Influence of leukocyte adhesion on partitioning of healthy and diabetic red blood cells at vascular bifurcations

Abstract

Hemorheological changes associated with diabetes include alteration in red blood cell (RBC) deformability, shape and volume, and an increased number of adhered leukocytes (WBCs). How such changes affect RBC distribution in capillary vessel networks is not fully known. Here, we undertake a 3D high-fidelity computational study of diabetic and healthy RBC partitioning in vascular bifurcations without and with an adherent WBC. We predict that without a WBC, the difference between healthy and diabetic RBC partitioning is small but subtle. While both exhibit disproportionate but regular partitioning, the sigmoid curves depicting diabetic RBCs could become concave up. An adherent WBC is shown to cause a significant asymmetry with RBCs exhibiting reverse partitioning when it is located closer to the flow-dominant branch and highly disproportionate regular partitioning when it is located closer to the non-dominant branch. The effect of a WBC is more pronounced on diabetic RBCs. Using numerical data and a reduced-order theoretical model, it is found that such asymmetry in RBC partitioning strikingly differs from WBC-induced asymmetry in the flow rate partitioning—the adhesion of a WBC is shown to increase the flow fraction in the flow-dominant daughter vessel. Interesting differences are also predicted for the time-dependent partitioning: Without the WBC, partitioning data of healthy cells show more temporal scatter than for diabetic cells. This trend is then reversed and increased by several folds with the WBC present. Physical mechanisms underlying these differences are elucidated using diabetic and healthy RBC flow patterns and their interaction with the WBC. Additionally, we predict that the presence of a WBC can cause flow reversal in the non-dominant branch for certain pressure conditions, a phenomenon not observed without the WBC.

INTRODUCTION

When flowing red blood cells (RBCs) encounter a vascular bifurcation, the way they distribute (or partition) to the downstream branches is dictated by cell-cell and cell-vessel interactions. This partitioning is controlled by vessel-diameter-to-cell-size ratio, bifurcation geometry, hematocrit, flow rate, and RBC deformability, among other parameters (1–6). In large vessels, RBCs’ behavior may follow that of passive tracers, and hence their partitioning follows how the flow rate is partitioned. In capillary vessels, the size and deformability effects are dominant, and RBC partitioning may not follow the flow rate partitioning, i.e. their partitioning may be disproportionate (5,6). This affects distribution of RBCs in the capillary vessel networks, as has been observed in vivo (2,4,5,7–10). The distribution of RBCs in the capillary network is critical to healthy functioning of the body and in disease progression, as it affects oxygen distribution as well as the hemodynamic forces responsible for diverse physiological functions such as endothelial cell response, blood flow regulation, and immune cell adhesion (1,11–15).

Extensive literature exists on RBC partitioning studied using various approaches. Among these, in vivo studies considered RBC partitioning at microvascular bifurcations of live animals (2,4,5,7–9). Using in vivo data, empirical relationships between the RBC flux ratio and the blood flow rate ratio were developed (2,9,16) and later applied in tissue-scale modeling of the microvascular blood flow (17). In vitro studies using microfluidic bifurcations and RBCs or representative particles have been considered, and the effects of hematocrit and vessel geometry have been addressed (3,18–27). Computational studies that model individual RBC deformation in 2D or 3D have also been considered that provided quantitative analysis and elucidated the underlying physical mechanisms dictating the partitioning behavior (28–34). Similar computational models have also been extended to study RBC partitioning in microvascular networks comprising of multiple vessels/bifurcations and representing healthy/abnormal conditions (35–43).

Diabetes can cause significant hemorheological disorders including morphological and mechanical changes in RBCs (44–50). A healthy RBC under resting conditions assumes a disk-like shape with concave surfaces. In contrast, using RBCs from diabetic patients, several studies have reported a reduction of the RBC’s surface concavity and the appearance of a convex surface, increased cell volume, reduced cell diameter, and a reduced cell surface-to-volume ratio (51–53). Additionally, studies have reported a reduction in RBC deformability (54–61). Increased RBC aggregation has also been reported by many studies (62–67). These hemorheological alterations are known to increase blood viscosity (68) and vascular resistance, which in turn, may cause reduced tissue perfusion and oxygen delivery (69–73).

The influence of deformability and shape/size of RBCs and representative particles on their partitioning behavior has been considered by several studies (18,19,23,26,27,29,34,37,41,43,74–81). While a number of them have shown that deformability influences partitioning, others have shown a negligible effect (3,42). RBC lingering at the bifurcation apex which tends to affect the partitioning was shown to decrease with reduced deformability (37,41). Regarding the particle size effect, studies have shown larger particles’ increased tendency to enter the lower flow rate branch thereby altering the partitioning behavior in comparison to smaller particles (34). The combined influence of stiffness and shape/volume as in sickle RBCs is also shown to have a significant impact on partitioning (43). When networks of multiple capillary vessels and bifurcations are considered, stiffer RBCs are shown to be distributed more heterogeneously because of the differences in their partitioning and lingering behavior compared to healthy RBCs (37,41).

Another consequence of diabetes is an increased number of leukocytes adhering to the blood vessel wall (82–87). Firm adhesion of leukocytes is one step of the leukocyte recruitment cascade where they initially slowly roll along and then adhere to the vascular wall (118–121). With their relatively large size and high stiffness, adhered leukocytes are known to significantly affect the blood flow pattern in microvessels (14,15,88–102). Preferential adhesion of leukocytes at microfluidic and vascular bifurcations even in non-diabetic conditions has been reported and the local flow dynamics at the bifurcation has been implicated as the primary cause (100). One study has also reported increased rigidity of leukocytes in diabetes which could facilitate further aggravation of leukocyte-mediated blood flow alterations (103).

Flow behavior of diabetic RBCs and the resulting hemodynamic alterations have been quantified recently by a few investigations using high-fidelity cell-resolved simulations. These include quantifying diabetic RBC dynamics in linear shear and tube flow (104), platelet margination in the presence of diabetic RBCs (105), and modeling of thrombus formation in diabetic blood (106) – all of which take into consideration the altered cell stiffness and size/shape. Additionally, diabetic RBC flow in retinal microaneurysms has also been considered (107).

Despite a growing interest in diabetic RBC hemodynamics and its significance, there appears to be no study that quantifies their partitioning behavior and contrasts against healthy cells. More importantly, no study exists that addresses the partitioning of diabetic RBCs in the presence of adhered leukocytes mimicking the disease-like conditions. Given their large size and nearly rigid nature, leukocytes partially blocking the vessel upstream a bifurcation can significantly and differentially alter the partitioning behavior of healthy and diabetic RBCs and thereby alter the RBC distribution in downstream capillaries.

To fill the knowledge gap, here we undertake a 3D, high-fidelity, cell-resolved modeling study of RBC partitioning in single isolated bifurcations without and with an adhered leukocyte in the vessel upstream. We predict that in the absence of a leukocyte, the difference between healthy and diabetic RBC partitioning is small but subtle. In contrast, the presence of the leukocyte significantly alters the partitioning behavior of both cells, but more so for the diabetic cells, creating a large difference between the two cell types. Additionally, the adhered leukocyte is shown to cause a flow reversal in the low flow rate branch of the bifurcation under certain conditions. These influences persist even when the leukocyte is adhered relatively far from the bifurcation as shown here by simulations and accompanying simple theoretical models. Taken together, this study provides novel results and insights with implications for microvascular disorders in diabetes.

MATERIALS AND METHODS

Fig. 1(a) gives a snapshot of one simulated bifurcation with an adhered leukocyte (WBC). The vessels are of circular cross-section, and the bifurcations modeled are symmetric so that both “daughter” branches have equal diameter and 45 degree branching angle with respect to the “mother” vessel. The mother vessel diameter $D_0$ is varied as 7.5, 10, 15 and $18\mu \text{m}$, representing capillaries and pre-capillary arterioles. The daughter vessel diameter is fixed as $D_0/\sqrt{2}$ (i. e, the total inlet and outlet areas are the same). Flow is driven by specifying pressure as the boundary condition at the inlet and outlets.

Figure 1.

(a) A representative simulation of diabetic RBCs in a bifurcation with an adhered WBC. (b), (c) undeformed shape of a healthy and diabetic RBC respectively. $R$ is the cell radius, $h$ is thickness, and $h_c$ is concave depth.

Each RBC is modeled as a sac of viscous fluid (hemoglobin) enclosed by a thin hyperelastic membrane that resists deformation against shearing, area dilatation, and bending. The shear and area dilatation resistances are implemented using Skalak et al’s model (108), for which the principal elastic tensions $\left({\tau }_1,{\tau }_2\right)$ in the RBC membrane are written in terms of the stretch ratios $\left({\epsilon }_1,{\epsilon }_2\right)$ as

$${\tau }_1=\frac{G_S}{{\epsilon }_1{\epsilon }_2}\left\{{\epsilon }_1^2\left({\epsilon }_1^2-1\right)+C_A{\left({\epsilon }_1{\epsilon }_2\right)}^2\left[{\left({\epsilon }_1{\epsilon }_2\right)}^2-1\right]\right\}$$ (1)

(and similarly, for ${\tau }_2$) where $G_S$ and $G_S{C}_A$ are shear and area dilation moduli, respectively. The bending resistance is modeled following Helfrich’s formulation (109) in terms of a bending force density ${\overrightarrow{f}}_B$ as

$${\overrightarrow{f}}_B=E_B\left[2k\left(2k^2-2k_g\right)+2{\Delta }_{S}k\right]\overrightarrow{n}$$ (2)

where $E_B$ is the bending resistance modulus, $k$ and $k_g$ are the mean and Gaussian curvatures, and ${\Delta }_s$ the surface Laplacian. Values of the moduli are listed in Table I. Notably, the shear modulus of diabetic RBCs (d-RBC, henceforth) is taken as twice that of healthy RBCs (h-RBC) following Refs. (53,104). The bending modulus is taken to be the same as healthy RBCs. The viscosity of the RBC interior fluid is taken as five times that of the outside fluid (plasma) viscosity for both cell types.

Table I.

Shape and deformability parameters of healthy and diabetic RBCs used in the current model. The centerline velocity in the mother vessel in absence of the cells is taken to be 5 mm/s.

	Healthy RBC	Diabetic RBC
RBC radius $(R,\mu \text{m})$	3.91	3.6
Thickness $(h,\mu \text{m})$	2.6	3.3
Volume $\left(\mu {\text{m}}^3\right)$	94.1	112.3
Surface area/volume $\left(\mu {\text{m}}^{-1}\right)$	1.425	1.117
Concave depth $\left(h_c,\mu \text{m}\right)$	0.878	0.24
$C_0$	0.207	0.78
$C_2$	2.003	1.132
$C_4$	−1.123	−0.352
$G_S(\mu \text{N}/\text{m})$	5.0	10.0
$C_A$	100	100
$E_B\left(\times {10}^{19}\text{J}\right)$	7.64	7.64

The undeformed shape of RBCs for both cell types are shown in Fig. 1 and prescribed as

$$z^{\prime }=\frac{1}{2}\sqrt{1-{r^{\prime }}^2}\left(C_0+C_2{r}^{\prime 2}+C_4{r}^{\prime 4}\right)$$ (3)

where $\left(x^{\prime },y^{\prime },z^{\prime }\right)=(x/R,y/R,z/R)$, $r^{\prime }=\sqrt{x^{\prime 2}+y^{\prime 2}},R$ is the cell radius, $C_0,C_2$ and $C_4$ are constants, and $(x,y,z)$ are centered at the cell center. The volume/shape for h- and d-RBCs used in the current study are listed in Table I and representative of those reported in the literature (51–53,104). As seen, a d-RBC has higher cell volume and reduced surface-area-to-volume ratio, slightly reduced cell radius, increased cell thickness $\left(h\right)$, and reduced concavity $\left(h_c\right)$. The coefficients $C_0,C_2$ and $C_4$ for h-RBCs are taken from (117) and those for d-RBCs are estimated to match their volume/shape used here.

Leukocytes are much stiffer than RBCs and maintain a nearly spherical shape. As noted before, leukocyte deformability is further reduced in diabetes. While a tear-drop shape of adhered leukocytes has been reported (88–90), for the present study the effect of such deformation on RBC partitioning is deemed negligible. As such, following prior works (92,93,101,110), the leukocyte is modeled as a rigid sphere sticking to the wall of the mother vessel upstream of the bifurcation. The leukocyte diameter is varied as 7, 12, 13 and $15\mu \text{m}$ while its distance from the bifurcation is varied from 12 to $140\mu \text{m}$.

A WBC can also slowly roll near a bifurcation, which is not considered in the current study. Both in vivo and in vitro studies have shown that WBC rolling velocity is on the order of tens of $\mu \text{m}/\text{s}$ (118,119), which is orders of magnitude slower than the velocity of RBCs (on the order of mm/s). A slowly rolling WBC can therefore be approximated as a stagnant WBC in comparison to the flowing RBCs. Furthermore, the WBC slow rolling occurs as an intermittent stop-and-go motion; each ‘stop’ can last for approximately one second (120,121), which is sufficient to allow several hundred RBCs to partition through the bifurcations considered, justifying our selection of a firmly adhered WBC.

The complete details of the numerical methodology and validations are given in our previous publications (35–37,127,128,132). Briefly, a finite volume/spectral method-based solver for fluid motion is used along with a finite-element method for cell membrane deformation, and the immersed boundary methods for fluid/structure interaction (FSI). The bifurcations are built using a CAD program and the vessel surfaces are meshed using triangular elements. The computation domain is a box containing the bifurcation model and is discretized using a staggered grid rectangular Eulerian mesh. The vessels are non-deformable and a sharp interface ghost node immersed boundary method is used to enforce the no-slip boundary condition on the vessel surface. The fluid motion is governed by the unsteady Stokes equations. The two-way coupling between the cells and the fluid is modeled using a continuous forcing immersed boundary method whereby the cell membrane tensions are distributed to the neighboring fluid. The velocity of a material point on the cell surface is interpolated back from the surrounding fluid velocity which is then used to update the deformed shape. Each RBC surface is discretized by 5120 triangular elements while the Eulerian mesh size considered is $0.2\mu \text{m}$, and the time step used for flow and cell updates is $1\mu \text{s}$, all of which were shown to be sufficient to resolve individual RBC or capsule deformation in our prior studies (127,132). Each simulation is performed for 0.5–1 seconds of quasi-steady physical blood flow, excluding an initial period of flow development, to provide enough data for accurately predicting average behavior.

The model does not explicitly consider close-range interaction between a WBC and RBCs, such as the glycocalyx on the RBC surface and electrostatic repulsion. The closest RBC-WBC distance noted in the simulations is ~200 nm, which is one order larger than the length (~20 nm) over which the electrostatic repulsion interaction occurs (122) and two orders larger than the thickness (~6 nm) of the glycocalyx (123,124). As discussed in Peng et al (125), these close-range interactions are important for RBCs squeezing through very narrow slits (e.g. $<1\mu \text{m}$, as in ref. 126), while for most cases the WBC-vessel gap is $3\mu \text{m}$. Another issue is the hydrodynamic lubrication between a rigid boundary and a deforming surface. The model includes a lubrication type force $\left(~h^{-3}\right),h$ being the distance between surfaces. In our prior studies, we validated our model against numerous examples of such close-range hydrodynamic interactions. These include micropipette aspiration of an RBC, a deformable capsule flowing through a rectangular gap, and RBCs partitioning in a narrow channel (127). The near-wall RBC-free layer in $5-6\mu \text{m}$ diameter vessels was also validated (128). Moreover, RBC lingering at the bifurcation apex, which is also a close-range interaction and first predicted by our model, was later validated by in vivo studies (75,41).

Given a relatively larger gap, sufficient distance between a WBC and RBCs, and enough distance (> tens of $\mu \text{m}$) for an RBC to recover after passing the WBC before partitioning, inclusion/improvement of a close-range interaction model is unlikely to significantly affect the partitioning results.

RBCs with a deformed shape, which are obtained from separate simulations, are introduced at the inlet of the mother vessel. The inlet hematocrit is a parameter that is varied as 0.25, 0.2, and 0.15 and is held constant for h- and d-RBCs for comparison purposes. The actual number of cells may be different for the two cell types due to their different volume. The hematocrit used here refers to “tube” hematocrit (which is calculated as the RBC volume fraction) and is less than the discharge or systemic hematocrit, as is the case due to the Fahraeus effect.

Partitioning is quantified in terms of two parameters: the RBC flux ratio $N^{*}$ and the flow rate ratio $Q^{*}$, defined as

$$N^{*}=\frac{N_1\left(\Delta t\right)}{N_0\left(\Delta t\right)}\text{and}{Q}^{*}=\frac{Q_1\left(t\right)}{Q_0\left(t\right)}$$ (4)

where the suffixes 0 and 1 refer to the mother and a daughter vessel, respectively, and $\Delta t$ is a time window. The inlet/outlet pressures are controlled to yield different values of $Q^{*}$ which is then used to present $N^{*}\left(Q^{*}\right)$. Time averages of $N^{*}$ and $Q^{*}$, denoted as $\overline{N^{*}}$ and $\overline{Q^{*}}$, and obtained by averaging over the entire data collection period, are used to quantify the average (quasi-steady) partitioning behavior. The RBC flux ratios for h- and d-cells are denoted as $\overline{N_H^{*}}$ and $\overline{N_D^{*}}$.

RESULTS

Average partitioning without a WBC

Fig. 2 presents the results for average partitioning in absence of a WBC. As seen, both h- and d-RBCs show disproportionate partitioning. The small but subtle distinction between h- and d-RBCs’ partitioning is best revealed by fitting the numerical data with Pries et al’s correlation (2), where the logit function is used to model the sigmoidal relationship between $\overline{N^{*}}$ and $\overline{Q^{*}}$ as

$$\text{logit}\left(\overline{N^{*}}\right)=A+B\text{logit}\left(\frac{\overline{Q^{*}}-X_0}{1-2X_0}\right)$$ (5)

where

$$\text{logit}(x)=\text{ln}\left(\frac{x}{1-x}\right)$$ (6)

Figure 2.

Time-averaged partitioning without a WBC. (a) and (b) $\overline{N^{*}}$ vs. $\overline{Q^{*}}$ for $D_0=10$ and $7.5\mu \text{m}$ respectively. Sigmoid curves obtained by fitting the present numerical data are shown. The numerical data is shown for one case only (red triangles). Red: h-RBC; blue: d-RBC. In both (a) and (b), solid and dash-dot curves are for hematocrit 0.25 and 0.15, respectively. All other cases, and all numerical data are provided in the supplement.

The coefficients $A,B$, and $X_0$ are determined from our numerical data. The coefficient $X_0$ is related to the flow rate ratio $\overline{Q_c^{*}}$ above which all RBCs enter the flow-dominant branch, that is, for $\overline{Q^{*}}>1-X_0,\overline{N^{*}}=1$. Pries et al provided a correlation for $X_0$ as a function of hematocrit and vessel diameter. However, the use of this correlation for the present parameters resulted in a significant overestimation in $\overline{Q_c^{*}}$ which we predict as much less than 1. Instead, we perform simulations at small increments of $\overline{Q^{*}}$ to more accurately obtain $\overline{Q_c^{*}}$ and hence $X_0$. Once $X_0$ is determined, $A$ and $B$ are found by fitting to our data. The coefficient $A$ determines the horizontal shift of the sigmoid away from the point $\left(\overline{Q^{*}},\overline{N^{*}}\right)=\left(\mathrm{0.5,0.5}\right)$ and is zero for symmetric bifurcations without a WBC. The coefficient $B$ represents the concavity of the sigmoid curves.

The sigmoidal fits thus obtained are shown in Fig. 2 for a few bifurcations and hematocrits. To avoid clutter, the numerical data is overlaid with the curve for only one case. Additional geometries, hematocrits, and data from all cases overlayed onto the sigmoid fits are provided in the supplement. The following differences between h- and d-RBC partitioning can be noted.

• $\overline{N^{*}}>\overline{Q^{*}}$ irrespective of RBC type, meaning that the flow-dominant branch receives an even higher fraction of both cells. $\overline{N^{*}}>\overline{Q^{*}}$ has been referred to as regular partitioning.

• For larger mother vessel diameter $\left(D_0=15\right.\text{and}\left.10\mu \text{m}\right),\overline{N_D^{*}}>\overline{N_H^{*}}$ at the same $\overline{Q^{*}}$, with the difference between the two cell types growing larger with decreasing hematocrit. Hence, d-RBCs show more disproportionate (though, regular) partitioning than h-RBCs.

• For smaller bifurcations at higher hematocrit, the sigmoid curves for d-RBCs become concave-up in contrast to concave-down. As such, the slope of the sigmoid curves increases with increasing $\overline{Q^{*}}$ for d-RBCs, while the opposite is the case for h-RBCs. In other words, d-RBC partitioning is more sensitive to the changes in the flow partitioning than that of h-RBCs. Because of the upward concavity, $\overline{N_D^{*}}<\overline{N_H^{*}}$ at $\overline{Q^{*}}$ sufficiently below $\overline{Q_c^{*}}$, but $\overline{N_H^{*}}>\overline{N_D^{*}}$ at higher $\overline{Q^{*}}$.

• For most cases, $\overline{Q_c^{*}}$ for d-RBCs is less than that of h-RBCs, which occurs due to the increased disproportionality of partitioning and increased sensitivity to the flow partitioning for d-RBCs.

As noted above, the coefficient $B$ in Eq. (5) represents the concavity of the sigmoid: it is > 1 for concave-down and < 1 for concave-up. Pries et al provided a correlation for $B$ in terms of hematocrit and diameter which resulted in $B>1$ (2). For our numerical data, $B>1$ for h-RBCs, but $B<1$ is a possibility for d-RBCs (e.g., $B=0.76$ for $D_0=7.5\mu \text{m},H=0.25$).

Mechanisms of partitioning differences without a WBC

Different partitioning behavior of h- and d-RBCs can be explained based on their flow dynamics. For this, we consider two groups:

• $\overline{N_D^{*}}>\overline{N_H^{*}}$ (for the same flow partitioning $\overline{Q^{*}}$) which occurs either at (1) increasing vessel diameter, or (2) decreasing hematocrit, or (3) as $\overline{Q^{*}}$ nears $\overline{Q_c^{*}}$; and,

• $\overline{N_D^{*}}<\overline{N_H^{*}}$ which occurs at decreasing diameter and increasing hematocrit as evident from Fig. 2.

Fig. 3(a–d) explains the first situation for $D_0=10\mu \text{m}$ at lower hematocrit. Two mechanisms can be identified in this case leading to $\overline{N_D^{*}}>\overline{N_H^{*}}$. First, although both types of cells flow in single file, h-RBCs flow in a more zig-zag pattern leading to a distribution in the mother vessel that is wider than that of d-RBCs, as seen in the PDFs in Fig. 3(b). This is likely caused by the presence of a higher number of h-RBCs than d-RBCs at the same hematocrit due to the difference in their volume. As such, h-RBCs are distributed on both sides of the separating streamline resulting in some of them entering the non-dominant branch. In contrast, d-RBCs in single-file tend to flow on the dominant-branch side of the separation surface, causing more of such cells to enter the flow-dominant branch. Second, h-RBCs are often seen to flow in groups of two to three closely following cells. Such cell clusters can fill the entire cross-section of the dominant branch, forcing the upstream cells to go to the non-dominant branch. In contrast, d-RBCs are more uniformly spaced allowing one cell after another to enter the dominant branch.

Figure 3.

Mechanisms of h- and d-RBC partitioning without a WBC. In (a,b,e,f,i,j) red is for h-RBCs and blue is for d-RBCs. (a)-(d) $\overline{N_D^{*}}>\overline{N_H^{*}},D_0=10\mu \text{m},H=0.15$. (b) shows the PDF of the RBC centroid distribution. The dotted line represents the mother vessel centerline. The image sequence in (c1-c3) and (d1-d3) show h- and d-RBC dynamics, respectively. H-RBC clustering in c2 blocks the dominant branch sending a follower RBC (in blue) to the other branch. (d) d-RBCs show no partitioning. * indicates the flow-dominant branch with arrows indicating direction of cell entry. (e)-(h) $\overline{N_D^{*}}>\overline{N_H^{*}},D_0=10\mu \text{m},H=0.25$. The h-RBC PDF is more spread, and the dominant branch blocking persists at this higher hematocrit. h-RBCs partition in (g), but d-RBCs do not in (h). (i)-(l) $\overline{N_D^{*}}<\overline{N_H^{*}},D_0=7.5\mu \text{m},H=0.25$. In image sequence (k1-k3), the leading h-RBC (blue) lingers at the apex but deforms enough to still allow the follower RBC (green) to enter the dominant branch. In sequence (l1-l3), the leading d-RBC (blue) completely blocks the dominant branch, sending the following cell (green) to the other branch.

Fig. 3(e)–(h) illustrates what happens at higher hematocrit and high $\overline{Q^{*}}$ for the same bifurcation. In this case, the two mechanisms described above are further amplified. Even at higher hematocrit, d-RBCs flow in single-file, but h-RBCs flow with an even wider distribution enhancing the first mechanism. The dominant branch is occasionally blocked by h-RBC clusters which amplifies the second mechanism as well sending some follower cells to the other branch.

Fig. 3(i)–(l) explains the mechanism for $\overline{N_D^{*}}<\overline{N_H^{*}}$ which occurs in the smallest bifurcation. In this case, both h- and d-RBCs flow in single-file. An h-RBC is seen to linger at the bifurcation, but it also deforms significantly such that there is enough space in the dominant branch for the follower RBC to deform and enter. In contrast, a d-RBC deforms much less and completely fills the dominant branch cross-section causing the upstream cell to enter the non-dominant branch.

Influence of an adherent WBC

The presence of an adherent WBC significantly affects the partitioning behavior as shown in Fig. 4. Even at the same pressure boundary conditions, the flow partitioning may be different in the presence of a WBC compared to that without a WBC. This effect also depends on whether the WBC is located on the same side as the dominant vessel or not. As such, the entire range of $\overline{Q^{*}}(~0\text{to}1)$ is investigated, unlike $0.5<\overline{Q^{*}}<1$ in absence of a WBC. To present the results, $\overline{Q^{*}}$ of the branch that is closer to the WBC is used, regardless of whether it is the flow-dominant branch or not.

Figure 4.

Average partitioning in the presence of a WBC for hematocrit 0.15 (a) and 0.25 (b). Red and blue curves are for h-RBCs and d-RBCs respectively. Solid curves are sigmoid fitted to the simulation data in the presence of the WBC. Dash curves are sigmoids in absence of the WBC. Bottom panel: * indicates the dominant daughter branch; $\overline{Q^{*}}$ in the horizontal axis is for the daughter branch on the same side as the WBC. The schematics on the right and left indicate a WBC located closer to the flow-dominant $\left(\overline{Q^{*}}>0.5\right)$ and non-dominant branch $(\overline{Q^{*}}<0.5)$ respectively.

As seen in Fig. 4, the WBC causes an asymmetry in RBC partitioning for both cell types such that $\overline{N^{*}}\left(\overline{Q^{*}}\right)\ne n^{*}\left(q^{*}\right)$ where $n^{*}=1-\overline{N^{*}}$ and $q^{*}=1-\overline{Q^{*}}$. The results without the WBC are also shown to highlight the differences. The asymmetry is higher at lower hematocrit for both cells. More importantly, the asymmetry is significantly more pronounced for d-RBCs than for h-RBCs.

Nearly all $\overline{N^{*}}$ data with a WBC lies below those without a WBC. When the WBC is situated on the same side as the flow-dominant branch, it prevents some RBCs from going toward the dominant vessel; this results in $\overline{N^{*}}$ with a WBC to be less than $\overline{N^{*}}$ without a WBC. When the WBC is located opposite to the flow-dominant branch, it causes even more RBCs to go to the dominant branch by altering their trajectories, resulting in a further reduction in $\overline{N^{*}}$ compared to that without the WBC.

The asymmetry in RBC partitioning caused by the WBC also causes $\overline{Q_c^{*}}$ to be different depending on the WBC location relative to the dominant branch: $\overline{Q_c^{*}}$ for $\overline{N^{*}}=1$ is higher than $(1-\overline{Q_c^{*}})$ for $\overline{N^{*}}=0$, with this difference being more pronounced for d-RBCs. Also, the difference in $\overline{Q_c^{*}}$ values between h- and d-RBCs is higher at $\overline{N^{*}}=0$ than at $\overline{N^{*}}=1$. Additionally, data for different hematocrits shows a pronounced difference as $\overline{N^{*}}\to 0$, but tends to converge as $\overline{N^{*}}\to 1$.

logit function in presence of a WBC

While a non-zero value of the parameter $A$ in Eq. (5) can represent an asymmetric sigmoid, the asymmetry in the presence of a WBC is severe enough so that no single value of $X_0$ can be used to represent $\overline{Q_c^{*}}$ for both daughter branches. Instead, two parameters $X_{\mathit{\text{near}}}$ and $X_{\mathit{\text{far}}}$ are introduced representing the flow rate thresholds for the branch nearer to and further away from the WBC. These thresholds are obtained directly from our simulations run at finer increments of $\overline{Q^{*}}$. Thus, Eq (5) now becomes:

$$\text{logit}\left(\overline{N^{*}}\right)=A+B\text{logit}\left(\frac{\overline{Q^{*}}-X_{far}}{X_{\mathit{\text{near}}}-X_{far}}\right)$$ (7)

This functional form allows for a sigmoid to be defined in the modified range $X_{\mathit{\text{far}}}<\overline{Q^{*}}<X_{\mathit{\text{near}}}$. The values of $A,B,X_{\mathit{\text{near}}}$, and $X_{\mathit{\text{far}}}$ are given in the supplement for a few cases.

WBC-induced reverse partitioning

Over a range of $\overline{Q^{*}}$ above 0.5, the presence of the WBC causes $\overline{N^{*}}$ to be less than $\overline{Q^{*}}$, meaning the flow-dominant daughter branch now receives a lower RBC fraction. When $\overline{N^{*}}<\overline{Q^{*}}$, it has been referred to as reverse partitioning (2,4,21,23). Notably, the reverse partitioning is not observed without the WBC for the bifurcation geometries considered here for both RBC types. As seen in Fig. 4, most data for $\overline{Q^{*}}>0.5$ lies below the line $\overline{N^{*}}=\overline{Q^{*}}$ meaning the presence of the leukocyte biases both h- and d-RBCs enough to cause the reverse partitioning. Strikingly, d-RBCs exhibit more reverse partitioning than h-RBCs. In contrast, the regular partitioning is seen only for h-RBCs at high values of $\overline{Q^{*}}$. For $\overline{Q^{*}}<0.5$, both cell types exhibit regular partitioning, with d-RBCs showing more disproportionality.

Influence of WBC location

Fig. 5 shows the effect of WBC location from the bifurcation $\left(L_{WBC}\right)$ on the RBC partitioning. As seen, even a WBC located as far as $L_{\mathit{\text{WBC}}}=140\mu \text{m}$ upstream (> 9 vessel diameters) causes the $\overline{N^{*}}$ vs. $\overline{Q^{*}}$ plot to be asymmetric, suggesting that the effect of the WBC persists for a long distance. As the WBC is located increasingly further away from the bifurcation, the results tend to approach towards that without the WBC. However, the approach is different for different values of $\overline{Q^{*}}$ as seen from Fig. 5(b) where $\overline{N^{*}}$ is plotted as a function of $L_{\mathit{\text{WBC}}}$ for specific values of $\overline{Q^{*}}$. An approach to the no-WBC results occurs earlier for higher $\overline{Q^{*}}$ values than at lower ones, which is a result of the way asymmetry in the partitioning curve occurs. Also, the approach is faster for h-RBCs than for d-RBCs as seen in Fig. 5(b). For h-RBCs, the data corresponding to $L_{\mathit{\text{WBC}}}=140\mu \text{m}$ fully recover to the no-WBC case at $\overline{Q^{*}}>0.5$, but not at $\overline{Q^{*}}<0.5$. For d-RBCs, the $L_{\mathit{\text{WBC}}}=140\mu \text{m}$ results are close (but not fully recovered) to the no-WBC case at $\overline{Q^{*}}>0.5$, while they are far from the no-WBC results at $\overline{Q^{*}}<0.5$. Thus, even at larger distances, the WBC has a more severe effect on d-RBC partitioning than on that of h-RBCs.

Figure 5.

Effect of WBC distance $\left(L_{WBC}\right)$ from the bifurcation apex. In both (a) and (b), red and blue correspond to h- and d-RBCs respectively, and dash lines represent no-WBC curves. (a) $L_{\mathit{\text{WBC}}}=60\mu \text{m}$ (dash-dot curves, diamond symbols) and $140\mu \text{m}$ (solid curve, square symbols). (b) $\overline{N^{*}}$ versus $L_{\mathit{\text{WBC}}}$ for different $\overline{Q^{*}}$. Solid lines represent results with WBC, dash lines without WBC.

Anomaly in WBC-mediated flow partitioning

Two possible factors that can cause the strong bias in RBC partitioning in the presence of a WBC are considered: (1) the presence of a WBC creates an asymmetry in the flow rate partitioning which in turn affects the RBC partitioning, and (2) RBC trajectories are affected by the presence of the WBC. Next, we seek to isolate these two factors.

First, we investigate the asymmetry in the flow rate partitioning caused by the WBC. To do this, we simulate flow of RBC-free plasma fluid in the presence of the WBC as well as without the WBC, both subject to the same pressure boundary conditions. We compute the plasma flow rate ratio $Q_{pl,WBC}^{*}$ in the presence of a WBC and plot it as a function of the plasma flow rate ratio $Q_{pl}^{*}$ without the WBC in Fig. 6(a). The flow rate ratio is considered for the vessel closer to the WBC. As seen, $Q_{pl,WBC}^{*}>Q_{pl}^{*}$ for $Q_{pl}^{*}>0.5$, and $Q_{pl,WBC}^{*}<Q_{pl}^{*}$ for $Q_{pl}^{*}<0.5$. This means, surprisingly, that the WBC alone causes the flow-dominant branch to receive an even higher fraction of the incoming flow, irrespective of whether the branch is closer to the WBC or not. The presence of a WBC is expected to deflect the fluid away from the daughter branch located on the same side, which indeed occurs when the same-side daughter branch is not the flow-dominant branch. However, when the same-side daughter branch is the flow-dominant branch, the presence of the WBC further increases the flow fraction in that branch, which is counterintuitive. Therefore, the presence of a WBC causes the flow rate partitioning to be more ‘regular,’ unlike the RBC partitioning that may become ‘reverse’ as seen earlier.

Figure 6.

Influence of WBC on RBC-free plasma flow partitioning. (a) plasma flow rate ratio $Q_{pl,WBC}^{*}$ in the presence of a WBC as a function of the plasma flow rate ratio $Q_{pl}^{*}$ without a WBC. Data presented is for the daughter branch closer to the WBC. In the associated schematic bifurcations, * represents the flow-dominant branch. The WBC at different distances from the bifurcation apex is considered. (b) An electrical circuit analogy to explain the effect of the WBC on the flow partitioning.

Fig. 6(a) also presents the data for different WBC locations. Even for the furthest distance considered here $\left(140\mu \text{m}\right)$, the asymmetry in the flow rate partitioning is still significant with the recovery of $Q_{pl,WBC}^{*}$ to $Q_{pl}^{*}$ being rather slow.

Reduced-order theory explaining flow rate partitioning

The above numerical result, though counterintuitive, can be mathematically supported using an electrical circuit analogy of a vascular bifurcation as shown in Fig. 6(b). Here, resistances of the daughter branches are denoted by $R_1$ and $R_2$, and flow rates by $Q_1$ and $Q_2$. The mother vessel resistance is $R_0$ with flow rate $Q_0$. The inlet and exit pressures are $P_0,P_1$ and $P_2$ respectively. Then,

$$P_1=P_0-Q_0{R}_0-Q_1{R}_1$$ (8)

$$P_2=P_0-Q_0{R}_0-Q_2{R}_2$$ (9)

and

$$Q_0=Q_1+Q_2$$ (10)

Defining $\Delta P^{*}=\left(P_0-P_1\right)/\left(P_0-P_2\right)$, one gets

$$Q_1^{*}=\frac{Q_1}{Q_0}=\frac{\left(R_2+R_0\right)\Delta P^{*}-R_0}{R_1+R_2\Delta P^{*}}$$ (11)

The condition for branch 1 to be flow-dominant is clearly $Q_1^{*}>0.5$. Applying this to Eq. (11) and solving for $\Delta P^{*}$, we obtain

$$\Delta P^{*}>\frac{R_0+0.5R_1}{R_0+0.5R_2}$$ (12)

Upon setting $R_1=R_2=R$ as in the simulation setup, the condition reduces to $\Delta P^{*}>1$ for the flow dominant branch and vice versa. Furthermore, differentiating Eq. (11), one gets

$$\frac{\partial Q_1^{*}}{\partial R_0}=\frac{\Delta P^{*}-1}{\left(\Delta P^{*}+1\right)R}$$ (13)

Eq. (13) shows that if $\Delta P^{*}>1,\frac{\partial Q_1^{*}}{\partial R_0}>0$; that is, the flow-dominant branch would receive an even higher flow fraction if the resistance of the mother vessel is increased. Since the presence of a WBC causes the mother vessel flow resistance to increase, this theory supports the numerical findings that the flow-dominant branch would receive a higher flow fraction even when the location of the WBC partially blocks that side of the mother vessel.

If the daughter branches have unequal resistances, Eq. (13) becomes

$$\frac{\partial Q_1^{*}}{\partial R_0}=\frac{\Delta P^{*}-1}{R_1+R_2\Delta P^{*}}$$ (14)

Together with Eq. (12), it shows the possibility for $\Delta P^{*}<1$ while branch 1 can still be dominant, meaning that the flow-dominant branch may receive a reduced flow fraction as $R_0$ is increased. However, as long as $\Delta P^{*}>1$, the flow fraction in that branch will increase as $R_0$ increases.

The results in Fig. 6 are a result of imposing pressure boundary conditions. Imposing a flow rate boundary condition leads to the trivial case where $Q_{pl}^{*}$ is the same with and without a WBC.

The above result for the flow partitioning is in stark contrast to the RBC partitioning as seen before in Fig. 5 where the branch closer to the WBC receives a much less fraction of RBCs even when it receives a higher fraction of flow. This results in a reverse partitioning when $\overline{Q^{*}}>0.5$, but a more disproportionate regular partitioning for $\overline{Q^{*}}<0.5$.

The above argument, however, does not explain the more severe effect of a WBC on d-RBCs. Hence, it must be related to different flow dynamics of h- and d-RBCs under the influence of a WBC, which is considered next.

Mechanisms of WBC-mediated h-RBC and d-RBC partitioning

The physical mechanism for this effect, explored in Fig. 7, is due to differences between h- and d-RBCs in terms of their trajectory recovery upon traversing the WBC. Figs. 7(a) and (b) show the h- and d-RBC distributions in the bifurcation at one time instant, and Fig. 7(e) shows the PDFs of the RBC distribution taken at three different locations downstream of the WBC. The h-RBC distribution recovers relatively faster resulting in a less skewed profile, while the d-RBC distribution remains highly skewed toward the lower vessel wall. Two main mechanisms cause this difference: (1) After passing the WBC, h-RBCs ‘recover’ their trajectories much faster than d-RBCs. This is due to faster recovery from extreme deformation that occurs as they squeeze ‘under’ the WBC, and due to a faster rate of migration towards the vessel center. (2) h-RBCs can laterally (i.e., side-ways) traverse the WBC due to their ability to deform more as shown in Fig. 7(c). This allows some h-RBCs to maintain their trajectory without getting deflected towards the bottom of WBC and hence resulting in a less skewed cell distribution. In contrast, d-RBCs do not exhibit lateral traversing; they only pass under the WBC, as shown in Fig. 7(d).

Figure 7.

Mechanism of WBC influence on RBC partitioning. (a,c) and (b,d) show h- and d-RBC distributions in the presence of a WBC $(L_{\mathit{\text{wbc}}}=140\mu \text{m})$. In (c), arrows indicate h-RBCs laterally traversing (flowing around) the WBC which is not seen for d-RBCs as shown in (d). (e) PDFs of RBC distribution taken at three locations along the mother vessel as marked in (a) and (b). In (e), red indicates h-RBCs and blue indicates d-RBCs. Solid, dash-dot, and dash curves correspond to PDFs at different locations as indicated by similar line patterns in (a) and (b). (f,g) D-RBC flow pattern at different hematocrit values to show hematocrit effect on partitioning in the presence of a WBC.

As seen earlier in Fig. 4, the WBC-induced bias is more severe at lower hematocrit. The physical mechanism explaining this effect is presented by Figs. 7(f) and (g). The skewed distribution of RBCs causes intermittent blockage of the lower daughter branch at higher hematocrit while also being conducive to crowding of cells near the bifurcation region. This in turn causes a few RBCs to enter the upper branch. Such cell crowding and blockages do not occur at lower hematocrit allowing most RBCs to flow through the lower branch resulting in a more asymmetric partitioning. This effect is exacerbated in the case of d-RBCs due to the more biased distribution as was discussed above.

WBC size effect

The effect of varying WBC diameter $\left(d_{wbc}=\mathrm{10,12,13}\mu \text{m}\right)$ is shown in Fig. 8(a). Here, the mother vessel diameter is kept fixed ${(D}_0=15\mu \text{m})$. The partitioning becomes more asymmetric with increasing WBC size while the difference between cell types increases with d-RBCs showing more asymmetric partitioning than h-RBCs.

Figure 8.

WBC size effect. (a)-(c): Effect of varying WBC diameter $(d_{wbc}=\mathrm{10,\; 12,\; 13}\mu \text{m})$ for a fixed mother vessel diameter $\left(D_0=15\mu \text{m}\right)$. (b) For $d_{wbc}=10\mu \text{m}$, lateral traversing is present for both h- and d-RBCs. (c) For $d_{wbc}=13\mu \text{m}$, only h-RBCs can pass, but not d-RBCs. (d) Simultaneously varying $D_0$ and $d_{wbc}$ keeping the maximum gap size at $3\mu \text{m}$. Only the fitted sigmoids are shown here. Plots showing the numerical data are given in S.I. Hematocrit is fixed at 0.25.

The mechanisms underlying the WBC size effect are explored as follows. For $d_{wbc}=10\mu \text{m}$, the lateral traversing mechanism as described before is present for both h- and d-RBCs due to a larger space between the vessel wall and the WBC as seen in Fig. 8(b). Also, this increased space causes less RBC deformation and faster trajectory recovery thereby reducing the skewness in both h-RBC and d-RBC distributions. As such, the difference between h- and d-RBC partitioning is minimal for the smallest WBC size. For $d_{wbc}=12\mu \text{m}$, the lateral traversing is present only for h-RBCs but not for d-RBCs due to the reduced space. The reduced gap also causes more deformation and slower trajectory recovery of the d-RBCs. Similarly, the d-RBC distribution becomes more skewed leading to greater asymmetry in their partitioning. For the largest WBC size, $d_{wbc}=13\mu \text{m}$, d-RBCs are no longer able to pass the WBC, and they pile upstream as seen in Fig. 8(c). In contrast, h-RBCs are able to pass but only under the ‘bottom’ of the WBC with no lateral traversal being seen. This causes a highly skewed partitioning even for h-RBCs, while d-RBCs cease to flow.

Fig. 8(d) presents the effect of simultaneously varying the mother vessel diameter and WBC diameter as $\left(D_0,d_{wbc}\right)=\left(\mathrm{10,\; 7}\right),\left(\mathrm{15,\; 12}\right)$ and $\left(\mathrm{18,15}\right)\mu \text{m}$, such that the maximum gap between the WBC bottom and vessel wall remains constant at $3\mu \text{m}$. This gap size is selected based on the above result of $\left(D_0,d_{wbc}\right)=\left(\mathrm{15,12}\right)\mu \text{m}$ for which the lateral traversing is present only for h-RBCs but not for d-RBCs. For $\left(D_0,d_{wbc}\right)=\left(\mathrm{10,7}\right)$ the partitioning curves of h-RBCs and d-RBCs show relatively small difference. This is because for this combination, neither h-RBCs nor d-RBCs can laterally traverse the WBC; they can only pass through the bottom gap resulting in a similar degree of skewness in cell distribution. For $\left(D_0,d_{wbc}\right)=\left(\mathrm{15,12}\right)$, the difference between the partitioning curves is large, with the d-RBC data showing more asymmetric partitioning. As noted above, this happens because h-RBCs can pass through the bottom gap as well as laterally traverse the WBC resulting in reduced skewness in their distribution, while no lateral traversing is possible for d-RBCs resulting in more bias in their distribution. For $\left(D_0,d_{wbc}\right)=\left(\mathrm{18,15}\right)$, the difference between the partitioning curves increases even more, with h-RBCs now showing a highly skewed curve and d-RBCs showing only reverse partitioning for nearly the entire range of $\overline{Q^{*}}>0.5$. This happens because the relative cross-sectional area blockage is now higher (70%) compared to the previous cases (64% and 49%) which reduces the lateral traversing of h-RBCs. This also severely constrains d-RBCs while they squeeze only through the bottom gap, resulting in a severe bias in their distribution and significant reverse partitioning.

Additionally, while keeping gap size constant, partitioning of d-RBCs becomes more disproportionate and reverse with increasing vessel diameter, while the behavior of h-RBCs is more subtle. The behavior of d-RBCs is clear as they were unable to laterally traverse across all three combinations of ${(D}_0,d_{wbc})$, being increasingly biased toward the bottom as the relative cross-sectional area is reduced forcing them all to flow under the WBC. For h-RBCs in going from $\left(D_0,d_{wbc}\right)=\left(\mathrm{10,7}\right)$ to (15,12) and despite this reduction in relative cross-sectional area, they gain the ability to laterally traverse. Going from ${(D}_0,\left.d_{wbc}\right)=\left(\mathrm{15,12}\right)$ to (18,15), only the area reduction effect is present. That is, there are two competing effects for h-RBCs: downward bias due to decreased relative cross-sectional area, and a reverse of this bias from gaining the ability to laterally traverse. This explains the $\overline{Q^{*}}$-dependent trend of disproportionality with vessel diameter seen in h-RBCs in Fig.8(d) as an explicit effect of increased deformability.

Flow reversal by WBC

We find that the presence of a WBC can cause flow reversal in the non-dominant daughter branch beyond a certain value of the pressure ratio $\Delta P^{*}=\left(P_0-P_1\right)/\left(P_0-P_2\right)$ for which no such flow reversal exists in the absence of a WBC. Letting $Q_1^{*}$ correspond to the branch closer to the WBC and also the flow-dominant branch, the flow reversal in the non-dominant branch would mean $Q_1^{*}>1$. This can happen with and without RBCs and is shown in Fig. 9(a) for RBC-free plasma flow where $Q_1^{*}$ versus $\Delta P^{*}$ is plotted. The symbols in the figure are the simulated data, which show that above a certain $\Delta P^{*}(~1.79)$, $Q_1^{*}$ becomes greater than unity in the presence of a WBC, but not without the WBC.

Figure 9.

Flow reversal by an adhered WBC. (a) $Q_1^{*}$ versus $\Delta P^{*}$ for plasma flow (no RBC) only. Symbols represent simulation data, and the curves represent the theoretical results: the continuous curve represents the presence of the WBC (Eq. (15)); the dash-dot curve is without the WBC (Eq. (11)). (b) Simulation data of $Q_1^{*}$ versus $\Delta P^{*}$ for RBC flow. Red symbols represent h-RBCs and blue represents d-RBCs. The theoretical curves that are the same as in (a) are replotted. (c), (d) Pressure contours and streamlines without and with the WBC, respectively, for the same values of inlet and exit pressure. * indicates the flow-dominant branch.

The theoretical analysis presented earlier also predicts this flow reversal phenomenon. This can be readily seen from Eq. (15), which is plotted (using curves) in Fig. 9(a) and overlayed with the simulated data. Plotting the theoretical curve requires knowing the resistances of the vessels ${(R}_0,R_1,R_2)$. For the flow resistances without the WBC, the Poiseuille’s formula is used, e.g., $R_0=\left(128\mu L_0\right)/\left(\pi D_0^4\right)$, and similarly for $R_1$ and $R_2$. When a WBC is present, the resistance in the mother vessel is increased above $R_0$ and written as $R_0+R_{\mathit{\text{wbc}}}$ where $R_{\mathit{\text{wbc}}}$ is the additional resistance caused by the WBC. Then, Eq. (11) becomes

$$Q_1^{*}=\frac{\left(R_2+R_0+R_{wbc}\right)\Delta P^{*}-\left(R_0+R_{wbc}\right)}{R_1+R_2\Delta P^{*}}$$ (15)

$R_{\mathit{\text{wbc}}}$ is calculated a posteriori from one point in the $(\Delta P^{*},Q_1^{*})$ parameter space using the simulated data and can be shown to be independent of $\Delta P^{*}$. With this, the theoretical curve can be plotted. As seen in Fig. 9(a), both theoretical curves, Eq. (11) for no WBC and Eq. (15) with WBC, nicely agree with the simulated data and predict that the flow reversal occurs in the presence of a WBC above a certain $\Delta P^{*}$.

The physical mechanism of this flow reversal is the wake effect of a WBC as illustrated by Figs. 9(c) and (d). In low inertia flows, the pressure immediately behind a bluff body subjected to an incoming uniform flow in an unbounded domain drops significantly and attains the lowest value (111). Extrapolating this result to the present case of the adhered WBC inside a vessel, there could be a locally reduced pressure after the WBC. This reduced pressure can then create a negative pressure gradient in the non-dominant daughter vessel which then results in the reverse flow. In contrast, the dominant vessel exit pressure is sufficiently low so that fluid continues to move out of that branch. Note that the wake pressure in low inertia flows recovers as $r^{-2}$ where $r$ is the downstream distance from the WBC (111). Thus, the flow reversal can happen even when the WBC is adhered relatively far from the bifurcation, provided the local pressure near the bifurcation is still less than $P_2$.

Fig. 9(b) presents the numerical data of $Q_1^{*}$ versus $\Delta P^{*}$ when RBC flow is considered, along with the theoretical curves of 9(a) replotted. It shows that flow reversal is present with RBCs also, and for both h- and d- cells. The numerical data for both h-RBCs and d-RBCs coincide with each other as well as with the theoretical curve when no WBC is present. However, with the WBC present, the numerical data for h-RBCs deviate from the theoretical curve at $Q_1^{*}$ close to unity, but those for d-RBCs deviate significantly even at lower $Q_1^{*}$. This is because the partitioning becomes more skewed with a WBC as seen earlier, resulting in one daughter branch getting significantly more RBCs than the other, causing the flow resistances in the daughter vessels to differ more. With the presence of RBCs, the fluid behaves as non-Newtonian and hence the flow resistance calculation using the Poiseuille’s formula is no longer accurate. With d-RBCs, the partitioning becomes highly skewed, affecting the flow resistances more disproportionately in the daughter vessels which causes the departure from the theoretical curve even at lower $Q_1^{*}$. Also, the increased stiffness of d-RBCs likely causes more deviation from the Poiseuille’s formula for the resistance calculation. Nonetheless, the numerical results show that the WBC-mediated flow reversal is present even when RBC flow is considered, and the threshold $\Delta P^{*}$ above which it occurs is less than that for RBC-free plasma fluid as evident in Fig. 9(b).

Time-dependent partitioning and RBC dynamics

Because of the discrete nature of RBC flow in small vessels, their partitioning behavior is time-dependent so that the RBC flux ratio and flow rate ratio varies with time. Fig. 10 shows such time dependent partitioning using the scatter plot of $N^{*}\left(t\right)$ vs $Q^{*}\left(t\right)$ for the flow-dominant branch. For all cases shown here, the average partitioning is of regular type. However, the scatter plot shows that the time-dependent partitioning can fluctuate between the regular and reverse types. The nature of such time-dependency is controlled by vessel size and RBC flexibility, as well as the presence of a WBC. Figs. 10(a) and (b) correspond to $D_0=15$ and $7.5\mu \text{m}$, respectively, without the WBC, while (f) and (g) are in the presence of a WBC for $L_{WBC}=12$ and $140\mu \text{m}$ respectively.

Figure 10.

Time-dependent partitioning and RBC dynamics. (a), (b) $N^{*}\left(t\right)$ vs. $Q^{*}\left(t\right)$ scatter plot in absence of the WBC for $D_0=15$ and $7.5\mu \text{m}$ respectively. In all scatter plots, red represents h-RBCs and blue represents d-RBCs. (c), (d) Lingering of h-RBCs blocking the non-dominant and dominant branches, respectively, as indicated by black arrows. In all snapshots, * indicates the flow-dominant branch. (e) Absence of lingering in d-RBC flow. (f), (g) $N^{*}\left(t\right)$ vs. $Q^{*}\left(t\right)$ scatter plot in presence of the WBC for $L_{\mathit{\text{WBC}}}=12$ and $140\mu \text{m}$ respectively. (h) Continuous train of h-RBCs even when the WBC is present. (i), (j) Discontinuous flow of d-RBCs as discrete clumps (blue arrows) and formation of plasma gaps in between.

In the absence of a WBC, the spread of the data scatter is larger for h-RBCs than for d-RBCs, as seen in Figs. 10(a) and (b). The underlying mechanism for this difference is explained by Fig. 10(c)–(e) and associated with the lingering behavior of h-cells. An h-RBC tends to linger at the bifurcation apex and partially block one daughter branch. This can cause other cells behind to pile up at the bifurcation and further block the branch. This can happen to both the dominant and non-dominant branches. In Fig. 10(c), the non-dominant branch is blocked which causes an increase in $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ in the flow-dominant branch, whereas in Fig. 10(d), the dominant branch is blocked leading to a decrease. Due to their reduced deformability and inflated volume, d-RBCs do not exhibit the lingering behavior as seen in Fig. 10(e), resulting in less time-dependent variation of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$.

As the vessel size decreases, more frequent instances of lingering-induced blockage occur, with even a single h-RBC lingering at the apex, as in the case of $D_0=7.5\mu \text{m}$, causing a temporary blockage. This results in more time-dependent variations in $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ as seen in Fig. 10(b) in comparison to 10(a).

Increasing hematocrit of h-RBCs also causes more fluctuations in $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ as it increases the likelihood of lingering-induced blockages.

The exception to the above results is found to occur for the smallest bifurcation at high values of $\overline{Q^{*}}$. As noted earlier for time-averaged partitioning, nearly all h-RBCs from the feeding vessel can enter the dominant branch while d-RBCs continue to partition to both branches. This leads to higher fluctuations of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ for d-RBCs, unlike what is observed for all other situations without the WBC.

In the presence of a WBC, two key changes to the above results are observed as shown in Figs. 10(f) and (g). First, the fluctuations of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ now increase significantly for both types of RBCs. Second, the fluctuations are more for d-RBCs than for h-RBCs, in reversal of what is seen in the absence of a WBC. The standard deviations of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ in absence of the WBC are up to 0.02, whereas in the presence of a WBC they are up to about 0.06 for h-RBCs and 0.13 for d-RBCs. Even when the WBC is placed further away from the bifurcation, the time-dependency remains similar as evident from the scatter spread in Fig. 10(g).

In the presence of a WBC, h-RBC lingering is no longer the dominant underlying mechanism for fluctuations. Rather, the passage of individual RBCs through the small gap between the WBC and the vessel wall is what primarily dictates the fluctuations. In the case of h-RBCs, multiple cells can pass through the gap causing a relatively continuous movement of cells. Their flexibility and flat shape also allow h-RBCs to slide past each other through the bifurcation region further enabling the continuous cell flow (Fig. 10(h)). In contrast, fewer d-RBCs (often only one) can pass through the gap resulting in discontinuous cell movement. The higher stiffness of d-RBCs and the intermittent blockage of the gap also slow down the fluid, which leads to an accumulation of such cells at the bifurcation forming ‘clumps’. Reduced flexibility and swelled volume hinder the d-RBCs’ ability to slide past each other. The resulting flow of d-RBCs in the daughter branches occurs as a flow of discrete cell packs with plasma gaps in between as shown in Figs. 10(i) and (j). This results in a significant increase in time variations of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ for d-RBCs.

Since the time variations of $Q^{*}\left(t\right)$ and $N^{*}\left(t\right)$ in the presence of a WBC are primarily dictated by the passage of RBCs through the gap between the WBC and vessel wall and the formation of d-RBC clumps, the high amplitude fluctuations occur even when the WBC is moved further away from the bifurcation as shown by the data scatter in Fig. 10(g).

Instantaneous flow reversal

Additionally, $Q^{*}\left(t\right)$ for d-RBCs may instantaneously be greater than unity as seen in the presence of a WBC as shown in Figs. 10(f)–(g). This corresponds to instantaneous flow reversals in the non-dominant daughter branch, even though the time averaged flow is not reversed. As demonstrated earlier, the presence of a WBC causes a reduction of pressure in its wake and in the bifurcation region. The flow of RBCs causes time-dependent variations in the wake pressure, and d-RBCs affect the pressure more than h-RBCs because of the more discrete nature of flow of such cells (due to a smaller number of them at the same hematocrit because of their increased volume). Even though the WBC wake pressure remains above the non-dominant branch exit pressure in a time-averaged sense, the instantaneous pressure in the presence of d-RBCs can drop below the exit pressure causing a flow reversal. As was seen earlier in Fig. 9(a), (b), the flow partitioning is highly sensitive to changes in $\Delta P^{*}$ and hence the exit pressure in the presence of a WBC. As such, small variations in the WBC wake pressure caused by the intermittent flow of d-RBCs can lead to these instantaneous flow reversals.

DISCUSSION AND CONCLUSION

Below, we first summarize the key findings, followed by discussion of the results within the context of the previous studies, and their physiological significance.

In absence of a WBC, both h- and d-RBCs exhibited regular partitioning. Whether h-RBCs or d-RBCs exhibit more disproportionality depends on vessel diameter, hematocrit, and flow rate ratio. For the smallest vessel, the resulting sigmoids for d-RBCs are concave-up. These differences are rooted in the deformability and size/shape difference between the RBC types. Three mechanisms related to cell dynamics are identified causing the observed differences in partitioning: (i) narrower distribution of d-RBCs over the vessel cross-section, (ii) lingering-assisted cell cluster-induced blockage of a branch by h-RBCs, and reduced lingering of d-RBCs, and (iii) blockage of smaller capillaries by d-RBCs.

An adherent WBC is shown to cause a significant asymmetry in the partitioning curve, with reverse partitioning when it is located closer to the flow-dominant branch and highly disproportionate regular partitioning when it is located closer to the non-dominant branch. The effect is more pronounced for d-RBCs. The logit function representing the sigmoid relation is modified to account for the severity of the WBC effect. It is shown, using numerical data and a reduced-order model, that such asymmetry in the RBC partitioning strikingly differs from WBC-induced asymmetry in the flow rate partitioning, as the adherence of a WBC increases the flow fraction in the flow-dominant daughter vessel. The difference in h- and d-RBC partitioning arises due to the more skewed d-RBC distribution downstream of the WBC. Two underlying mechanisms leading to this are the slower recovery of d-RBCs as they squeeze out of the WBC-vessel wall gap and the absence of lateral traversal. These mechanisms also dictate the WBC size effect on RBC partitioning.

Interesting differences are also predicted for the time-dependent partitioning. Without the WBC, partitioning of h-RBCs is more scattered than that of d-RBCs which is due to lingering of h-RBCs and its absence for d-RBCs. The WBC causes the scatter to increase by several folds for both RBC types, but more so for d-RBCs, in a reversal to the result without the WBC. With the WBC, lingering is no longer the dominant cause for the scatter; instead, the RBC flow pattern altered by the WBC dictates. D-RBCs flow in a more discontinuous manner in the presence of the WBC, while h-RBCs tend to flow more continuously.

Additionally, we predict that the presence of a WBC can cause flow reversal in the non-dominant branch for certain pressure conditions. Such flow reversal is not seen without the WBC, which was verified using the reduced-order model. Furthermore, even when the time-average flow is not reversed, the time-dependent flow may intermittently be reversed.

Current findings in the context of previous studies are now discussed. Doyeux et al. investigated partitioning of both rigid and deformable spheres linking their upstream distribution in the feeder vessel to behavior at the bifurcation (34). Specifically, larger particles tended to enter the lower flow rate branch due to being above/below a separating surface with higher probability. Others also reported a similar behavior with malaria-infected stiffer RBCs (33) and less deformable 2D-cells (29,80). Additionally, Audet and Olbricht found the distribution of cells upstream the bifurcation has a significant impact on partitioning (28). Recently, Stathoulopoulos et al. also observed a much sharper and centered distribution of stiffer cells in the mother vessel (27). Our results agree in general with the above works with subtle discrepancies. In our case, we report more disproportionate partitioning of d-RBCs as a direct consequence of the much wider distribution of h-RBCs in the mother vessel, allowing more h-RBCs to enter the lower flow rate branch. However, this is despite h-RBCs being smaller than d-RBCs in volume, contradicting the above works. Only at small diameter do we see less disproportionality in larger d-RBCs. In another numerical study of h-RBCs flowing in capillary networks with WBCs, Ostalowski and Tan (102) observed dense grouping of RBCs and obstruction-induced flow reversal, as also observed here.

Previous studies have also identified cell lingering near the bifurcation apex as a key mechanism dictating partitioning. Rashidi et al. found that reduced lingering of stiffer RBCs causes more disproportionately regular partitioning and that increased lingering of deformable cells can lead to both positive and negative deviations from the Zweifach-Fung effect (41). This is explained by partial blockages of both daughter vessels by lingering cells, also observed by others (75) and in our previous works (36,37). In the present work, we too observe decreased lingering and increased disproportionality for d-RBCs’ partitioning. However, with decreasing vessel diameter d-RBCs can block a vessel even without lingering. It is here that we see less disproportionate partitioning of d-RBCs in a departure from previous works in which stiffer cells partition more disproportionately. This is also where we observe the novel concave up partitioning curve, implying an increased sensitivity of d-RBCs to flow conditions—something that Pries’ and others’ correlations (2,9,16) do not predict.

A recent work from Cheng et al. investigated partitioning of both stiffened and sickle cells (43). The authors found that sickle RBC margination upstream a bifurcation causes preferential reverse partitioning. In our current study, the WBC-induced bias and subsequent reverse partitioning is much more pronounced for d-RBCs. Together, both our results and those of Cheng et al. show the impact of cell trajectory on reverse partitioning in stiffer diseased cells with abnormal shapes.

In another in silico work, Enjalbert et al. extensively studied the effect of vessel compression on partitioning (42). Our adhered leukocyte model near the bifurcation similarly blocks the upstream vessel. Together, both results show a constriction-induced abnormality in partitioning that is both hematocrit-dependent and persists even as the blockage is placed far upstream. While Enjalbert et al. predict no difference between h- and stiffer cells, and very little effect from an asymmetric constriction, in the present study the starkest difference between h- and d-RBCs arises in the presence of the adhered WBC which causes an asymmetric constriction.

The current study did not consider WBC-RBC receptor-ligand bridging. While WBC-endothelium adhesion is common, mechanically strong, and can withstand fluid shear, RBC-WBC bridging is not common under normal conditions. Sickle RBCs can bind to an adhered WBC, and diabetic RBCs can bind to the endothelium (129,130). However, information on WBC-RBC binding in diabetes or normal conditions is apparently unavailable. Also, RBCs encounter very high shear rate as they flow through the WBC-vessel gap, which makes their binding unlikely.

To our knowledge, this work is the first to consider the effect of adhered WBCs on partitioning of RBCs under altered shape and deformability. The physiological significance of these findings is immense. The severe impact of the WBC on RBC partitioning predicted here suggests very different RBC distribution in the microcirculation upon firm adhesion of WBCs, likely altering the tissue oxygenation. Similarly, the large difference between h- and d-RBC partitioning in the presence of a WBC found here suggests very different microvascular hematocrit distributions and oxygen delivery to surrounding tissue under healthy and diabetic conditions. The increased clumping of d-RBCs after traversing a WBC may provide conditions amenable to RBC aggregation which also has adverse downstream effects, such as impaired oxygen delivery and altered shear stress on endothelial cells. Similar downstream effects can also be caused by the WBC-induced flow reversal predicted here. Altered RBC distribution is also expected to result in different hemodynamic forces, and hence, a number of physiological functions including vasoconstriction, under healthy and diabetic conditions. Another future development could be an “effective partitioning” model for diabetic RBCs, in a similar spirit to the study of effective rheology of blood flow with rigidified cells (131). Reduced-order models as discussed here exist to predict healthy RBC partitioning (16,42) and can be extended to abnormal RBCs, keeping in mind that the finite RBC size effect and RBC-bifurcation geometry interaction affect partitioning.

Significance.

Diabetes is becoming a global epidemic with more than half a billion people diagnosed in 2024. It is a major risk factor in cardiovascular, neurodegenerative, renal, and retinal diseases. At the microvascular level, diabetes can cause morphological and mechanical changes in red blood cells and increased frequency of leukocyte adhesion to the vascular wall. The impact of such changes on the distribution of red cells in the microcirculation is not fully known. Here we show that an adhered WBC severely affects the partitioning of red cells, and more so of diabetic red cells, at vascular bifurcations. The findings suggest very different distribution of red cells, and hence oxygen delivery as well as hemodynamic forces, in the microcirculation under healthy and diabetic conditions.

Data Availability Statement

Data will be shared upon reasonable request.