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. Author manuscript; available in PMC: 2012 Mar 1.

Published in final edited form as: J Magn Magn Mater. 2011 Mar 1;323(6):651–668. doi: 10.1016/j.jmmm.2010.09.008

The three prototypical behaviors: A) magnetic force dominated case (Ψ = 10^-3, = 1), B) velocity dominated case (Ψ = 10^-5, = 10^-3), and C) boundary layer formation (Ψ = 10^-2, = 10^-3). (A) The magnetic force dominated case shows a cross-sectional concentration of the magnetic nano-particles for three times at t = 0.03 seconds (early), 0.3 seconds (middle), and at equilibrium, at Pe = 333. Particles are pulled towards the magnet and out through the bottom of the tissue resulting in a constant concentration equal to the blood inlet concentration. Here the tissue diffusion is set to equal the diffusion in the endothelial membrane. (B) Velocity dominated shows a cross-sectional concentration of the magnetic nano-particles for three times at t = 0.03 seconds (early), 18 seconds (middle), and at equilibrium, at Pe = 333. Particles are washed out before they generate a significant boundary layer along the vessel wall. At long times diffusion equilibrates the concentration between tissue and blood. Here the tissue Renkin number is set at = 10 which means it is ten times as easy for particles to diffuse through tissue than through the endothelial membrane. (C) Boundary layer formation shows a cross-sectional magnetic nano-particle concentration for three times at t = 0.03 seconds (early), 30 seconds (middle), and at equilibrium, at Pe = 333. (i) The steady state profile for Ψ = 10^-2. Here the particle concentration is shown on the same linear scale as in other time snap shots. (ii) The steady state profile for a higher magnetic-Richardson number, for Ψ = 10^-1. Here both the particle conentration and the cross-sectional plot are shown on a log scale. In both boundary layer cases (Ψ = 10^-2 and 10^-1) the particles build-up along the vessel membrane on both the vessel side and within the membrane. The boundary layer forms very rapidly. In (ii) the membrane particle concentration is sufficiently high to cause a concentration in the tissue greater than the vessel inlet concentration. In both (i) and (ii) the tissue Renkin number is set at = 10 which means it is ten times as easy for particles to diffuse through tissue than through the endothelial membrane.

Inline graphic — The three prototypical behaviors: A) magnetic force dominated case (Ψ = 10^-3, = 1), B) velocity dominated case (Ψ = 10^-5, = 10^-3), and C) boundary layer formation (Ψ = 10^-2, = 10^-3). (A) The magnetic force dominated case shows a cross-sectional concentration of the magnetic nano-particles for three times at t = 0.03 seconds (early), 0.3 seconds (middle), and at equilibrium, at Pe = 333. Particles are pulled towards the magnet and out through the bottom of the tissue resulting in a constant concentration equal to the blood inlet concentration. Here the tissue diffusion is set to equal the diffusion in the endothelial membrane. (B) Velocity dominated shows a cross-sectional concentration of the magnetic nano-particles for three times at t = 0.03 seconds (early), 18 seconds (middle), and at equilibrium, at Pe = 333. Particles are washed out before they generate a significant boundary layer along the vessel wall. At long times diffusion equilibrates the concentration between tissue and blood. Here the tissue Renkin number is set at = 10 which means it is ten times as easy for particles to diffuse through tissue than through the endothelial membrane. (C) Boundary layer formation shows a cross-sectional magnetic nano-particle concentration for three times at t = 0.03 seconds (early), 30 seconds (middle), and at equilibrium, at Pe = 333. (i) The steady state profile for Ψ = 10^-2. Here the particle concentration is shown on the same linear scale as in other time snap shots. (ii) The steady state profile for a higher magnetic-Richardson number, for Ψ = 10^-1. Here both the particle conentration and the cross-sectional plot are shown on a log scale. In both boundary layer cases (Ψ = 10^-2 and 10^-1) the particles build-up along the vessel membrane on both the vessel side and within the membrane. The boundary layer forms very rapidly. In (ii) the membrane particle concentration is sufficiently high to cause a concentration in the tissue greater than the vessel inlet concentration. In both (i) and (ii) the tissue Renkin number is set at = 10 which means it is ten times as easy for particles to diffuse through tissue than through the endothelial membrane.