# Singular perturbation theory for predicting extravasation of Brownian particles

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## Abstract

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.

## Keywords

Brownian dynamics Extravasation Law of additive resistances Singular perturbation## Notes

### Acknowledgments

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.

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