^{1,a)}and B. Shapiro

^{2,b)}

### Abstract

The transport of particles (diameter 0.56 μm) by magnetic forces in a small blood vessel (diameter *D* = 16.9 μm, mean velocity *U* = 2.89 mm/s, red cell volume fraction *H* _{ c } = 0.22) is studied using a simulation model that explicitly includes hydrodynamic interactions with realistically deformable red blood cells. A biomedical application of such a system is targeted drug or hyperthermia delivery, for which transport to the vessel wall is essential for localizing therapy. In the absence of magnetic forces, it is seen that interactions with the unsteadily flowing red cells cause lateral particle velocity fluctuations with an approximately normal distribution with variance σ = 140 μm/s. The resulting dispersion is over 100 times faster than expected for Brownian diffusion, which we neglect. Magnetic forces relative to the drag force on a hypothetically fixed particle at the vessel center are selected to range from Ψ = 0.006 to 0.204. The stronger forces quickly drive the magnetic particles to the vessel wall, though in this case the red cells impede margination; for weaker forces, many of the particles are marginated more quickly than might be predicted for a homogeneous fluid by the apparently chaotic stirring induced by the motions of the red cells. A corresponding non-dimensional parameter Ψ^{′}, which is based on the characteristic fluctuation velocity σ rather than the centerline velocity, explains the switch-over between these behaviors. Forces that are applied parallel to the vessel are seen to have a surprisingly strong effect due to the streamwise-asymmetric orientation of the flowing blood cells. In essence, the cells act as low-Reynolds number analogs of turning vanes, causing streamwise accelerated particles to be directed toward the vessel center and streamwise decelerated particles to be directed toward the vessel wall.

Support from National Science Foundation (NSF) (CBET 09-32607) is gratefully acknowledged.

I. INTRODUCTION

II. SIMULATION MODEL

A. Flow parameters

B. Flow solver

C. Simulation procedure

III. RESULTS

A. Red-cell flow statistics

B. Force-free particle advection

C. Magnetic forces perpendicular to the flow direction

D. Magnetic forces parallel to the flow direction

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Cell communication
- 19.0
- Haemodynamics
- 17.0
- Viscosity
- 16.0
- Plasma transport properties
- 9.0
- Diffusion
- 7.0

## Figures

Visualization of the initial condition showing the red cells and magnetic particles (enhanced online). [URL: http://dx.doi.org/10.1063/1.4718752.1]10.1063/1.4718752.1

Visualization of the initial condition showing the red cells and magnetic particles (enhanced online). [URL: http://dx.doi.org/10.1063/1.4718752.1]10.1063/1.4718752.1

Mean velocity versus : ——— simulations and – – – – the corresponding Poiseuille-flow for the same pressure gradient. The simulation results show overlaying results for three cases: , (0, 0, 0.1), and (0, 0, −0.1).

Mean velocity versus : ——— simulations and – – – – the corresponding Poiseuille-flow for the same pressure gradient. The simulation results show overlaying results for three cases: , (0, 0, 0.1), and (0, 0, −0.1).

(a) Red cells with their respective fitted ellipsoids. (b) The principal eigenvector **e** _{3} associated with the smallest eigenvalue λ_{3} is used to quantify the tilt angle of each cell. As shown, both cells have negative tilt angles. (c) The average distribution of tilt angles for the cases with the magnetic forces as labeled.

(a) Red cells with their respective fitted ellipsoids. (b) The principal eigenvector **e** _{3} associated with the smallest eigenvalue λ_{3} is used to quantify the tilt angle of each cell. As shown, both cells have negative tilt angles. (c) The average distribution of tilt angles for the cases with the magnetic forces as labeled.

Particles trajectories without magnetic forces: (a) projected onto a *x*–*y* plane and (b) maximum instantaneous distance from the tube centerline .

Particles trajectories without magnetic forces: (a) projected onto a *x*–*y* plane and (b) maximum instantaneous distance from the tube centerline .

Probability density function (m/s) of *x* velocity for particles with *r*(*t*) ⩽ 6 μm averaged over the period of the simulation. The symbols ○ correspond to data accumulated in bins from the simulation data and the line ——— is a Gaussian fit (see text).

Probability density function (m/s) of *x* velocity for particles with *r*(*t*) ⩽ 6 μm averaged over the period of the simulation. The symbols ○ correspond to data accumulated in bins from the simulation data and the line ——— is a Gaussian fit (see text).

Particle trajectories for increasing Ψ_{ x } as labeled. The corresponding Ψ_{ x } = 0 case is shown in Figure 4(a).

Particle trajectories for increasing Ψ_{ x } as labeled. The corresponding Ψ_{ x } = 0 case is shown in Figure 4(a).

Particle radial location histories for increasing Ψ_{ x } as labeled: ——— simulated particles and predictions for s sphere in unbounded fluid with the plasma viscosity μ⋯ ⋯ and bulk blood viscosity μ_{ b } – – – – .

Particle radial location histories for increasing Ψ_{ x } as labeled: ——— simulated particles and predictions for s sphere in unbounded fluid with the plasma viscosity μ⋯ ⋯ and bulk blood viscosity μ_{ b } – – – – .

Particle radial location histories for Ψ_{ x } = 0 and the Ψ_{ z } as labeled.

Particle radial location histories for Ψ_{ x } = 0 and the Ψ_{ z } as labeled.

The asymmetry of the cells directs the particles toward or away from the wall depending upon the sense of the force relative to the flow direction.

The asymmetry of the cells directs the particles toward or away from the wall depending upon the sense of the force relative to the flow direction.

Travel distance for , (0, 0, 0), and (0, 0, −0.2) cases as labeled.

Travel distance for , (0, 0, 0), and (0, 0, −0.2) cases as labeled.

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