Journal of Engineering Mathematics

, Volume 84, Issue 1, pp 155–171 | Cite as

Singular perturbation theory for predicting extravasation of Brownian particles

  • Preyas Shah
  • Sean Fitzgibbon
  • Vivek Narsimhan
  • Eric S. G. Shaqfeh


Motivated by recent studies on tumor treatments using the drug delivery of nanoparticles, we provide a singular perturbation theory and perform Brownian dynamics simulations to quantify the extravasation rate of Brownian particles in a shear flow over a circular pore with a lumped mass transfer resistance. The analytic theory we present is an expansion in the limit of a vanishing Péclet number (\(P\)), which is the ratio of convective fluxes to diffusive fluxes on the length scale of the pore. We state the concentration of particles near the pore and the extravasation rate (Sherwood number) to \(O(P^{1/2})\). This model improves upon previous studies because the results are valid for all values of the particle mass transfer coefficient across the pore, as modeled by the Damköhler number (\(\kappa \)), which is the ratio of the reaction rate to the diffusive mass transfer rate at the boundary. Previous studies focused on the adsorption-dominated regime (i.e., \(\kappa \rightarrow \infty \)). Specifically, our work provides a theoretical basis and an interpolation-based approximate method for calculating the Sherwood number (a measure of the extravasation rate) for the case of finite resistance [\(\kappa \sim O(1)\)] at small Péclet numbers, which are physiologically important in the extravasation of nanoparticles. We compare the predictions of our theory and an approximate method to Brownian dynamics simulations with reflection–reaction boundary conditions as modeled by \(\kappa \). They are found to agree well at small \(P\) and for the \(\kappa \ll 1\) and \(\kappa \gg 1\) asymptotic limits representing the diffusion-dominated and adsorption-dominated regimes, respectively. Although this model neglects the finite size effects of the particles, it provides an important first step toward understanding the physics of extravasation in the tumor vasculature.


Brownian dynamics Extravasation Law of additive resistances Singular perturbation 



One of the coauthors of this article, E.S.G.S., was a graduate student at Stanford during the days when Milton Van Dyke was a “giant” on campus. He was proud to have taken all available advanced courses from Prof. Van Dyke, including his perturbation theory course. The course was a revelation, and E.S.G.S. remembers the humorous and incisive lectures that introduced the subject. E.S.G.S. is eternally grateful for that experience. The perturbation theory in this manuscript is just a small example of the preparation that E.S.G.S. credits in large part to the introduction by Prof. Van Dyke. The authors are also thankful for the many fruitful discussions with Prof. Andreas Acrivos and the critical feedback they received from him. The authors are grateful for the funding support provided by the National Institutes of Health National Cancer Institute Grant U54 CA 151459-02, Stanford Graduate Engineering Fellowship, and NSF-MRI2 Award 0960306 for providing computing resources that contributed to the research. V.N. is supported by the National Science Foundation through a graduate research fellowship.


  1. 1.
    Smith BR, Kempen P, Bouley D, Xu A, Zhuang L, Melosh N, Dai H, Sinclair R, Gambhir SS (2012) Shape matters: intravital microscopy reveals surprising geometrical dependence for nanoparticles in tumor models of extravasation. Nano Lett 12(7):3369–3377ADSCrossRefGoogle Scholar
  2. 2.
    Zhao H, Shaqfeh ESG, Narsimhan V (2012) Shear-induced particle migration and margination in a cellular suspension. Phys Fluids 24:011902ADSCrossRefGoogle Scholar
  3. 3.
    Zhao H, Shaqfeh ESG (2011) Shear-induced platelet margination in a microchannel. Phys Rev E 83:061924ADSCrossRefGoogle Scholar
  4. 4.
    Heldin CH, Rubin K, Pietras K, Östman A (2004) High interstitial fluid pressure: an obstacel in cancer therapy. Nat Rev Cancer 4:806–813CrossRefGoogle Scholar
  5. 5.
    Stroher M, Boucher Y, Stangassinger M, Jain RK (2000) Oncotic pressure in solid tumors is elevated. Cancer Res 60:4251–4255Google Scholar
  6. 6.
    Phillips CG (1990) Heat and mass transfer from a film into steady shear flow. Q J Mech Appl Math 43(1):135–159CrossRefMATHGoogle Scholar
  7. 7.
    Ackerberg RC, Patel RD, Gupta SK (1978) The heat/mass transfer to a finite strip at small Péclet numbers. J Fluid Mech 86(1):49–65ADSCrossRefMATHGoogle Scholar
  8. 8.
    Stone HA (1989) Heat/mass transfers from surface films to shear flows at arbitrary Péclet numbers. Phys Fluids A1:1112–1122ADSCrossRefGoogle Scholar
  9. 9.
    Lévêque MA (1928) Les lois de la transmission de chaleur par convection. Ann Mines Mem. Series 12, 13:201–299, 305–362, 381–415Google Scholar
  10. 10.
    Vink H (1996) Duling BR identification of distinct luminal domains for macromolecules, erythrocytes and leukocytes within mammalian capillaries. Circ Res 79:581–589CrossRefGoogle Scholar
  11. 11.
    Smith ML, Long DS, Damanio ER, Ley K (2003) Near-wall \(\mu \)-PIV reveals a hydrodynamically relevant endothelial surface layer in venules in vivo. Biophys J 85:637–645CrossRefGoogle Scholar
  12. 12.
    Pries AR, Secomb TW (2005) Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol Heart Circ Physiol 289:H2657–H2664CrossRefGoogle Scholar
  13. 13.
    Batchelor GK (1979) Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution. J Fluid Mech 95:369–400ADSCrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Acrivos A (1980) A note on the rate of heat or mass transfer from a small particle freely suspended in a linear shear field. J Fluid Mech 98:229–304CrossRefGoogle Scholar
  15. 15.
    Churchill SW, Usagi R (1972) A general expression for the correlation of rates of transfer and other phenomenon. AIChE 8(6):1121–1128CrossRefGoogle Scholar
  16. 16.
    Gladwell GML, Barber JR, Olesiak Z (1983) Thermal problems with radiation boundary conditions. Q J Mech Appl Math 36(3):387–401CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Duffy DG (2008) Mixed boundary value problems. In: Chen G, Bridges TJ (eds) Applied Mathematics & Nonlinear Science, vol 15. Chapman & Hall/CRC, Boca Raton, pp 272–273Google Scholar
  18. 18.
    Gladwell GML (1980) Contact problems in classical theory of elasticity. Sijthoff and Noordhoff, Alphen aan den RijnMATHGoogle Scholar
  19. 19.
    Kalumuck KM (1983) A theory for the performance of hot-film shear-stress probes. Dissertation, Massachusetts Institute of TechnologyGoogle Scholar
  20. 20.
    Lamm G, Schulten K (1983) Extended brownian dynamics II. Reactive, nonlinear diffusion. J Chem Phys 78:2713–2734ADSCrossRefGoogle Scholar
  21. 21.
    Reiss LP, Hanratty TJ (1963) An experimental study of the unsteady nature of the viscous sublayer. AIChE 9(2):154–160CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Preyas Shah
    • 1
  • Sean Fitzgibbon
    • 2
  • Vivek Narsimhan
    • 2
  • Eric S. G. Shaqfeh
    • 1
    • 2
    • 3
  1. 1.Department of Mechanical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Chemical EngineeringStanford UniversityStanfordUSA
  3. 3.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA

Personalised recommendations