Tank-treading dynamics of red blood cells in shear flow: On the membrane viscosity rheology

Abstract

In this article, extensive three-dimensional simulations are conducted for tank-treading (TT) red blood cells (RBCs) in shear flow with different cell viscous properties and flow conditions. Apart from recent numerical studies on TT RBCs, this research considers the uncertainty in cytoplasm viscosity, covers a more complete range of shear flow situations of available experiments, and examines the TT behaviors in more details. Key TT characteristics, including the rotation frequency, deformation index, and inclination angle, are compared with available experimental results of similar shear flow conditions. Fairly good simulation-experiment agreements for these parameters can be obtained by adjusting the membrane viscosity values; however, different rheological relationships between the membrane viscosity and the flow shear rate are noted for these comparisons: shear thinning from the TT frequency, Newtonian from the inclination angle, and shear thickening from the cell deformation. Previous studies claimed a shear-thinning membrane viscosity model based on the TT frequency results; however, such a conclusion seems premature from our results and more carefully designed and better controlled investigations are required for the RBC membrane rheology. In addition, our simulation results reveal complicate RBC TT features and such information could be helpful for a better understanding of in vivo and in vitro RBC dynamics.

Significance

Three-dimensional simulations are conducted for red blood cell deformation in shear flow, and the simulated cell deformation, rotation frequency, and inclination angle are compared with experimental measurements under different suspension viscosity and shear rate. Our results show that membrane viscosity has a direct influence on the cell dynamics; however, currently available experimental data are not sufficient to determine the membrane viscosity accurately. More accurate and controlled experiments are required in the future for more reliable measurement of red cell membrane properties.

Introduction

Blood consists of plasma, erythrocytes (usually called red blood cells [RBCs]), white blood cells, and platelets. RBCs, with a volume concentration of $\sim$ 45% in a healthy human body, vastly outnumber white blood cells and platelets. The large number density of RBCs along with their other distinctive properties, such as their biconcave shape and high membrane deformability, make them the most influential component for hemodynamics and hemorheology (1, 2, 3). Over the past several decades, substantial research has been devoted to investigate the RBC mechanics using various techniques, such as micropipette aspiration (4), atomic force microscopy (5), rheoscopy (6), ektacytometry (7), optical tweezers (8), and magnetic twisting cytometry (9). Due to the system complexity from the small RBC size, neo-Hookean membrane elasticity and complicated cell-flow interaction, direct measurements are not possible; instead, RBC parameters were retrieved by fitting theoretical models or numerical calculations to experimental results (8,10, 11, 12, 13, 14). As a matter of fact, reported values in the literature for RBC mechanical properties show large variations, and this can be attributed to the measurement uncertainty and inaccuracy in experiments, the theoretical assumptions and numerical errors, and the inter- and intraindividual variations in cell size, shape, and other cell properties among blood samples being tested (3,10,15,16). Take the RBC cytoplasm viscosity as an example. Tran-Son-Tay et al. (10) used 10 cP for young RBCs and 18 cP for old cells. For a normal population of human RBCs, the hemoglobin concentration of the cytoplasm is in the range of 290–330 g/mL, and the measured viscosity at 37°C varies significantly, from 4.1 to 23.9 cP (17). Furthermore, McClain et al. (14) found that the cytoplasm viscosity at a hemoglobin concentration of 335 g/mL as 46 $\pm$ 6 cP. In addition, the viscous characteristics of RBC membrane has been substantiated in numerous experiments (9,10,12,18,19); however, the reported membrane viscosity quantities differ from each other significantly, from 0.02 to 1 $\mu$mPa (20,21).

Under a simple shear flow with sufficiently high shear rate and/or exterior fluid viscosity, the RBC membrane rotates around the elongated cell in an approximately stable shape, and the interior cytoplasm performs an eddy-like circulation flow. This phenomenon is called the tank-treading (TT) motion and it provides us a convenient avenue for RBC property measurements (20, 21, 22, 23, 24). Several parameters have been introduced to characterize the TT behavior, including the deformation index, inclination angle, and TT frequency. Of particular interests are the experimental studies conducted by Fischer and co-workers (16,25, 26, 27). Two techniques were employed in these experiments: the rheoscopy and the capillary tube methods. In rheoscopy, a transparent cone-plate chamber is adapted to a microscope with interference contrast optics. The RBC suspension is introduced in the gap space between the cone and plate. When the cone and plate rotate in opposite directions, a stationary layer exists with no translational velocity but the same shear rate as at other locations in the gap. Cells suspended in this layer are observed via the microscope along the gradient of the undisturbed shear flow. Cell deformation can be directly measured from the microscope images. To visualize the membrane motion and thus obtain the TT frequency, a Latex sphere (diameter 0.7 $\mu$m) is attached on the RBC surface as a membrane marker. By following the marker positions in a series of microscope images of a TT cell, the TT frequency can be calculated. More detailed descriptions on the rheoscopy measurements can be found in the literature (22,26). From these rheoscopy experiments, extensive data on the TT frequency (26) and cell deformation (16,26) have been collected for RBCs suspended in dextran solutions of different viscosity and with different shear rates applied. Since the microscope images were taken in the direction along the shear gradient in rheoscopy, the cell orientation during the TT motion cannot be assessed. To overcome this difficulty, Fischer and Korzeniewski (27) took images of RBCs flowing in a glass capillary tube of diameter 975 $\mu$m in the midplane of the capillary. Although the flow situation here was not an exact shear flow as in rheoscopy or ektacytometry, the shear rate gradient should be negligible considering the large difference between RBC size ($\sim 8\mu$m in diameter and $\sim 2\mu$m in thickness) and the capillary tube diameter. In a recent study for capsule migration in tube flow (28), it was found that cell rotation, deformation, and inclination during the migration process can be approximated as those in the shear flow of the local shear rate in the tube flow even with the capsule-tube diameter ratio at 1:4. In these experimental studies, the exterior fluid viscosity changed in the range of 10.7–109.3 cP, and the imposed shear rates varied from about 1 to 260 s⁻¹ (see Fig. 5). These experimental results are valuable for estimating RBC properties (21,24,29,30).

Figure 5.

A chart to show the shear flow conditions considered in present simulations (red$+$) and the two recent studies by Tsubota (21) (magenta$\ast$) and Matteoli et al. (24) (blue$\times$). Also shown as smaller gray and black symbols are the flow situations utilized in experiments by Fischer and co-workers (16,26,27). To see this figure in color, go online.

The TT motion of RBCs has been also studied theoretically and numerically. An early theoretical study was performed by Keller and Skalak (31), where the cell was modeled as an ellipsoid with its surface rotating around the interior cytoplasm fluid. However, this theoretical approach cannot incorporate the particular biconcave RBC shape and the complex membrane mechanics. To simulate RBC dynamics in flow, several numerical techniques have been utilized, including the boundary integral method (32,33), the immersed boundary method (IBM) (34,35), and the dissipative particle dynamics method (36, 37, 38). Previous simulations usually considered a pure elastic membrane according to the neo-Hooken or Skalak constitutive relationships (29,30,32,35,37), and studies on the membrane viscosity effect on the RBC TT behaviors are far from adequate. Recently, Li and Zhang (39) proposed a finite difference method to include the membrane viscosity in immersed boundary simulations. This method has been subsequently used to simulate the TT dynamics in shear flow (40) and lateral migration process in tube flow (28) for spherical capsules with viscoelastic membranes, and the RBC relaxation process after being stretched in shear flow or by tensile load as in optical tweezers experiments (13). More recently, Matteoli et al. (24) simulated the TT motion of RBCs and found that different membrane viscosity values are required for different shear flow situations (i.e., different exterior fluid viscosity and different shear rate values) to match the TT frequency results from Fischer’s experiments (26). Their results suggested a shear-thinning dependence of the membrane viscosity on the imposed shear rate: a higher membrane viscosity value at a lower shear flow rate, and vice versa. However, this conclusion was drawn based on the TT frequency only, and the other equally important TT characteristic parameters, namely the deformation index and the inclination angle, have been completely ignored. Moreover, for substances with non-constant viscosity values, such as non-Newtonian fluids and nonlinear viscoelastic materials, the viscosity value should be related to the local, simultaneous strain status in the materials (41,42), instead of assigning different viscosity values to the entire material under different external loading conditions as in (24). Another recent numerical study was performed by Tsubota (21), which also suggested that the RBC membrane should be treated as a shear-thinning viscoelastic sheet for a good match to Fischer’s TT frequency data (26). In this work, the Carreau model (43), which expresses the membrane viscosity as a function of the membrane strain rate, was adopted. This treatment is more realistic compared with that in (24). The cell deformation has been also considered in (21); however, the comparison for deformation index was conducted with the ektacytometry measurements by Baskurt and Meiselman (44) using rat RBCs, while the TT frequency from the same calculation was compared with Fischer’s experimental results for human RBCs (26). In ektacytometry, the cell deformation is measured from the light diffraction patterns, which might be different from the real cell deformation. As for the elastic membrane stress, Tsubota (21) employed Fung’s modification of the commonly used Skalak law (45). It is interesting to note the relatively poor agreement in RBC deformation to optical tweezers results using either the original or modified Skalak law in (21) (see Fig. 11 a), while good agreement for such a comparison has been reported by Guglietta et al. (13) and reproduced in our current study (see Fig. 3), both using the original Skalak law for the membrane strain energy function. Please note that, in these optical tweezers measurement simulations, the fluid and membrane viscosities have no effect on the cell deformation, since the cell deformation is obtained from the final stable cell shape. The relatively poor simulation-experiment agreement for cell deformation in optical tweezers measurements may imply that the membrane mechanics is not accurately represented in the numerical model in (21). Furthermore, few other concerns exit for both studies. Despite the evident influence of the interior viscosity on the TT frequency shown in previous research (46,47) and these two recent studies (21,24), the authors had fixed the interior cytoplasm viscosity (10.78 cP in (24) and 28.9 cP in (21)) with no justifications. In addition, although the experimental results from Fischer and co-workers (16,26,27) are available for exterior fluid viscosity from 10.7 to 109.3 cP, the simulations in (21,24) were limited to the low exterior viscosity range (see Fig. 5), and the cell inclination angle was not considered at all. This may open questions to the validity and accuracy of the suggested membrane viscosity models in these studies for other TT parameters and high viscosity fluids. Finally, it appears inappropriate to pursue a neatly perfect simulation-experiment agreement for one specific system parameter (the TT frequency) without considering the model performance in other parameters (cell deformation and inclination angle).

Figure 11.

Comparison of the simulated inclination angle $\theta$ (color symbols) to experimental measurements by Fischer and Korzeniewski (27) (gray symbols) under different shear conditions. For the color symbols, we adopt different fill colors for the interior viscosity and different symbol shapes for the membrane viscosity. The error bars show the $\theta$ variation during a TT period for the simulation results or the distributions of the measured $\theta$ in experiments. For ${\mu }_o=14.6$ cP (a) and 28.9 cP (b), experimental data at these exterior viscosity values are not available and hence we display those from the close viscosity values for comparison. The error bars indicate the data uncertainty in measurement for experiments or the angle variation during a TT period for simulations. To see this figure in color, go online.

Figure 3.

Comparisons of (a) the axial length $L$ and transverse width $W$ of the deformed RBC under load F and (b) the deformation index $D_{xz}$ obtained from present simulations and optical tweezers experiments by Suresh et al. (8). Also shown in (b) as green symbols are the simulation results by Tsubota (21). The error bars indicate the data uncertainty in measurement for experiments, or the value variation when using different cell length definitions for simulations. To see this figure in color, go online.

To address these concerns, a more comprehensive analysis of the TT motion of RBCs in shear flow is deemed necessary. In this study, we first simulate the RBC deformation in optical tweezers experiments, and the good agreement indicates that the membrane elasticity model and parameters used in our calculations are appropriate. Then extensive calculations are carried out for TT RBCs under various situations. Fundamental differences exist between this current research and recent studies by Tsubota (21) and Matteoli et al. (24) on several aspects: first, (21,24) used specific values for the unknown cytoplasm viscosity, which may vary in a large range in human RBCs (10,14,17). The blood samples used in experiments by Fischer and co-workers (16,26,27) were obtained from different donors, and assuming a same cytoplasm viscosity value for all these experiments could be debatable. In this work, we will test RBC dynamics with the cytoplasm viscosity varying from 6 to 20 cP. Also, these two studies only considered a subset of the suspending viscosities that employed in experiments, while our simulations cover the same suspending viscosity range in experiments (see Fig. 5). Moreover, the two recent studies only compared the TT rotation frequency data, and completely ignored other important TT characteristics such as the inclination angle and the RBC deformation, for which experimental data are available from Fischer and co-workers (16,27). On the other hand, our present study will examine all these RBC TT dynamics parameters and compare them with available experimental measurements. Our results show that the general trends between the TT parameters and the shear flow condition agree well with experiments; however, it is difficult to determine the precise values for the cytoplasm and membrane viscosities based on available experimental results. The TT behaviors, including the rotation frequency, deformation index, and inclination angle, respond to membrane viscosity change more dramatically than to the cytoplasm viscosity change. In terms of the possible nonlinear dependence of membrane viscosity on the membrane strain condition, our TT frequency results imply a shear-thinning correlation as reported in (21,24); on the contrary, an improved agreement for the cell deformation can be expected if a shear-thickening model is adopted for the membrane viscosity. As for the inclination angle, a constant membrane viscosity of 1–5 × 10⁻⁷ mPa yields a fairly good agreement to the experimental data for all shear flow situations. This study further reveals the complexity of the RBC TT motion and the relations among cell properties, flow condition, and the TT dynamics; and more accurate, comprehensive, and controlled measurements are required to determine the cytoplasm viscosity and the rheology model and parameter values for the membrane viscosity for RBCs.

Materials and methods

Governing equations and numerical methods for fluid flow and flow-membrane interaction

The continuity and Navier-Stokes equations governing the flow field of an incompressible fluid with density $\rho$ are given as:

$$\nabla ·\mathbf{u}=0;\rho \left(\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}·\nabla \mathbf{u}\right)=-\nabla p+\mu {\nabla }^2\mathbf{u}+\mathbf{f};$$ (1)

where $\mathbf{u}$ denotes the fluid velocity, $p$ is the pressure, and $\mu$ indicates the fluid viscosity. $\mathbf{f}$ represents the body force acting on the fluid induced by the flow-membrane interaction or an external force such as gravity. In this work, the lattice Boltzmann method with the D3Q19 (19 lattice velocities in 3 dimensions) lattice model is used to solve the flow field by considering a uniform Eulerian grid over the entire flow domain, including the exterior and interior fluids (48). This numerical method has been detailed and validated in our previous publications (28,29,39,40).

The RBC membrane is considered as a flexible, thicknessless two-dimensional (2D) sheet, which may deform when subjected to an external flow field. Internal stresses are then developed in the cell membrane with respect to the membrane constitutive law (see RBC membrane mechanics). To couple the dynamic cell deformation and the fluid motion, we use the IBM (29,49). In the framework of IBM, a membrane is represented by a finite number of Lagrangian nodes ${\mathbf{x}}_m$. The deformation-induced membrane force $\mathbf{F}$ acting on node ${\mathbf{x}}_m$ can be distributed to the neighboring fluid nodes ${\mathbf{x}}_f$ as body forces $\mathbf{f}$:

$$\mathbf{f}\left({\mathbf{x}}_f\right)=\sum _{{\mathbf{x}}_m}D({\mathbf{x}}_f-{\mathbf{x}}_m)\mathbf{F}\left({\mathbf{x}}_m\right),$$ (2)

$$\begin{array}{l}D\left(\mathbf{x}\right)=\frac{1}{{\left(4\delta x\right)}^3}\left(1+\cos \frac{\pi x}{2\delta x}\right)\left(1+\cos \frac{\pi y}{2\delta x}\right)\left(1+\cos \frac{\pi z}{2\delta x}\right),\hfill \\ x|\le 2\delta x,|y|\le 2\delta x,\text{and}|z|\le 2\delta x\hfill \\ D\left(\mathbf{x}\right)=0,\text{otherwise},\hfill \end{array}$$ (3)

where, $x$, $y$, and $z$ are the three Cartesian components of location vector $\mathbf{x}$, $\delta x$ denotes the Euler grid resolution, and $D\left(\mathbf{x}\right)$ is discrete delta function. The subscripts _m and _f indicate the Cartesian position for a membrane node and a fluid node, respectively. The membrane velocity ${\mathbf{u}}_m\left({\mathbf{x}}_m\right)$ can also be computed from the local surrounding flow $\mathbf{u}$ through a velocity interpolation operation using the function $D\left(\mathbf{x}\right)$:

$${\mathbf{u}}_m\left({\mathbf{x}}_m\right)={(\Delta x)}^3\sum _{{\mathbf{x}}_f}D({\mathbf{x}}_f-{\mathbf{x}}_m)\mathbf{u}\left({\mathbf{x}}_f\right).$$ (4)

In both Eqs. 2 and 4, the summation runs over all fluid nodes with $D\left(\mathbf{x}\right)>0$. Furthermore, the membrane node position can be calculated from its velocity, for example, via the Euler scheme. With the cell motion and deformation, the fluid velocity for fluid nodes near the membrane needs to be updated accordingly. This is accomplished via the phase index, which indicates the relative position of a fluid node to the membrane. More details for this treatment can be found in the literature (23,28).

RBC membrane mechanics

The biconcave shape of an undeformed human RBC can be described by (18):

$$\overline{z}=0.5{\left(1-{\overline{r}}^2\right)}^{0.5}\left(c_0+c_1{\overline{r}}^2+c_2{\overline{r}}^4\right),$$ (5)

where $\overline{z}$ and $\overline{r}$ are dimensionless cylindrical coordinates based on the RBC radius $R$, and the shape parameters are $c_0=0.207$, $c_1=2.002$, and $c_2=-1.122$. The RBC membrane is treated as a 2D viscoelastic sheet following the Kelvin-Voigt model with the total membrane stress $\mathbf{\tau }$ consisting of the elastic part ${\mathbf{\tau }}^e$ and the viscous part ${\mathbf{\tau }}^v$ (33,50, 51, 52). For the elastic part, in addition to the in-plane shear and dilatation elasticity, the RBC membrane also exhibits resistance to bending deformation. Furthermore, surface and volume energies are often considered to penalize and diminish area and volume deviations in numerical simulations. To incorporate these aspects, the total strain energy $W$ is defined as (29,53).

$$W=W_s+W_b+W_A+W_V,$$ (6)

where $W_s$, $W_b$, $W_A$, and $W_V$ are the corresponding terms associated with strain, bending, area, and volume energies, respectively. Details on these energetic terms are presented below. Accordingly, the membrane force ${\mathbf{F}}_m$ at location ${\mathbf{x}}_m$ on the membrane can be obtained via the virtual work principle as

$${\mathbf{F}}_m\left({\mathbf{x}}_m\right)=-\frac{\partial W}{\partial {\mathbf{x}}_m}.$$ (7)

On top of the elastic force from Eq. 7, the force due to the membrane viscosity is also calculated from the shear strain rate as to be described in membrane viscosity.

Membrane in-plane elasticity

Following the Skalak constitutive law, which was developed for biological membranes (54), the strain energy function for the elastic stress induced by the membrane in-plane deformation can be described as

$$W_s=\frac{E_s}{8}\int \left[\left(I_1^2+2I_1-2I_2\right)+C{I}_2^2\right]dA$$ (8)

where $E_s$ is the shear modulus, and $C=E_a/E_s$ is dilatation coefficient. The RBC membrane is nearly area incompressible with a large dilatation module $E_a$ and, hence, the value of $C$ is large. The strain invariants for a 2D material, $I_1$ and $I_2$, can be computed from the principle stretch ratios ${\mathrm{ϵ}}_1$ and ${\mathrm{ϵ}}_2$ as:

$$I_1={\mathrm{ϵ}}_1^2+{\mathrm{ϵ}}_2^2-2;I_2={\mathrm{ϵ}}_1^2{\mathrm{ϵ}}_2^2-1.$$ (9)

The associated stress tensor for the in-plane membrane elasticity ${\mathbf{\tau }}_s^e$ can then be obtained from the energy function Eq. 8 as (54,55).

$${\mathbf{\tau }}_s^e=\frac{1}{{\mathrm{ϵ}}_2}\frac{\partial W_s}{\partial {\mathrm{ϵ}}_1}{\mathit{e}}_1\otimes {\mathit{e}}_1+\frac{1}{{\mathrm{ϵ}}_1}\frac{\partial W_s}{\partial {\mathrm{ϵ}}_2}{\mathit{e}}_2\otimes {\mathit{e}}_2,$$ (10)

where

$${\tau }_1^e=\frac{1}{{\mathrm{ϵ}}_2}\frac{\partial W_s}{\partial {\mathrm{ϵ}}_1};{\tau }_2^e=\frac{1}{{\mathrm{ϵ}}_1}\frac{\partial W_s}{\partial {\mathrm{ϵ}}_2};$$ (11)

are the two principle stresses. ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ are the unit vectors for the principle directions, which can be obtained as the unit eigenvectors of the left Cauchy-Green deformation tensor $\mathbf{G}=\mathbf{F}{\mathbf{F}}^T$ (55). Here, $\mathbf{F}$ is the deformation gradient matrix, and it can be calculated by comparing the deformed and undeformed configurations of each element of the triangulated membrane (53,56).

Membrane bending elasticity

The RBC membrane also exhibits resistance to bending deformation, particularly in regions with large local curvatures. The simplified Helfrich expression is applied to model the bending resistance on the triangular mesh (53):

$$W_b=\frac{E_b}{2}\sum _j{\left({\beta }_j-{\beta }_{j,0}\right)}^2.$$ (12)

Here, $E_b$ is the bending modulus and ${\beta }_j$ denotes the angle between the outward normal vectors of the two neighboring face elements sharing the same edge $j$. ${\beta }_{j,0}$ is the initial value of ${\beta }_j$ and the summation is taken over all edges of the membrane elements. Details of the derivation of the bending forces can be found in (53).

Membrane surface and volume energies

To enhance the conservation in total volume and membrane area in numerical simulations, the following penalty energy terms are usually employed (29,55):

$$W_A=\frac{E_A}{2}\frac{{\left(A-A_0\right)}^2}{A_o},W_V=\frac{E_V}{2}\frac{{\left(V-V_0\right)}^2}{V_o}.$$ (13)

Here, $E_A$ and $E_V$ are the penalty moduli for changes in volume $V$ and total area $A$. $V_0$ and $A_0$ are the initial volume and total area of the undeformed RBC, respectively. Please note that these two artificial energetic terms are introduced to prevent large deviations in surface area and volume from the original values caused by numerical errors during calculations, and this practice is typical in IBM simulations of RBC dynamics (20,29,55,57).

Membrane viscosity

The viscous stress ${\mathbf{\tau }}^v$ in membranes is typically decomposed into the shear viscous stress ${\mathbf{\tau }}_s^v$ and the dilatational viscous stress ${\mathbf{\tau }}_d^v$ (33,50, 51, 52):

$${\mathbf{\tau }}^v={\mathbf{\tau }}_s^v+{\mathbf{\tau }}_d^v={\mu }_s\left[2\mathbf{D}-\mathrm{t}r\left(\mathbf{D}\right)\mathbf{I}\right]+{\mu }_d\mathrm{t}r\left(\mathbf{D}\right)\mathbf{I}.$$ (14)

Here, ${\mu }_s$ represents the shear membrane viscosity, ${\mu }_d$ denotes the dilatational membrane viscosity, and $\mathbf{I}$ is the 2 $\times$ 2 unit matrix. The matrix $\mathbf{D}$ is the strain rate tensor that can be computed from the strain tensor $\mathbf{E}$ via a backward finite difference approach, while another method is using the gradient of membrane velocity (50, 51, 52,58).

$$\mathbf{D}=\frac{1}{2}\left[{\nabla }_m{\mathbf{u}}_m+{\left({\nabla }_m{\mathbf{u}}_m\right)}^T\right],$$ (15)

where ${\nabla }_m$ and ${\mathbf{u}}_m$ are, respectively, the gradient operator and the membrane velocity, both in the local membrane plane.

As mentioned previously, the total nodal force at each Lagrangian membrane node needs to be spread to neighboring Eulerian fluid nodes in IBM. The calculations for nodal forces from the energy terms $W_s$, $W_b$, $W_A$, and $W_V$ are well documented in the literature (53), and the finite difference approach for the membrane viscous stresses (39,40) is described in the next section.

Finite-difference method for membrane viscosity calculation

The elastic and viscous stresses in the RBC membrane are induced by the same membrane deformation and the total membrane stress is the sum of the elastic and viscous stresses (17,21,39,57), and thus the viscoelastic strain-stress relationship in RBC membranes can be visualized with the Kelvin-Voigt model in Fig. 1 a: the spring element $k$ and the dashpot element $\mu$ represent the elastic and viscous aspects of the membrane, respectively; and they sustain the same amount of deformation and strain. An artificial spring $k^{\prime }$, which has no physical implications relevant to RBC mechanics, is added in series to the dashpot element $\mu$ (Fig. 1 b) to circumvent the numerical instability when considering the Kelvin-Voigt model directly in calculations (39,57). The combination of the dashpot $\mu$ and the artificial spring $k^{\prime }$ forms a Maxwell viscoelastic element. In this fashion, the original Kelvin-Voigt model is changed into the standard-linear-solid model (59). It should be noted that the system behavior becomes insensitive to the stiffness of the artificial spring when it is adequately large (39). Nevertheless, incorporating the artificial spring in the model prevents an explicit expression of the Maxwell element’s stress for a given strain history $\mathrm{ϵ}\left(t\right)$, and a numerical method is required. Following recent publications (20,28,39,46), the finite difference method (39) is adopted here to calculate the viscous stress in the Maxwell element.

Figure 1.

Schematic representations of (a) the Kelvin-Voigt model for the original membrane viscoelasticity and (b) the standard linear solid model with an artificial spring added to improve numerical stability. See text for details.

The strain of the Maxwell element is the sum of those for the dashpot $\mu$ and artificial spring $k^{\prime }$, and their internal stresses are identical regarding the serial connection of $\mu$ and $k^{\prime }$:

$${\sigma }_M={\sigma }_{\mu }={\sigma }_{k^{\prime }};$$ (16)

$${\mathrm{ϵ}}_M={\mathrm{ϵ}}_{\mu }+{\mathrm{ϵ}}_{k^{\prime }}.$$ (17)

Here, $\sigma$ and $\mathrm{ϵ}$ denote the stresses and strains, respectively; and subscripts are adopted to represent the corresponding components for these properties: _M for the Maxwell element, _μ for the dashpot $\mu$, and _k′ for the spring $k^{\prime }$. The relations between the stresses and strains in the dashpot $\mu$ and spring $k^{\prime }$ are expressed using the coefficients for the viscous dashpot and the linear spring as

$${\sigma }_{\mu }=\mu {\dot{\mathrm{ϵ}}}_{\mu };{\sigma }_{k^{\prime }}=k^{\prime }{\mathrm{ϵ}}_{k^{\prime }},$$ (18)

where $\mu$ and $k^{\prime }$ are also employed to denote, respectively, the viscosity of the dashpot $\mu$ and the Hookean coefficient of the spring $k^{\prime }$. Taking the time rate of Eq. 17 and using relations in Eqs. 16 and 18 yield

$${\dot{\mathrm{ϵ}}}_M=\frac{{\sigma }_M}{\mu }+\frac{{\dot{\sigma }}_M}{k^{\prime }}.$$ (19)

This represents the dynamic constitutive relation for the Maxwell element. Furthermore, by applying the central finite difference approximations for the time rate terms ${\dot{\mathrm{ϵ}}}_M$ and ${\dot{\sigma }}_M$ and considering the stress at $t-\Delta t/2$ as the average of those at $t-\Delta t$ and $t$, an expression for the stress ${\sigma }_M\left(t\right)$ can be established as (39,40):

$${\sigma }_M\left(t\right)\approx \frac{(2\mu -k^{\prime }\Delta t){\sigma }_M(t-\Delta t)+2\mu k^{\prime }[{\mathrm{ϵ}}_M\left(t\right)-{\mathrm{ϵ}}_M(t-\Delta t)]}{2\mu +k^{\prime }\Delta t}.$$ (20)

With the initial condition specified (for example, ${\mathrm{ϵ}}_M=0$ and ${\tau }_M=0$ at $t=0$ for a stress-free initial state in typical calculations) and an appropriate time interval $\Delta t$ selected, the above equation can be employed to calculate the stress in the Maxwell element ${\sigma }_M\left(t\right)$ from the strain function ${\mathrm{ϵ}}_M\left(t\right)$.

Please note that Fig. 1 a is only a 1D, conceptual representation of the real membrane mechanics described in RBC membrane mechanics, and it is helpful for clarity in the algorithm description. For the triangular mesh decomposition of the RBC membrane, the strain and stress terms in Eq. 20 should be replaced by the shear/dilatational strain and stress $2\times 2$ tensors for each membrane element. In practice, we only need to apply Eq. 20 three times for each membrane element, each for one of the three independent strain/stress tensor components. A detailed step-by-step description of the implementation of this finite-difference scheme for the membrane viscosity in general IBM simulations is provided in the supporting material. The elastic part of the membrane mechanics is not involved in the viscous stress calculation.

Validation: RBC deformation under external stretching load

Before simulating the RBC TT motion in shear flow with various cytoplasm and membrane viscosities, validation tests are first performed for the stretching process of RBCs in optical tweezers experiments (8) to confirm that appropriate values have been used for the membrane elastic moduli discussed in RBC membrane mechanics. It should be mentioned that our computer programs have been validated for capsule TT dynamics in shear flows with various membrane resolutions, capillary numbers, interior-exterior fluid viscosity ratios, and membrane Boussinesq numbers in our previous publications (28,29,39).

Stretching an RBC in optical tweezers is accomplished by exerting external mechanical load on the opposite sides of the cell by optical trapping (8). Under this external load, the RBC elongates gradually and attains a steady shape at the end. A benefit of this experiment for our current study is that the fluid and membrane viscosities are not involved in the final elongated cell shape. Therefore, a good agreement between the numerical and experimental results can help to justify the modulus values for $E_s$, $E_b$, and $E_a$ utilized in our following simulations for TT RBCs in shear flow with various cytoplasm and membrane viscosity combinations.

The tensile stretching of RBCs in optical tweezers is simulated as follows. The RBC, which is initially at rest, is placed at the center of a cubic domain with size of $24.7\times 24.7\times 24.7\mu$m ($76\times 76\times 76$ in lattice unit $\delta x$) (13). The cell membrane is discretized into 5120 triangular elements and the membrane elastic parameters are listed in Table 1. Two external forces with equal values and in opposite directions along the x axis are exerted to the RBC surface nodes at the left and right edges, as displayed schematically in Fig. 2. The force $\mathbf{F}$ is distributed evenly on the nodes on the periphery of the contact area of the microbeads with the cell surface. After the external forces are imposed, the cell gradually deforms to reach an equilibrium stretched state. The transverse length $W$ can be readily obtained from the maximum and minimum positions of the membrane nodes along the transverse direction $Z$. For the axial length $L$, different methods had been used in previous studies, such as the distance between the periphery of the force contact areas from two sides (34), the distance between the middle positions of the force contact areas (21), and the maximum distance between the membrane node positions along the axial direction (60). These distances are indicated by, respectively, $L_{\mathit{min}}$, $L_{middle}$, and $L_{\max }$ in Fig. 2 b. Here, to provide better comparisons, all of these distances are considered and the differences between them are presented as error bars. The deformation index can then be defined as

$$D_{xz}=\frac{L-W}{L+W}.$$ (21)

Table 1.

Parameters for the RBC radius and membrane moduli utilized in this study

Radius $R$	3.91 $\mu$m
Elastic shear modulus $E_s$	5.3 $\mu$N/m
Elastic dilatational modulus $E_a$	50 $E_s$
Bending modulus $E_b$	2 $\times {10}^{-19}$ Nm

These values are taken from a recent study by Guglietta et al. (20) and the original sources for these property values can be found therein.

Figure 2.

Schematic illustrations of (a) the undeformed RBC at rest with silica microbeads attached to the cell surface from opposite sides and (b) the elongated cell under external load $\mathbf{F}$. $W$ is the transverse length of the deformed cell, and L_min, $L_{middle}$, and $L_{\mathit{max}}$ denote three measurements of the cell length using different methods in the literature. See text for details. To see this figure in color, go online.

The RBC stretching simulations are conducted with different external forces and the results are compared with those from experimental measurements (8) in Fig. 3. It can be seen that there is a good agreement between our simulations and experiments, and it thus confirms that the mechanical properties listed in Table 1 are appropriate for further simulations of RBC TT dynamics. The simulation results obtained by Tsubota (21) are also depicted in Fig. 3 b to show the relatively poor agreement from that study, particularly at high external forces. In addition, simulations are also performed with bending modulus of $E_b=1.2\times {10}^{-18}$ Nm (six times of the value in Table 1) to confirm that the simulation results are relatively insensitive to the utilized bending modulus.

Simulation setup for RBC TT dynamics

With the appropriate agreement achieved in the above validation tests, the mechanical properties of the RBC membrane listed in Table 1 are used in the following simulations. The simulation setup is depicted in Fig. 4. The undeformed RBC is placed at the center of a 3D domain with size of $37.3\times 24.9\times 24.9\mu$m ($180\times 120\times 120$ in lattice unit $\delta x$), and the cell membrane is represented by 5120 triangular elements. These mesh settings for the fluid domain and the cell membrane are determined based on previous studies from our group (28,29,39,46) and others (20,55,57,61). A finer mesh with 20,480 triangular elements has been tested and there is no apparent difference observed in simulation results (39). To impose a simple shear flow, the top and bottom domain boundaries are treated as solid walls moving at velocity $U_0$; however, in opposite horizontal directions. This setup generates a simple shear flow with shear rate of $\dot{\gamma }=2U_0/H$ around the cell, where $H$ is the distance between the top and bottom boundaries. Periodic boundary conditions are imposed along the $X$ and $Y$ directions. The viscosities of the interior and exterior fluids of the RBC are denoted by ${\mu }_i$ and ${\mu }_o$, respectively.

Figure 4.

Illustration of the deformed RBC under shear flow. The projected length $L_p$ and width $B$ of the cell in the $XY$ plane are used to characterize the cell deformation as $L_p/B$. The inclination angle $\theta$ is defined as the angle between the flow direction and the major axis, and the membrane rotation (blue arrows) is measured by $\phi$, the angular position of a membrane marker (blue circle). To see this figure in color, go online.

When subjected to shear flow, the RBC elongates and aligns itself with an angle $\theta$ from the flow direction, while the membrane rotates around the interior fluid. The inclination angle $\theta$ is obtained by approximating the deformed RBC as an equivalent ellipsoid based on the inertia tensor (1). The TT frequency $TTF$ (in s⁻¹) can be calculated from the trajectory of a membrane marker point in the middle shear plane using the rotation angle $\phi$. Following the rheoscopy experiments (16,26), the cell deformation index is defined as the ratio of its projected length $L_p$ to the projected width $B$ in the $XY$ plane (Fig. 4). It should be mentioned that, at low shear rates or with low exterior fluid viscosity, compressive stresses may develop on the cell surface, which may lead to membrane buckling and formation of wrinkles (62). In addition, including the membrane viscosity can compound this issue with oscillations in the cell deformation and orientation (46,57,63). To mitigate these difficulties and improve the stability of the simulations, a larger value of the bending modulus is necessary. Please note that this has negligible impact on the overall cell behaviors (see Fig. 3) and that this technique has been previously adopted in the literature (24,29,30).

In this study, the simulation results are provided in physical units to enable direct comparisons with experimental results from Fischer and co-workers (16,26,27). The shear flow conditions, include the suspension viscosity and shear rate, utilized in these experiments are shown as gray and black symbols in Fig. 5. In the same chart, also shown are the flow conditions considered in the two recent studies by Matteoli et al. (24) (blue $\times$’s) and Tsubota (21) (magenta $\ast$’s). It can be seen that (21,24) have only covered a subset of available experimental data with relatively low suspension viscosity values. In this study, to have a more complete coverage of the available experimental data, we simulate all 12 shear flow situations in (16), with 4 different exterior viscosities of 14.6–104 cP and three shear rates for each (red $+$’s in Fig. 5). Please note that these ${\mu }_o$ and $\dot{\gamma }$ values to be used in the next simulations are set to be identical to those in Fischer and Korzeniewski’s experiments for cell deformation in shear flow (16). On the other hand, the reported membrane viscosity of RBCs in the literature is in the range of (0.2–10) × 10⁻⁷ mPa (20,21). For the large dilatation modulus (i.e., strong area conservation) of RBC membranes, only the shear viscosity ${\mu }_s$ is included in this research, as in previous studies (20,21,24,46). Accordingly, four membrane viscosity levels are considered: ${\mu }_s=0$ (pure elastic membrane), ${10}^{-7}$, $5\times {10}^{-7}$, and ${10}^{-6}$ mPa. In addition, the literature values for the interior fluid viscosity ${\mu }_i$ range from 5 to 55 cP (21,29). In this study, we also consider four different values of ${\mu }_i$: 6, 10, 15, and 20 cP. Using further higher interior viscosity values can induce numerical instability (and hence a finer mesh resolution and a smaller time step will be required), and also may change the cell motion to the swinging or tumbling modes. Therefore, we consider $4\times 4=16$ cell conditions of different cytoplasm and membrane viscosity values. In total, in this study we have $12\times 16=192$ cases to simulate for the RBC TT motion under different flow and cell conditions, and all the key TT characteristic parameters, including the rotation frequency $TTF$, deformation index $L_p/B$, and inclination angle $\theta$, will be collected and compared with the experimental results.

Results and discussions

For the system considered in this work, three non-dimensional parameters can be introduced (39,57,58,61), namely the interior-exterior viscosity ratio $\lambda ={\mu }_i/{\mu }_o$, the capillary number $Ca={\mu }_o\dot{\gamma }a/E_s$, and the Boussinesq number $B{q}_s={\mu }_s/{\mu }_{o}a$. Here, $a=2.82\mu$m is the equivalent sphere radius based on an RBC volume of 94 $\mu$m³. The values of these non-dimensional parameters are listed in Table 2 for all the cases calculated in this study; however, in our next result presentation and discussion, we mainly use physical values for the convenience of direct comparisons to experimental data.

Table 2.

Non-dimensional control parameters for systems calculated in this study

(a) Interior-exterior viscosity ratio $\lambda$
${\mu }_o$ (cP)	${\mu }_i$ (cP)
	6	10	15	20
14.6	0.41	0.68	1.03	1.37
28.9	0.21	0.35	0.52	0.69
55.9	0.11	0.18	0.27	0.36
104	0.06	0.10	0.14	0.19

(b) Boussinesq number $B{q}_s$
${\mu }_o$ (cP)	${\mu }_s$ ($\times {10}^{-7}$ mPa)
	1	2	5	10
14.6	2.43	4.86	12.1	24.3
28.9	1.23	2.45	6.14	12.3
55.9	0.63	1.27	3.17	6.34
104	0.34	0.68	1.70	3.41

(c) Capillary number $Ca$
${\mu }_o$ (cP)	$\dot{\gamma }$ (s−1)
	3.75	6.25	7.5	12.5	15	25	37.5	50	75	150
14.6	–	–	–	–	–	–	0.29	–	0.58	1.17
28.9	–	–	–	0.19	–	0.38	–	0.77	–	–
55.9	–	0.19	–	0.37	–	0.74	–	–	–	–
104	0.21	–	0.42	–	0.83	–	–	–	–	–

The RBC TT motion in simple shear flow is a dynamic process. For the illustrative purpose, several cell shapes are displayed in Fig. 6 for the cases with ${\mu }_o=55.9$ cP, $\dot{\gamma }=25s^{-1}$, ${\mu }_i$ = 10 cP and four different membrane viscosities of ${\mu }_s=0\sim {10}^{-6}$ mPa (labeled on the left edge of Fig. 6 a). The shapes are collected during a period of the TT motion and the corresponding time instant for each shape is labeled underneath. The initial undeformed shape of the cell with an inclination angle of 45° is also shown at the top row. Obviously, under the shear flow, the cell has elongated along the flow and tilted toward the flow direction with its membrane rotating around the interior fluid, as shown by the membrane marker position (blue circles). Comparing the shapes at different membrane viscosities ${\mu }_s$ indicates that the cell is less stretched and more tilted toward the flow direction at higher ${\mu }_s$ with smaller deformation index $L_p/B$, as can be observed in Fig. 6 b. In addition, increasing ${\mu }_s$ intensifies dissipation in the membrane, which retards the membrane rotation (see the time intervals in Fig. 6 a). These behaviors have also been reported for spherical capsules in shear flow (46).

Figure 6.

Representative RBC shapes during a TT period in (a) the $XZ$ plane (front view) and (b) the $XY$ plane (top view). The position of a membrane marker is indicated by the blue points to illustrate the TT rotation of the cell. Here, four cases of different membrane viscosities (labeled at left) are shown, and other simulation parameters, the exterior fluid viscosity, shear rate, and interior fluid viscosity, are held the same as 55.9 cP, 25 s⁻¹, and 10 cP, respectively. To see this figure in color, go online.

During the TT motion, the cell deformation and inclination angle still oscillate about their mean values (20,29,46). Therefore, before presenting detailed results on the TT behaviors of the 192 cases considered, we take the case with ${\mu }_o=55.9$ cP, $\dot{\gamma }=25s^{-1}$, ${\mu }_i$ = 10 cP, and ${\mu }_s={10}^{-7}$ mPa as an example to describe the procedure for obtaining the cell dynamics parameters from the simulation results. Fig. 7 depicts variations of the rotation angle $\phi$ of the membrane marker, the projected cell length $L_p$ and width $B$, the deformation index $L_p/B$, and the inclination angle $\theta$ with time. After a relatively short initial transient phase, the cell starts to exhibit periodic deformation and motion. The number of cycles in variations of $L_p$, $B$, $L_p/B$, and $\theta$ are twice of that for $\phi$ because of the system symmetry. Since the cell elongation in $L_p$ is associated with the shrinkage in the cell width $B$, a relatively large variation amplitude is evident for the deformation index $L_p/B$. For the cell deformation and inclination angle, we take the maximum, mean, and minimum values over the last three periods of $\phi$ (see the horizontal dashed lines in Fig. 7) for further analysis. These maximum, mean, and minimum values are presented as error bars for comparison with experiments in the following sections. The TT frequency $TTF$ is calculated directly from the elapsed time of the last three $\phi$ cycles. All simulations have been run until at least five $\phi$ cycles have been covered.

Figure 7.

Temporal variations of (a) the rotation angle $\phi$, (b) the projected cell length $L_p$ and width $B$, (c) the deformation index $L_p/B$, and (d) the inclination angle $\theta$ for the illustrative case with ${\mu }_o=55.9$ cP, $\dot{\gamma }$ = 25 s⁻¹, ${\mu }_i$ =10 cP, and ${\mu }_s={10}^{-7}$ mPa. In (c) and (d), the dashed lines indicate the maximum, mean, and minimum values of the parameter variation over the last three $\phi$ periods. To see this figure in color, go online.

TT frequency

Variations of TT frequency $TTF$ as a function of shear rate for the four different exterior viscosities are illustrated in Fig. 8 in the logarithmic scale. For the experimental results from Fischer (26), the $TTF$ values of the cells for all three donors are considered and shown as gray symbols with error bars. In Fig. 8 a, c, and d) for ${\mu }_o=$ 14.6, 55.9, and 104 cP utilized in our calculations, the experimental results at these ${\mu }_o$ values are not available, and hence the experimental results at close ${\mu }_o$ values are presented for approximate comparisons. At the qualitative level, the linear relationship between the logarithmic values of $TTF$ and $\dot{\gamma }$ observed from experiments (dashed lines) is well reproduced from our simulations. The dashed lines connecting experimental data points pass through the color symbol clusters from our simulations, indicating that, by tuning the cytoplasm viscosity ${\mu }_i$ and membrane viscosity ${\mu }_s$, one can obtain the exact experimental $TTF$ results, as reported in recent studies (21,24).

Figure 8.

Comparison of the simulated $TTF$ (color symbols) to experimental measurements by Fischer (26) (gray symbols with error bars for data uncertainty) under different shear conditions. For the color symbols, we adopt different fill colors for the interior viscosity and different symbol shapes for the membrane viscosity. The error bars show the distributions of the measured $TTF$. For ${\mu }_o=14.6$ cP (a), 55.9 cP (c), and 104 cP (d), experimental data at these exterior viscosity values are not available and hence we display those from the close viscosity values for comparison. To see this figure in color, go online.

We now look at the individual influences from ${\mu }_i$ and ${\mu }_s$ for a certain flow condition (i.e., fixed ${\mu }_o$ and $\dot{\gamma }$). To show the $TTF\sim ({\mu }_i,{\mu }_s)$ dependence more clearly, in addition to the side strips on the right of each panel in Fig. 8, we plot the simulated $TTF$ results at the intermediate shear rate for each exterior viscosity considered in this study in Fig. 9. The dashed line is the interpolated value from the experimental results in (26) for that shear flow condition $({\mu }_o,\dot{\gamma })$. In general the simulated $TTF$ reduces with both membrane viscosity ${\mu }_s$ and interior fluid viscosity ${\mu }_i$ (see Fig. 9 b, c, and d). This reduction is caused by the additional dissipation in the membrane and/or the interior fluid, and this observation is consistent with previous studies (21,24). For the values considered here, it appears that the membrane viscosity ${\mu }_s$ is more influential on the frequency $TTF$ than the interior viscosity ${\mu }_i$. This might be associated with the low interior-exterior fluid viscosity ratio $\lambda ={\mu }_i/{\mu }_o$: previous studies showed that TT parameters do not respond to the change in interior viscosity much when $\lambda <1$ (28,46,47,62,64). Exceptions are noted for cases with lowest exterior fluid viscosity of ${\mu }_o=14.6$ cP and highest membrane viscosity of ${\mu }_s={10}^{-6}$ mPa in Fig. 9 a: at a fixed interior viscosity ${\mu }_i$, $TTF$ reduces with ${\mu }_s$ from 0 to $5\times {10}^{-7}$ mPa; however, it starts to increase at ${\mu }_s={10}^{-6}$ mPa. This is caused by the high membrane viscosity, which strongly restricts the cell deformation and thus slows down the cell rotation. Similar variation behaviors of the rotation frequency with interior and membrane viscosities have been noted in our previous study for spherical capsules (46). The membrane viscosity effect for capsule and droplet dynamics is often represented by the Boussinesq number $B{q}_s={\mu }_s/{\mu }_{o}a$, where $a$ is the radius of the capsule or droplet. Using the equivalent radius $a=2.82\mu$m for an RBC, the corresponding $B{q}_s$ value increases from 2.43 for ${\mu }_s={10}^{-7}$ mPa to 24.3 for ${\mu }_s={10}^{-6}$ mPa for the relatively low suspending viscosity ${\mu }_o$ = 14.6 cP in Fig. 9 a. The larger suspending viscosity ${\mu }_o$ values reduce the Boussinesq number $B{q}_s$ to 1.23–12.3 in Fig. 9 b, 0.63–6.34 in Fig. 9 c, and 0.34–3.41 in Fig. 9 d (see Table 2b). Li and Zhang (46) had shown that, with the increase of the Boussinesq number, the TT rotation frequency of a spherical capsule decreases first and then starts to increase. It seems that, for the RBC system considered here, the cases in Fig. 9 b–d, fall before the turning point and we see monotonic decrease of $TTF$ with ${\mu }_s$. However, in Fig. 9 a, the turning point is reached and we observe a minimum $TTF$ at ${\mu }_s\sim 5\times {10}^{-7}$ mPa. In addition, it is also noted in (46) that the turning point depends on the interior-exterior viscosity ratio $\lambda ={\mu }_i/{\mu }_o$: the minimum $TTF$ location shifts rightward (i.e., to larger $B{q}_s$ values) as $\lambda$ decreases. In Fig. 9, using the intermediate cytoplasm viscosity ${\mu }_i=10$ cP, we have $\lambda =0.68$ for ${\mu }_o=14.6$ cP, 0.35 for ${\mu }_o=28.9$ cP, 0.18 for ${\mu }_o=55.9$ cP, and 0.10 for ${\mu }_o=104$ cP. The smaller $\lambda$ values imply that the decreasing part of the $TTF\sim {\mu }_s$ curve extends further to a higher membrane viscosity; and this might also be responsible for the monotonic $TTF$ decrease in Fig. 9 b–d. Finally, the shear rate $\dot{\gamma }$ and the biconcave RBC shape could also affect the RBC TT dynamics, and more detailed investigations would be necessary in the future. Nevertheless, without considering the membrane viscosity (i.e., ${\mu }_s$ = 0), the simulated $TTF$ is always higher than the experimental data, even using a high cytoplasm viscosity ${\mu }_i=20$ cP. This suggests that the membrane viscosity cannot be neglected for a good agreement of $TTF$ in Figs. 8 and 9.

Figure 9.

Dependence of the rotation frequency $TTF$ on the cytoplasm viscosity ${\mu }_i$ and membrane viscosity ${\mu }_s$ at the intermediate shear rates for different exterior fluid viscosities. The dashed lines represent the interpolated $TTF$ values from experimental data from (26) (gray symbols in Fig. 8). To see this figure in color, go online.

When comparing the simulation results (color symbols) with the interpolated experimental values (dashed line) in Fig. 9, it is not difficult to select cell viscous property values for a good agreement to the experimental data for $TTF$. For example, we can choose the cytoplasm viscosity as ${\mu }_i=10$ cP and assign different membrane viscosity values for different shear conditions: ${\mu }_s=2$ mPa for $\dot{\gamma }=75$ s⁻¹ in Fig. 9 a, ${\mu }_s=3$ mPa for $\dot{\gamma }=25$ s⁻¹ in Fig. 9 b, ${\mu }_s=5$ mPa for $\dot{\gamma }=12.5$ s⁻¹ in Fig. 9 c, and ${\mu }_s=9$ mPa for $\dot{\gamma }=7.5$ s⁻¹ in Fig. 9 d. The fitting membrane viscosity ${\mu }_s$ is smaller for a higher shear rate $\dot{\gamma }$, the so-called shear-thinning behavior discussed in the recent studies by Tsubota (21) and Matteoli et al. (24). This apparent shear-thinning ${\mu }_s\sim \dot{\gamma }$ relation is also evident in Fig. 10, where all the simulation and experimental results are plotted together in a single ${\mu }_s\sim \dot{\gamma }$ graph. The dashed lines connecting experimental data points intersect with individual vertical simulation symbol clusters at the lower ends for lower shear rates $\dot{\gamma }<30$ s⁻¹, and the intersection location moves to the central region of the clusters for $\dot{\gamma }>30$ s⁻¹. Recall that the color symbols in each vertical cluster correspond to different $({\mu }_i,{\mu }_s)$ combinations under the same shear flow condition, with the upper end of high $TTF$ from low ${\mu }_s$ (see Fig. 9). Accordingly, a better agreement can be expected by assigning a higher membrane viscosity at the low shear rate region and vice versa—the shear-thinning dependence for membrane viscosity. Please note that all these observations and discussions are based on the $TTF$ data and that the other two TT characteristic parameters, the deformation index and the inclination angle, have not been considered yet.

Figure 10.

A collective $TTF\sim \dot{\gamma }$ plot for all 192 cases simulated in this research (color symbols) and those measured in (26) (gray symbols with error bars for data uncertainty). The dashed lines connect experimental data points of a same exterior viscosity. For the color symbols, we adopt different fill colors for the interior viscosity and different symbol shapes for the membrane viscosity. To see this figure in color, go online.

RBC orientation

Next, we look at the inclination angle $\theta$ for TT RBCs of different cytoplasm and membrane viscosities and under different flow conditions (Fig. 11). For the four exterior fluid viscosity values utilized in our simulations, experimental results for ${\mu }_o=55.9$ cP (Fig. 11 c) and 104 cP (Fig. 11 d) are available from Fischer and Korzeniewski (27); and hence they can be compared with our simulation results directly. For the other two ${\mu }_o$ values, we show the experimental data obtained with close suspending viscosities as labeled in Fig. 11 a and b. To have a clear view of the effects of cytoplasm and membrane viscosities on the cell orientation, we also enlarge the simulation symbol clusters on the right of each panel in Fig. 11 and plot the $\theta$ variations with ${\mu }_s$ and ${\mu }_i$ for the intermediate shear rate for each exterior viscosity ${\mu }_o$ in Fig. 12. The relatively large variation in $\theta$ from simulations is due to the sensitivity of the calculated $\theta$ value on the deformed cell shape via the equivalent inertia tensor method (47). This is evident by looking at the cell shapes at ${\mu }_s={10}^{-7}$ mPa in Fig. 6 and the $\theta$ variation in Fig. 7 d: the cell orientation is visually constant during the TT period; however, the calculated $\theta$ value still varies between 16.30 and 22.08°.

Figure 12.

Dependence of the inclination angle $\theta$ on the cytoplasm viscosity ${\mu }_i$ and membrane viscosity ${\mu }_s$ at the intermediate shear rates for different exterior fluid viscosities. The dashed lines represent the interpolated $TTF$ values from experimental data from (27) (gray symbols in Fig. 11). The error bars indicate the angle variation during a TT period for simulations. To see this figure in color, go online.

It can be seen from Fig. 11 that the inclination angle $\theta$, in general, decreases with shear rate for a same exterior viscosity ${\mu }_o$; and this is consistent with findings in previous studies for spherical capsules of elastic membranes (29,46,47,64,65). This behavior is reasonable since a higher shear rate generates stronger shear stresses on the cell membrane, and thus aligns the deformed cell orientation closer to the flow direction. Overall, the interior viscosity ${\mu }_i$ and the membrane viscosity ${\mu }_s$ have similar suppressing effects on the inclination angle; however, some irregularity does exist in several cases. For example, in Fig. 12 a, for ${\mu }_i=6$ and 20 cP, $\theta$ decreases with ${\mu }_s$ first up to ${\mu }_s=5\times {10}^{-7}$ mPa and then starts to increase as ${\mu }_s$ further increases to ${10}^{-6}$ mPa; and in Fig. 12 a–c, for ${\mu }_s=5\times {10}^{-7}$ mPa, the minimum $\theta$ values are associated with ${\mu }_i=6$ cP. Li and Zhang (46) recently studied the TT behaviors of a spherical capsule with viscoelastic membrane and their results revealed complex relationships between the inclination angle $\theta$ and the interior and membrane viscosities. In specific, they found that, as the membrane viscosity increases, the inclination angle $\theta$ decreases first, gradually reaches the minimum point, and then starts to increase back. Similar to our discussions above for the TT frequency in Fig. 9, the smaller Boussinesq number $B{q}_s$ values in Fig. 12 b–d, only cover the early decreasing part of $\theta$, and the turning point is not reached in our simulations. On the other hand, with a relatively large range of $B{q}_s$ in Fig. 12 a, the turning point of the minimum $\theta$ becomes visible at ${\mu }_s\sim 5\times {10}^{-7}$ mPa. Compared with the spherical shape and the same capillary number $Ca=0.3$ in (46), the biconcave RBC shape and the different shear flow conditions (different capillary numbers and different interior-exterior viscosity ratios in dimensionless analysis) considered in our current simulations complicate the $\theta$ behavior further; however, a thorough and detailed investigation on this topic is beyond the scope of this study.

Similar to the observations for $TTF$ in TT frequency, the experimental $\theta$ results from (27) are covered by our simulations with different cytoplasm and membrane viscosities; and simply increasing the interior viscosity ${\mu }_i$ without considering the membrane viscosity cannot generate a favorable comparison with experiments. The quick reduction in $\theta$ with membrane viscosity ${\mu }_s$ makes it possible to reproduce the experimental results by tuning the ${\mu }_s$ values. According to Fig. 12, a fairly good agreement can be established when setting ${\mu }_i=10$ cP and ${\mu }_s=4\sim 5\times {10}^{-7}$ mPa. The shear-thinning ${\mu }_s\sim \dot{\gamma }$ relationship observed in $TTF$ is not necessary here for the inclination angle $\theta$.

RBC deformation

Finally, we examine the RBC deformation index $L_p/B$ from various cytoplasm and membrane viscosity values and flow conditions. Our simulation results are compared with experimental measurements from Fischer (26) and Fischer and Korzeniewski (16) in Fig. 13. As mentioned in TT frequency, the experiments in (26) employed dextran solutions of different viscosities from those adopted in our current simulations, and thus we display the experimental results of close exterior viscosity ${\mu }_o$ from (26) for comparison. For the results from (16), since we use the same set of ${\mu }_o$ and $\dot{\gamma }$ values in simulations, a direct comparison can be made; however, this poses difficulties in the result presentation in Fig. 13: the simulation and experimental symbols overlap each other since they are of the same shear rates. For this reason, the experimental data points from (16) (black squares) are shifted slightly to the right in Fig. 13; however, the solid lines are still constructed based on the real (not shifted) experimental values. Overall, for both the numerical and experimental results, $L_p/B$ increases with the shear rate $\dot{\gamma }$ since the cell is more elongated due to the higher shear stress exerted on membrane at higher shear rates. Large variations are noted for both the simulated and experimental cell deformation index. As Fischer and Korzeniewski explained in (16), the wide range of the measured $L_p/B$ could be attributed to the unsteady shear rate in the rheoscope, the intraindividual distribution of the cell properties, such as the membrane shear modulus and membrane viscosity, as well as the oscillations in cell inclination and shape during the TT motion. For the simulation results, those uncertainty factors in shear rate and cell properties are excluded, and the error bars indicate the variation limits of the deformation index during a TT cycle as shown in Fig. 7. The $L_p/B$ variation amplitude is indicated by the vertical distance between the error bars and the mean value symbol in Fig. 13, and this can be seen more clearly in Fig. 14 where the results for the intermediate shear rate of each exterior fluid viscosity are plotted. It is interesting to note that the $L_p/B$ variation amplitude is larger for low membrane viscosity and becomes smaller for high membrane viscosity. This might be related to the relatively higher inclination angle $\theta$ for low membrane viscosity values (see Fig. 12): the projected length $L_p$ can be approximated as $L_p\approx L\cos \theta$, where $L$ is the cell length along its elongated axis. At a lower shear rate, the inclination angle ($\theta$ = 20–25°) is relatively large (see Fig. 11) and the $\theta$ oscillation can generate a relatively large disturbance in the projected length $L_p$. Obviously variations in the cell length $B$ also play an important role in determining the deformation index $L_p/B$.

Figure 13.

Comparison of the simulated deformation index $L_p/B$ (color symbols) to experimental measurements by Fischer (26) (black symbols) and Fischer and Korzeniewski (16) (gray symbols) under different shear conditions. For the color symbols, we adopt different fill colors for the interior viscosity and different symbol shapes for the membrane viscosity. The error bars show the $L_p/B$ variation during a TT period for the simulation results or the distributions of the measured $L_p/B$ in experiments. Experimental data at the four exterior viscosity values considered in our simulations are not available in (26) and hence we display those from the close viscosity values for comparison. The error bars indicate the data uncertainty in measurement for experiments or the deformation index variation during a TT period for simulations. To see this figure in color, go online.

Figure 14.

Dependence of the deformation index $L_p/B$ on the cytoplasm viscosity ${\mu }_i$ and membrane viscosity ${\mu }_s$ at the intermediate shear rates for different exterior fluid viscosities. The dashed lines represent the interpolated $L_p/B$ values from experimental data from (26) (gray symbols in Fig. 13), and the solid lines for the interpolated values from (16) (black symbols in Fig. 13). The error bars indicate the deformation index variation during a TT period for simulations. To see this figure in color, go online.

As for the individual effects from ${\mu }_i$ and ${\mu }_s$ on cell deformation $L_p/B$, we can look at the side symbol strips in Fig. 13 and more clearly the $L_p/B$ plots in Fig. 14. The decrease in $L_p/B$ with ${\mu }_s$ can be attributed to the reduced cell length because of the stronger constraint on cell deformation for larger ${\mu }_s$ values (see Fig. 6 b). The effect of the interior fluid viscosity ${\mu }_i$ is relatively weak; especially for ${\mu }_i$ = 10–20 cP. It is interesting to note that, in general, ${\mu }_i$ has an opposite influence on $L_p/B$ than ${\mu }_s$: the deformation index $L_p/B$ increases with the interior viscosity ${\mu }_i$, and the ${\mu }_i$ influence is more significant at higher shear rates. Few exceptional cases are noted at high membrane viscosity ${\mu }_s={10}^{-6}$ in Fig. 14 a; and this should be related to the different behaviors of these cases in $TTF$ (Fig. 9 a) and $\theta$ (Fig. 12 a). The enhancing effect on cell deformation from ${\mu }_i$ seems contradictory to results for spherical capsules in previous studies (28,62,66), which found that increasing ${\mu }_i$ reduces the capsule deformation. It should be noted that fundamental differences exist between our current study and these previous publications. First, in previous studies, the cell deformation is quantified by the Taylor deformation index $D_{xz}$ given in Eq. 21, which uses the cell length $L$ and width $W$ in the shear plane (the $XZ$ plane). However, here for a direct comparison with experimental results, we adopt the ratio of the projected cell length $L_p$ to the width $B$ in the $XY$ plane. While the projected length $L_p$ might be approximately related to the cell length $L$ via the inclination angle $\theta$, $B$ and $W$ are measured in different directions and they can be largely different for RBCs. For example, for the undeformed RBC at the beginning of the simulation (see the first images in Fig. 6 a and b), B = 7.91 $\mu$m (the cell diameter) and the cell thickness $W$ varies from 0.819 $\mu$m at the center to 2.520 $\mu$m at the rim. In addition, we believe that the biconcave RBC shape must have a dramatic impact on the cell dynamics. For example, the tumbling or swinging motions are not a concern for spherical capsules, but the particular cell geometry greatly complicates the RBC motion and deformation in shear flow, especially when the membrane viscosity is involved (61).

Despite the large uncertainty and wide variations for measured and simulated $L_p/B$ in Fig. 13, one can see that the experimental lines pass through the simulation result clusters at approximately the middle for lower shear rates with ${\mu }_o=14.6$ and 28.9 cP, and the intersection location shifts upward for higher shear rates with ${\mu }_o=55.9$ and 104 cP. Since the simulated $L_p/B$ is larger for low ${\mu }_s$ values and smaller for high ${\mu }_s$ values, this observation implies that one should adopt a larger ${\mu }_s$ for the higher shear rate cases in Fig. 13 a, and adopt a smaller ${\mu }_s$ for the lower shear rate cases in Fig. 13 d. In contrary to the observation in TT frequency for $TTF$, this actually suggests a shear-thickening relationship between ${\mu }_s$ and $\dot{\gamma }$: the membrane exhibits a larger viscosity when it undergoes a higher shear rate. The apparent shear-thickening membrane viscosity can also be inferred from Fig. 14. Here two straight lines are displayed for each (${\mu }_o$, $\dot{\gamma }$) flow condition: the solid line is based on the experimental results of nearby shear rates but the same exterior viscosity from (16), while the dashed line is obtained by a bi-linear interpolation of experimental results of nearby shear rates and exterior viscosities from (26). Although it is difficult to find the best value for each flow condition in Fig. 14, and the experimental straight lines pass above the numerical $L_p/B$ curves with no intersection in Fig. 14 c and d, (it happens that, at the intermediate shear rates for ${\mu }_o=55.9$ and 104 cP, the experimental lines go beyond our simulation clusters in Fig. 13 c and d), it appears that a higher interior viscosity ${\mu }_i$ is required to match the experimental values at $\dot{\gamma }=12.5$ and 7.5 s⁻¹, and the membrane viscosity value should increase with the shear rate $\dot{\gamma }$. There is another way to see the unfitness of the shear-thinning relation from $TTF$ for the cell deformation $L_p/B$: for the very low shear rate case with ${\mu }_o=104$ cP and $\dot{\gamma }=3.75$ s⁻¹, the $TTF$ result in Fig. 9 d suggests a high membrane viscosity ${\mu }_s={10}^{-6}$ mPa, but such a high membrane viscosity value gives the largest numerical-experimental difference for cell deformation $L_p/B$.

Summary and further discussion

In this study, extensive 3D simulations have been performed for the TT dynamics of RBCs in shear flows by considering various cytoplasm and membrane viscosities and flow conditions, and the representative TT characteristics, including the rotation frequency, cell deformation $L_p/B$, and inclination angle $\theta$, have been compared with experimental measurements available in the literature. Validation tests have also been conducted for the stretching process of an RBC in optical tweezers, and the good agreement to experiments confirms that appropriate membrane elasticity modulus values have been utilized in our RBC model. Similar to recent studies (21,24), the $TTF$ results from our simulations suggest a shear-thinning dependence of the membrane viscosity on the shear rate. However, when looking at the other TT parameters, a constant membrane viscosity value at $4\sim 5\times {10}^{-7}$ mPa can yield a good agreement for the inclination angle, while a shear-thickening relation between the membrane viscosity and flow shear rate is required for a reasonable simulation-experiment match for the cell deformation. The different membrane rheological behaviors noted in our results could be due to several factors, including the variations among the RBCs being tested in experiments, uncertainties and errors in measurements, and inaccurate membrane properties used in simulations. Also, due to the actin filaments in the cytoplasm fluid, it may exhibit viscoelastic behaviors (67,68) instead of a pure Newtonian fluid as considered in our simulations. Nevertheless, claiming the shear-thinning membrane rheology based only on the $TTF$ results without considering other cell dynamics parameters, as in (21,24), appears premature. In addition, even with fixed cell geometry and membrane elasticity parameters, fairly complicated RBC behaviors have been observed in our simulations in responding to changes in cytoplasm viscosity, membrane viscosity, suspension viscosity, and shear rate.

We are aware that several concerns and limitations exist in this study. The experimental results used in our comparisons were obtained from blood samples from different individuals, and large interindividual and $TTF$ intraindividual variations may exist in RBC shape, size, and all other cellular properties. However, our simulations have assumed that the same set of cell parameters can apply to these different samples. To improve, it would be desirable to perform all such experiments using RBCs from a same blood sample. The empirical measurements require image analysis of microscope photographs, and considerable errors could be introduced for the low image resolution. Nevertheless, there is no argument about the invaluable role of experimental measurements in validating theoretical models and numerical simulations; however, considering the inherent uncertainty in system parameters and inevitable measurement errors in RBC experiments, one should not pursue a perfect match for one particular aspect of the system performance between simulations and experiments. Based on the simulation results, no conclusive remarks can be made for the ${\mu }_s\sim \dot{\gamma }$ relationship at present. In this study, we have been focusing on the TT regime of RBCs in shear flow; however, the tumbling-to-TT transition process (27,61,69,70) could be valuable for further investigations of the membrane mechanics and RBC dynamics. We would like to clarify that we are not attempting to oppose or deny the possible shear-thinning rheology of RBC membrane viscosity; instead, we believe that more carefully designed investigations are necessary to establish an appropriate ${\mu }_s\sim \dot{\gamma }$ model and to determine its parameter values.

Author contributions

A.R. developed the computer programs, carried out all simulations, collected and analyzed the data, and drafted the manuscript. J.Z. initialized and supervised the research and revised the manuscript.

Editor: Padmini Rangamani