Modeling of Biomechanics and Biorheology of Red Blood Cells in Type 2 Diabetes Mellitus

Abstract

Erythrocytes in patients with type-2 diabetes mellitus (T2DM) are associated with reduced cell deformability and elevated blood viscosity, which contribute to impaired blood flow and other pathophysiological aspects of diabetes-related vascular complications. In this study, by using a two-component red blood cell (RBC) model and systematic parameter variation, we perform detailed computational simulations to probe the alteration of the biomechanical, rheological, and dynamic behavior of T2DM RBCs in response to morphological change and membrane stiffening. First, we examine the elastic response of T2DM RBCs subject to static tensile forcing and their viscoelastic relaxation response upon release of the stretching force. Second, we investigate the membrane fluctuations of T2DM RBCs and explore the effect of cell shape on the fluctuation amplitudes. Third, we subject the T2DM RBCs to shear flow and probe the effects of cell shape and effective membrane viscosity on their tank-treading movement. In addition, we model the cell dynamic behavior in a microfluidic channel with constriction and quantify the biorheological properties of individual T2DM RBCs. Finally, we simulate T2DM RBC suspensions under shear and compare the predicted viscosity with experimental measurements. Taken together, these simulation results and their comparison with currently available experimental data are helpful in identifying a specific parametric model—the first of its kind, to our knowledge—that best describes the main hallmarks of T2DM RBCs, which can be used in future simulation studies of hematologic complications of T2DM patients.

Introduction

Diabetes mellitus (DM), the fastest growing chronic disease worldwide, is a metabolic dysfunction characterized by elevated blood glucose levels (hyperglycemia) (1). Type 2 DM (T2DM) is the most common form of diabetes. People with T2DM demonstrate significantly higher mortality rates relative to nondiabetics due to an increased risk of developing macro- and microvascular complications (2). Hemorheological abnormalities that emerge in diabetic patients play a key role in the pathogenesis and progression of life-threatening coronary and peripheral artery diseases. One of the hemorheological determinants is the impaired deformability of red blood cells (RBCs) involved in T2DM (3). A healthy human RBC is a nucleus-free cell; it is primarily comprised of a fluid-like lipid bilayer contributing to the bending resistance, an attached spectrin network (cytoskeleton) maintaining cell shape and facilitating its motion, and transmembrane proteins bridging the connections between lipid and spectrin domains (4). Owing to the fluid nature of the lipid bilayer and elastic nature of the cytoskeleton, the RBC is capable of dramatic deformations when passing through narrow capillaries as small as 3 μm in diameter without any damage.

For RBCs in pathological conditions, the alterations in cell geometry and membrane properties of diseased RBCs could lead to impaired functionality including loss of deformability (5, 6). For example, the shape distortion and membrane stiffening of RBCs induced by parasitic infectious diseases like malaria (7, 8) and certain genetic blood disorders like sickle cell disease (9) cause increased cell rigidity and decreased cell deformability. Impairment of RBC deformability has also been demonstrated in T2DM. For example, using micropipette aspiration and filtration techniques, McMillan et al. (10) and Kowluru et al. (11) showed that T2DM RBCs are less deformable and more fragile compared to nondiabetic subjects. Agrawal et al. (12) found that the size of a T2DM RBC is larger than that of a normal RBC because of the possible metabolic disturbances. Babu and Singh (13) showed that the development of irregularity in the contour of the T2DM RBCs under hyperglycemia would cause a significant reduction in cell deformability. In addition, several studies using atomic force microscopy (AFM) directly examined the biomechanical properties of diabetic RBCs and demonstrated that they are less deformable than normal RBCs (Fig. 1), which could be due to the oxidation and glycosylation of hemoglobin and proteins on cell membrane under abnormal glycemic conditions (14, 15, 16).

Figure 1.

(a) Young’s modulus of normal and diabetic RBCs measured in experiments, with data as follows: crosses, Fornal et al. (14); triangles, Ciasca et al. (15); squares, Zhang et al. (16); circles, Lekka et al. (90). (b) Sketch of the RBC models with equilibrium biconcave (S/V = 1.44) and near-oblate (S/V = 1.04) shapes. To see this figure in color, go online.

RBC deformability is an important hemorheological parameter in determining whole-blood viscosity and, hence, blood-flow resistance in the microcirculation (17, 18, 19). Increased blood viscosity has been demonstrated in patients with T2DM, inducing insufficient blood supply and vascular damage, and eventually leading to diabetic microangiopathy and other circulation problems. For example, Skovborg et al. (17) revealed that the blood viscosity of diabetic subjects is ∼20% higher than that of controls. Ercan et al. (20) suggested that elevated plasma cholesterol contributes to increased blood viscosity by an additional effect of hyperglycemia in T2DM patients. In long-standing DM with nonproliferative retinopathy, Turczyński et al. (21) showed that blood viscosity is positively correlated with retinopathy severity, which is attributed to the decrease of RBC deformability.

Along with the aforementioned experimental studies, recent advances in computational modeling and simulation enable investigation of a broad range of biomechanical and rheological blood-related problems at different length scales (22, 23, 24, 25, 26, 27, 28, 29). In recent years, mesoscopic-particle-based RBC models, which treat both the fluid and the membrane as particulate materials, are increasingly popular as a promising tool for modeling the structural, mechanical, and rheological properties of RBCs in normal and pathological conditions (30, 31, 32, 33). Two different particle-based RBC models using dissipative particle dynamics (DPD) (34), i.e., one-component RBC models (35, 36) and two-component RBC models (37), have been developed. In general, one-component RBC models depict the cell membrane as a single shell with effective properties that represent the combined effects of the lipid bilayer and the spectrin network (38, 39, 40, 41, 42, 43, 44); two-component RBC models treat the lipid bilayer and the cytoskeleton as two distinct components but also explicitly incorporate bilayer-cytoskeletal interactions. Therefore, two-component RBC models are capable of modeling the biophysics of RBCs arising from the interactions between the lipid bilayer and the cytoskeletal network (37). Such models have been successfully applied to quantify molecular-level mechanical forces involved in bilayer-cytoskeletal dissociation (37, 45), predict mechanical properties of defective RBCs in hereditary spherocytosis with different cytoskeleton connectivity (46), and discover new mechanisms responsible for the stiffening of malaria-infected RBCs (8, 46).

In this study, we extend the two-component whole-cell model to T2DM RBCs and investigate the morphological and biomechanical characteristics of T2DM RBCs and hemorheological properties of T2DM blood. Specifically, we test and validate the whole-cell model through rigorous comparisons with experimental data from five different sets of independent measurements that probe different aspects of biomechanical and rheological properties of T2DM RBCs, including RBC stretching deformation and shape relaxation, membrane fluctuations, RBC dynamics in shear flow, and blood viscosity in T2DM. The rest of the article includes a brief description of the development of diabetic RBC models, detailed simulation results and discussion, and a summary of major findings and conclusion.

Methods

In this study, we probe the biomechanics, rheology, and dynamics of T2DM RBCs with the help of the two-component RBC model based on the DPD method. The intracellular and extracellular fluids are modeled by collections of free DPD particles and their separation is enforced by bounce-back reflections at the RBC membrane surface. The RBC membrane interacts with the fluid and wall particles through DPD forces, and the system temperature is maintained by the DPD thermostat. For completeness, we review briefly the two-component RBC model below, whereas for a detailed description of the DPD method and the RBC model, we refer to (34, 35, 36, 37).

Two-component RBC model

In the two-component whole-cell model, the cell membrane is modeled by two distinct components, i.e., the lipid bilayer and the cytoskeleton, and each component is constructed by a two-dimensional triangulated network with $N_{\text{v}}$ vertices. In general, the two-component RBC model takes into account the elastic energy, bending energy, bilayer-cytoskeleton interaction energy, and constraints of the fixed surface area and enclosed volume, given by

$$V\left({\mathbf{x}}_i\right)=V_{\text{s}}+V_{\text{b}}+V_{\text{a}}+V_{\text{v}}+V_{\text{int}}$$ (1)

where $V_{\text{s}}$ is the elastic energy that mimics the elastic spectrin network, given by

$$V_{\text{s}}=\sum _{j\in 1\dots N_{\text{s}}}\left[\frac{k_{\text{B}}T{l}_{\text{m}}\left(3x_j^2-2x_j^3\right)}{4p\left(1-x_j\right)}+\frac{k_{\text{p}}}{\left(n-1\right)l_j^{n-1}}\right],$$ (2)

where $l_j$ is the length of the spring j, $l_{\text{m}}$ is the maximum spring extension, $x_j=l_j/l_{\text{m}}$, p is the persistence length, $k_{\text{B}}T$ is the energy unit, $k_{\text{p}}$ is the spring constant, and n is a specified exponent. The bending resistance from the lipid bilayer of the RBC membrane is modeled by

$$V_{\text{b}}=\sum _{j\in 1\dots N_{\text{s}}}{k}_{\text{b}}\left[1-\text{cos}\left({\theta }_j-{\theta }_0\right)\right],$$ (3)

where $k_{\text{b}}$ is the bending constant, ${\theta }_j$ is the instantaneous angle between two adjacent triangles having the common edge j, and ${\theta }_0$ is the spontaneous angle.

Constraints on the area and volume conservation of RBCs are imposed to mimic the area-preserving lipid bilayer and the incompressible interior fluid. The corresponding energy is given by

$$V_{\text{a}}=\sum _{j\in 1\dots N_t}\frac{k_{\text{d}}{\left(A_j-A_0\right)}^2}{2A_0}+\frac{k_{\text{a}}{\left(A_{\text{cell}}-A_{\text{cell},0}^{\text{tot}}\right)}^2}{2A_{\text{cell},0}^{\text{tot}}},$$ (4)

$$V_{\text{v}}=\frac{k_{\text{v}}{\left(V_{\text{cell}}-V_{\text{cell},0}^{\text{tot}}\right)}^2}{2V_{\text{cell},0}^{\text{tot}}},$$ (5)

where $N_{\text{t}}$ is the number of triangles in the membrane network, $A_0$ is the triangle area, and $k_{\text{d}}$, $k_{\text{a}}$, and $k_{\text{v}}$ are the local area, global area, and volume constraint coefficients, respectively. The terms $A_{\text{cell},0}^{\text{tot}}$ and $V_{\text{cell},0}^{\text{tot}}$ represent the specified total area and volume, respectively.

The bilayer-cytoskeleton interaction potential, $V_{\text{int}}$, is expressed as a summation of harmonic potentials given by

$$V_{\text{int}}=\sum _{j,j^{\prime }\in 1\dots N_{\text{bs}}}\frac{k_{\text{bs}}{\left(d_{j{j}^{\prime }}-d_{j{j}^{\prime },0}\right)}^2}{2},$$ (6)

where $k_{\text{bs}}$ and $N_{\text{bs}}$ are the spring constant and the number of bond connections between the lipid bilayer and the cytoskeleton, respectively. $d_{jj\prime }$ is the distance between vertex j of the cytoskeleton and the corresponding projection point $j^{\prime }$ on the lipid bilayer, with the corresponding unit vector ${\mathbf{n}}_{j{j}^{\prime }}$; $d_{j{j}^{\prime },0}$ is the initial distance between the vertex j and the point $j^{\prime }$, which is set to zero in the simulations presented here.

Mechanical properties of the RBC membrane

It is known that the membrane elasticity of RBCs characterizes their resistance to deformation, and membrane viscosity characterizes the viscous resistance of the cell membrane to shear deformation (47). Following the linear analysis for a regular hexagonal network by Dao et al. (48), we correlate the model parameters and the network macroscopic elastic properties, i.e., shear modulus (μ), which is determined by

$$\begin{array}{c}\mu =\frac{\sqrt{3}{k}_{\text{B}}T}{4p{l}_{\text{m}}{x}_0}\left(\frac{x_0}{2{\left(1-x_0\right)}^3}-\frac{1}{4{\left(1-x_0\right)}^2}+\frac{1}{4}\right)\\ +\frac{\sqrt{3}{k}_{\text{p}}\left(n+1\right)}{4l_0^{n+1}},\end{array},$$ (7)

where $l_0$ is the equilibrium spring length and $x_0=l_0/l_{\text{m}}$. In addition, the corresponding area compression is given by K = 2μ + $k_{\text{a}}$ + $k_{\text{d}}$ (36, 49).

The relation between the modeled bending constant, $k_{\text{b}}$, and the macroscopic bending rigidity, $k_{\text{c}}$, of the Helfrich model can be derived as $k_{\text{b}}=\left(2/\sqrt{3}\right)k_{\text{c}}$ for a spherical membrane (50). This expression describes the bending contribution of the energy in Eq. 3 but may not fully represent actual bending resistance of the RBC membrane. In the triangulated membrane network, the bending elasticity is expressed in terms of the angle ${\theta }_j$ between neighboring triangles, and the edges of the triangles are modeled as springs. Thus, when the angle ${\theta }_j$ is adjusted, the springs may be stretched as well, which could result in local in-plane strains (deformation). To evaluate such an effect, we estimate the ratio between the bending and shear moduli of RBC membrane, $k_{\text{c}}\text{/}\left({\mu }_0{R}^2\right)$ = $\mathcal{O}\left({10}^{-3}\right)$, where R is the radius of a sphere having the same surface area as that of an RBC. Thus, bending contributes little to the RBC deformation.

In addition, membrane viscosity also plays an important role in the dynamic behavior of RBCs in physiological and pathological conditions. To estimate the effective membrane viscosity, ${\eta }_{\text{m}}$, we simply combine the viscous contributions from the lipid bilayer, ${\eta }_{\text{b}}$, and cytoskeleton, ${\eta }_{\text{s}}$. Following prior work (37), these two viscous terms can be calculated as ${\eta }_i=\sqrt{3}{\gamma }_i^T+\left(\sqrt{3}{\gamma }_i^C/4\right)$ $\left(i=\text{s,}\text{b}\right)$, where ${\gamma }_i^T$ and ${\gamma }_i^C$ are dissipative parameters of the DPD model. It is worth noting that the lipid bilayer is modeled as a two-dimensional triangulated network in this RBC model. Therefore, it cannot capture the diffusion of lipid molecules. However, a higher-fidelity model that includes the two components of the RBC membrane plus transmembrane proteins could address this issue (51, 52). For example, Li et al. (53) have shown that a decrease in lipid bilayer diffusivity (or an increase in lipid bilayer viscosity) does not change the shear modulus of the RBC membrane. We have recently developed a whole-cell RBC model at protein resolution using OpenRBC (33), which could be potentially applied to investigate the diffusion of lipid molecules in the RBC membrane at the whole-cell level.

Parameter estimation

Under physiological conditions, a normal RBC has a distinctive biconcave shape with a large surface area/volume $\left(S/V\right)$ ratio, which facilitates oxygen transport through the cell membrane and contributes to the remarkable cell deformability. In this study, we model a normal RBC (N-RBC) with the following parameters: total number of vertices, $N_{\text{v}}$ = 500; RBC membrane shear modulus, ${\mu }_0$ = 4.73 μN/m; bending modulus, $k_{\text{c}}$ = 2.4 × 10⁻¹⁹ J; and effective membrane viscosity, ${\eta }_{\text{m}}$ = 0.128 Pa·s. We note that ${\eta }_{\text{m}}$ here is the three-dimensional membrane viscosity derived from the geometric relationship ${\eta }_{\text{m}}$= ${\eta }_{\text{m}}^{2\text{D}}\text{/}R$ (54), where ${\eta }_{\text{m}}^{2\text{D}}$ = 0.416 μN s/m is the two-dimensional membrane viscosity approaching the measured value of 0.36 μN s/m in experiments (47). According to Berk et al. (55), the cytoskeleton viscosity is at least one or two orders of magnitude greater than the bilayer viscosity, and therefore we set ${\eta }_{\text{b}}$ = 0.0027 Pa·s and ${\eta }_{\text{s}}$ = 0.125 Pa·s, respectively. In addition, RBC surface area, $A_{\text{cell},0}^{\text{tot}}$ = 132.87 $\mu {\text{m}}^2$, cell volume $V_{\text{cell},0}^{\text{tot}}$ = 92.45 $\mu {\text{m}}^3$, and surface area/volume ratio, $S/V$ = 1.44, are given to our N-RBC model.

Different from the N-RBC, the diabetic RBCs have decreased cell deformability and increased cell volume. Here, we have developed three potential T2DM RBC models (D-RBC1, D-RBC2, and D-RBC3) based on previous in vitro experiments. First, we selected experimentally determined shear modulus data (56) and set the shear modulus of a diabetic RBC model (D-RBC1) to μ = 2.0${\mu }_0$. Second, previous AFM measurements suggest that there is a visible change in cell shape from the normal biconcave shape to a near-oblate shape with a reduced $S/V$ ratio in the pathophysiology of T2DM RBCs (57). Considering this fact, here, we propose a modified diabetic RBC model (D-RBC2) with an oblate shape ($S/V$ $\sim$ 1.04) (Fig. 1 b). Third, lower membrane fluidity and increased membrane viscosity for T2DM RBCs compared to controls have been reported and suggested to be the consequences of nonenzymatic glycation-induced changes in the RBC membrane (58). In addition, previous studies also suggest a comparable shape recovery time $\left(t_{\text{c}}\right)$ between normal and diabetic RBCs (10, 59). It is known that the recovery time, $t_{\text{c}}={\eta }_{\text{m}}\text{/}\mu$, is primarily determined by the viscoelastic properties of the RBC membrane (47, 60). In combination with the above experimental results, we have developed a third diabetic RBC model (D-RBC3) by considering the aberrant cell shape and impaired deformability, as well as adjusted effective membrane viscosity. In summary, the D-RBC1 model holds a biconcave shape ($S/V$ = 1.44), as does that of an N-RBC with an increased μ; the D-RBC2 model takes a near-oblate shape with a reduced $S/V$ ratio ($\sim$ 1.04) accompanied by an increase in μ; and the D-RBC3 model has the same characteristics as the D-RBC2 model but with the additional trait of enhanced ${\eta }_{\text{m}}$. Parameters related to the membrane properties among the different RBC models, including the shear modulus, μ, effective membrane viscosity, ${\eta }_{\text{m}}$, and surface area/volume ratio $\left(S/V\right)$, are summarized in Table 1.

Table 1.

Model Parameters Extracted from Available Measured Data for RBCs in Normal and T2DM Conditions

	N-RBC	D-RBC1	D-RBC2	D-RBC3
μ$\left(\mu \text{N/m}\right)$	4.73 (47)	9.46 (56)	9.46 (56)	9.46 (56)
$S/V$$\left(\mu {\text{m}}^2\text{/}\mu {\text{m}}^3\right)$	1.44 (89)	1.44	1.04 (57)	1.04 (57)
${\eta }_{\text{m}}$$\left(\text{Pa}\cdot \text{s}\right)$	0.128 (37)	0.128	0.128	0.256 (59)

Numbers in parentheses indicate the literature source for the given parameter.

The simulations were performed using a modified version of the atomistic code LAMMPS. The time integration of the motion equations is computed through a modified velocity-Verlet algorithm with $\lambda =0.50$ and time step $\text{Δ}t$ = 1.0 $\times$ 10^–4 τ $\approx$ 0.66 $\mu \text{s}$. It takes (∼1.0–5.0) × 10⁶ time steps for a typical simulation performed in this work. Compared to the one-component RBC model, the two-component RBC model takes about the same computational time for the prediction of a single-cell dynamic response. However, for a blood-flow system with a large number of RBCs (for example, for a hematocrit level at 45.0%), we found that the computational costs are increased by a factor of 3–5 for the two-component RBC model compared to those for the one-component RBC model.

Results and Discussion

In this section, we employ the two-component RBC model to investigate the mechanics, rheology, and dynamics of T2DM RBCs. First, we probe the static and dynamic responses of T2DM RBCs subject to tensile forcing and quantify the cell deformation. Second, we simulate the dynamic behavior of T2DM RBCs in shear flow and investigate the effect of membrane viscosity on the tank-treading frequency. Finally, we study the biorheological properties of individual T2DM RBCs and predict the shear viscosity of T2DM RBC suspensions.

Mechanical properties of T2DM RBCs

To investigate the elastic response of RBCs in normal and T2DM conditions, we subject the RBC to stretching deformation by imposing an external tensile force at diametrically opposite directions analogous to that in optical tweezers experiments (61). In our simulations, the total stretching force, $F_{\text{s}}$, is applied in the opposite direction to $\mathit{ϵ}{N}_{\text{v}}$ ($\mathit{ϵ}$ = 0.05) vertices of the lipid bilayer component of the RBC membrane. The stretching response of the RBC is then characterized by the variation of axial $\left(D_{\text{A}}\right)$ and transverse $\left(D_{\text{T}}\right)$ diameters of the RBC (Fig. 2). For D-RBC1, we find a significant decrease in $D_{\text{A}}$ when compared with that of N-RBC, which is due to the increase in shear stiffness of T2DM RBCs. For D-RBC2 with a near-oblate shape, we find a further decrease in $D_{\text{A}}$, indicating a further reduction in cell deformability. This result confirms that the decrease in $S/V$ ratio from $\sim$ 1.44 to $\sim$ 1.04 causes a reduction in RBC deformability, which is consistent with our recent computational results that the $S/V$ ratio is one of the main dominant factors in cell deformability (43). For D-RBC3, we find that the $D_{\text{A}}$ values are almost exactly the same as those obtaiend for D-RBC2 (Fig. 2). This result demonstrates that the influence of the membrane viscosity on RBC stretching deformation is negligible, because it is performed under the equilibrium stretched state at a given $F_{\text{s}}$. Hence, we conclude that the membrane shear stiffness and $S/V$ ratio are the two most important parameters in static RBC deformation.

Figure 2.

Stretching response of normal and T2DM RBC membrane at different values of the stretching force. The error bars are obtained by increasing or decreasing μ by 20% from the default values (Table 1). The experimental data are adopted from Suresh et al. (61), and the different stretched RBCs at stretching force $F_{\text{s}}$ = 100 pN are presented on the right. To see this figure in color, go online.

In addition, instantaneous membrane fluctuations of RBCs, also called “membrane flickering” (62), are commonly used to characterize the membrane stiffness. Following the work of Fedosov et al. (63) and Peng et al. (37), we then examine the membrane fluctuations of T2DM RBCs by computing instantaneous fluctuation height of the RBC membrane surface (Fig. 3). From this plot, we find a good agreement between the fluctuation distributions in experiments (red line) and simulations (black squares) for normal RBCs. For T2DM RBCs, we find narrower distributions or smaller full-width half-maximum (FWHM) values compared to those for normal RBCs, which shows the influence of local membrane curvature and effective geometry on the membrane fluctuations. These results are consistent with previous simulations of malaria-infected RBCs at the schizont stage (63): When simulations employed a biconcave RBC shape, a wider distribution than that in experiment (62) for the schizont stage was observed; however, by employing a nearly spherical membrane (spherical RBCs at the schizont stage observed in experiments) in the RBC model, the distribution became much narrower and presented a better agreement with the experiments. These computational results show that both membrane shear modulus and shape alteration play important roles in RBC membrane fluctuations. Moreover, the membrane distributions obtained from D-RBC2 and D-RBC3 remain almost the same, indicating a small effect of membrane fluidity on the cell membrane fluctuations. It is interesting to see that our simulation results are qualitatively in accordance with an up-to-date experimental work by Lee et al. (64) showing that the membrane fluctuations of diabetic RBCs are significantly lower than those of nondiabetic RBCs.

Figure 3.

Membrane fluctuation distributions of normal and T2DM RBCs. The solid red line represents experimental data for membrane fluctuation distribution of normal RBCs (62). The full-width, half-maximum fluctuation value is ∼128 nm for N-RBC, 110 nm for D-RBC1, 66 nm for D-RBC2, and 65 nm for D-RBC3. To see this figure in color, go online.

Upon external tensile forces, a normal RBC undergoes elastic deformation, and it restores to its original state when the external force is released. Therefore, the RBC deformability can also be reflected on the shape relaxation of RBCs. It is well-accepted that the shape recovery time $t_{\text{c}}\approx 0.1$ s for normal RBCs (47, 60, 65). When the cell age was taken into account, Williamson et al. (59) found that $t_{\text{c}}=0.127\pm 0.011$ and $0.132\pm 0.018$ s for young RBCs in normal and diabetic conditions, respectively. It increases to $t_{\text{c}}=0.152\pm 0.010$ and $0.149\pm 0.014$ s for old RBCs in healthy and diabetic cells. Hence, the difference in $t_{\text{c}}$ between young and old cells is pronounced, whereas that between normal and diabetic cells is not significant. We then perform a stretching-relaxation test of normal and T2DM RBCs with different model parameters (Fig. 4). We find that the recovery processes are different from each other even though all of these modeled RBCs are able to recover their original shapes. In addition, assuming a pathological RBC state where the bilayer-cytoskeletal interaction is significantly lower, we find an apparent uncoupling between bilayer and cytoskeleton when RBCs are subjected to tensile forcing (Fig. 4 b). Herein, we want to emphasize that in most cases, the one-component and two-component RBC models do not differ too much; however, under extreme mechanical conditions or disease states, the two-component RBC model is needed.

Figure 4.

(a) Dynamic cell deformation and relaxation processes of different RBC models at the forces shown at top. (b) An apparent uncoupling between the lipid bilayer and cytoskeleton occurs by reducing the bilayer-cytoskeletal interaction of the D-RBC3 model in one or two orders of magnitude. (c) Estimated shape recovery time, $t_{\text{c}}$, of different RBC models. t = 0 is the time when the external force is released and the cell starts to recover its original shape. To see this figure in color, go online.

In fact, the dynamic recovery of the RBC, $R\left(t\right)=f\left(D_{\text{A}},D_{\text{T}}\right)$, can be described by a time-dependent exponential decay,

$$R\left(t\right)=\frac{{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_t-{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_{\infty }}{{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_t+{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_{\infty }}\cdot \frac{{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_0+{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_{\infty }}{{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_0-{\left(\frac{D_{\text{A}}}{D_{\text{T}}}\right)}_{\infty }}=\text{exp}\left[-{\left(\frac{t}{t_{\text{c}}}\right)}^{\alpha }\right],$$ (8)

where the subscripts 0 and $\infty$ correspond to the ratios at $t=0$ and $\infty$. The exponent $\alpha =1.0$ is a generalization for estimating $t_{\text{c}}$ (60, 65, 66), whereas $\alpha =0.7$ has been used by Fedosov et al. (36) for the purpose of better fitting their computational results. For comparison, we obtained different curves by fitting the simulation data to exponential functions with $\alpha =1.0$ and $\alpha =0.7$ (see Fig. S1). We found that both fitting curves are in fairly good agreement with the RBC recovery dynamics. In the following work, we adopted $\alpha =1.0$ for estimating $t_{\text{c}}$ of different RBC models.

Fig. 4 c shows the corresponding best fit to the relaxation dynamics in Fig. 4 a. The shape recovery time is estimated to be $t_{\text{c}}$ = 0.11, 0.08, 0.07, and 0.12 s for N-RBC, D-RBC1, D-RBC2, and D-RBC3, respectively. The comparable $t_{\text{c}}$ values for N-RBC and D-RBC3 show a correspondence with the experimental results (59), thus, we conclude that the effective membrane viscosity contributes largely to RBC relaxation. The underestimated $t_{\text{c}}$ value in D-RBC1 and D-RBC2 could attribute to the increased membrane shear stiffness and the reduced S/V ratio. In addition, there is no significant difference in $t_{\text{c}}$ between D-RBC1 and D-RBC2, which is consistent with the experimental results that the geometric structure of the RBC has little effect on the cell recovery process (60, 67). Our computational results on RBC mechanics from stretching behavior to recovery response demonstrate that the RBC membrane elasticity, S/V ratio, and effective membrane viscosity are all essential to probe RBC deformability in T2DM.

Dynamic behavior of individual T2DM RBC in flow

The RBCs in shear flow have been observed to exhibit two primary types of dynamics: tank-treading (TT) motion and tumbling (TB) motion, depending on cell geometry (S/V ratio), cell elasticity (capillary number $C{a}_{\mu }={\eta }_o\dot{\gamma }{R}_0/\mu$, where $R_0$ is the radius of a sphere with the same volume as an RBC), and fluid properties such as viscosity ratio ($\lambda ={\eta }_{\text{i}}\text{/}{\eta }_{\text{o}}$, where ${\eta }_{\text{i}}$ and ${\eta }_{\text{o}}$ are the cytoplasm and suspending fluid viscosities) (23, 68, 69, 70, 71). At low shear rate, $\dot{\gamma }$, the resistance to shear causes RBCs to tumble. As $\dot{\gamma }$ increases, the dynamics of RBCs changes from TB to TT motion (72). Physiologically, the TT motion of RBCs in flowing blood produces a lift force to prevent the RBCs from migration toward the peripheral blood vessel, allowing high flow efficiency and sufficient oxygen transfer in blood circulation (73). In this study, we simulate the TT motion of an RBC in shear flow by placing a single RBC in linear shear flow (a Couette flow) between two planar solid walls. To produce shear flow in the fluidic channel, two solid walls move with the same speed but in opposite directions, and $\dot{\gamma }$ can be mediated by changing the speed of the solid walls. According to previous experimental studies (74, 75), we set ${\eta }_{\text{o}}$ = 0.0289 Pa·s (74) and ${\eta }_{\text{i}}$ = 0.006 Pa·s (75).

Fig. 5 a shows the angular trajectory (θ) as a function of time for different RBC models at $\dot{\gamma }$ = 105 s⁻¹. It is evident that both D-RBC1 and D-RBC2 have TT motion faster than that of N-RBC, which is indicated by a shorter period for the marked particle on the cell membrane to complete a TT cycle, as shown in Fig. 5 b. The accelerated TT motion for D-RBC1 compared to that of N-RBC is probably due to the average elongation decrease with increasing cell membrane stiffness (70). For an oblate shape of D-RBC2, it rotates even faster than for D-RBC1, which may owe to an increased cell thickness, as reported in previous computational work on higher inclination and TT speed of osmotically swollen RBCs (76). In addition, D-RBC3 exhibits a slower rotation and an extended TT period. This result shows that TT motion is sensitive to the membrane viscous dissipation, and although the swollen cell tends to speed up the TT dynamics, the elevated effective membrane viscosity slows it down even more.

Figure 5.

TT motion of an RBC in shear flow. (a) Angular trajectory (θ) of a marked particle in the RBC membrane during the TT motion for different RBC models at shear rate $\dot{\gamma }$ = 105 ${\text{s}}^{-1}$. θ is the inclination angle between the position vector of the marked particle and the flow direction. (b) Corresponding snapshots of different RBC models at t = 0.20, 0.25, 0.30, 0.35, and 0.40 s. To see this figure in color, go online.

TT frequency (f), i.e., the number of TT cycles per second, is an important characteristic in the RBC TT motion. The TT frequency can be estimated from the time-dependent angular trajectories of a marked particle based on f = 1/$P_{\text{tt}}$, where $P_{\text{tt}}$ is the time for an oscillation cycle of the markered particle. In previous studies, Tran-Son-Tay et al. (77) and Williamson et al. (59) investigated the TT frequency of RBCs in normal and diabetic conditions. They obtained a linear relationship between f and $\dot{\gamma }$ and found a slight decrease in TT frequency of diabetic RBCs compared to normal RBCs. To gain further insight into the RBC TT motion, we investigate the TT frequency of an individual RBC in normal and T2DM conditions. As suggested by previous experimental and computational studies on the dynamics of individual RBC in shear flow, RBCs tumble at low shear rates (35, 36, 45, 73). Therefore, we collected and analyzed the RBC TT motion at relatively high shear rates, herein $\dot{\gamma }$ > 45 s⁻¹. Fig. 6 shows that f increases linearly with $\dot{\gamma }$ for all four RBC models. At the same $\dot{\gamma }$, D-RBC2 has the largest f value, N-RBC and D-RBC1 have moderate f value, whereas D-RBC3 has the smallest TT frequency. Comparing with experimental data, we find that our N-RBC and D-RBC3 have the corresponding TT frequency. These simulation results show that in addition to capturing the dynamic behavior of a normal RBC under shear flow, D-RBC3 with explicit consideration of increased effective membrane viscosity leads to an accurate model of the impaired RBC dynamics in T2DM.

Figure 6.

Functional dependence of RBC TT frequency with respect to shear rate $\dot{\gamma }$. The error bars were obtained by increasing or decreasing ${\eta }_{\text{m}}$ by 10% from their default values. Simulation results are compared with experimental data by Fischer (red circle) (74), by Tran-Son-Tay et al. (red cross) (77), and by Williamson et al. (green star) (59). The red and green lines show the linear fits for normal and diabetic RBCs, respectively, in experiments. To see this figure in color, go online.

Rheological properties of T2DM RBCs in shear flow

The rheological behavior of blood samples from nondiabetic and diabetic patients have been studied in experiments (17, 18, 19, 78). Decreased cell deformability and increased cell volume in T2DM have shown significant implications for microcirculatory alterations. Computational modeling and simulations help in predicting how RBCs behave in shear flow and providing insights into how viscous flow transforms RBC shapes, and vice versa, how deformed RBCs distort the surrounding flow (22, 39, 40, 79, 80, 81, 82, 83). Here, to address the effects of cell elasticity and shape on the biorheological behavior of individual T2DM RBCs, we investigate the dynamic behavior of T2DM RBCs in a microfluidic channel with constriction. The microfluidic channel, as illustrated in the inset of Fig. 7, contains a symmetric converging and diverging nozzle-shaped section. At its narrowest domain, the microchannel is 30.0 μm long and 2.7 μm high and has widths ranging from 4.0 to 6.0 μm. The narrow and wide domains are connected by walls formed by stationary DPD particles.

Figure 7.

RBC traversal across a microfluidic channel of 6.0 μm width at various values of pressure difference. The cell transit velocity, v, is defined as the transit distance divided by the average transit time for an RBC traversing the narrow channel, and $\text{Δ}P$ is a local pressure gradient in the channel. For better comparison, the normalized cell transit velocity ($v^{\text{∗}}$ = v/$v_{\text{s}}$) and local pressure difference ($\text{Δ}{P}^{\text{∗}}$ = $\text{Δ}P$/$\text{Δ}{P}_{\text{s}}$) are adopted, where $v_{\text{s}}$ = 1.6 mm/s and $\text{Δ}{P}_{\text{s}}$ = 0.1 kPa. Experiments (open circles) and simulations (solid circles) of a normal RBC traversing a 6-μm -wide channel from Quinn et al. (79) are shown. (Inset) Schematic of the microfluidic channel with a symmetric converging-diverging geometry. To see this figure in color, go online.

Fig. 7 and Movies S1, S2, and S3, show the typical dynamic processes of normal and T2DM RBCs flowing through the converging-diverging microfluidic channel. Our simulations indicate that the RBCs passing through the microchannel are sensitive to the S/V ratio (or cell volume). As shown in Fig. 7, a decrease in the S/V ratio from 1.44 to 1.04 results in a decrease in cell transit velocity when individual RBCs travel through a 6-μm-wide microchannel. Similar to previous computational studies of RBCs and tumor cells passing through narrow slits (43, 84, 85), the cell volume increase (S/V decrease) would slow down the passing process. The average cell transit velocity decreases more significantly, however, when the individual T2DM RBCs pass through a 4-μm-wide microchannel see Movie S3. These simulations reveal that the increase of flow resistance by T2DM RBCs is larger than the resistance by normal RBCs, hence pointing to the importance of the S/V ratio as a determinant of T2DM RBCs traversal across narrow capillaries.

Computational models have also been proven to be an important tool in predicting the macroscopic flow properties of an RBC suspension or blood flow (e.g., yield stress and shear viscosity) from the mesoscopic properties of individual RBCs (e.g., membrane viscosity and cell deformability). Next, we employ our RBC models to examine the blood viscosity over a range of $\dot{\gamma }$ from 1 to 500 s⁻¹. Fig. 8 and Movies S4 and S5 show the relative viscosity of RBC suspensions with the N-RBC and D-RBC3 models. The model predictions clearly capture the increased blood viscosity in T2DM, in good agreement with the experimental data. This abnormality in blood viscosity is attributed to the reduced cell deformability associated with the alterations in cell shape. Our model does not include cell-cell aggregation interactions. Hence, it fails to model the formation of rouleaux structures and a tremendous viscosity increase at low shear rates (38, 86); this can be a topic for future work. Nevertheless, our model with explicit description of the RBC structure and membrane properties could lead to useful insights into the cell mechanistic processes and guide future work for better understanding the correlation between metabolic dysfunction and hematological abnormalities in T2DM.

Figure 8.

Functional dependence of shear viscosity of T2DM RBC suspension on shear rate at hematocrit $H_t=45.0\text{\%}$. Experimental data are as follows: circles are from Skovborg et al. (17), crosses from Zingg et al. (18), and diamonds from Peduzzi et al. (78); red symbols are for normal RBC suspension and green symbols for diabetic RBC suspension. To see this figure in color, go online.

Conclusions

We have presented a two-component whole-cell model that seems to capture the biomechanical, rheological, and dynamic properties of RBCs in T2DM based on currently available data. The RBC membrane is modeled by two distinct networks for representing the spectrin network and the lipid bilayer. We have proposed and examined three potential T2DM RBC models (D-RBC1, D-RBC2, and D-RBC3) based on the alterations in cell shape and membrane viscoelasticity measured experimentally, including increased shear stiffness, altered cell shape, and elevated membrane viscosity.

Using these RBC models, we first studied the static and dynamic deformation of T2DM RBCs subject to different tensile forcing. The significant reduction in stretching response of our diabetic RBC models compared to N-RBCs at the same tensile force shows the large stiffness contribution from membrane shear resistance and a near-oblate structural constraint. However, only the D-RBC3 model can exhibit reasonable relaxation processes in comparison with the corresponding experiments, indicating that membrane viscosity plays an important role in RBC deformability. Second, we quantitatively probed the linear relationship between tank-treading frequency and shear-flow rate of RBCs under normal and diabetic states by our N-RBC and D-RBC3 models. The agreement of the simulation results with the currently available experiments shows that the RBC complex dynamics in shear flow can be predicted by taking into account the membrane elasticity, cell geometry, and viscous dissipation. Finally, we extended the simulation from individual cell dynamics to the collective dynamics of RBC suspensions. We predicted a higher viscosity of T2DM blood compared to that of normal subjects, and the results are also consistent with experimental data.

It is known that the main effect of T2DM on blood properties derives from the uncontrolled concentration of glucose. Glycated hemoglobin (HbA1c), which is formed by a simple chemical reaction between blood glucose and normal hemoglobin in RBCs, reflects the time-averaged blood glucose level in an individual and can be used for the diagnosis of T2DM. Experimental studies have shown that T2DM RBCs are less deformable and more fragile if there is an increased amount of HbA1c (87, 88). However, the precise effect of blood glucose level on cell deformability and blood rheology remains an open question. The current RBC models, including the two-component models used here, still have limitations in facilitating the detailed exploration of diverse biophysical and biomechanical aspects involved in such cases. There is, hence, a compelling need to develop a hybrid model constructed by combining a kinetic model for the formation of HbA1c at the subcelluar level with the particle-based RBC model. Such a model can be used to quantify the functional relationship between glucose (or HbA1c) sensitivity and the biomechanical properties of T2DM RBCs as well as the hemorheological properties of T2DM blood. With the help of more extensive clinical tests and experimental studies, unsolved questions concerning the correlation between the degree of hyperglycemia and the alterations in diabetic hematology could be possibly addressed in the near future.

Author Contributions

H.-Y.C., X.L., and G.E.K. conceived and designed research; H.-Y.C. and X.L. performed research; H.-Y.C., X.L., and G.E.K. analyzed data; H.-Y.C., X.L., and G.E.K. contributed new reagents/analytic tools; and H.-Y.C., X.L., and G.E.K. wrote the article.

Editor: Vivek Shenoy.