We give an account of the work of Robin Fåhraeus over the years 1917-1938, his contribution to our understanding of blood rheology, and its relevance to circulatory physiology. Fåhraeus published few original papers on this subject, yet he clearly understood the phenomena occurring in the tube flow of mammalian blood. 1) The concentration of cells in a tube less than 0.3 mm in diameter differs from that in the larger feed tube or reservoir, the Fåhraeus effect. This is due to a difference in the mean velocity of cells and plasma in the smaller vessel associated with a nonuniform distribution of the cells. 2) In tubes less than 0.3 mm in diameter, the resistance to blood flow decreases with decreasing tube diameter, the Fåhraeus-Lindqvist effect. We define and generalize the two effects and describe how red cell aggregation at low shear rates affects cell vessel concentration and resistance to flow. The fluid mechanical principles underlying blood cell lateral migration in tube flow and its application to Fåhraeus' work are discussed. Experimental data on the Fåhraeus and Fåhraeus-Lindqvist effects are given for red cells, white cells, and platelets. Finally, the extension of the classical Fåhraeus effect to microcirculatory beds, the Fåhraeus Network effect, is described. One of the explanations for the observed, very low average capillary hematocrits is that the low values are due to a combination of the repeated phase separation of red cells and plasma at capillary bifurcations (network effect) and the single-vessel Fåhraeus effect.

The aggregation of red blood cells in blood flowing through small tubes at very low shear rates leads to the two-phase flow of an inner core of rouleaux surrounded by a cell-depleted peripheral layer. The formation of this layer is known to be accompanied by a decrease in hydrodynamic resistance to flow. To quantitate this effect, we measured the pressure gradient, flow rate, and the radius of the red blood cell core in suspensions flowing through tubes of 172-microns radius at mean linear flow rates (U) from 50 to 0.15 tube diameters.sec-1. Washed red blood cells were suspended in 1.5% buffered dextran 110 at hematocrits of 34-52%. Using syringe pumps, blood flowed from a stirred reservoir through a vertical 12-cm length of tube in either the upward or downward direction. The pressure drop was measured with transducers. Mean values in distributions in the core radius were obtained by analyzing cine films of flow taken through a microscope with flow in the upward direction, measuring the core radius at five equally spaced axial positions of the tube in each of 100 frames. At 34% and 46% hematocrit, the hydrodynamic resistance increased as U decreased from 50 sec-1, reaching a maximum at U-2 sec-1. It then decreased to a minimum at U less than 0.5 sec-1 as the red blood cell core formed in the tube, and the mean core radius/tube radius ratio decreased from 0.98 to 0.74 with marked axial fluctuations at the lower U. At higher hematocrits, both the increase and decrease in hydrodynamic resistance were greater. In a red blood cell albumin-saline suspension, where there is no aggregation of red blood cells and no two-phase flow, hydrodynamic resistance increases linearly with decreasing U. The experimental results were compared with the predictions of a two-phase steady-flow model, assuming axisymmetric flow of a core surrounded by cell-free suspending medium. Two models were considered, one in which the core is solid, the other in which the rheological properties of the suspension in the core are given by the Quemada equation. The effects of sedimentation of the core resulting in a zero net flow pressure gradient were taken into account. Provided that an experimentally extrapolated value for the zero pressure gradient was used, the Quemada-fluid model gave good agreement with the experimentally observed core radius as a function of U and hematocrit.

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Hemoglobin solutions prepared from hemolyzed human erythrocyte packs have Newtonian flow properties. Diluted solutions are also Newtonian. All solutions have a viscosity lower than the apparent viscosity of erythrocyte suspensions of equal oxygen-carrying capacity. The presence of cell debris in hemoglobin solutions causes non-Newtonian (pseudoplastic or rheopectic) flow behavior.

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This paper deals with the theoretical analysis of the unloading of oxygen from a red cell. A scale analysis of the governing transport equations shows that the solutions have a boundary layer structure near the red-cell membrane. The boundary layer is a region of chemical nonequilibrium, and it owes its existence to the fact that the kinetic time scales are shorter than the diffusion time scales in the red cell. The presence of the boundary layer allows an analytical solution to be obtained by the method of matched asymptotic expansions. A very useful result from the analysis is a simple, lumped-parameter description of the oxygen delivery from a red cell. The accuracy of the lumped-parameter description has been verified by comparing its predictions with results obtained by numerical integration of the full equations for a one-dimensional slab. As an application, we calculate minimum oxygen unloading times for red cells.

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We have developed a mathematical model of microvascular network blood flow in which the nonlinear flow properties of blood and the nonuniform axial distribution of red blood cells in each vessel, as well as disproportionate cell partitioning at bifurcations, are all accounted for. The movements of red blood cells in the network are tracked; hence, the model is able to simulate temporal variations in local flow parameters in the network due to hemodynamic mechanisms. The model was applied to four rat mesenteric networks for which the topology, boundary conditions, blood velocity, and discharge hematocrit (Hctd) had been measured for each branch. Temporal variations in Hctd and blood velocity after simulation convergence were predicted. In some cases of the three vessels connected to a node, Hctd of one vessel fluctuates in a simple periodic form, Hctd of the second one oscillates in a more complex periodic form, whereas the Hctd of the third one does not oscillate at all. These variations were obtained with constant flow boundary conditions and, therefore, are due to hemodynamic factors alone. The temporal variations in flow parameters predicted by the model simulations are caused by hemorheological mechanisms and would be superimposed on variations caused by other mechanisms (e.g., vasomotion). The frequencies of the predicted fluctuations in blood velocity are in qualitative agreement with observed in vivo variations in dual-slit velocity in the arterioles of the cremaster muscle of anesthetized Golden hamster.