Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell

A two-dimensional computer simulation model was proposed for tank-treading and tumbling motions of an elastic biconcave red blood cell (RBC) under steady shear flow. The RBC model consisted of an outer cellular membrane and an inner fluid; the membrane’s elastic properties were modeled by springs for stretch/compression and bending to consider the membrane’s natural state in a practical manner. Membrane deformation was coupled with incompressible viscous flow of the inner and outer fluids of the RBC using a particle method. The proposed simulation model was capable of reproducing tank-treading and tumbling motions of an RBC along with rotational oscillation, which is the transition between the two motions. In simulations using the same initial RBC shape with different natural states of the RBC membrane, only tank-treading motion was exhibited in the case of a uniform natural state of the membrane, and a nonuniform natural state was necessary to generate the rotational oscillation and tumbling motion. Simulation results corresponded to published data from experimental and computational studies. In the range of simulation parameters considered, the relative membrane elastic force versus fluid viscous force was $\sim 1$ at the transition when the natural state nonuniformity was taken into account in estimating the membrane elastic force. A combination of natural state nonuniformity and elastic spring constant determined that change in the RBC deformation at the transition is that from a large compressive deformation to no deformation, such as rigid body.

Figure 1

(Color online) Computer simulation model of RBC motion under simple shear flow with particle method. (a) RBC and plasma as blood components, each of which is expressed by an assembly of computed particles. (b) Spring model of elastic RBC membrane. Stretch/compression and bending springs connect particles assigned to the RBC membrane. (c) Simulation model of a single RBC under shear flow. The model consists of RBC, blood plasma and rigid walls. (d) Definition of angle ${\phi }_{\mathrm{LA}}$ for longitudinal direction of the entire RBC and ${\phi }_{\text{Mem}}$ for certain material points on the RBC membrane. Angles ${\phi }_{\mathrm{LA}}$ and ${\phi }_{\text{Mem}}$ are used to quantify tank-treading and tumbling motions of the RBC in simulation results.

Figure 2

(Color online) Simulation results of RBC motions in case of uniform natural state of the elastic RBC membrane. Line graphs show changes in angle ${\phi }_{\mathrm{LA}}/\pi$ and ${\phi }_{\text{Mem}}/\pi$ for different sets of bending capillary number ${\text{Ca}}_{\text{B}}$ and uniform reference angle ${\theta }_I^0$, $\left({\text{Ca}}_{\text{B}},{\theta }_I^0\right)=\left(5,0\right)$, (0.05, 0) and $\left(0.5,2\pi /N\right)$, as a function of normalized time $\dot{\gamma }t$. RBC shapes during tank-treading motions are illustrated for these parameter sets in the line graphs.

Figure 3

(Color online) Simulation results of RBC motions in case of nonuniform natural state of the elastic RBC membrane. The membrane’s natural state is assumed to be the biconcave shape shown in Fig. 1b. (a) Rotational oscillation of the entire RBC with tank-treading motion exhibited for ${\text{Ca}}_{\text{B}}\ge 1$. Line graphs show changes in angle ${\phi }_{\mathrm{LA}}/\pi$ and ${\phi }_{\text{Mem}}/\pi$ for ${\text{Ca}}_{\text{B}}=5$, 1.7 and 1 as a function of normalized time $\dot{\gamma }t$. The time-course of RBC shape changes is illustrated for ${\text{Ca}}_{\text{B}}=1$, in which the marker shows a certain material point on the membrane. (b) Tumbling motion of the entire RBC exhibited for ${\text{Ca}}_{\text{B}}<1$. Line graphs show changes in angle ${\phi }_{\mathrm{LA}}/\pi$ and ${\phi }_{\text{Mem}}/\pi$ for ${\text{Ca}}_{\text{B}}=0.95$, 0.5 and 0.05 as a function of normalized time $\dot{\gamma }t$. The time course of RBC shape changes is illustrated for ${\text{Ca}}_{\text{B}}=0.95$, in which the marker shows a certain material point on the membrane.

Figure 4

(Color online) Average periods $\dot{\gamma }T$ and angles $\Phi /\pi$ that represent tank-treading and tumbling motions in case of a nonuniform natural state as a function of the logarithm of bending capillary number ${\text{Ca}}_{\text{B}}$. Black lines show normalized periods $\dot{\gamma }{T}_{\mathrm{LA}}$ on angle ${\phi }_{\mathrm{LA}}$ and $\dot{\gamma }{T}_{\text{Mem}}$ on angle ${\phi }_{\text{Mem}}$ for a rotation $\left(2\pi \right)$. Red (gray) inset in lower right shows angles for rotation amplitude ${\Phi }_{\mathrm{OA}}/\pi$ and mean inclination angle ${\Phi }_{\mathrm{LA}}/\pi$ of the entire RBC.

Figure 5

Phase diagram of RBC tank-treading and tumbling motions with respect to the logarithm of bending capillary number ${\text{Ca}}_{\text{B}}$ and natural state nonuniformity $\alpha$. Open circles show the parameter sets at the transition between tank-treading and tumbling predicted by computer simulations. Dashed and dotted lines show transition lines from simple estimation of elastic force relative to the fluid force $F_{\text{Max}}^{\ast }=1.17$ and 1.33, respectively. The force $F_{\text{Max}}^{\ast }$ is estimated under the assumption of steady tank-treading motion; a detailed explanation of $F_{\text{Max}}^{\ast }$ is found in Sec. 3D. The ${\text{Ca}}_{\text{B}}$ and $\alpha$ dependency of RBC deformation at the transition is also illustrated.

Figure 6

Elastic resistance force, $F_{\text{Mem}}$, relative to fluid viscous force, $F_{\text{Fld}}$, estimated for a given membrane tank-treading motion for a prescribed biconcave shape under shear flow as a function of normalized peripheral length $L^{\ast }=L/L_{\text{P}}$. Coordinate $L$ is taken along the membrane periphery, and $L_{\text{P}}$ is the total length of this periphery. The membrane configuration at the natural state completely matches the biconcave shape at $L^{\ast }=0$ (original position), 0.5 (half rotation) and 1 (full rotation). Solid and dotted lines show $F_{\text{Mem}}/F_{\text{Fld}}$ for $\left({\text{Ca}}_{\text{B}},\alpha \right)=\left(5,3.5\right)$ and (0.05, 0.05), respectively. $F_{\text{Max}}^{\ast }$ is defined as the maximum value of $F_{\text{Mem}}/F_{\text{Fld}}$ during tank-treading motion, and is used in Fig. 5 to explain the simulation results of Sec. 3C.

Figure 7

(Color online) Effects of viscosity on simulation results of average periods $\dot{\gamma }T$ and angles $\Phi /\pi$ of RBC motions. (a) Reynolds number $\text{Re}=\rho u_{0}D/{\mu }_{\text{Out}}$ of 0.02, 0.2, 2, and 20 and (b) viscosity ratio ${\mu }^{\prime }={\mu }_{\text{In}}/{\mu }_{\text{Out}}$ of 0.5, 1, 2.5, and 5 are examined as representative parameters of viscosity. Periods $\dot{\gamma }T$ and angles $\Phi /\pi$ are plotted as a function of the logarithm of bending capillary number ${\text{Ca}}_{\text{B}}$, as in Fig. 4. Black lines show normalized periods $\dot{\gamma }{T}_{\mathrm{LA}}$ on angle ${\phi }_{\mathrm{LA}}$ and $\dot{\gamma }{T}_{\text{Mem}}$ on angle ${\phi }_{\text{Mem}}$ for a rotation $\left(2\pi \right)$. Green (gray) lines with closed circles in (b) show $\dot{\gamma }{T}_{\mathrm{LA}}$. Red (gray) inset in lower right shows angles for rotation amplitude ${\Phi }_{\text{OA}}/\pi$ and mean inclination angle ${\Phi }_{\mathrm{LA}}/\pi$ of the entire RBC. The simulation results of $\text{Re}=0.2$ in (a) and ${\mu }^{\prime }=5$ in (b), indicated by solid lines, are the same as those used in Sec. 3B, and thus, these solid lines illustrate the same results as those illustrated in Fig. 4.

Figure 8

(Color online) Effects of number of computed particles on simulation results of average periods $\dot{\gamma }T$ and angles $\Phi /\pi$ of RBC motions. Periods $\dot{\gamma }T$ and angles $\Phi /\pi$, as a function of the logarithm of bending capillary number ${\text{Ca}}_{\text{B}}$, are plotted for three cases: the same number of particles as that used in Sec. 3B $\left(\times 1.0\right)$, 1.8 times more particles $\left(\times 1.8\right)$ and 4.0 times more particles $\left(\times 4.0\right)$. The solid lines for $\left(\times 1.0\right)$ illustrate the same result as that illustrated in Fig. 4. Black lines show normalized periods $\dot{\gamma }{T}_{\mathrm{LA}}$ on angle ${\phi }_{\mathrm{LA}}$ and $T_{\text{Mem}}$ on angle ${\phi }_{\text{Mem}}$ for a rotation $\left(2\pi \right)$. Red (gray) inset in lower right shows angles for rotation amplitude ${\Phi }_{\mathrm{OA}}/\pi$ and mean inclination angle ${\Phi }_{\mathrm{LA}}/\pi$ of the entire RBC.