Dynamics of blood flow: modeling of Fåhraeus and Fåhraeus-Lindqvist effects using a shear-induced red blood cell migration model

Abstract

Blood flow in micro capillaries of diameter approximately 15-500 μm is accompanied with a lower tube hematocrit level and lower apparent viscosity as the diameter decreases. These effects are termed the Fåhraeus and Fåhraeus-Lindqvist effects, respectively. Both effects are linked to axial accumulation of red blood cells. In the present investigation, we extend previous works using a shear-induced model for the migration of red blood cells and adopt a model for blood viscosity that accounts for the suspending medium viscosity and local hematocrit level. For fully developed hematocrit profiles (i.e., independent of axial location), the diffusion fluxes due to particle collision frequency and viscosity gradients are of equal magnitude and opposite directions. The ratio of the diffusion coefficients for the two fluxes affects both the Fåhraeus and Fåhraeus-Lindqvist effects and is found related to the capillary diameter and discharge hematocrit using a well-known data-fit correlation for apparent blood viscosity. The velocity and hematocrit profiles were determined numerically as functions of radial coordinate, tube diameter, and discharge hematocrit. The velocity profile determined numerically is consistent with the derived analytical expression and the results are in good agreement with published numerical results and experimental data for hematocrit ratio and hematocrit and velocity profiles.

Keywords: Apparent blood viscosity; Axial accumulation; Cell depletion; Microvessels; Red blood cells.

Fig. 1

Comparison of the Runge–Kutta numerical integration results for the hematocrit profile (using Eqs. (33) and (34)), the analytical solution, Eq. (30) in [14] (generalized Eq. (35) in the present work with n = 1.82) and the numerical solutions in [16, 17] for the case H_T = 0.45, using K_μ = 0.62, K_c = 0.41 (ζ = 1.512) and H_m = 0.67

Fig. 2

Comparison of the Runge–Kutta numerical integration results for the normalized velocity profile (using Eqs. (33) and (34)), the analytical solution, Eq. (41) and the numerical solutions in [16, 17] for the case H_T = 0.45, using K_μ = 0.62, K_c = 0.41 (ζ = 1.512) and H_m = 0.67

Fig. 3

Comparison of the normalized velocity profile results with the experimental data in [27] and the numerical results in [12] for the case H_D = 0.335, pressure gradient = 3732 dyn/cm³ and R = 27.1 μm

Fig. 4

Comparison of the hematocrit ratio results with the numerical results in [12] and the reported experimental data [–31] for the case HD = 0.405

Fig. 5

Results for ζ = K_μ/K_c versus vessel radius R for the case H_D = 0.405