# Deformation of a Red Blood Cell in a Narrow Rectangular Microchannel

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flow and RBC Model

#### 2.2. Numerical Simulation

## 3. Results

#### 3.1. Deformation of a Translating RBC in a Narrow Rectangular Microchannel

#### 3.2. Effects of Perturbations on Stable Membrane Configuration

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RBC | Red blood cell |

LBM | Lattice–Boltzmann method |

FEM | Finite element method |

IBM | Immersed boundary method |

GPU | Graphics processing unit |

## Appendix A. Sample Preparation and Observation

**Figure A1.**Detailed channel geometry. Magnified view of the channels with the cross-sections of (

**a**) 10 $\mathsf{\mu}$m × 3.5 $\mathsf{\mu}$m and (

**b**) 3.5 $\mathsf{\mu}$m × 10 $\mathsf{\mu}$m, which were used in Figure A2 and Figure A1, respectively. (

**c**) Schematic view of the whole channel, whose stream-wise length was 8000 $\mathsf{\mu}$m. (

**d**) Representative images of a flowing RBC in a microfluidic device. Flow direction is from left to right.

**Figure A2.**Representative images of the flowing RBC at frame numbers 0 (top), 20, 40 60, and 80 (bottom), respectively, where the speed of the cell is 1200 $\mathsf{\mu}$m/s (see also Video S1). A representative numerical result of a stable slipper-shaped RBC subjected to $Ca$ = 0.01 (calculated centroid velocity ${V}_{c}\sim $ 25 $\mathsf{\mu}$m/s) is also displayed (see also Video S2).

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**Figure 1.**Computational domain to reproduce a translating red blood cell (RBC) in the narrow rectangular microchannel. The domain mimicked a microfluidic device as shown in Figure A1. The domain cross-section was 10 $\mathsf{\mu}$m × 3.5 $\mathsf{\mu}$m along the wall-normal and span-wise directions, respectively, and the stream-wise distance was set to be 50 $\mathsf{\mu}$m. Flow direction is from left to right.

**Figure 2.**(

**a**) Typical snapshots of a deformed RBC subjected to $Ca$ = 0.15 at the initial state (left) and steady state (right). Two views, from the span-wise and stream-wise directions, are shown above and below, respectively. The markers represent node points. (

**b**) Superposition of the fully deformed RBC projected on the x-z plane at $Ca$ = 0.15. The two lines obtained with $\Delta $x = 125 nm (black) and 62.5 nm (red), respectively. The membrane position is normalized by the reference radius a.

**Figure 3.**(

**a**) Snapshots of a fully deformed RBC for different $Ca$. (

**b**) Typical snapshots of a RBC at $Ca$ = 0.01, where the blue plane denotes the center of the x-z-plane parallel to the flow direction, dividing the cell into the volume 1 ($Vo{l}_{1}$) and volume 2 ($Vo{l}_{2}$). (

**c**) The symmetry index $I{D}_{sym}$ as a function of $Ca$. The insets represent snapshots of deformed RBCs at specific $Ca$.

**Figure 4.**(

**a**) Time history of the RBC centroid velocity (${V}_{c}$) at $Ca$ = 0.01, where ${V}_{c}$ = 24.7 $\mathsf{\mu}$m/s at $\dot{\gamma}t$ = 200 corresponding to t = 12 s. The images represent snapshots of the deformed RBC at $\dot{\gamma}t$ = 70 (t = 4.2 s) and 200 (t = 12 s), respectively. The blue line denotes the center axis of the channel. (

**b**) Time average of the RBC centroid velocity ${V}_{c}$ and total fluid velocity ${V}_{total}$ as a function of $Ca$. The velocity ${V}_{i}$ is normalized by the characteristic fluid velocity without cell ${U}^{\infty}$, where ${V}_{i}$ represents ${V}_{c}$ and ${V}_{total}$ by the index i = “c” or “total”.

**Figure 5.**(

**a**) Time average of the deformation index ${L}_{i}/{L}_{i}^{ref}$ as a function of $Ca$, where the maximum, middle, and minimum axis lengths (${L}_{max}$, ${L}_{min}$, and ${L}_{mid}$, respectively) are normalized by each reference length ${L}_{i}^{ref}$ (i.e., no flow condition), where the reference major and minor axis lengths are ${d}_{0}$ and ${t}_{0}$ (thickness), respectively. (

**b**) Averaged first and second invariants ${\mathcal{I}}_{i}$ (i = 1 and 2) as a function of $Ca$. (

**c**) Averaged maximum and isotropic tensions; ${\mathcal{T}}_{max}$ and ${\mathcal{T}}_{iso}$, respectively. These values are normalized by the shear elastic modulus ${G}_{s}$. (

**d**) The average of these tensions, ${\mathcal{T}}_{i}$, as a function of the deformation index ${L}_{min}/{t}_{0}$, which is the ratio between the minimum axis length of the deformed RBC (thickness) and the reference thickness.

**Figure 6.**Time history of the RBC centroid velocity (${V}_{c}$) at (

**a**) low $Ca$ = 5 × 10${}^{-3}$, and (

**b**) high $Ca$ = 0.5 for different initial positions along the span-wise direction ${x}_{0}$, where one RBC is initially placed at the midline of the channel (${x}_{0}$ = 0, dashed line) and the other RBC is placed two fluid meshes away from the midline (${x}_{0}$ = −2$\Delta $x, red line). The images represent snapshots of the RBC with ${x}_{0}$ = −2$\Delta $x at the indicated times (see also Videos S5 and S6). Note that ${V}_{c}$ = 12.7 $\mathsf{\mu}$m/s for $Ca$ = 5 × 10${}^{-3}$ at $\dot{\gamma}t$ = 500 (t = 60 s), and ${V}_{c}$ = 1180 $\mathsf{\mu}$m/s for $Ca$ = 0.5 at $\dot{\gamma}t$ = 500 (t = 0.6 s).

**Figure 7.**(

**a**) The symmetry index $I{D}_{sym}$ as a function of $Ca$ for different values of bending modulus ${k}_{b}$ = 3.0 × 10${}^{-20}$ J (square), 1.2 × 10${}^{-19}$ J (triangle), and 2.4 × 10${}^{-19}$ J (inverted triangle). The results obtained with a viscosity ratio of unity (i.e., $\lambda $ = 1) are also displayed (diamond). The circular dot represents the result of ${x}_{0}$ = −2$\Delta x$ at low $Ca$ (= 5 × 10${}^{-3}$) and high $Ca$ (= 0.5) and $\lambda $ = 5. These results were obtained with ${k}_{b}$ = 1.2 × 10${}^{-19}$ J. (

**b**) Time average of the RBC centroid velocity (${V}_{c}$) as a function of $Ca$ for different viscosity ratios ($\lambda $ = 1 and 5). (

**c**) Averaged maximum and isotropic tensions, ${\mathcal{T}}_{max}$ and ${\mathcal{T}}_{iso}$, respectively. (

**d**) Averaged tension ${\mathcal{T}}_{i}$ as a function of the deformation index ${L}_{min}/{t}_{0}$.

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## Share and Cite

**MDPI and ACS Style**

Takeishi, N.; Ito, H.; Kaneko, M.; Wada, S. Deformation of a Red Blood Cell in a Narrow Rectangular Microchannel. *Micromachines* **2019**, *10*, 199.
https://doi.org/10.3390/mi10030199

**AMA Style**

Takeishi N, Ito H, Kaneko M, Wada S. Deformation of a Red Blood Cell in a Narrow Rectangular Microchannel. *Micromachines*. 2019; 10(3):199.
https://doi.org/10.3390/mi10030199

**Chicago/Turabian Style**

Takeishi, Naoki, Hiroaki Ito, Makoto Kaneko, and Shigeo Wada. 2019. "Deformation of a Red Blood Cell in a Narrow Rectangular Microchannel" *Micromachines* 10, no. 3: 199.
https://doi.org/10.3390/mi10030199