Flow of Red Blood Cells in Stenosed Microvessels

Abstract

A computational study is presented on the flow of deformable red blood cells in stenosed microvessels. It is observed that the Fahraeus-Lindqvist effect is significantly enhanced due to the presence of a stenosis. The apparent viscosity of blood is observed to increase by several folds when compared to non-stenosed vessels. An asymmetric distribution of the red blood cells, caused by geometric focusing in stenosed vessels, is observed to play a major role in the enhancement. The asymmetry in cell distribution also results in an asymmetry in average velocity and wall shear stress along the length of the stenosis. The discrete motion of the cells causes large time-dependent fluctuations in flow properties. The root-mean-square of flow rate fluctuations could be an order of magnitude higher than that in non-stenosed vessels. Several folds increase in Eulerian velocity fluctuation is also observed in the vicinity of the stenosis. Surprisingly, a transient flow reversal is observed upstream a stenosis but not downstream. The asymmetry and fluctuations in flow quantities and the flow reversal would not occur in absence of the cells. It is concluded that the flow physics and its physiological consequences are significantly different in micro- versus macrovascular stenosis.

Figure 1. Geometry of a stenosed microvessel considered in the study.

(I)–(VI) are streamwise locations where some flow quantities are measured for analysis.

Figure 2. Snapshots showing instantaneous RBC distribution from a few representative simulations for non-stenosed (left column) and 84% stenosed (right column) vessels at H_t ≈ 22–24%.

(a,b) β = 1, (c–h) β = 4.

Figure 3

(a,b) Cell distribution in the vicinity of the stenosis at two instances corresponding to local extrema in instantaneous flow rate Q(t) shown in (c). (d,e,f) are for a non-stenosed vessel.

Figure 4

(a) Representative time-dependent flow rate Q(t) in stenosed (—) and non-stenosed (- - - - -) vessels; (b) RMS of fluctuations of versus vessel diameter for β = 1 (O), 2 (Δ), 3 (◻), and 4 (∇); filled symbols are for stenosed vessels, and unfilled symbols for non-stenosed vessels. (c) Representative FFT of flow rate for stenosed (—) and non-stenosed (- - - - -) vessels.

Figure 5

(a) Representative time-dependent Eulerian velocity at a fixed radial distance 1.2 μm (near the edge of the CFL) from the wall but at three different streamwise locations: At a location far upstream (red thick line, location (I) as defined in Fig. 1), at the beginning of stenosis (black thin line, location II), and at the end of stenosis (blue dotted line, location VI). The Eulerian velocity has been scaled by the time-averaged velocity at the same location as . (b) RMS of fluctuations of for the cases shown in (a). Also added is the RMS at a location far downstream marked as (VII). (c) Spectra of for the three cases shown in (a). Here D = 25 μm, β = 3, and H_t = 24%.

Figure 6. Upstream flow reversal in presence of RBCs: Shown here are instantaneous velocity profiles up- and downstream a symmetric stenosis (D = 25 μm,  = 84%, β = 3, H_t = 24%).

(a,b,c) refer to three different time instants. Continuous lines represent upstream velocity profiles at locations (II) or (III) as indicated (see Fig. 1 for locations), and dash lines represent downstream profiles at locations (V) or (VI). Here r is the radial location.

Figure 7

Time- and azimuthally-averaged RBC distribution at β = 1 (a,d,g) and 4 (b,e,h), and spatial variation of CFL thickness δ/R (c,f,i) for D = 11 μm (a–c), 17 μm (d–f), and 25 μm (g–i). For the RBC distribution, contours are plotted from 0 (blue) to 0.5 (red) with 0.01 increment. For the CFL thickness, dotted lines are for non-stenosed vessels, continuous lines for stenosed vessels, β = 1 (blue) and 4 (red). Here  = 84%, and H_t = 24%. Here R is the vessel radius in non-stenosed section.

Figure 8

Average velocity profiles (ū vs. r) for symmetrically stenosed vessels (a–c) at different locations. D = 11 μm (a), 17 μm (b) and 25 μm (c). The dash line is the velocity profiles for pure plasma. (I) to (VI) correspond to different locations in the vessel as shown in Fig. 1. Here  = 84%, β = 1, and H_t = 24%.

Figure 9. Apparent viscosity of blood showing a significant enhancement of the Fahraeus-Lindqvist effect in the stenosed vessels.

Lines with unfilled symbols are for non-stenosed tubes with RBCs, and lines with filled symbols are for stenosed tubes ( = 84%, H_t  ≈ 22–24%) with RBCs, for various values of  = 1 (O, blue), 2 (Δ, green), 3 (◻, black), 4 (∇, red). Dash line is for stenosed tubes with plasma only. Dash-dot lines are for stenosed tubes with Newtonian fluids having viscosities same as the apparent viscosity of blood in non-stenosed tubes of the same diameters.

Figure 10

(a) Effect of mean pressure gradient β and (b) hematocrit H_t on the apparent viscosity for non-stenosed (continuous lines with unfilled symbols), and 84% stenosed tubes (continuous lines with filled symbols) for D = 11 (◻), 17 (Δ), and 25 μm (O). In (a) H_t = 24% is kept constant and β is varied; in (b) β = 3 is kept constant and H_t is varied.

Figure 11. Comparison of the Fahraeus effect in 84% stenosed (filled symbols) and non-stenosed (unfilled symbols) vessels.

Here hematocrit ratio H_t/H_D is shown as a function of tube diameter D for β = 1 (O, blue), and 3 (◻, black), for H_t = 24%. Error bars represent absolute errors.

Figure 12

(a) Effect of area blockage on RMS of flow rate oscillations for different vessels at constant β = 3. (b) Effect of on RMS(Q) for different β but at constant D = 17 μm. (c) Effect of on Eulerian velocity fluctuations at locations II (upstream) and VI (downstream). Here D = 17 μm and β = 3. For all cases, H_t = 23%.

Figure 13

(a) Effect of on the Fahraeus-Lindqvist effect. The dash lines are for plasma. (b) μ_rel versus for different vessels.

Figure 14

Effect of (a) β and (b) H_t on μ_rel for various . In (a) H_t = 23%, and in (b) β = 3. For all cases D = 17 μm.

Figure 15

(a) A snapshot of cell motion through an asymmetric stenosis (D = 25 μm,  = 1, β = 1, H_t = 24%). (b) Effect of on RMS of flow rate. (c,d,e) shows the effect of on time-dependent Eulerian velocity at locations II (continuous red line) and VI (dotted blue line).

Figure 16

(a) Average velocity profiles (ū vs. r) at different streamwise locations for  = 1, D = 17 μm. (b) Skewness of velocity profiles as a function of at locations II and VI. (c,d) RBC distribution in D = 17 μm for  = 0.3 and 1, respectively. (e,f) Same but in D = 25 μm.

Figure 17. Comparison of the Fahraeus-Lindqvist effect in symmetric and asymmetric stenosis.

(a) Variation of μ_rel with respect to D for  = 0 and 1. The dash-dot line is the relative viscosity obtained in the asymmetric stenosis for a Newtonian fluid having viscosity equal to the apparent blood viscosity in non-stenosed tubes. (b) Variation of μ_rel with respect to S for different vessels.

Figure 18. Summary of the findings.