Transport Characteristics of Small Molecules Diffusing near Deforming Blood Cells

Abstract

Understanding the transport of small molecules such as oxygen in biological systems requires knowledge about how molecules dynamically interact with these molecules. This study investigates how red blood cells influence the diffusion of small molecules in simple shear flow by coupling Brownian dynamics simulations with a finite element–boundary integral method to simulate particle transport near spherical and red blood cells. The simulation found that the presence of a rotating cell significantly reduces the effective diffusion rate of small molecules. Specifically, the circulatory flow induced by cell deformation during tank-treading motion leads to a diminished mean squared displacement of particles. Notably, a tumbling red blood cell produces a more pronounced effect on particle motion compared with a spherical cell under identical flow conditions. This research has broader implications for understanding complex diffusion processes in various biological systems, by highlighting the complex interactions between cellular motion and molecular transport.

1. Introduction

The transport of oxygen, carbon dioxide, and other small molecules is fundamental for the continued functioning of humans. The main purpose of red blood cells (RBCs) is the mass transport of oxygen throughout the body and of carbon dioxide to the lungs. This transport also has large effects in a variety of diseases, from the lack of oxygen during stroke as well as the reintroduction of oxygen post-stroke [1], to a more negative prognosis in illnesses such as cancer [2].

Experimental methods for determining the oxygen concentration in the microcirculation include microcathodes, phosphorescence quenching microscopy, oximetry, and resonance Raman microscopy [3]. Early research used microcathodes to measure oxygen concentration by sticking them directly into the surrounding tissue [4]; however, measuring oxygen concentration directly in the bloodstream was not possible due to damage to the blood vessels and undesired interations with plasma proteins. Indirect methods, such as phosphorescence quenching microscopy [5,6] and hemoglobin oximetry [7], were introduced later.

Previous theoretical research has examined the effect of the presence of RBCs on oxygen transport. Krogh [8] was the first to derive the diffusion constant for oxygen in muscles, where the capillaries have a regular structure. Taking the limit of RBCs as solid spheres, Federspiel and Popel [9] first solved the oxygen transport equations in capillaries compared with a continuous hemoglobin solution. Based on experimental results in muscle, Roca et al. [10] mathematically derived the relative effects of oxygen convection and diffusion, and found that both played important roles in oxygen transport.

Previous experimental research has shown that the diffusion of oxygen from plasma to the surrounding tissue in capillaries is inhomogeneous due to the presence of RBCs [11]. However, even recent numerical research on oxygen transport in tissue neglects the effect of the presence of RBCs [12]. Thus, in this study, we examine the effect that a single RBC exerts on the transport properties of small molecules such as oxygen. We show that a single cell reduces the mass transport of nearby diffusing molecules using a Brownian dynamics simulation of point particles coupled with the finite element–boundary integral method used to capture cell mechanics. By increasing the number of cells and applying them to tube flow, a more accurate representation of molecule dynamics in the blood can be obtained that is not dependent on time averaging.

2. Materials and Methods

2.1. Method Overview

Our simulation consists of a red blood cell (RBC) in an infinite simple shear flow with strength g, as shown in Figure 1, surrounded by small molecules undergoing Brownian motion. The coordinate system is chosen such that, for a point x in the domain, the unit vectors ${\underset{¯}{e}}_1$, ${\underset{¯}{e}}_2$, and ${\underset{¯}{e}}_3$ point in the directions of flow, vorticity, and velocity gradient, respectively. We chose an infinite flow in order to provide a simple theoretical basis that can be used in future studies to isolate effects like cell–cell and cell–wall interactions. The effect of cell–cell interactions could act to either increase the transport rate, due to an increase in mixing, or decrease it, due to increased steric interactions. The addition of the vessel wall further complicates molecule behavior, as both the endothelial glycocalyx and the diffusion through the endothelial cells as well as interstitial fluid must be considered.

This model assumes that the suspending fluid, the RBC membrane, and the internal contents of the RBC are all uniform, continuous materials. The molecular interactions between the diffusing molecules and the solvent are assumed to be captured in the diffusion constant, which is a macroscale compilation of various molecular effects. Thus, in this study, we do not take into account individual molecular effects, such as the van der Waals forces that exist transiently between diffusing molecules and solvent molecules, or steric effects between diffusing molecules and larger polymers that would exist in a plasma solvent.

In the case of oxygen diffusion after unbinding from hemoglobin, this also neglects the effect of transport through membrane proteins, which is why we have focused solely on diffusion outside the cell. For molecules only undergoing advection, the no-slip condition ensures that molecules do not enter the cell. For molecules also undergoing diffusion, it is theoretically possible that molecules will diffuse into the cell; in order to reduce this possibility, we have reduced the interval between time steps, such that membrane crossing is not observed on visual inspection.

This simulation consists of four parts: solving the mechanics of the RBC membrane for a given deformation state at a given time; coupling of the membrane and fluid using the finite element method (FEM); solving the velocity field using the boundary integral method (BIM); and solving the particle motion using Brownian dynamics. The first three parts have been described in previous work [13,14], so we will summarize briefly in the next sections.

2.2. Membrane Mechanics

The resting shape of the RBC membrane is found from experiments to be an oblate ellipsoid that is concave in the direction of the short axis [15]. The equation giving the shape of the RBC membrane for an RBC located at the origin in the general case is

$$\begin{array}{cc}\hfill D\left(r\right)& =\rho \left[C_0+C_2{\left({\displaystyle \frac{r}{R_0}}\right)}^2+C_4{\left({\displaystyle \frac{r}{R_0}}\right)}^4\right],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \rho & ={\left[1-{\left({\displaystyle \frac{r}{R_0}}\right)}^2\right]}^{1/2},\hfill \end{array}$$

(2)

where r is a point on the membrane given in cylindrical coordinate and at physiological values of the osmotic pressure, $C_0=0.81$, $C_2=7.83$, and $C_4=-4.39$. Our discretized RBC model consists of 2562 nodes and 5120 elements connected in an unstructured triangular mesh, with an example shown in Figure 2.

We take the RBC membrane to be a two-dimensional material with elastic and bending resistance. The membrane elasticity is given by the Skalak constitutive law, which includes terms for shear elasticity and resistance to areal expansion [16]. The strain energy function for a Skalakian membrane is given as follows.

$$w_s={\displaystyle \frac{G_s}{4}}[{({\lambda }_1^2-1)}^2+{({\lambda }_2^2-1)}^2+C{({\lambda }_1^2{\lambda }_2^2-1)}^2]$$

(3)

Here, ${\lambda }_1$ and ${\lambda }_2$ are the principal stretch ratios, $G_s$ is the shear elasticity modulus, and C is a constant that determines the degree of area incompressibility, such that the elastic modulus for area extension $K_s=G_s(1+2C)$. The membrane tension is given by the relation

$$\underset{¯}{\underset{¯}{T}}={\displaystyle \frac{1}{J_s}}\underset{¯}{\underset{¯}{F}}·{\displaystyle \frac{\partial w_s}{\partial \underset{¯}{\underset{¯}{e}}}}·{\underset{¯}{\underset{¯}{F}}}^T,$$

(4)

where $\underset{¯}{\underset{¯}{F}}$ is the deformation gradient tensor, $J_s=\det \underset{¯}{\underset{¯}{F}}$, and $\underset{¯}{\underset{¯}{e}}=({\underset{¯}{\underset{¯}{F}}}^T·\underset{¯}{\underset{¯}{F}}-\delta )/2$ is the strain tensor, where $\underset{¯}{\underset{¯}{\delta }}$ is the unit tensor. The corresponding elastic stress vector $\underset{¯}{q_s}$ is then given by the relation

$${\nabla }_s·\underset{¯}{\underset{¯}{T}}+{\underset{¯}{q}}_s=0.$$

(5)

The bending resistance is given by the Helfrich bending energy [17], which was derived to describe the resistance to bending within lipid bilayers. The corresponding stress vector for a membrane with a flat resting shape is given by

$${\underset{¯}{q}}_b=\kappa [2H(2H^2-2K)+2{\mathsf{\Delta }}_{s}H]\underset{¯}{n},$$

(6)

where $H=(c_1+c_2)/2$ is the mean curvature defined with respect to principal curvatures $c_1$ and $c_2$, $K=c_1{c}_2$ is the Gaussian curvature, ${\mathsf{\Delta }}_s$ is the Laplace–Beltrami operator, and n is the outward-pointing unit normal vector.

2.3. Flow Dynamics

We take the suspending flow to be a Stokes flow, as the Reynolds number for a single RBC in blood flow is much less than 1. The equations for Stokes flow are expressed in the general case in terms of the pressure p and velocity u by the following linearized Navier–Stokes equations and the equation of continuity.

$$\begin{array}{cc}\hfill -\nabla p+\mu \mathsf{\Delta }\underset{¯}{u}& =0\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \nabla ·\underset{¯}{u}& =0\hfill \end{array}$$

(8)

We also assume that the internal RBC contents are a Newtonian fluid and set the external fluid viscosity $\mu$ equal to the viscosity of this hemoglobin solution. When a force or distribution of forces exists in the fluid, it is placed on the right-hand side of Equation (7). In this case, the distribution of forces represents the forces acting on the external and internal fluids by the RBC membrane.

Then, the boundary integral equation, given by

$$u_i\left({\underset{¯}{x}}_0\right)=u_i^{\infty }\left({\underset{¯}{x}}_0\right)-{\displaystyle \frac{1}{8\pi \mu }}{\int }_A{J}_{ij}({\underset{¯}{x}}_0,\underset{¯}{x})q_j\left(\underset{¯}{x}\right)\mathrm{d}A,$$

(9)

can be used to solve the fluid velocity at any point ${\underset{¯}{x}}_0$ on the membrane from the total stress vector $\underset{¯}{q}={\underset{¯}{q}}_s+{\underset{¯}{q}}_b$. Here, J is the Ossen tensor, given by

$$J_{ij}({\underset{¯}{x}}_0,\underset{¯}{x})={\displaystyle \frac{{\delta }_{ij}}{r}}+{\displaystyle \frac{r_i{r}_j}{r^3}},$$

(10)

and the integral is solved at points x on the membrane surface A using a Gaussian quadrature. The positions of nodes ${\underset{¯}{x}}_0$ on the membrane are updated using a second-order Runge–Kutta scheme.

The background flow is given by a simple shear flow with a shear rate g, expressed as follows.

$${\underset{¯}{u}}^{\infty }\left({\underset{¯}{x}}_0\right)=g{x}_3{\underset{¯}{e}}_1$$

(11)

2.4. Small Molecule Dynamics

Point particles are placed just outside of the surface of the cell. For each node on the RBC membrane, a corresponding particle is placed at a position on the line connecting the node to the origin that is 1.1 times the distance between the node and the origin. This distance is more than an order of magnitude larger than the thickness of the glycocalyx of the RBC membrane, so its steric and charge effects can be neglected. Advection of the small molecules is solved at each time step by the following equation:

$$\underset{¯}{x}(t+\mathsf{\Delta }t)=\underset{¯}{x}\left(t\right)+\underset{¯}{u}\mathsf{\Delta }t+{\underset{¯}{x}}^B,$$

(12)

where ${\underset{¯}{x}}^B$ is a random displacement given by

$${\underset{¯}{x}}^B=\sqrt{2D\mathsf{\Delta }t}\underset{¯}{\xi },$$

(13)

 D is the diffusion constant, and $\underset{¯}{\xi }$ is a vector with components chosen randomly from a Gaussian distribution with a mean of 0 and a variance of $2D\mathsf{\Delta }t$.

2.5. Non-Dimensional Numbers

In this study, we use two non-dimensional numbers to investigate the flow dynamics of particles. The first is the capillary number,

$$\mathrm{Ca}={\displaystyle \frac{\mu ga}{G_s}},$$

(14)

which expresses the relative effects of the viscosity and membrane elasticity and can be derived by non-dimensionalization of the boundary integral equation. Here, a is the characteristic length of the RBC. An increase in the capillary number signifies an increase in the flow strength, which leads to larger deformation of the RBC.

The other non-dimensional number is the Peclet number for mass transport,

$$\mathrm{Pe}={\displaystyle \frac{g{a}^2}{D}}.$$

(15)

This represents the relative effects of advective and diffusive transport. As the Peclet number increases, the flow strength also increases, so suspended particles move with the flow rather than diffusing evenly in all directions. The Peclet number can be derived from the non-dimensionalization of Equation (12).

3. Results and Discussion

3.1. Validation of Brownian Motion in Shear Flow

The mean squared displacement (MSD) in three-dimensional space for a particle undergoing Brownian motion in a shear flow with velocity gradient g and diffusion constant D is given by [18]

$$\langle r^2\rangle =6Dt+(2/3)D{g}^2{t}^3,$$

(16)

with the displacement in the vorticity and velocity gradient directions given by the solution in a stationary fluid,

$$\langle x_2^2\rangle =\langle x_3^2\rangle =2Dt,$$

(17)

and the displacement in the flow direction by

$$\langle x_1^2\rangle =2Dt+(2/3)D{g}^2{t}^3.$$

(18)

We validated the Brownian dynamics portion of our code by simulating 2560 particles placed initially at $\underset{¯}{x}=\underset{¯}{0}$ and allowed to freely diffuse from $t=0$ without any cells present. The simulation results are shown in Figure 3. The values of the MSD in the vorticity and velocity gradient directions are equal for constant $\mathrm{Pe}$ and increase proportionally with D, or $1/\mathrm{Pe}$. The MSD in the flow direction is also proportional to $2Dt$ for small values of t and transitions to the super-diffusion regime for $gt>2$. While it may seem unintuitive that particles with larger MSDs are in a lower $\mathrm{Pe}$ regime, the larger diffusion constant for the same flow rate allows particles to access regions with a larger velocity further away from the origin. On the other hand, particles with lower diffusion constants stay closer to the origin, where the velocity is closer to zero. As the fluid only flows in the $x_1$ direction, the particle motion is independent of $\mathrm{Pe}$ in the other directions.

3.2. Behavior of Particles near a Spherical Cell

Next, we examine the transport characteristics near a cell with a spherical resting shape in Figure 4. Here, we see that the presence of a cell leads to more complex time-dependent behavior. At the start of the simulation, diffusing particles show the same tendencies as those in the flow with no cell; that is, the particles in a lower $\mathrm{Pe}$ environment show higher MSD values for a given time point. Then, as the simulation progresses, particles with low values of $\mathrm{Pe}$, or high diffusion constants, show the same super-diffusion tendencies as the case with no cells, while particles with low diffusion constants have a slope between the diffusion and advection regimes. This occurs because of the flow circulation generated by the tank-treading motion of the cell in shear flow, in which the cell membrane revolves around the centroid of the cell. Due to the continuity of the fluid velocity vectors, the flow near the cell also similarly revolves around the cell instead of being transported in the flow direction by the shear flow. A fraction of particles with low diffusion constants become trapped in this circulatory flow instead of diffusing into regions of higher velocity, leading to a relative decrease in the MSD compared with particles with higher diffusivity.

When $\mathrm{Ca}$ is small, the cell is nearly spherical even as it undergoes shear flow. As $\mathrm{Ca}$ increases, the cell is stretched into an ellipsoid oriented at approximately a 45° angle with respect to the flow direction. For the same $\mathrm{Pe}$, particles that are suspended in a lower $\mathrm{Ca}$ flow are pushed into regions of a higher velocity due to the larger height of the cell with smaller deformation. The capillary number can be rearranged and substituted into the equation for the Peclet number, leading to the following relation:

$$\mathrm{Pe}={\displaystyle \frac{G_{s}a}{\mu D}}\times \mathrm{Ca}.$$

(19)

A back of the envelope calculation using rough physiological values of $\mu$, a, and $G_s$, and using the diffusion constant for oxygen in plasma [19], this equation can be approximated as

$$\mathrm{Pe}=8\times \mathrm{Ca}.$$

(20)

Comparing the two figures, it can be seen that an increase in flow strength with constant diffusion leads paradoxically to a small decrease in the MSD for long time periods.

3.3. Behavior of Particles near a Red Blood Cell

Here, we examine the transport characteristics of diffusing particles near a red blood cell. In addition to tank-treading motion, red blood cells show an additional rotational mode, tumbling. During tumbling motion, which occurs at low $\mathrm{Ca}$, RBCs keep their biconcave shape and rotate around the cell centroid like a solid body instead of deforming into an ellipsoid, as seen in tank-treading motion at high $\mathrm{Ca}$. Additional information about the rotational modes of RBCs and other fluid-filled bodies can be found in Abkarian and Viallat [20]. An example of this motion, taken at $\mathrm{Ca}=0.1$ and $\mathrm{Pe}=\infty$ for clarity, is shown in Figure 5.

The dependence of MSD on $\mathrm{Ca}$ and $\mathrm{Pe}$ can be found in Figure 6. The effect of tumbling can be seen in the graph for $\mathrm{Ca}=1$, in which MSD values show more fluctutations with respect to time than the other cases. This can be explained by the fact that tumbling is an unsteady motion, whereas tank-treading is a quasi-steady motion, excluding the initial transient convergence to the steady deformation. The period of the tumbling and tank-treading motion has been estimated independently to be around $gt\approx 20$ [21], so approximately five rotations/revolutions have been captured here.

4. Conclusions

In conclusion, we calculated the motion of particles undergoing diffusion near a spherical cell and a red blood cell using a boundary integral method coupled with Brownian dynamics. We found that the presence of a rotating cell near diffusing small molecules in simple shear flow reduced the effective diffusion rate of these molecules. We showed that the circulatory flow induced by a cell deformed in shear flow and undergoing tank-treading motion led to a reduction in the mean squared displacement of the particles. The effect of a tumbling RBC inducing unsteady flow had a larger effect than a spherical cell under the same flow conditions.

These results are important as a step in understanding the complex unsteady dynamics of diffusion-dominant oxygen transport in the microcirculation, as much current research takes the oxygen concentration in the blood flow to be homogeneous. These results can also be applicable to a variety of systems outside of oxygen diffusion in which small molecule motion is important for understanding the system, such as understanding the dynamics of drug delivery systems and platelet recruitment at a wound.