Going beyond 20 μm-sized channels for studying red blood cell phase separation in microfluidic bifurcations

Abstract

Despite the development of microfluidics, experimental challenges are considerable for achieving a quantitative study of phase separation, i.e., the non-proportional distribution of Red Blood Cells (RBCs) and suspending fluid, in microfluidic bifurcations with channels smaller than 20 μm. Yet, a basic understanding of phase separation in such small vessels is needed for understanding the coupling between microvascular network architecture and dynamics at larger scale. Here, we present the experimental methodologies and measurement techniques developed for that purpose for RBC concentrations (tube hematocrits) ranging between 2% and 20%. The maximal RBC velocity profile is directly measured by a temporal cross-correlation technique which enables to capture the RBC slip velocity at walls with high resolution, highlighting two different regimes (flat and more blunted ones) as a function of RBC confinement. The tube hematocrit is independently measured by a photometric technique. The RBC and suspending fluid flow rates are then deduced assuming the velocity profile of a Newtonian fluid with no slip at walls for the latter. The accuracy of this combination of techniques is demonstrated by comparison with reference measurements and verification of RBC and suspending fluid mass conservation at individual bifurcations. The present methodologies are much more accurate, with less than 15% relative errors, than the ones used in previous in vivo experiments. Their potential for studying steady state phase separation is demonstrated, highlighting an unexpected decrease of phase separation with increasing hematocrit in symmetrical, but not asymmetrical, bifurcations and providing new reference data in regimes where in vitro results were previously lacking.

FIG. 1.

Typical images of RBC flows in micro-bifurcations. From left to right and top to bottom: microbifurcation of type A, B, C, and D. RBCs are flowing from left to right in the inlet (horizontal) channel. The tube hematocrit in the inlet channel is 3.9%, 5.5%, 13%, and 8%, respectively.

FIG. 2.

Typical optical density profile OD(x), obtained from an image sequence of RBC flowing in a $20\hspace{0.17em}\mu \mathrm{m}\times 20\hspace{0.17em}\mu \mathrm{m}$ microchannel. The vertical dotted lines indicate the location of the microchannel walls.

FIG. 3.

Optical density as a function of H_t determined by counting, for 20 μm × 20 μm cross-section channels. Dots: data points; dashed line: best linear fit to the data, $H_t=101.01\times OD$. The dominant uncertainty in the use of Equation (1) comes from counting error, with a relative uncertainty $\delta N/N\approx 15\%$. The resulting error-bars on H_t are shown for a few data points only, for the sake of clarity.

FIG. 4.

RBC maximal velocity profiles in a 20 μm × 20 μm (a) and 10 μm × 10 μm (b) microchannel. Dots: velocity data normalized by V₀; Solid line: best fitting adjustment to Equation (2). For sake of clarity, the uncertainty on each velocity measurement by dual-slit is indicated on a single isolated point (not a real data point). The uncertainty on the V₀ value obtained is $\pm 0.05\times V_0$. The best-fitting values of V₀ and B are indicated on the graph, as well as the tube hematocrit H_t.

FIG. 5.

RBC flow rate obtained from Equation (4) as a function of a reference RBC flow rate value obtained by counting. Circles: measurements in 10 μm × 10 μm microchannels; Dots: measurements in 20 μm × 20 μm microchannels; Dashed line: identity function. The error bars are shown for a limited set of data points only, for sake of clarity. The error bars on $Q_{ref,RBC}$ come from the RBC-counting error and are $\pm 0.15\times Q_{ref,RBC}$. The error bars on Q_RBC are mainly caused by experimental errors on the determination of V₀ and H_t, see Equation (4) and are $\pm 0.17\times Q_{\mathit{R}\mathit{B}\mathit{C}}$.

FIG. 6.

(a) Sum of the RBC flow rates measured in the daughter branches of a bifurcation, $Q_{\mathit{R}\mathit{B}\mathit{C}}^1+Q_{\mathit{R}\mathit{B}\mathit{C}}^2$, compared with the RBC flow rate measured in the feeding branch, $Q_{\mathit{R}\mathit{B}\mathit{C}}^e$. (b) Sum of the suspending flow rates measured in the daughter branches of a bifurcation, $Q_f^1+Q_f^2$, compared with the suspending fluid flow rate measured in the feeding branch, $Q_f^e$. The different labels correspond to different types of micro-bifurcations.

FIG. 7.

Relation between the tube hematocrit H_t and the discharge hematocrit $H_d=Q_{\mathit{R}\mathit{B}\mathit{C}}/Q_{\mathit{b}\mathit{l}\mathit{o}\mathit{o}\mathit{d}}$. Circles: data points obtained in $10\hspace{0.17em}\mu \mathrm{m}\times 10\hspace{0.17em}\mu \mathrm{m}$ microchannels; Black squares: data points for $20\hspace{0.17em}\mu \mathrm{m}\times 20\hspace{0.17em}\mu \mathrm{m}$ microchannels. Lines: empirical law adapted from Pries et al., $H_t/H_d=H_d+(1-H_d)(1+1.7\hspace{0.17em}\hspace{0.17em}\text{exp}(-0.483D_h)-0.6\hspace{0.17em}\hspace{0.17em}\text{exp}(-0.013D_h))$, with $D_h=10\hspace{0.17em}\mu \mathrm{m}$ (dotted line) and $D_h=20\hspace{0.17em}\mu \mathrm{m}$ (dashed line).

FIG. 8.

Deviation from mass conservation: ratio between incoming and outgoing RBC flow rate at a bifurcation, $Q_{\mathit{R}\mathit{B}\mathit{C}}^e/(Q_{\mathit{R}\mathit{B}\mathit{C}}^1+Q_{\mathit{R}\mathit{B}\mathit{C}}^2)$, compared with the ratio of incoming and outgoing whole blood flow rate, $Q_{\mathit{b}\mathit{l}\mathit{o}\mathit{o}\mathit{d}}^e/(Q_{\mathit{b}\mathit{l}\mathit{o}\mathit{o}\mathit{d}}^1+Q_{\mathit{b}\mathit{l}\mathit{o}\mathit{o}\mathit{d}}^2)$. Black symbols: in vitro data for our four types of microbifurcations, grey symbols: in vivo data compiled by Cokelet et al.

FIG. 9.

Phase separation in symmetrical bifurcations: influence of inlet tube hematocrit (left: below 5%; right: above 5%). Upper row: bifurcation A; Middle row: bifurcation B, Lower row: Bifurcation C. Main panels: fractional RBC flow as a function of fractional blood flow in the same branch. Insets: ratio of discharge hematocrit in daughter branch relative to inlet branch as function of fractional blood flow in the same daughter branch. Filled symbols: outlet channel 1; Open symbols: outlet channel 2; Circles: original data; Squares: images of original data by central symmetry around (0.5, 0.5). The error bars correspond to $E/2$, as given by Equation (7). Continuous line: Pries et al. empirical law in the limit of small hematocrits. Dotted line: Proportional repartition of blood and RBCs (no phase separation). Dashed line in insets: maximal phase separation (all RBCs flowing in the channel with highest flow).

FIG. 10.

Phase separation in symmetrical bifurcations for inlet tube hematocrits below 5%. Upper row: bifurcation A; Middle row: bifurcation B, and Lower row: Bifurcation C. Same conventions as in Figure 9 except that experimental data have been corrected following Eqs. (2)–(12) in Ref. so that blood and RBC mass conservation are satisfied at each bifurcation and that the images of corrected data by central symmetry around (0.5, 0.5) are not shown because they are exactly superimposed to the corrected data.

FIG. 11.

Phase separation in the asymmetrical bifurcation D: influence of hematocrit. Same conventions as in Figure 9, except that bold lines correspond to predictions of phase separation laws for outlet branch 1 and thin lines for outlet branch 2, and that dashed lines represent Gould and Linninger ad hoc phase separation law.