Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T18:12:22.727Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle trapping by an external body force in the limit of large Peclet number: applications to magnetic targeting in the blood flow

Published online by Cambridge University Press:  04 January 2010

G. RICHARDSON ,
K. KAOURI  and
H. M. BYRNE
G. RICHARDSON
Affiliation:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK e-mail: [email protected]
K. KAOURI
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
H. M. BYRNE
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the technology of magnetically targeted drug and gene delivery, in which a magnetic field is used to direct magnetic carrier particles from the circulation to a target site, we develop a continuum model for the motion of particles (magnetic carriers) subject to an external body force (magnetic field) in a flow of a concentrated suspension of a species of neutrally buoyant particles (blood). An advection–diffusion equation describes the evolution of the carrier particles as they advect in the flow under the action of an external body force, and diffuse as a result of random interactions with the suspension of neutrally buoyant particles (shear-induced diffusion). The model is analysed for the case in which there is steady Poiseuille flow in a cylindrical vessel, the diffusive effects are weak and there is weak carrier uptake along the walls of the vessel. The method of matched asymptotic expansions is used to show that carriers are concentrated in a boundary layer along the vessel wall and, further, that there is a carrier flux along this layer which results in a sub-layer, along one side of the vessel, in which carriers are even more highly concentrated. Three distinguished limits are identified: they correspond to cases for which (i) the force is sufficiently weak that most particles move through the vessel without entering the boundary layers along the walls of the vessel and (ii) and (iii) to a force which is sufficiently strong that a significant fraction of the particles enter the boundary layers and, depending upon the carrier absorption from the vessel walls, there is insignificant/significant axial carrier flux in these layers.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 1 , February 2010 , pp. 77 - 107
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ahluwalia, D. S., Keller, J. B. & Knessl, C. (1998) Advection–diffusion around a curved obstacle. J. Math. Phys. 39, 3694–3710.CrossRefGoogle Scholar
[2]Alexiou, C. et al. (2001) Magnetic mitoxantrone nanoparticle detection by histology, x-ray and MRI after magnetic tumour targeting. J. Magn. Magn. Mater. 225, 187–193.CrossRefGoogle Scholar
[3]Alexiou, C., Jurgons, R., Schmid, R., Hilpert, A., Bergmann, C., Parak, F. & Iro, H. (2005) In vitro and in vivo investigations of targeted chemotherapy with magnetic nanoparticles. J. Magn. Magn. Mater. 293, 389–393.CrossRefGoogle Scholar
[4]Bishop, J. J., Popel, A. S., Intaglietta, M. & Johnson, P. C. (2002) Effect of aggregation and shear rate on the dispersion of red blood cells flowing in venules. Am. J. Physiol. Heart Circ. Physiol. 283, H1985–H1996.CrossRefGoogle ScholarPubMed
[5]Davis, R. H. & Acrivos, A. (1985) Sedimentation of noncolloidal particles at low Reynolds numbers. Annu. Rev. Fluid Mech. 17, 91–118.CrossRefGoogle Scholar
[6]Dobson, J. (2006) Magnetic nanoparticles for drug delivery. Drug Dev. Res. 67, 55–60.CrossRefGoogle Scholar
[7]Eckstein, E. C., Bailey, D. G. & Shapiro, A. (1977) Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191–208.CrossRefGoogle Scholar
[8]Forbes, Z. G., Yellen, B. B., Barbee, K. A. & Friedman, G. (2005) An approach to targeted drug delivery based on uniform magnetic fields. IEEE Trans. Magn. 10, 158–166.Google Scholar
[9]Friedman, G. & Yellen, B. (2003) Magnetic separation and assembly of solid phase in fluids. Curr. Opin. Colloid Interface Sci. 39, 3372–3377.Google Scholar
[10]Goldman, A. J., Cox, R. G. & Brenner, H. (1967) Slow viscous motion of a sphere parallel to a plane wall—II Couette flow. Chem. Eng. Sci. 22, 653–660.CrossRefGoogle Scholar
[11]Goldsmith, H. L. (1971) Red cell motions and wall interactions in tube flow. Fed. Proc. 30, 1578.Google ScholarPubMed
[12]Goodwin, S., Peterson, C., Hoh, C. & Bittner, C. (1999) Targeting and retention of magnetic targeted carriers (MTCs) enhancing intra-arterial chemotherapy. J. Magn. Magn. Mater. 194, 132–139.CrossRefGoogle Scholar
[13]Grief, A. D.Results of unpublished computational code.Google Scholar
[14]Grief, A. D. & Richardson, G. (2005) Mathematical modelling of magnetically targeted drug delivery. J. Magn. Magn. Mater. 293, 455–463.CrossRefGoogle Scholar
[15]Knessl, C. (2001) On two-dimensional convection-diffusion past a circle. SIAM J. Appl. Math. 62, 310–335.CrossRefGoogle Scholar
[16]Leighton, D. & Acrivos, A. (1987) The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
[17]Levick, J. R. (2000) An Introduction to Cardiovascular Physiology, Arnold, UK.Google Scholar
[18]Lubbe, A. S. et al. (1996) Clinical experiences with magnetic drug targeting: A phase I study with 4'-epidoxorubicin in 14 patients with advanced solid tumours. Cancer Res. 56, 4686–4693.Google Scholar
[19]Mah, C., Fraites, T. J., Zolutukhin, I., Song, S. H., Flotte, T. R., Dobson, J., Batich, C. & Byrne, B. J. (2002) Improved method of recombinant AAV2 delivery for systemic targeted gene therapy. Mol. Ther. 6, 106–112.CrossRefGoogle ScholarPubMed
[20]Muthana, M., Scott, S. D., Farrow, N., Morrow, F., Murdoch, C., Grubb, S., Brown, N., Dobson, J. & Lewis, C. E. (2008) A novel magnetic approach to enhance the efficacy of cell-based gene therapies. Gene Therapy 15, 902–910. doi:10.1038/gt.2008.57.CrossRefGoogle ScholarPubMed
[21]Pankurst, Q. A., Connolly, J., Jones, S. K. & Dobson, J. (2003) Applications of magnetic nanoparticles in biomedicine. J. Phys. D: Appl. Phys. 36, R167–R181.CrossRefGoogle Scholar
[22]Pich, J. (1972) Theory of gravitational deposition of particles from laminar flows in channels. Aerosol Sci. 3, 351–361.CrossRefGoogle Scholar
[23]Reboux, S., Richardson, G. & Jensen, O. E. (2008) Bond tilting and sliding friction in a model of cell adhesion. Proc. R. Soc. A 464, 447–467.CrossRefGoogle Scholar
[24]Taylor, G. I. (1953) Dispersion of soluble matter in a solvent flowing slowly through a tube. Proc. Roy. Soc. A 219, 186–203.Google Scholar
[25]Shraiman, B. I. (1987) Diffusive transport in a Rayleigh–Benard convection cell. Phys Rev. A 36, 261–267.CrossRefGoogle Scholar
[26]Uijttewaal, W. S. J., Nijhof, E.-J., Bronkhorst, P. J. H., Hartog, E. D. & Heethaar, R. M. (1993) Near-wall excess of platelets induced by lateral migration of erythrocytes in flowing blood. Am. J. Physiol. 33, H1239–H1244.Google Scholar
[27]Voltairas, P. A., Fotiadis, D. I. & Michalis, L. K. (2002) Hydrodynamics of magnetic targeted drug targeting. J. Biomech. 35, 813–821.CrossRefGoogle ScholarPubMed
[28]Wilson, M. W., Kerlan, R. K., Findleman, N. A., Venook, A. P., LaBerge, J. M., Koda, J. & Gordon, R. L. (2004) Hepatocellular carcinoma: Regional therapy with a magnetic targeted carrier bound to doxorubicin in a dual MR imaging/conventional angiography suite initial experience with four patients. Radiology 230, 287–293.CrossRefGoogle Scholar
[29]Womersley, J. R. (1955) Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–563.CrossRefGoogle ScholarPubMed
[30]Wolfram website http://functions.wolfram.com/HypergeometricFunctions/Google Scholar
[31]Zhang, W., Stone, H. A. & Sherwood, J. D. (1996) Mass transfer at a microelectrode in channel flow. J. Chem. Phys. 100, 9462–9464.CrossRefGoogle Scholar
[32]Zydney, A. L. & Colton, C. K. (1988) Augmented solute transport in the shear flow of a concentrated suspension. Physicochem. Hydrodyn. 10, 77–96.Google Scholar