# Axial and Nonaxial Migration of Red Blood Cells in a Microtube

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Flow and RBC Model

#### 2.2. Numerical Simulation

#### 2.3. Analysis

## 3. Results

#### 3.1. Effect of Capillary Number $Ca$ on RBC Shapes

#### 3.2. Effect of Viscosity Ratio $\lambda $ on RBC Shapes

## 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 |

(non-)TT motion | (non-)tank-treading motion |

## Appendix A. Deformation of a Spherical Capsule

**Figure A1.**Time-averaged Taylor parameters ${D}_{12}$ of an SK spherical capsule as a function of $Ca$ for different viscosities $\lambda $ (0.2, 1, 5, and 10); previous numerical results of Foessel et al. [48] are also displayed. The inset represents a tank-treading spherical capsule at $Ca$ = 1.0 and $\lambda $ = 1. The results were obtained with $R{e}_{p}$ = 0.2.

## Appendix B. Effect of Initial Position of RBC on Stable Deformed Shapes

**Figure A2.**Snapshots of flowing RBCs in steady state at $Ca$ = 1.2 for different initial positions ${r}_{0}$ (0, 1 $\mathsf{\mu}$m, 1.5 $\mathsf{\mu}$m, 2 $\mathsf{\mu}$m, and 3 $\mathsf{\mu}$m). The results were obtained with $Re$ = 0.2 and $\lambda $ = 5.

## Appendix C. Behavior of a Spherical Capsule in a Circular Channel

**Figure A3.**Time history of radial position of capsule centroid for different $Ca$ (0.05 and 1.2), where the insets are the snapshots of flowing spherical capsule at steady state for each $Ca$ (see also Videos S5 and S6). The results were obtained with $Re$ = 0.2 and $\lambda $ = 5.

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**Figure 1.**Simulation setup: A single RBC with radius of 4 $\mathsf{\mu}$m is placed with random orientation in a circular channel with diameter of 15 $\mathsf{\mu}$m and length of 80 $\mathsf{\mu}$m. Periodic boundary conditions are imposed on the flow (z-direction) and no-slip conditions are employed for the wall (x- and y-direction). Green dots represent material points at the initial concave node point, and blue dots at the initial edge node point.

**Figure 2.**Snapshots of flowing RBCs in steady state (${\dot{\gamma}}_{m}t$ = 800) for (

**a**,

**c**) $Ca$ = 0.05 (see also Video S1 and S3) and (

**b**,

**d**) $Ca$ = 1.2 (see also Video S2 and S4). The right side is the axial view and the left side is the lateral view; the flow direction is from left to right. For each $Ca$, the initial position of the RBC centroid is set to be (

**c**,

**d**) ${r}_{0}/R$ = 0 and (

**a**,

**b**) ${r}_{0}/R$ = 0.4. (

**e**) Time history of the radial position of the RBC centroid $r/R$ for different $Ca$ and ${r}_{0}/R$. The results were obtained with $\lambda $ = 5.

**Figure 3.**Time average of the radial position of the RBC centroid $\langle r\rangle /R$ as a function of $Ca$ for initial position ${r}_{0}/R$ = 0 (triangles) and ${r}_{0}/R$ = 0.4 (inverse triangles), where $\langle \xb7\rangle $ denotes time average. The error bars represent standard deviations on the time axis. The results at $Re$ = 10 for low $Ca$ (0.05) and high $Ca$ (1.2) are also plotted, with black squares for ${r}_{0}/R$ = 0.4 and orange circles for ${r}_{0}/R$ = 0. The results were obtained with $\lambda $ = 5.

**Figure 4.**(

**a**) Time average of the volumetric flow rate $\langle Q\rangle /{Q}^{\infty}$, (

**b**) projected area of the RBC to the cross-sectional area of the channel (x-y plane) $\langle {A}_{xy}\rangle /\left(\pi {a}^{2}\right)$, and (

**c**) powers associated with membrane deformations $\langle \delta {W}_{mem}^{*}\rangle $ as a function of $Ca$ for different initial positions ${r}_{0}/R$ (0 and 0.4). (

**d**) Replotted data of $\langle \delta {W}_{mem}^{*}\rangle $ as a function of equilibrium radial position $\langle r\rangle /R$ for different $Ca$. The error bars represent standard deviations on the time axis. The results were obtained with $\lambda $ = 5.

**Figure 5.**(

**a**) Snapshots of flowing RBCs in steady state at $Ca$ = 0.05 (bottom row) and $Ca$ = 1.2 (top row) for different $\lambda $. (

**b**) Time average of the radial position of the RBC centroid $\langle r\rangle /R$ as a function of $\lambda $ for $Ca$ = 0.05 (blue inverse triangles) and $Ca$ = 1.2 (red triangles). The results were obtained with an initial off-centered position at ${r}_{0}/R$ = 0.4.

**Figure 6.**(

**a**) Time average of the volumetric flow rate $\langle Q\rangle /{Q}^{\infty}$, (

**b**) projected area of the RBC $\langle {A}_{xy}\rangle /\left(\pi {a}^{2}\right)$, and (

**c**) powers associated with membrane deformations $\langle \delta {W}_{mem}^{*}\rangle $ as a function of $\lambda $ for different $Ca$ (0.05 and 1.2). (

**d**) Replotted data of $\langle \delta {W}_{mem}^{*}\rangle $ as a function of equilibrium radial position $\langle r\rangle /R$ for different $\lambda $. The results were obtained with an initial off-centered position ${r}_{0}/R$ = 0.4.

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**MDPI and ACS Style**

Takeishi, N.; Yamashita, H.; Omori, T.; Yokoyama, N.; Sugihara-Seki, M.
Axial and Nonaxial Migration of Red Blood Cells in a Microtube. *Micromachines* **2021**, *12*, 1162.
https://doi.org/10.3390/mi12101162

**AMA Style**

Takeishi N, Yamashita H, Omori T, Yokoyama N, Sugihara-Seki M.
Axial and Nonaxial Migration of Red Blood Cells in a Microtube. *Micromachines*. 2021; 12(10):1162.
https://doi.org/10.3390/mi12101162

**Chicago/Turabian Style**

Takeishi, Naoki, Hiroshi Yamashita, Toshihiro Omori, Naoto Yokoyama, and Masako Sugihara-Seki.
2021. "Axial and Nonaxial Migration of Red Blood Cells in a Microtube" *Micromachines* 12, no. 10: 1162.
https://doi.org/10.3390/mi12101162