Cytoplasmic viscosity-dependent margination of red blood cells (RBC) for flow inside microchannels was studied using numerical simulations, and the results were verified with microfluidic experiments. Wide range of suspension volume fractions or hematocrits was considered in this study. Lattice Boltzmann method for fluid-phase coupled with spectrin-link method for RBC membrane deformation was used for accurate analysis of cell margination. RBC margination behavior shows strong dependence on the internal viscosity of the RBCs. At equilibrium, RBCs with higher internal viscosity marginate closer to the channel wall and the RBCs with normal internal viscosity migrate to the central core of the channel. Same margination pattern has been verified through experiments conducted with straight channel microfluidic devices. Segregation between RBCs of different internal viscosity is enhanced as the shear rate and the hematocrit increases. Stronger separation between normal RBCs and RBCs with high internal viscosity is obtained as the width of a high aspect ratio channel is reduced. Overall, the margination behavior of RBCs with different internal viscosities resembles with the margination behavior of RBCs with different levels of deformability. Observations from this work will be useful in designing microfluidic devices for separating the subpopulations of RBCs with different levels of deformability that appear in many hematologic diseases such as sickle cell disease (SCD), malaria, or cancer.

## Introduction

Red blood cells (RBCs) are often the target of hematologic diseases such as sickle cell disease (SCD), malaria, spherocytosis, and some types of cancers, which alter biomechanical properties such as morphology, size, and stiffness of red blood cells. Enriching RBC subpopulations based on their biomechanical properties have the potential to lead to better assessment of disease progression and treatment of these diseases. Current techniques for diseased cell sample enrichment are time consuming, expensive, and need well-trained professionals [13]. Microfluidic cell enrichment devices are currently the subject of considerable interest due to their low cost, high throughput, easy operation, and the potential to provide enrichment within the physiological flow condition. The main motivation of this work came from the need to design a microfluidic device that can separate RBC subpopulations based on their deformability. Understanding of deformability-dependent RBCs migration in a straight microchannel is a key to designing such a device.

In one of the earliest studies on lateral migration of particles in suspension flow, Segré and Silberberg [4,5] discovered that particles in tube flow migrate to circular equilibrium position at approximately 0.6 tube radii. A particle in a Poiseuille flow, as shown in Fig. S1(a) of the supplementary document which is available under “Supplemental Data” tab for this paper on the ASME Digital Collection, experiences two opposing forces. One is $FS$ (Fig. S1(b)), a force acting away from the centerline and toward the wall, is created due to the nonlinear velocity gradient (hence, shear gradient) and to the negative slip velocity [2,68]. The other force, known as wall force $FW$ [7,915], acting in the opposing direction to $FS$ (shown in supplementary Fig. S1(c)), is created due to the presence of the wall. The equilibrium lateral position of particles, as observed by Segré and Silberberg [4,5], is a result of the balance between the two opposing forces,$FS$ and $FW$.

Several authors [7,1622] have studied migration of deformable particles in noninertial flow (flow with Reynolds number $Rec≤1$). Mortazavi and Tryggvason [7] reported that in the range of $Rec≤1$, migration of droplets depends strongly on the ratio of internal viscosity of the droplets to the viscosity of the suspending fluid ($λ$) and, a droplet with $λ≥0.125$ moves away from the centerline and somewhat stabilizes at an equilibrium location between the wall and the centerline. These findings are in agreement with the results of experimental study of Hiller and Kowalewski [22] and the numerical studies of Zhou and Pozrikidis [20] and of Doddi and Bagchi [17]. Several authors [2,7,2326] have studied migration of deformable particles in inertial flow (flow with $Rec>1$) and have found that the more deformable particles tend to concentrate around the central region of the channel, and the less deformable ones get dispersed in span-wise direction.

Goldsmith and Marlow [27] observed that normally deformable individual RBCs migrated to locations closer to the channel center as compared to the stiffer RBCs. In their numerical study, Dupin et al. [28] confirmed the existence of deformed tumbling, of undeformed tumbling and of tank treading motion (terms explained in the supplementary document which is available under “Supplemental Data” tab for this paper on the ASME Digital Collection) of RBCs that were also observed by Goldsmith and Marlow [27,29]. Using lattice Boltzmann method in conjunction with spectrin-link (SL) method, Reasor et al. [30,31] reproduced the parachuting behavior of individual RBCs (RBC deforming to parachute-like shape) in a capillary flow and the shear thinning behavior of blood that matched the results of experimental study of Tsukada et al. [32]. More recently, Mehrabadi et al. [3335] studied platelet margination for a mixture of platelet and RBCs and found that platelet margination is mainly driven by RBC-enhanced diffusion (RESID) of platelets toward the RBC-free layers next to the channel walls. A scaling relation was developed showing margination length ($LD$) to vary cubically with channel height.

This paper will report on the margination behavior of deformable RBCs based on their cytoplasmic viscosity. The results are presented for two broad classes of simulations: (1) individual RBCs of different internal viscosities where there were no cell-to-cell collisions, and (2) moderately dense ($φ=10%$) to dense ($φ=30%$) suspensions of RBCs where cell-to-cell collisions were significant. The second group of simulations focused on the impact of channel cross section size, of the shear rate ($γ˙$) and of the volume fraction ($φ$) on the margination behavior of RBCs with different internal viscosity in straight rectangular microchannels. Preliminary experiments were conducted to validate the results obtained from the simulations. Information about the numerical methods used in this work is provided in the supplementary document's Methodology section.

## Simulation Setup

Flows of RBC suspensions between two parallel plates with spacing $W$ have been simulated for multiple values of the parameters $W$, Rec, and $φ$. Table S1 in supplementary document, shows a list of all the cases that were considered. Figure 1 shows the geometrical configuration of the setup. Flow is driven by a constant body force in the axial direction X. Periodic boundary condition is applied in the length-wise direction (X-axis) and in the height-wise direction (Y-axis) and no-slip in the span-wise direction (Z axis). The setup resembles the Hele-Shaw flow. As there are walls only in one axis orientation (Z), the shear gradient and the wall force will be acting only along the Z axis, and hence, the cell migration will take place only along the Z orientation. In previous studies [2,36], it was shown that if the channel aspect ratio is greater than 2 $((H/W)>2)$, the cell migration will be dominant only along the width of the channel (Z axis). Therefore, the simulation setup used in this work is sufficient to mimic the flow inside a straight microchannel with a high aspect ratio.

Fig. 1
Fig. 1
Close modal

Internal viscosity of normal RBC was set to five times higher $(λ=(νin/νplasma)=5)$ than that of plasma ($μ=1.2×10−3 N s/m2$) [30,35]. Two different types of RBCs were used for the simulations, normal RBCs and RBCs with higher internal viscosity. For this work, the former will be referenced as normal RBCs, and the latter will be referenced as sickle RBCs. The term sickle is used to refer to the high internal viscosity RBCs, because the internal viscosity value used matches that of sickle RBCs [37]. For this study, the only difference between normal and sickle RBCs will be their internal viscosity value, as cytoplasmic viscosity has been shown to be major determinant of RBC deformability [37,38]. Initial shapes of both types of RBCs are same. Studying morphological changes that are observed for severely SCD affected RBCs is beyond the scope of this work. Prescribed mechanical properties of the two different cell types are summarized in supplementary Table S2. All these values were chosen from previous works [30,31,35,37,39].

To mimic the dilute suspension cases, the migration of single RBCs in a microchannel was studied to make sure that there were no cell-to-cell collisions. Eight single-cell simulations were run: four for $Rec=4$ and four for $Rec=8$. The four $Rec$ cases were based on two cell types (normal and sickle) and on two initial positions (near-center and near-wall) for each of the cell types. This was done to investigate the effect of the shear rate and the initial cell locations upon the internal viscosity-dependent migration pattern of the RBCs in the absence of cell-to-cell collisions. Figure 2(a) and supplementary Fig. S3, show the initial locations of all cells. For the moderate ($φ=10%$) to dense suspension ($φ=30%$) studies, appropriate number of cells (based on the value of $φ$) were seeded randomly in the whole domain at the beginning, with half of these being normal and the other half being sickle. Exploratory experiments with straight channel microfluidic devices showed that for volume fraction above 20%, the microchannels clogged very frequently. Hence, we determined that running simulations beyond 30% was not necessary and thus were not done.

Fig. 2
Fig. 2
Close modal
For the dilute suspension cases, nondimensional lateral displacement of individual cells was observed. Lateral displacement is defined as
$D(t)=z(t)−W2$
(1)

where $z(t)$ is the individual cell location at time $t$. So, the nondimensional quantity is $((D(t))/(W/2))$.

For the dense suspension cases, $((D¯(t))/(W/2))$ is used as the measure of margination, where average lateral displacement of cells from centerline $D¯(t)$ is defined as
$D¯=|Z(t)−W2¯|$
(2)

The difference between the average trajectories of stiff RBCs and of normal RBCs is given by $((D¯sickle−D¯normal)/(W/2))$ and is used as a measure of strength of separation between stiff RBCs and normal RBCs.

## Results

### Effects of Internal Viscosity on Red Blood Cells Deformation.

Earlier studies showed that vesicles [7,17,20,22] and RBCs [37] with higher internal viscosity are less deformable compared to vesicles and RBCs having lower internal viscosity. To investigate the deformation characteristics of RBCs based on the internal viscosity, benchmark cases from previous works [30,32] that studied the parachuting behavior of RBCs in microcapillary were simulated. Figure 3(a) shows snapshots of parachuting for the two cell types at different time steps and the sickle RBC demonstrated less deformation when compared to the normal RBC. Also, the normal RBC moved further in the channel when compared to the sickle RBC within the same time frame due to less resistance arising from the smaller frontal area. Variations of the deformation index $(Γ=(L/D))$, a measure of deformability [30,32], against the flow velocity for the two cell types are shown in Fig. 3(b). This shows that sickle RBCs are less deformable when compared to the normal RBCs. For normal RBCs, the obtained $Γ$ values match well with those from the previous studies [30,32]. To the best of our knowledge, there is no experimental data on parachuting experiments for RBCs with internal viscosity that is four times higher than for the normal RBCs. However, our simulation code has produced a different set of results that are very closely matching with the results of a set of exploratory experiments conducted by us. These results are presented in the Experimental Validation section. Hence, we argue that the $Γ$ values obtained for high internal viscosity RBCs are reliable.

Fig. 3
Fig. 3
Close modal

### Dilute Suspension.

Snapshots of the initial time-step and at a time-step after the RBCs have reached their equilibrium positions are shown in Fig. 2 and supplementary Fig. S3, for the two Rec cases. Figure 4 shows the nondimensional trajectories of the individual cells. Both normal and sickle RBCs laterally migrated from the initial locations to the equilibrium positions somewhere between the central plane $((D/(W/2))=0)$ and the wall $((D/(W/2))=1)$. The sickle RBC stabilized closer to the wall, and the normal RBC stabilized closer to the central plane of the channel even when both had the same initial transverse locations (i.e., initial $| (D/(W/2)) |$ is same for both types). These observations are similar to the earlier findings for deformable particles [2,7,2326] and RBCs [27]. Comparing Fig. 4(a) with Fig. 4(b), it is clear that both normal and sickle RBCs use shorter channel length to reach equilibrium at higher $Rec$. For the range of $Rec$ values considered for this work, the equilibrium positions for both cell types moved toward the wall as the $Rec$ increased. As explained in the Introduction section, in the absence of any cell-to-cell collisions, a deformable particle migrates laterally to an equilibrium location where $FS$ and $FW$ balances each other. Increasing the $Rec$ resulted in the increase of $FS$ values, which pushed the RBCs further toward the wall and the individual RBCs ended up in equilibrium locations closer to the walls.

Fig. 4
Fig. 4
Close modal

From Fig. 4, it appears that in the absence of any cell-to-cell collisions, equilibrium locations of sickle RBCs are almost overlapping despite the initial seeding locations. However, the normal RBCs have a wider equilibrium zone around the centerline. This variation is due to the differences in the deformability between normal and sickle RBCs. In the absence of any cell-to-cell collisions, cell deformability is the only factor that influences equilibrium locations for RBCs for a particular $Rec$. Cells with a lower deformability will cause a lower disturbance in the immediate area and hence will migrate to nearly the same location at equilibrium regardless of where these were seeded initially. Normal RBCs can deform more and can assume a wider range of shapes and thus create higher disturbances in fluid around the cells. Hence, normal RBCs originally seeded at different locations will migrate to different equilibrium locations giving rise to a wider equilibrium zone. Less deformable RBCs will have a narrower equilibrium zone compared to the more deformable RBCs and depending on the deformability of the RBCs the equilibrium zone may appear to be a single plane. Also, even after reaching equilibrium location/zone, RBCs do not follow smooth straight path and rather keep on oscillating with small amplitude due to continuous deformation. Hence, trajectories of some RBCs may not appear as smooth as others even after reaching equilibrium.

### Semidense to Dense Suspension.

The effects of channel cross section sizes, of shear rates and of hematocrit on internal viscosity-dependent margination of RBCs were studied. In addition, the effects of cell mixture type and of the combination of shear rates and hematocrits are presented in the supplementary document.

#### Effects of Red Blood Cells Internal Viscosity.

To study the effect of RBC internal viscosity on margination of RBCs when cell-to-cell collisions are significant, the case of RBCs flowing through a $60 μm×60 μm×30 μm$ channel with $Rec=10$ at hematocrit of $φ=10%$ was considered. Figure 5 and supplementary Fig. S4 showed that, at equilibrium the normal RBCs stay closer to the centerline, whereas the sickle RBCs diffuse to the outer region next to the walls following the RESID mechanism [35]. The observed $λ$-dependent margination pattern for dense RBC suspension is the same as that observed for dilute suspension in this work and earlier works [2,7,1622,2427].

Fig. 5
Fig. 5
Close modal

#### Cell-Free Layer.

The snapshots presented in the supplementary document (Figs. S4, S5, S9, and S11) show that for all the cases in the semidense to dense category; cell-free layers are formed next to the channel walls which resembles the observations from previous works [30,31,35]. Mehrabadi et al. [35] in their study of RBC and platelet mixture observed that platelets modeled as oblate solid particles diffuse to the RBC-free layer, and hence, it is an important parameter for the study of platelet margination. This study focused on the margination of RBCs and since both RBC types are deformable, there was no migration of sickle or less deformable RBCs to the cell-free layers. From Fig. 6, the nondimensional cell-free layer thickness $((δ/aRBC))$ was observed to follow a power-law relation with the RBC capillary number $CaG$; this resembled the observations reported in previous studies [35,40].

Fig. 6
Fig. 6
Close modal

#### Effects of Confinement Ratio.

Confinement ratio, a nondimensional measure of channel cross section, is defined as $(2 aRBC/W)$. We studied RBC margination in channels with different confinement ratios. From the average lateral displacement plot in Fig. 7(a) and the plot of $( (D¯sickle−D¯normal)/(W/2))$ in Fig. 7(b), the segregation between the cell types becomes stronger for channels with smaller cross section regardless of what the hematocrit value may be (compare Fig. 7 and supplementary Fig. S6). This finding is in agreement with the observations of Mehrabadi et al. [35], and it implies that, channels with smaller cross section will provide a stronger enrichment of cell types. However, due to practical factors such as expected throughput, pressure withstanding capability of the device and channel clogging, there are limits on how small the cross section can be. We checked the applicability of the scaling law proposed by Mehrabadi et al. [35] for $λ$-dependent margination of RBCs. The results are presented in the Scaling Law Validation section of the supplementary document.

Fig. 7
Fig. 7
Close modal

#### Effects of Shear Rate.

Effects of the shear rate ($γ˙w$) on margination of RBCs has been studied. Figure 8 and supplementary Fig. S8, show that the segregation between normal and sickle RBCs becomes stronger as the shear rate (and hence $Rec$) increases regardless of what the hematocrit value and channel width may be. From the figures, it is observed that mainly the trajectories of sickle RBCs get influenced with shear rate, while those of the normal RBCs are not as strongly affected. Snapshots of the relevant simulation cases can be found in supplementary Fig. S9. Shear rate affects the margination of deformable sickle RBCs more than the margination of platelets that were modeled as oblate solid particles by Mehrabadi et al. [35]. We propose that deformation of sickle RBCs is the primary reason behind the observed enhanced margination of the sickle RBCs under higher shear rate. Earlier studies [15,41,42] showed that rigid particles do not migrate laterally in noninertial flow, whereas deformable droplets always migrate laterally regardless of whether the flow is inertial or not [7,1622].

Fig. 8
Fig. 8
Close modal

#### Effects of Volume Fraction.

Figure 9 and supplementary Fig. S10, shows that the segregation between normal and sickle RBCs is enhanced as the hematocrit is increased regardless of what the shear rate may be. This enhancement of separation is due to the increased cell-to-cell collisions at higher hematocrit. Supplementary Fig. S11, shows snapshots of cells in a channel cross section view at initial time-step and at the equilibrium time-step for the relevant cases.

Fig. 9
Fig. 9
Close modal

## Experimental Validation

A straight channel microfluidic device was designed and fabricated using insights obtained from the simulations. Preliminary experiments were conducted to verify the margination phenomenon observed in the simulations, and the results are reported in this section. The device schematic drawing is shown in supplementary Fig. S13. A metric known as enrichment ratio (ER) [1,2] is used as a measure of enrichment efficiency and is defined for this work as
$ERstiff = (% stiff% normal)outlet(% stiff% normal)inlet$
(3)

For stiff RBCs, it is the ratio of the number of stiff RBCs to the number of normal RBCs at the outlet divided by the same quantity at the inlet. Experimentally, these ratios for inlet and each of the outlets are measured using a flow cytometer. For the device shown in Fig. S13, outlet2 is expected to have more of the stiffer RBCs compared to outlet1. Details about the experimental setup are provided in the supplementary document.

## Results

After conducting 15 repetitions of the experiment with different parameters as described in Experimental Setup section of the supplementary document, we calculated the average $ERstiff$ value for outlet1 and for outlet2, which were $ERstiffoutlet2=1.57± 0.05$ and $ERstiffoutlet1=0.62±0.02$ respectively. These $ERstiff$ values indicated that, for outlet2, stiff RBCs were dominant, and for outlet1, normal RBCs dominant were dominant. This confirmed that the deformability-based cell enrichment worked.

A validation simulation was run for the channel size of $40 μm×40 μm×30μm$ with $φ=10%$ and $Rec=5$, matching all parameters with those from the experiments. $ERstiffoutlet2$ was calculated to be 1.60, which agrees well with the experimental results. $ERstiffoutlet2$ for simulation was computed by calculating population distribution of sickle RBCs and of normal RBCs as percentages of the total cell population in the different outlet segments of the channel cross section as shown in Fig. 10. Additional simulations and experiments were conducted for $Rec=10$ to study the effects of the shear rate on the $ERstiff$. The values are plotted in Fig. 11. The error bars reflect a 95% confidence interval for a population mean when using a normal distribution. The simulation values were slightly higher than the experimental values. These differences could be attributed to the natural variations in the stiffness of the real RBCs, to the minor variations in the channel dimensions due to fabrication errors and to errors caused by disturbances from the surrounding environments during flow experiments.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

## Conclusion

In this study, we looked at transverse margination of RBCs with different internal viscosity for a wide range of cell volume fractions or hematocrits. In the absence of cell-to-cell collisions, deformable RBCs migrated transversely to various equilibrium positions. At equilibrium, deformable normal RBCs stayed closer to the central core of the channel while the RBCs with higher internal viscosity stayed closer to the walls. For the high hematocrit cases, cell-to-cell collisions played a significant role in margination in addition to the internal viscosity of the cells. Stronger separation between normal RBCs and RBCs with high internal viscosity was obtained as the width of a high aspect ratio channel was reduced. Normal RBCs tended within a very short time scale to gather in a region closer to the channel center. Through interactions with the normal RBCs, the RBCs with higher internal viscosity tended to diffuse to the outer regions closer to the walls following the RESID mechanism [35]. Segregation between the two cell types became stronger with higher flow Reynolds number or shear rate and with higher hematocrit. Deformable RBCs with different internal viscosities also followed the margination scaling law as proposed by Mehrabadi et al. [35]. The average enrichment ratios that were obtained from the preliminary experiments agree well with those that were obtained from the simulations. The results from this study will aid in the design of microfluidic devices for separating RBC subpopulations based on the RBC deformability.

## Acknowledgment

Simulations for this study were run on National Science Foundation's XSEDE and Georgia Institute of Technology's PACE supercomputing environment.

## Funding Data

• National Science Foundation (Grant Nos. 1229954 and 1441388).

## Nomenclature

• $aRBC$ =

•
• CaG =

capillary number based on shear modulus

•
• $d$ =

diameter of individual deformed RBC

•
• $D$ =

lateral displacement

•
• ER =

enrichment ratio

•
• $FS$ =

force due to nonlinear shear gradient

•
• $FW$ =

wall force

•
• $G$ =

shear modulus

•
• $H$ =

channel height

•
• $l$ =

length of individual deformed RBC

•
• $L$ =

channel length

•
• $LD$ =

margination length

•
• PD =

population distribution

•
• $Q$ =

flow rate

•
• $Re$ =

Reynolds number

•
• $Rec$ =

channel Reynolds number

•
• $ReRBC$ =

Reynolds number based on RBC dimension

•
• RBC =

red blood cell

•
• SCD =

sickle cell disease

•
• $t$ =

time

•
• $VRBC$ =

RBC sphere volume

•
• $W$ =

channel width

•
• $x¯$ =

average distance travelled by cells in flow direction

•
• $z(t)$ =

instantaneous lateral cell location

•
• $Γ$ =

deformation index

•
• $γ˙$ =

shear rate

•
• $γ˙w$ =

shear rate

•
• $λ$ =

ratio of internal viscosity to the viscosity of the suspending fluid

•
• $μ$ =

dynamic viscosity

•
• $ν$ =

kinematic viscosity

•
• $φ$ =

volume fraction

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