Estimation of viscosity profiles using velocimetry data from parallel flows of linearly viscous fluids: application to microvascular haemodynamics

Abstract

An approach is presented that uses velocimetry data to estimate accurately the spatial distribution of viscosity in steady laminar parallel flows of incompressible linearly viscous fluids. The approach is generally applicable to Newtonian fluids with spatially varying viscosity or to particle-suspension flows where a non-uniform distribution of the particles contributes to spatial variations in the local effective viscosity of the suspension. Emphasis is placed on the application of these methods to steady axisymmetric blood flow in cylindrical glass capillary tubes and microvessels. In this context, the spatial variations in viscosity over the vessel cross-section are predicted where it is assumed that the rheological properties associated with a heterogeneous red blood cell suspension can be well approximated by a continuous generalized linearly viscous fluid having a spatially non-uniform viscosity. For such a fluid, an expression for the viscosity profile over the vessel cross-section is derived that satisfies the conservation principles of mass and momentum and depends upon the a priori determined velocity distribution, which is extracted from fluorescent micro-particle image velocimetry data obtained from microvessels in vivo. These profiles provide useful information about dynamic, kinematic and rheological properties of the flow that include expressions for the axial pressure-gradient component, the local shear stress distribution, and the relative apparent viscosity. In microvessels, the effect of the glycocalyx surface layer on the vessel wall is also accounted for in the analysis by modelling the layer as a uniformly thick porous medium. Velocimetry data are presented from in vivo measurements made in venules after the application of a light-dye treatment to degrade the glycocalyx. Results reveal that these methods are sufficiently sensitive to detect a reduction in glycocalyx thickness of ∼ 0.3 µm, which represents a fractional decrease in thickness of ∼ 60-70% when compared with results from a separately published data set obtained from venules having an intact glycocalyx.

Microviscometry reveals reduced blood viscosity and altered shear rate and shear stress profiles in microvessels after hemodilution

“…To apply the microviscometric method to blood flow in microvessels in vivo, a generalization is introduced (20) to account for the hemodynamic influence of the ESL (17,18,27,28). Expressions for (r) and dp͞dz apply in microvessels (see Supporting Text) if the tube radius, R, is replaced by a, where a is the radial location of the effective hydrodynamic interface between the blood in the lumen and the ESL (17,20).…”

“…Expressions for (r) and dp͞dz apply in microvessels (see Supporting Text) if the tube radius, R, is replaced by a, where a is the radial location of the effective hydrodynamic interface between the blood in the lumen and the ESL (17,20). It is assumed that red cells and particle tracers do not invade the ESL (7,17,(28)(29)(30) and that plasma flow through the ESL can be well approximated with the Brinkman equation (17,20,29,31,32), where the hydraulic resistivity, K, of the ESL is taken to be more than Ϸ10 9 dyn⅐s͞cm 4 (1 dyn ϭ 10 N) (7,17,29,30). The thickness, R Ϫ a, of the ESL is estimated by following the methods described in ref.…”

“…The thickness, R Ϫ a, of the ESL is estimated by following the methods described in ref. 20, in which the value of a is determined by minimizing the least-squares error in the fit to the -PIV data. The minimum least-squares error for the 12 microvessels that we analyzed (i.d., 34.2 Ϯ 1.7 m) occurred over ESL thicknesses ranging 0.29-0.71 m, with an average thick- ness of Ϸ0.51 Ϯ 0.04 m for K ϭ 10 9 dyn⅐s͞cm 4 (see Table 1 and Figs.…”

“…The analysis (20) on which the microviscometric method is based invokes the continuum approximation and regards the heterogeneous red-cell suspension as a homogeneous, continuously varying, and linearly viscous incompressible fluid that has a spatially nonuniform viscosity distribution (23) over the vessel cross section. Cokelet (2) found support for the continuum approximation in his studies by using physiological concentrations of red blood cells suspended in plasma flowing at physiological shear rates in glass tubes as small as 20 m in diameter.…”

“…The central tenet of our approach (20), which we hereafter shall refer to as the microviscometric method, is that distributions in the local viscosity, (r), as a function of the radial position, r, over the cross section of glass capillary tubes and microvessels, can be determined analytically from the crosssectional axial velocity distribution, v z (r), where v z (r) can be extracted from particle tracers in the flow by using intravital fluorescent microparticle image velocimetry ( -PIV) (17,21,22). In addition to the viscosity distribution, (r), we can use the microviscometric method to predict quantitatively various flow parameters in microvessels in vivo, including axial pressure gradient, dp͞dz; volume flow rate, Q; and the relative apparent blood viscosity, rel , defined as the ratio of steady volume-flow rates per unit pressure drop of blood plasma relative to whole blood.…”