# Numerical Investigation of the Effects of Red Blood Cell Cytoplasmic Viscosity Contrasts on Single Cell and Bulk Transport Behaviour

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Wheeler Test

#### 3.2. Cross-Stream Migration in Straight Circular Channel Flow

#### 3.3. Periodic Cell Stretching

#### 3.4. Platelet-RBC Collisions

#### 3.5. Many-Cell Experiments

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Numerical cell meshes used in the simulations: (

**Left**) a red blood cell; and (

**Right**) a platelet.

**Figure 2.**Visualisation of the lattice nodes in presence of an RBC. The relaxation parameters of the inner nodes are increased to represent the viscosity contrast. The green coloured nodes are those that are determined to be inside the RBC by our algorithm and are assigned ${\tau}_{\mathrm{int}}$.

**Figure 3.**Wheeler test done with and without viscosity contrast and compared to experimental values reported by Yao et al. [46]. The shaded regions represent the uncertainty in the simulation results arising from the mesh discretisation.

**Figure 4.**(

**Left**) Migration of two RBCs starting at $5\phantom{\rule{4pt}{0ex}}$$\mathsf{\mu}$m and $9\phantom{\rule{4pt}{0ex}}$$\mathsf{\mu}$m away from the channel wall, respectively. (

**Right**) Non-dimensional trajectories and their linear fit to approximate Olla’s prediction (Equation (4)).

**Figure 5.**Stretching results with a period of 0.35 s: (

**Top left**) The elongation of the longest axis of the RBC ($\Delta L$) over time. (

**Bottom left**) The magnitude of the force applied on opposing membrane areas over time. (

**Right**) The applied force magnitude versus the corresponding elongation of the cell.

**Figure 6.**Stretching results with a period of 10 ms: (

**Top left**) The elongation of the longest axis of the RBC ($\Delta L$) over time. (

**Bottom left**) The magnitude of the force applied on opposing membrane areas over time. (

**Right**) The applied force magnitude versus the corresponding elongation of the cell.

**Figure 7.**Average y-position of the platelet cell centre for different shear rates after a single collision event with a red blood cell. The shaded regions represent the variations of the trajectories due to the different incoming cell orientations: (

**Left**) $\dot{\gamma}=4000$ s${}^{-1}$; and (

**Right**) $\dot{\gamma}=400$ s${}^{-1}$.

**Figure 8.**Average y-position of the platelet cell centre for different shear rates after repeated collisions with a red blood cell: (

**Left**) $\dot{\gamma}=4000$ s${}^{-1}$; and (

**Right**) $\dot{\gamma}=400$ s${}^{-1}$.

**Figure 9.**The average platelet displacement after a collision with a RBC depending on the initial collision distance between the cells, h: (

**Left**) $\dot{\gamma}=4000$ s${}^{-1}$; and (

**Right**) $\dot{\gamma}=400$ s${}^{-1}$.

**Figure 10.**${E}_{yy}$ component of the rate of strain tensor just before a collision event in bulk shear rate of $1000\phantom{\rule{0.166667em}{0ex}}$ s${}^{-1}$: (

**Left**) with viscosity contrast; and (

**Right**) without viscosity contrast. Note that the elongational strain is higher in the case of added viscosity contrast.

**Figure 11.**Side- and front-view visualisation of a circular channel flow simulation with a discharge haematocrit of ${H}_{d}=33\%$.

**Figure 12.**(

**Top left**) The relative apparent viscosity of blood in the straight circular channel flow simulation with a discharge haematocrit of $20\%$ and a channel diameter of 70 $\mathsf{\mu}$m. (

**Bottom left**) The relative apparent viscosity of blood in the straight circular channel flow simulation with a discharge haematocrit of $28\%$ and a channel diameter of 70 $\mathsf{\mu}$m. (

**Bottom right**) The relative apparent viscosity of blood in the straight circular channel flow simulation with a discharge haematocrit of $33\%$ and a channel diameter of 70 $\mathsf{\mu}$m. (

**Top right**) The variation of the relative apparent viscosity with respect to increasing Reynolds number for a straight circular channel flow with a discharge haematocrit of $20\%$, with and without the use of a viscosity contrast between internal and external fluids.

**Figure 13.**The haematocrit distribution in a straight circular channel flow simulation with a discharge haematocrit of $20\%$, 142 ms after the start of the simulation. $r/R$ denotes the ratio of the distance to the centre of the channel with the radius of the channel. The shaded regions indicate the standard deviations of the haematocrit over the previous 30 ms.

**Figure 14.**Evolution of the CFL-width over time in the performed pressure driven channel flow simulations with a tube diameter of D = 70 $\mathsf{\mu}$m and Re = 1.3 for different haematocrit values. The dashed black lines indicate values for this tube diameter from [25]

**Table 1.**Fitting parameters for the non-dimensional data represented in the right image of Figure 4.

${\mathit{z}}_{0}$ | $3\mathit{U}$ | b | ${\mathit{R}}^{2}$-Value | ${\mathit{UR}}_{\mathit{eff}}^{3}$ | $\mathsf{\Lambda}$ |
---|---|---|---|---|---|

5.000 | 0.328 | −0.143 | 0.990 | 10.627 | 1 |

9.000 | 0.389 | 27.845 | 0.996 | 12.6036 | 1 |

5.000 | 0.215 | 1.997 | 0.991 | 6.966 | 5 |

9.000 | 0.265 | 29.580 | 0.998 | 8.586 | 5 |

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**MDPI and ACS Style**

De Haan, M.; Zavodszky, G.; Azizi, V.; Hoekstra, A.G.
Numerical Investigation of the Effects of Red Blood Cell Cytoplasmic Viscosity Contrasts on Single Cell and Bulk Transport Behaviour. *Appl. Sci.* **2018**, *8*, 1616.
https://doi.org/10.3390/app8091616

**AMA Style**

De Haan M, Zavodszky G, Azizi V, Hoekstra AG.
Numerical Investigation of the Effects of Red Blood Cell Cytoplasmic Viscosity Contrasts on Single Cell and Bulk Transport Behaviour. *Applied Sciences*. 2018; 8(9):1616.
https://doi.org/10.3390/app8091616

**Chicago/Turabian Style**

De Haan, Mike, Gabor Zavodszky, Victor Azizi, and Alfons G. Hoekstra.
2018. "Numerical Investigation of the Effects of Red Blood Cell Cytoplasmic Viscosity Contrasts on Single Cell and Bulk Transport Behaviour" *Applied Sciences* 8, no. 9: 1616.
https://doi.org/10.3390/app8091616