domain with boundary and the coupling is provided by the boundary condition. For each of the examples considered, numerical integrations are provided.
Mixed Systems: ODEs – Balance Laws
, 2011
"... We prove the well posedness of mixed problems consisting of a system of ordinary differential equations coupled with systems of balance laws in domains with moving boundaries. The interfaces between the systems are provided by the boundary data and boundary positions. Various situations that fit int ..."
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Cited by 9 (2 self)
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We prove the well posedness of mixed problems consisting of a system of ordinary differential equations coupled with systems of balance laws in domains with moving boundaries. The interfaces between the systems are provided by the boundary data and boundary positions. Various situations that fit into this framework are studied, both analytically and numerically. We consider a piston moving in a pipe full of fluid, a model for fluid–particle interaction and a traffic model. References to other examples in the literature are provided.
Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries
 SIAM Journal on Applied Dynamical Systems
, 2005
"... Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given timedependent pressure drop between the inlet and the outlet boundary. The pressure drop is assumed to be small, thereby introducing creeping flow in the tube. St ..."
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Cited by 9 (5 self)
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Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given timedependent pressure drop between the inlet and the outlet boundary. The pressure drop is assumed to be small, thereby introducing creeping flow in the tube. Stokes equations for incompressible viscous fluid are used to model the flow, and the equations of a curved, linearly elastic membrane are used to model the wall. Due to the creeping flow and to small displacements, the interface between the fluid and the lateral wall is linearized and supposed to be the initial configuration of the membrane. We study the dynamics of this coupled fluidstructure system in the limit when the ratio between the characteristic width and the characteristic length tends to zero. Using the asymptotic techniques typically used for the study of shells and plates, we obtain a set of Biottype viscoelastic equations for the effective pressure and the effective displacements. The approximation is rigorously justified through a weak convergence result and through the error estimates for the solution of the effective equations modified by an outlet boundary layer. Applications of the model problem include blood flow in small arteries. We recover the wellknown law of Laplace and obtain new improved models that hold in cases when the shear modulus of the vessel wall is not negligible and the Poisson ratio is arbitrary.
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.
Fluidstructure interaction in blood flow allowing nonzero longitudinal structure displacement. Under revision
"... The study of flow of a viscous incompressible fluid through a compliant tube has many applications. One major application is blood flow through human arteries. Understanding wave propagation in arterial walls, local hemodynamics, and temporal wall shear stress gradient is important in understanding ..."
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Cited by 8 (0 self)
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The study of flow of a viscous incompressible fluid through a compliant tube has many applications. One major application is blood flow through human arteries. Understanding wave propagation in arterial walls, local hemodynamics, and temporal wall shear stress gradient is important in understanding the mechanisms leading to various complications in cardiovascular function. Many clinical treatments can be studied in detail only if a reliable model describing the response of arterial walls to the pulsatile blood flow is considered. It has been well accepted that in mediumtolarge arteries, blood can be modeled as a viscous, incompressible Newtonian fluid. Although blood is a suspension of red blood cells, white blood cells, and platelets in plasma, its nonNewtonian nature due to the particular rheology is relevant in small arteries (arterioles) and capillaries where the diameter of the arteries becomes comparable to the size of the cells. In mediumtolarge arteries, such as the coronary arteries (medium) and the abdominal aorta (large), the Navier
A twodimensional effective model describing fluidstructure interaction in blood flow: analysis, simulation and experimental validation. Comptes Rendus Mechanique Acad
 Sci. Paris
, 2005
"... This work is motivated by the study of blood flow in compliant arteries. In medium to large vessels such as the human aorta and iliac arteries, blood can be modeled as a viscous, incompressible Newtonian fluid, [27, 19]. Driven by a timeperiodic pressure pulse caused by the contractions and ..."
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Cited by 8 (3 self)
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This work is motivated by the study of blood flow in compliant arteries. In medium to large vessels such as the human aorta and iliac arteries, blood can be modeled as a viscous, incompressible Newtonian fluid, [27, 19]. Driven by a timeperiodic pressure pulse caused by the contractions and