Blood Flow In Elastic Vessels Research Articles

Computational analysis of one-dimensional models for simulation of blood flow in vascular networks

The paper is devoted to the comparison of one-dimensional models, used for the simulation of blood flow in elastic vessels. The inviscid, Newtonian, and non-Newtonian models of blood are considered. The one-dimensional non-Newtonian models, corresponding to widely-used rheological models with shear-dependent viscosity (Power Law, Carreau, Carreau–Yasuda, Cross, simplified Cross, modified Cross, Powell–Eyring, modified Powell–Eyring, Yeleswarapu, modified Yeleswarapu, and Quemada) are presented.The attention is focused on the investigation of the influence of viscosity, shape factor, velocity profile, non-Newtonian viscosity, and hematocrit. In the numerical experiments, the models of arterial systems with twenty and thirty-seven vessels are considered. It is demonstrated, that for the vessels near the heart the effect of viscosity can be considered as inessential, and an inviscid model can be used. The main effect on the relative deviations of solutions, obtained by different models, is realized by the velocity profile. The effect of non-Newtonian viscosity is not so essential, but can be taken into account for the models with peripheral vessels.

Numerical solutions for blood flow in elastic vessels

We consider the differential–algebraic system for the blood flow and pressure in the systemic arteries. By the operator splitting method, we transform the system into the hyperbolic one, introduce the bicharacteristics, and perform the time–space nonuniform discretization, obtaining the innovative difference scheme. Our results are illustrated with numerical experiments.

Formulation of exactly balanced solvers for blood flow in elastic vessels and their application to collapsed states

In this work, numerical solvers based on extensions of the Roe and HLL schemes are adapted to deal with test cases involving extreme collapsing conditions in elastic vessels. To achieve this goal, the system is transformed to provide a conservation–law form, allowing to define Rankine–Hugoniot conditions. The approximate solvers allow to describe the inner states of the solution. Therefore, source term fixes can be used to prevent unphysical values of vessel area and, at the same time, the eigenvalues of the system control stability. Numerical solvers of different order are tested using a wide variety of Riemann problems, including extreme vessel collapse and blockage. In all cases, the robustness of the approximate solvers presented here is checked using first and third order methods in time and space, using the WENO reconstruction scheme in combination with the TVDRK3 method.

The application of the Lattice Boltzmann method to the one-dimensional modeling of blood flow in elastic vessels

We consider mass conservation and momentum equations for inviscid blood flow in elastic vessels. It is shown that the Lattice Boltzmann(LB) models for non-ideal gas in the limit of low Knudsen numbers are asymptotically equivalent to the blood flow equations. The equation of state for non-ideal gas becomes vessel’s pressure-area response when LB models are applied for blood motion modeling. Two test case problems are considered: a propagation of a sole nonlinear wave and pulse wave propagation in a stented vessel.

Asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry

The asymptotic derivation of a new family of one-dimensional, weakly nonlinear and weakly dispersive equations that model the flow of an ideal fluid in an elastic vessel is presented. Dissipative effects due to the viscous nature of the fluid are also taken into account. The new models validate by asymptotic reasoning other non-dispersive systems of equations that are commonly used, and improve other nonlinear and dispersive mathematical models derived to describe the blood flow in elastic vessels. The new systems are studied analytically in terms of their basic characteristic properties such as the linear dispersion characteristics, symmetries, conservation laws and solitary waves. Unidirectional model equations are also derived and analysed in the case of vessels of constant radius. The capacity of the models to be used in practical problems is being demonstrated by employing a particular system with favourable properties to study the blood flow in a large artery. Two different cases are considered: A vessel with constant radius and a tapered vessel. Significant changes in the flow can be observed in the case of the tapered vessel.

Well-balanced high-order numerical schemes for one-dimensional blood flow in vessels with varying mechanical properties

We construct well-balanced, high-order numerical schemes for one-dimensional blood flow in elastic vessels with varying mechanical properties. We adopt the ADER (Arbitrary high-order DERivatives) finite volume framework, which is based on three building blocks: a first-order monotone numerical flux, a non-linear spatial reconstruction operator and the solution of the Generalised (or high-order) Riemann Problem. Here, we first construct a well-balanced first-order numerical flux following the Generalised Hydrostatic Reconstruction technique. Then, a conventional non-linear spatial reconstruction operator and the local solver for the Generalised Riemann Problem are modified in order to preserve well-balanced properties. A carefully chosen suit of test problems is used to systematically assess the proposed schemes and to demonstrate that well-balanced properties are mandatory for obtaining correct numerical solutions for both steady and time-dependent problems.

On the Mechanics Underlying the Reservoir-Excess Separation in Systemic Arteries and their Implications for Pulse Wave Analysis

Several works have separated the pressure waveform p in systemic arteries into reservoir p r and excess p exc components, p = p r + p exc, to improve pulse wave analysis, using windkessel models to calculate the reservoir pressure. However, the mechanics underlying this separation and the physical meaning of p r and p exc have not yet been established. They are studied here using the time-domain, inviscid and linear one-dimensional (1-D) equations of blood flow in elastic vessels. Solution of these equations in a distributed model of the 55 larger human arteries shows that p r calculated using a two-element windkessel model is space-independent and well approximated by the compliance-weighted space-average pressure of the arterial network. When arterial junctions are well-matched for the propagation of forward-travelling waves, p r calculated using a three-element windkessel model is space-dependent in systole and early diastole and is made of all the reflected waves originated at the terminal (peripheral) reflection sites, whereas p exc is the sum of the rest of the waves, which are obtained by propagating the left ventricular flow ejection without any peripheral reflection. In addition, new definitions of the reservoir and excess pressures from simultaneous pressure and flow measurements at an arbitrary location are proposed here. They provide valuable information for pulse wave analysis and overcome the limitations of the current two- and three-element windkessel models to calculate p r.

Placental transfusion insult in the predisposition for SIDS: A mathematical study

A difference has been observed between the newborn hearing screening tests of thirty-one SIDS cases versus control infants that survived the first year of life [Rubens DD, Vohr BV, Tucker R, O'Neil CA, Chung W. Newborn oto-acoustic emission hearing screening tests. Preliminary evidence for a marker of susceptibility to SIDS. Early Hum Dev 2008;84(4);225–9]. This study is motivated by the hypothesis that the predisposition for SIDS may be caused by inner ear and brainstem damage from a high venous pressure insult at birth that disrupts an infant's ability to detect rising CO 2 levels following the first month of life. The injury is not immediately lethal due to the persistence of fetal physiological responses during the early postnatal period [Guntheroth WG. Crib death, the Sudden Infant Death Syndrome. Armonk NY: Futura Publishing Co.; 1995. p. 291]. Elastic vessels are assumed in the umbilical vein and newborn venous circulation at the time of a potential high pressure placental transfusion insult and pulse wave propagation is simulated using the nonlinear one-dimensional equations of blood flow in elastic vessels. Peak pressures in the auricular veins increase with the amplitude and length of the umbilical surge, reaching over 60 mm Hg when two consecutive surges separated by 100 ms, of a peak pressure of 100 mm Hg, and a pulse interval of 200 ms are propagated in a network with low peripheral reflections. Our findings support the proposed mechanism for inner ear damage in SIDS and the potential benefit of a newborn hearing screening test in identifying susceptibility and early preventative measures following birth.

Ein Lattice-Boltzmann-Modell für pulsierenden Blutfluss in elastischen Gefäßen

Lattice Boltzmann Models (LBM) are widely used to solve fluid mechanical problems in engineering applications. In this work a brief introduction of LBM is given and a new boundary condition is proposed for the cardiovascular domain to support elastic walls in order to simulate blood flow in elastic vessels. The flow field is calculated in two spatial dimensions revealing characteristic flow patterns and geometrical changes of the arterial walls for different time dependent input contours of pressure and flow. For steady flow the results are compared to the predictions of the model proposed by Fung which is an extension of Poiseuille's theory. For unsteady flow the model was validated with the solution given by Womersley. The results are very promising for relevant Reynolds and Womersley numbers.

Lattice BGK Simulations of the Blood Flow in Elastic Vessels

The lattice Boltzmann method is applied to study the flow in elasticblood vessels. The volume–flow rate increases considerably when thecompliance constant of the blood vessel is below a critical value. Thereis a region of the compliance constant in which the average volume–flowrate is dramatically enhanced. A harmonic perturbation of the pressuredoes not change the behaviour of the average volume–flow rate while theharmonic wave attenuates very quickly along the tube when the resonantperiod is close to that of the input wave. The model, together with thesimulation results, is expected to be helpful to understand themechanism of the blood volume–flow rate related to the complianceconstant of the blood vessel, especially on the dependence of the fluxof human blood vessel under weather changes, which has medicalsignificance.

Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries

Flow and pressure waves, originating due to the contraction of the heart, propagate along the deformable vessels and reflect due to tapering, branching, and other discontinuities. The size and complexity of the cardiovascular system necessitate a “multiscale” approach, with “upstream” regions of interest (large arteries) coupled to reduced-order models of “downstream” vessels. Previous efforts to couple upstream and downstream domains have included specifying resistance and impedance outflow boundary conditions for the nonlinear one-dimensional wave propagation equations and iterative coupling between three-dimensional and one-dimensional numerical methods. We have developed a new approach to solve the one-dimensional nonlinear equations of blood flow in elastic vessels utilizing a space-time finite element method with GLS-stabilization for the upstream domain, and a boundary term to couple to the downstream domain. The outflow boundary conditions are derived following an approach analogous to the Dirichlet-to-Neumann (DtN) method. In the downstream domain, we solve simplified zero/one-dimensional equations to derive relationships between pressure and flow accommodating periodic and transient phenomena with a consistent formulation for different boundary condition types. In this paper, we also present a new boundary condition that accommodates transient phenomena based on a Green’s function solution of the linear, damped wave equation in the downstream domain.

The flow of blood in tubes: theory and experiment

The paper is devoted to the comparison of one-dimensional models, used for the simulation of blood flow in elastic vessels. The inviscid, Newtonian, and non-Newtonian models of blood are considered. The one-dimensional non-Newtonian models, corresponding to widely-used rheological models with shear-dependent viscosity (Power Law, Carreau, Carreau–Yasuda, Cross, simplified Cross, modified Cross, Powell–Eyring, modified Powell–Eyring, Yeleswarapu, modified Yeleswarapu, and Quemada) are presented.The attention is focused on the investigation of the influence of viscosity, shape factor, velocity profile, non-Newtonian viscosity, and hematocrit. In the numerical experiments, the models of arterial systems with twenty and thirty-seven vessels are considered. It is demonstrated, that for the vessels near the heart the effect of viscosity can be considered as inessential, and an inviscid model can be used. The main effect on the relative deviations of solutions, obtained by different models, is realized by the velocity profile. The effect of non-Newtonian viscosity is not so essential, but can be taken into account for the models with peripheral vessels.