Simulated Two-dimensional Red Blood Cell Motion, Deformation, and Partitioning in Microvessel Bifurcations

Abstract

Movement, deformation, and partitioning of mammalian red blood cells (RBCs) in diverging microvessel bifurcations are simulated using a two-dimensional, flexible-particle model. A set of viscoelastic elements represents the RBC membrane and the cytoplasm. Motion of isolated cells is considered, neglecting cell-to-cell interactions. Center-of-mass trajectories deviate from background flow streamlines due to migration of flexible cells towards the mother vessel centerline upstream of the bifurcation and due to flow perturbations caused by cell obstruction in the neighborhood of the bifurcation. RBC partitioning in the bifurcation is predicted by determining the RBC fraction entering each branch, for a given partition of total flow and for a given upstream distribution of RBCs. Typically, RBCs preferentially enter the higher-flow branch, leading to unequal discharge hematocrits in the downstream branches. This effect is increased by migration toward the centerline but decreased by the effects of obstruction. It is stronger for flexible cells than for rigid circular particles of corresponding size, and decreases with increasing parent vessel diameter. For unequally sized daughter vessels, partitioning is asymmetric, with RBCs tending to enter the smaller vessel. Partitioning is not significantly affected by branching angles. Model predictions are consistent with previous experimental results.

Appendix

Finite elements are used to express Eqs. (5–6) as a system of linear equations. This system is coupled with the 2(n + 1) equations arising from the force and moment balances for each node and the appropriate boundary conditions. The coupled linear system is solved to obtain the instantaneous velocities of the nodes on the cell. The finite element package FlexPDE (version 5.0.19) was used. In the simulations, the RBC membrane is discretized using n = 20 nodes. Between 100 and 400 fluid elements are used. The finite element package uses adaptive meshing to resolve narrow gaps between cells and walls, according to a user-specified tolerance. On a 1.66 GHz processor machine, computing the instantaneous velocities takes 10–20 s.

Given cell nodal positions at time t ⁿ, the nodal positions at some subsequent time t ⁿ⁺¹ = t ⁿ + dt are found using an explicit trapezoidal, order-2 method.1 In this scheme, the instantaneous velocities, u ⁿ _i , are obtained using the finite element package (superscript corresponds to time t ⁿ). The nodal positions at time t ⁿ⁺¹ are predicted using a forward Euler estimate:

$$\tilde{\user2{x}}^{{n + 1}}_{i} = {\user2{x}^{n}_{i} + dt{\user2{u}^{n}_{i} ;}} {{\quad 0 \le i \le n.}}$$

(8a)

These new nodal positions are used to obtain the instantaneous velocities $\tilde{\user2{u}}^{{n + 1}}_{i}$ of the cell nodes at time t ⁿ⁺¹. The final estimates of the nodal positions at time t ⁿ⁺¹ are given by:

$$ \begin{array}{*{20}c} {{\user2{x}^{{n + 1}}_{i} = \user2{x}^{n}_{i} + (dt/2)(\user2{u}^{n}_{i} + \tilde{\user2{u}}^{{n + 1}}_{i} );}} & {{0 \le i \le n.}} \\ \end{array} $$

(8b)

A time step of 1 ms was found to give numerical stability as well as good convergence, whereas the forward Euler method required a time step of around 0.2 ms for convergence. Generating a cell trajectory with the given method and time step usually takes 10–40 min on a 1.66 GHz processor machine.

To find y _c for a given vessel geometry and Ψ₁ using a bisection algorithm, quantities y _below and y _above are found such that y _c is in the interval [y _below, y _above]. The particle trajectory for (y _below + y _above)/2 is then found, and the interval is halved, so that y _c is in the new interval. The algorithm is repeated until the size of the interval was less than ε _bisection = 0.05w ₀. The midpoint of this final interval is then used as an estimate for y _c; ε _bisection can be viewed as an estimate of the error on this y _c. If a particle starting within 0.05 μm of either the top or bottom mother vessel wall enters branch 1 or branch 2 (respectively), all cells are presumed to enter only one branch and y _c(Ψ₁) is taken as y _t or y _b, as appropriate.

To compute the upstream particle velocity profile, the instantaneous velocities of circular particles placed at nine locations across the channel are computed. The arithmetic average of the external node velocities is fitted with a quartic function (r ² > 0.98) to obtain an estimate for u _d(y), for use in Eq. (7).