Impact of the membrane viscosity on the tank-treading behavior of red blood cells

Numerical simulations supported by previously published experimental data are used to compare the impact of the internal fluid viscosity and the membrane viscosity on a tank-treading red blood cell. The method used is based on a continuum framework both for the fluid and the membrane, their interaction being ensured thanks to the immersed boundary method. The finite-strain model implemented to account for membrane viscosity assumes a free energy form that leads to an additive decomposition of the equilibrium and nonequilibrium stresses. It also assumes that the stress tensor can be separated into a deviatoric part and a hydrostatic part, which are independent. Only the deviatoric part is accounted for in this study. Both the viscosities of the cytoplasm and of the membrane yield a decrease of the tank-treading frequency, with only moderate changes in the deformation of the red blood cell. However, it is shown that the values of tank-treading frequency from the literature cannot be explained by the internal fluid viscosity only. On the contrary, adding the membrane dissipation produces a good agreement with experimental results when using acceptable values of the internal fluid and membrane viscosities. In addition, this study proposes a direct inference of the value of the membrane viscosity as a function of the shear rate from the comparison between simulations and experiments and confirms experimental results that highlighted the shear-thinning behavior of the red blood cell membrane.

Figure 1

Rheological model of the behavior law used in this study. The curved springs represent the Skalak hyperelastic law and the straight spring represents the Hooke law. (a) Viscoelastic rheological model for the deviatoric part and (b) purely hyperelastic model for the hydrostatic part.

Figure 2

Representation of a capsule under shear flow. (a) Undeformed capsule at $t=0$; (b) deformed capsule at $t>0$ due to the shear flow, enabling the measurement of $D_{\mathrm{caps}}=\frac{L-l}{L+l}$.

Figure 3

Deformation index as a function of nondimensional time $\dot{\gamma }t$ for spherical capsules under shear with different ${\mathrm{Bq}}_{\mathrm{ext}}$, at $\mathrm{Ca}=0.6$ and $\lambda =1$. Comparison of present results (yales2bio) with those from Li and Zhang [38] and Guglietta et al. [17].

Figure 4

Representation of the tank-treading behavior. (a) Side view, enabling the measurement of the inclination angle $\theta$. Translating walls are located at the top and bottom boundaries. (b) Top view, for the measurement of the deformation index, $D=\frac{L_p-W}{L_p+W}$.

Figure 5

Impact of the internal viscosity and the membrane viscosity on the tank-treading characteristics of a single RBC in shear flow as a function of Ca. (a) Inclination of the RBC, (b) deformation index, and (c) the nondimensional frequency. Only chosen data are represented on (b) to improve clarity.

Figure 6

RBC shape comparison at $\mathrm{Ca}=2$ for (a) L1B0 and (b) L02B3, corresponding to $\lambda =1$, ${\mathrm{Bq}}_{\mathrm{int}}=0$ and $\lambda =0.2$, ${\mathrm{Bq}}_{\mathrm{int}}=3$, respectively. Corresponding values of $D$ are 0.48 for L1B0 and 0.46 for L02B3. Corresponding $f_{TT}^{*}$ is about 0.38 for both. Shape differences can mostly be seen on the extremities of the RBC. Different combinations of $\lambda$ and ${\mathrm{Bq}}_{\mathrm{int}}$ yield almost identical cells, but for a given value of Ca only.

Figure 7

Comparison between computational and experimental results. (a) Impact of the internal fluid viscosity on tank-treading frequencies for an external viscosity of ${\mu }_{\mathrm{ext}}=53.9$ cP. (b),(c) Impact of the membrane viscosity on frequencies for two different external viscosities, (b) ${\mu }_{\mathrm{ext}}=53.9$ cP and (c) ${\mu }_{\mathrm{ext}}=12.9$ cP. Error bars represent two times the standard deviation observed experimentally.

Figure 8

Comparison between the experimental frequencies [27] and the simulation frequencies obtained from the inferred membrane viscosity values. (a) Comparison with data at ${\mu }_{\mathrm{ext}}=53.9$ cP, (b) comparison with data at ${\mu }_{\mathrm{ext}}=12.9$ cP. (c) Inferred shear-thinning behavior of the membrane viscosity; values from Tran-Son-Tay [35] are included for comparison. Bars in (a) and (b) represent the standard deviation of the experimental data. Bars in (c) represent the range of values of the membrane viscosity in order to reproduce the dispersion of the experimental frequencies.