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 ($$P$$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 $$O(P^{1/2})$$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 ($$\kappa $$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., $$\kappa \rightarrow \infty $$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 [$$\kappa \sim O(1)$$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 $$\kappa $$κ. They are found to agree well at small $$P$$P and for the $$\kappa \ll 1$$κ≪1 and $$\kappa \gg 1$$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.

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预测布朗粒子外渗的奇异微扰理论

受最近使用纳米粒子药物递送的肿瘤治疗研究的启发，我们提供了一种奇异扰动理论并进行布朗动力学模拟，以量化布朗粒子在具有集总传质阻力的圆形孔上的剪切流中的外渗率。我们提出的分析理论是对消失 Péclet 数 ($$P$$P) 极限的扩展，它是孔隙长度尺度上对流通量与扩散通量的比率。我们将孔隙附近的颗粒浓度和外渗率（舍伍德数）表示为 $$ O (P ^ {1/2}) $$ O (P1 / 2)。该模型改进了先前的研究，因为结果对通过孔的颗粒传质系数的所有值均有效，如由 Damköhler 数 ($$ \ kappa $$ κ) 建模，它是反应速率与边界处扩散传质速率的比值。以前的研究集中在吸附主导的状态（即 $$ \ kappa \ rightarrow \ infty $$ κ → ∞）。具体来说，我们的工作提供了理论基础和基于插值的近似方法，用于计算有限阻力情况下的舍伍德数（外渗率的度量）[$$ \ kappa \ sim O (1) $$ κ∼O (1)] 在小 Péclet 数下，这在纳米粒子的外渗中具有生理学重要性。我们将我们的理论和近似方法的预测与布朗动力学模拟与反射 - 反应边界条件进行比较，如 $$ \ kappa $$ κ 建模。他们被发现在小 $$ P $$ P 和 $$ \ kappa \ ll 1 $$ κ≪1 和 $$ \ kappa \ gg 1 $$ κ≫1 渐近极限代表扩散主导和分别为吸附主导状态。尽管该模型忽略了颗粒的有限尺寸效应，但它为了解肿瘤血管系统中的外渗物理学迈出了重要的第一步。