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.

A Two-Zone Shear-Induced Red Blood Cell Migration Model for Blood Flow in Microvessels

“…In contrast to the previous model [16], not including the cell-free layer, the present one includes two zones as in the marginal zone theory: an annular cell-free layer in consistency with experimental observations, and a core region. However, in contrast to models based on Haynes model [5,[11][12][13] including a core region of uniform hematocrit level and a CFL, the core region has a non-uniform hematocrit level in the present model, with both the shear-induced migration model and the Krieger-Dougherty model of viscosity used to find the velocity profile and the hematocrit-level variations.…”

“…The developments in Chebbi [16] and Phillips et al [17] are only valid in the core region. The core zone extends from the centerline (r = 0) to r = R-δ, where δ is the cell-free layer thickness.…”

“…The core zone extends from the centerline (r = 0) to r = R-δ, where δ is the cell-free layer thickness. The RBC conservation equation for the fully developed hematocrit level and velocity profiles case yields [16]…”

“…(i) The shear-induced migration model in Weert [14] Mansour et al [15], and Chebbi [16] using the model of Phillips et al [17], which is an extension of a previous model by Leighton and Acrivos [18] to analyze blood flow (ii) The elastic-stress induced migration model in Moyers-Gonzalez et al [19], Moyers-Gonzalez and Owens [20] and Dimakopoulos et al [21], which is an extension of a previous model by Mavrantzas and Beris [22,23] to examine blood flow.…”

“…The models in Weert [14] and Mansour et al [15] use the Quemada model of blood viscosity [24] and were solved using finite element methods. In contrast, the treatment in Chebbi [16] extends the Krieger-Dougherty viscosity model of viscosity, used by Phillips et al [17] to analyze blood flow while considering analytic and numerical solutions of the ordinary differential equations governing velocity and hematocrit level gradients for the fully developed profiles case. The results are in good agreement with published velocity profiles and hematocrit ratio, and match closely with the results obtained in Weert [14] and Mansour et al [15] using finite element methods and considering the same model of viscosity (Krieger-Dougherty viscosity model).…”