Adélia Sequeira

On the shear-thinning and viscoelastic effects of blood flow under various flow rates

The aim of this paper is to describe and discuss the results of numerical comparative study performed in order to demonstrate and quantify some of the most relevant non-Newtonian characteristics of blood flow in medium-sized blood vessels, namely its shear-thinning and viscoelastic behavior.The models studied in this work are the classical Newtonian and Oldroyd-B models, as well as their generalized (shear-thinning) modifications. Numerical tests are performed on three-dimensional geometries, namely an idealized axisymmetric stenosis and a realistic stenosed carotid bifurcation reconstructed from medical images. The numerical solution of the system of governing equations is obtained by a finite-volume method on a structured grid. Model sensitivity tests are achieved with respect to the characteristic flow rate to evaluate its impact on the observed non-Newtonian effects.

Advances in Mathematical Fluid Mechanics

Edited by Galdi, G.P., University of Pittsburgh, USA Heywood, J.G., University of British Columbia, Vancouver, Canada Rannacher, R., University of Heidelberg, Germany Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematial Fluid Mechanics. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics.

Numerical simulation of the coagulation dynamics of blood

The process of platelet activation and blood coagulation is quite complex and not yet completely understood. Recently, a phenomenological meaningful model of blood coagulation and clot formation in flowing blood that extends existing models to integrate biochemical, physiological and rheological factors, has been developed. The aim of this paper is to present results from a computational study of a simplified version of this coupled fluid-biochemistry model. A generalized Newtonian model with shear-thinning viscosity has been adopted to describe the flow of blood. To simulate the biochemical changes and transport of various enzymes, proteins and platelets involved in the coagulation process, a set of coupled advection–diffusion–reaction equations is used. Three-dimensional numerical simulations are carried out for the whole model in a straight vessel with circular cross-section, using a finite volume semi-discretization in space, on structured grids, and a multistage scheme for time integration. Clot formation and growth are investigated in the vicinity of an injured region of the vessel wall. These are preliminary results aimed at showing the validation of the model and of the numerical code.

Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology

Newtonian and generalized Newtonian mathematical models for blood flow are compared in two different reconstructions of an anatomically realistic geometry of a saccular aneurysm, obtained from rotational CTA and differing to within image resolution. The sensitivity of the flow field is sought with respect to geometry reconstruction procedure and mathematical model choice in numerical simulations. Taking as example a patient specific intracranial aneurysm located on an outer bend under steady state simulations, it is found that the sensitivity to geometry variability is greater, but comparable, to the one of the rheological model. These sensitivities are not quantifiable a priori. The flow field exhibits a wide range of shear stresses and slow recirculation regions that emphasize the need for careful choice of constitutive models for the blood. On the other hand, the complex geometrical shape of the vessels is found to be sensitive to small scale perturbations within medical imaging resolution. The sensitivity to mathematical modeling and geometry definition are important when performing numerical simulations from in vivo data, and should be taken into account when discussing patient specific studies since differences in wall shear stress range from 3% to 18%.

A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries

The mathematical modelling and numerical simulation of the human cardiovascular system is playing nowadays an important role in the comprehension of the genesis and development of cardiovascular diseases. In this paper we deal with two problems of 3D modelling and simulation in this field, which are very often neglected in the literature. On the one hand blood flow in arteries is characterized by travelling pressure waves due to the interaction of blood with the vessel wall. On the other hand, blood exhibits non-Newtonian properties, like shear-thinning, viscoelasticity and thixotropy. The present work is concerned with the coupling of a generalized Newtonian fluid, accounting for the shear-thinning behaviour of blood, with an elastic structure describing the vessel wall, to capture the pulse wave due to the interaction between blood and the vessel wall. We provide an energy estimate for the coupling and compare the numerical results with those obtained with an equivalent fluid-structure interaction model using a Newtonian fluid.

Rheological models for blood

Rheology is the science of the deformation and flow of materials. It deals with the theoretical concepts of kinematics, conservation laws and constitutive relations, describing the interrelation between force, deformation and flow. The experimental determination of the rheological behaviour of materials is called rheometry. The object of haemorheology is the application of rheology to the study of flow properties of blood and its formed elements, and the coupling of blood and the blood vessels in living organisms. This field involves the investigation of the macroscopic behaviour of blood determined in rheometric experiments, its microscopic properties in vitro and in vivo and studies of the interactions among blood cellular components and between these components and the endothelial cells that line blood vessels.

A DIRECTOR THEORY APPROACH FOR MODELING BLOOD FLOW IN THE ARTERIAL SYSTEM: AN ALTERNATIVE TO CLASSICAL 1D MODELS

It remains computationally infeasible to model the full three-dimensional (3D) equations for blood flow in large sections of the circulatory system. As a result, one-dimensional (1D) and lumped parameter models play an important role in studies of the arterial system. A variety of 1D models are used, distinguished by the closure approximations employed. In this paper, we introduce a nine-director theory for flow in axisymmetric bodies as an alternative to the 1D models. Advantages of the director theory include (i) the theory makes use of all components of linear momentum; (ii) the flow is not assumed to be uni-directional; (iii) the theory is hierarchical; (iv) there is no need for closure approximations; and (v) wall shear stress enters directly as a dependent variable. In order to simplify the equations for mathematical analysis, for this work, attention is focused on cases where it is appropriate to model the flow as quasi-steady and the wall motion does not have a significant impact on bulk flow parameters. This work lays the foundation for future applications of the theory to unsteady flows in flexible walled vessels. For the geometries considered here, the nine-director theory has the same advantage as 1D models in providing a relatively simple relation between flow rate and average pressure drop. Conditions for existence, uniqueness and local stability of steady solutions are determined for both the 1D and nine-director equations. The predictive capability of classical 1D models found in the recent literature and a nine-director model7,15 are carefully evaluated through comparison with analytical and computational solutions to the axisymmetric, steady Navier–Stokes equations in geometries relevant to blood flow. For these benchmark problems over the range of Reynolds numbers considered, the nine-director theory is found to provide better results than the classical 1D models. A novel approach for parameter identification is in the 1D model is given and shown to substantially improve its predictive capability in these test cases.

Numerical Study of the Significance of the Non-Newtonian Nature of Blood in Steady Flow Through a Stenosed Vessel

In this paper we present a comparative numerical study of non-Newtonian shear-thinning and viscoelastic blood flow models through an idealized stenosis. Three-dimensional numerical simulations are performed using a finite volume semidiscretization in space, on structured grids, and a multistage Runge-Kutta scheme for time integration, to investigate the influence of combined effects of inertia, viscosity and viscoelasticity in this particular geometry. This work lays the foundation for future applications to pulsatile flows in stenosed vessels using constitutive models capturing the rheological response of blood, under relevant physiological conditions.

Numerical simulations of shear dependent viscoelastic flows with a combined finite element-finite volume method

A hybrid combined finite element-finite volume method has been developed for the numerical simulation of shear-dependent viscoelastic flow problems governed by a generalized Oldroyd-B model with a non-constant viscosity function. The method is applied to the 4:1 planar contraction benchmark problem, to investigate the influence of the viscosity effects on the flow and results are compared with those found in the literature for creeping Oldroyd-B flows, for a range of Weissenberg numbers. The method is also applied to flow in a smooth stenosed channel. It is shown that the qualitative behavior of the flow is influenced by the rheological properties of the fluid, namely its viscoelastic and inertial effects, as well as the shear-thinning viscosity. These results appear in the framework of a preliminary study of the numerical simulation of steady and pulsatile blood flows in two-dimensional stenotic vessels, using this hybrid finite element-finite volume method.

Mesoscopic simulations of unsteady shear-thinning flows

The capability of the lattice Boltzmann method as an accurate mesoscopic solver for unsteady non-Newtonian flows is shown by investigating pulsatile shear-thinning blood flow in a three-dimensional idealised vessel. The non-Newtonian behaviour of blood flow is modelled by the Carreau-Yasuda viscosity model. Higher velocity and shear stress magnitudes, relative to Newtonian cases, are observed for the shear-thinning simulations in response to changes in the shear-rate dependent Womersley parameter. We also investigate the flexibility of the method through the shear-thinning behaviour of the lattice Boltzmann relaxation parameter at different Deborah numbers.