Dependence of cell-free-layer width on rheological parameters: Combining empirical data on flow separation at microvascular bifurcations with geometrical considerations

When blood flows through vessel networks, red blood cells (RBCs) are typically concentrated close to the vessel center line, creating a lubrication layer near the vessel wall. As RBCs bind oxygen, the width of this cell-free layer (CFL) impacts not only the blood rheology inside the vasculature, but also oxygen delivery to the tissues they perfuse and, hence, their function. Existing attempts to relate the width of the CFL to the rheological properties of the blood and the geometrical properties of the vessel are based on an analysis of the forces acting on RBCs suspended in the blood. However, the complexity of interactions in the blood makes this a challenging task. Here, we propose an alternative, two-step approach to derive such a functional relationship. First, we extend widely accepted empirical fits describing the minimum flow fraction needed for RBCs to enter a daughter vessel downstream of a microvascular bifurcation so that it depends not only on the diameter and discharge haematocrit of the parent vessel, but also on its average shear rate. Second, we propose a simple geometrical model for the minimum flow fraction based on the cross-sectional blood flow profile within the parent vessel upstream of the bifurcation—considering uniform, parabolic, and blunt velocity profiles—and derive the leading-order approximation to this model for small CFL widths. By equating the functional relationships obtained using these two approaches, we derive expressions relating the CFL width to the vessel diameter, discharge haematocrit, and mean shear rate. The resulting expressions are in good agreement with available in vivo data and represent a promising basis for future research