Singular perturbation theory for predicting extravasation of Brownian particles

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(P1/2). 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 (κ), 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., κ → ∞). 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 [κ ~ 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 κ. They are found to agree well at small P and for the κ ≪ 1 and κ ≫ 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.

Extravasation of Brownian Spheroidal Nanoparticles through Vascular Pores

“…Thus, the P eclet number can be as large as Oð10Þ, which implies that bulk diffusion and convection timescales are comparable and thus, they both influence extravasation. As per Shah et al (25), we estimate that the adsorption coefficient at a physical pore is k ¼ Oð1Þ, because of the fact that the pore length is on the order of the pore radius (26,27). The strength of suction flow is related to the pressure difference via the fluid viscosity, and for the values of oncotic pressures encountered in tumors (15), the dimensionless suction strength can be as large as Q ¼ Oð10Þ.…”

“…We first do this for point particles analytically and verify our result using BD simulations. Both the analytical theory and the BD simulations are improvements on, and a logical completion of, the work first developed by Shah et al (25). The agreement between the analytical solution and point-particle BD gives us the confidence to generalize our point-particle BD simulations to simulations of finite-sized particles, because no analytical solutions are available for that case.…”

“…In many tumors observed in tumor models, the pore size is $100 À 200 nm in radius (24). Microcirculatory shear rates are Oð1000Þs À1 , and particle diffusivity is Oð10 À11 Þm 2 /s (25). Thus, the P eclet number can be as large as Oð10Þ, which implies that bulk diffusion and convection timescales are comparable and thus, they both influence extravasation.…”

“…Previous analytical (29) and computational studies (30) of the same problem assumed instant adsorption (k [ 1 and Q ¼ 0). The analysis by Shah et al (25) lifts this restriction as the Sherwood number was expressed in terms of the P eclet number to OðP 1=2 Þ, with FIGURE 3 Particle geometries considered: prolate spheroids, oblate spheroids, and spheres. To see this figure in color, go online.…”

“…First, we describe the BD algorithm for point particles. This algorithm is a major improvement over the one in Shah et al (25) because one can now easily simulate Robin boundary conditions, coupled with any flow condition. Then, we introduce the BD algorithm for a general spheroidal particle, which reduces to the point-particle BD in the limit of vanishing particle size.…”