The resistance to blood flow is a function of the rheological properties of blood as well as the geometric features of blood vessels. While circulatory investigations generally aim at studying the functions of the heart and blood vessels, there is an increasing realization that blood rheology can also play a significant role in circulatory regulation in health and disease (10, 11). The principal factors determining blood viscosity are hematocrit, plasma viscos­ ity, cell aggregation, and cell deformability (8). This review focuses on the role of the deformability of the red blood cell (RBC) in circulatory regulation under physiological and pathophysiological conditions. 1 Deformability is a relatively general term that describes the ability of a body (RBCs in this case) to change its shape in response to a deforming force. RBC deformation (or strain) can involve a change in cell curvature, a uniaxial deformation, or area expansion (3, 33). The deforming stress (force per unit area) is usually applied from outside the cell, e.g. by fluid shear stress or local membrane aspiration, but it can also originate from within the cell, e.g. RBC swelling in hypotonic medium or fiber formation in deoxygenated sickle cells. The extent, rate, and mode of deformation depend on the magnitude, rate, and direction of the stress. Therefore, the stress-strain (and stress-strain rate) relationships are complex and cannot be described by a single deformability parameter.

Red Cell Deformability and its Relevance to Blood Flow

Etude des debits liquidiens a travers les vaisseaux cylindriques, du comportement de debit du sang, des concepts de base du transport de masse dans le sang. Rappel de la nomenclature

Rheological aspects of thrombosis and haemostasis: basic principles and applications. ICTH-Report--Subcommittee on Rheology of the International Committee on Thrombosis and Haemostasis.

In this review, we describe and analyze a mesoscale simulation method for fluid flow, which was introduced by Malevanets and Kapral in 1999, and is now called multi-particle collision dynamics (MPC) or stochastic rotation dynamics (SRD). The method consists of alternating streaming and collision steps in an ensemble of point particles. The multi-particle collisions are performed by grouping particles in collision cells, and mass, momentum, and energy are locally conserved. This simulation technique captures both full hydrodynamic interactions and thermal fluctuations. The first part of the review begins with a description of several widely used MPC algorithms and then discusses important features of the original SRD algorithm and frequently used variations. Two complementary approaches for deriving the hydrodynamic equations and evaluating the transport coefficients are reviewed. It is then shown how MPC algorithms can be generalized to model non-ideal fluids, and binary mixtures with a consolute point. The importance of angular-momentum conservation for systems like phase-separated liquids with different viscosities is discussed. The second part of the review describes a number of recent applications of MPC algorithms to study colloid and polymer dynamics, the behavior of vesicles and cells in hydrodynamic flows, and the dynamics of viscoelastic fluids.

Multi-Particle Collision Dynamics -- a Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids

The shape dynamics of fluid vesicles is governed by the coupling of the flow within the two-dimensional membrane to the hydrodynamics of the surrounding bulk fluid. We present a numerical scheme which is capable of solving this flow problem for arbitrarily shaped vesicles using the Oseen tensor formalism. For the particular problem of simple shear flow, stationary shapes are found for a large range of parameters. The dependence of the orientation of the vesicle and the membrane velocity on shear rate and vesicle volume can be understood from a simplified model.

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Fluid vesicles in shear flow

Red blood cells are known to change shape in response to local flow conditions. Deformability affects red blood cell physiological function and the hydrodynamic properties of blood. The immersed boundary method is used to simulate three-dimensional membrane-fluid flow interactions for cells with the same internal and external fluid viscosities. The method has been validated for small deformations of an initially spherical capsule in simple shear flow for both neo-Hookean and the Evans-Skalak membrane models. Initially oblate spheroidal capsules are simulated and it is shown that the red blood cell membrane exhibits asymptotic behavior as the ratio of the dilation modulus to the extensional modulus is increased and a good approximation of local area conservation is obtained. Tank treading behavior is observed and its period calculated.

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Large deformation of red blood cell ghosts in a simple shear flow

A theoretical model is developed for the motion of a human red blood cell in a shear field. The model consists of a tank-treading ellipsoidal membrane encapsulating an incompressible Newtonian liquid immersed in a plane shear flow of another incom- pressible Newtonian liquid. Equilibrium and energy considerations lead to a solution for the motion of the particle that depends on the ellipsoidal-axis ratios and the ratio of the inner- to outer-liquid viscosities. The effect of variation in these parameters is explored and it is shown that, depending on their values, one of two types of overall motion is exhibited: a steady stationary-orientation motion or an unsteady flipping motion. A qualitative agreement of the predicted behaviour of the model with experi- mental observations on red blood cells is found.

Stuart R. Keller

Papers

Motion of a tank-treading ellipsoidal particle in a shear flow

Mechanics of flagellar motion with an application to a conical spiral flagellate.