Mechanics of red-cell motion through very narrow capillaries

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Abstract
In a recent paper, Lighthill (1968) has formulated a model of compressible pellet movement through narrow, fluid-filled, elastic tubes; the model involves a thin lubricating layer of fluid between the pellet and the wall, and the results obtained are expressed in the form of relationships between various velocity, resistance, clearance, and lubrication-film thickness parameters. An important biological motivation for this work was the problem of the movement of red blood cells through narrow capillaries. The present paper examines in detail the approximations and assumptions of Lighthill's model, and extends the investigation in several directions. First, the elastic behaviour of the pellet is discussed with particular relevance to the red cell, which is considered to consist of a fluid-filled flexible membrane. It is shown, in particular, that as the velocity of such a red cell in a capillary increases, the consequent increase in viscous drag on the rim tends to 'bow' the cell, and allow it to fit more easily into a capillary of given diameter. Further, the modulus of elastic response to the non-uniform lubrication pressures around the rim is shown to depend on the membrane's 'resistance to deformation' and the geometry of the cell. Next, a model of axisymmetric pellet flow through a tube is set up. Results are expressed in terms of clearance and resistance experienced by the pellet, for various velocities and film thicknesses typical of capillary flows. Much greater resistances than those estimated by Lighthill are obtained; for normal flow in capillaries of 5 to 7 $\mu $m diameter, these range from 4.5 to 7 times that expected for a corresponding Poiseuille fluid flow with whole-blood viscosity. It is not suggested that these results are true of all capillaries, or of a capillary network, but only of those very narrow capillaries in the diameter range being considered. Of equal importance is the nonlinear dependence of resistance on velocity. The red cell velocity falls off very rapidly as the pressure gradient is reduced; this is discussed in detail with reference to the phenomenon of auto-regulation. Finally, the effects of asymmetry and tube porosity are considered. It is shown that deviations from axisymmetry tend to be reduced by pressures developed in the fluid; this is due to the compressible nature of the pellets, and means that the results obtained for capillary flow are reasonably valid even for configurations in which the red cells may not be axisymmetric. Leakage of plasma through the walls of the capillary is shown not to affect the flow characteristics of a capillary to any great extent, or to significantly increase the possibility of 'seizing-up' of the flow due to failure of lubrication at small film thicknesses.

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