Finite platelet size could be responsible for platelet margination effect

Abstract

Blood flows through vessels as a segregated suspension. Erythrocytes distribute closer to the vessel axis, whereas platelets accumulate near vessel walls. Directed platelet migration to the vessel walls promotes their hemostatic function. The mechanisms underlying this migration remain poorly understood, although various hypotheses have been proposed to explain this phenomenon (e.g., the available volume model and the drift-flux model). To study this issue, we constructed a mathematical model that predicts the platelet distribution profile across the flow in the presence of erythrocytes. This model considers platelet and erythrocyte dimensions and assumes an even platelet distribution between erythrocytes. The model predictions agree with available experimental data for near-wall layer margination using platelets and platelet-modeling particles and the lateral migration rate for these particles. Our analysis shows that the strong expulsion of the platelets from the core to the periphery of the blood vessel may mainly arise from the finite size of the platelets, which impedes their positioning in between the densely packed erythrocytes in the core. This result provides what we believe is a new insight into the rheological control of platelet hemostasis by erythrocytes.

Figure 1

Illustration of the concept of the “excluded and available volumes”. Further away from the wall (from left to right), the erythrocyte volume fraction increases, causing a rapid decrease in the volume fraction that is available to the platelets.

Figure 2

(A) Dependence of the available volume fraction for platelet-modeling spheres on the erythrocyte volume fraction at a_P = 0.01 (●), 1 (■), 1.25 (▾), and 2.5 μm (♦). (Solid and dashed lines) Plotted according to Eq. 2 with p = 1, 3, 4, and 12, respectively, and f = 0. (Dotted line) Plotted with p = 4, f = 0.1. (Dash-dotted line) Intererythrocyte volume fraction 1 − Φ_RBC. (B) Dependence p(a_P) and its approximation by the function p = exp(a_P).

Figure 3

Lateral platelet distribution (thin lines, left axes) calculated using Eqs. 1 and 2 at various values of the parameters p (A) and f (B). The erythrocyte volume fraction distribution (thick curves, right axes) was assigned using Eq. 3 with Φ₀ = 0.4.

Figure 4

Hematocrit dependence for the NWE of platelets (A) and platelet-modeling particles (B) calculated at p = 3.5, f = 0.06 (lines) and from in vitro experiments (markers). (A) The distributions of OsO₄-fixed platelets in the presence of erythrocyte ghosts were detected using the laser-Doppler velocimetry method in glass tubes of 3 mm internal diameter and averaged over the near-wall and central zones (20), as described in Mathematical Model (see main text). Wall shear rates were 240 s⁻¹ (open triangle), 760 s⁻¹ (open circle), and 1200 s⁻¹ (open square). Fixing the platelets did not change their behavior in the flow. Theoretical calculations used Eqs. 1–3; the calculated platelet profiles were averaged similarly to the experimental profiles. (B) The distributions of platelet-sized particles (2a_P = 2.2 ÷ 2.5 μm) in the presence of washed erythrocytes were obtained by stroboscopic epifluorescence microscopy in Tilles and Eckstein (13) (open hexagon), Eckstein et al. (14) (white and black hexagon), and Eckstein et al. (15) (black and white hexagon) at ${\dot{\gamma }}_w$ = 1630 s⁻¹ and by using the freeze-capture method in Bilsker et al. (16) (crossed box, vertically divided box, quartered box, and horizontally divided box at ${\dot{\gamma }}_w$ = 400, 410, 500, and 900 s⁻¹, respectively), Koleski and Eckstein (17) (solid star, ${\dot{\gamma }}_w$ = 400 s⁻¹) and Yeh et al. (18) (open star, ${\dot{\gamma }}_w$ = 555 s⁻¹; and white and black star, 700 ÷ 900 s⁻¹). The errors are indicated if they were reported in the original studies. At Φ₀ = 0.15 and 0.4, the markers are shifted by ±0.01 to reduce overlapping. Eq. 4 (with Eqs. 2 and 3b) were used for the theoretical calculations.

Figure 5

Migration of a individual platelet-modeling sphere across the flow. (A) The experimental paths for two latex spheres with a 2-μm diameter in a erythrocyte ghost suspension that flows through a tube with a radius R₀ = 38.25 μm (3). (Upper bound) Flow axis; (bottom bound) tube wall. The erythrocyte volume fraction at the inflow was Φ₀ = 0.44, and the wall shear rate was 16 s⁻¹. (B) The radial distribution of the probability density was calculated using Eq. 7 (with Eqs. 2, 3, 5, and 6) at successive time points (solid lines). The tube radius, wall shear rate, and hematocrit corresponded to the experimental conditions; p- and f-values were the same as in Fig. 4, and t₀ was set to 20 s. The steady-state profile was calculated using Eq. 1 and is shown (dashed line). (C) Time dependence of the probability density at r = 0, 1/2, and 1. The characteristic times for the change in probability density (0.37 and 0.78 for r = 1/2 and 1, respectively) were determined from the decrease in $\left(p-p_{t\to \infty }/1-p_{t\to \infty }\right)$ values by a factor of e.

Figure 6

Radial distribution of the platelet-modeling spheres that are 2.5 μm in diameter from experiment (A) and the calculated result (B) at various distances from the tube entrance. (A) The steady-state erythrocyte distribution was established at the beginning of each experiment (19). Next, the flow source was switched to a similar erythrocyte-containing reservoir with a small added mix of spheres. After a few seconds, the flow was rapidly frozen, and the radial profiles for the sphere distribution (spheres/μm²) were measured using fluorescence microscopy. The profiles from a few separate trials were averaged. The axial coordinates for the cross sections are presented as a percentage of the length of an ideal paraboloid that contains spheres at any given time. This length was L₀ = 40 μm; the tube length was 50 μm; the tube radius was R₀ = 110 μm; and the hematocrit was 40%. (B) The calculation was performed using Eq. 8 (with Eqs. 2, 3, 5, and 6) at Φ₀ = 0.4; the p- and f-values were the same as in Fig. 4.