Modeling of diffusive nanoparticle transport to porous vasculature [electronic resource]
Keywords: Mechanical Engineering.

Recent studies on strategies for tumor treatment focus on drug delivery via nanoparticle carriers that are now available in various shapes and sizes. These nanoparticles pass or 'extravasate' through pores in tumor vasculature that form during angiogenesis. Motivated by the need to improve efficiency and, thus, reduce the side effects of these treatments, we provide an analytical and simulation-based and experimentally supported (in vivo and in vitro) study of the extravasation rate of NPs through pores. We quantify this rate as a function of nanoparticle shape, size, and flow properties in a model that is representative of the microscale region where extravasation occurs. We model the mass transport problem by the advection-diffusion of point and finite sized particles to a flat planar surface embedded with pores. The planar surface can have finite porosity and specific to the application, the porous regions can be modeled as first-order reactive patches where the reaction can be viewed as a lumped resistance to mass transfer at the pore. Such porous media are ubiquitous in nature and engineering. The fluid flow near the surface is modeled as a bulk shear flow, along with a pressure-driven `Sampson' flow through the pores. The objective is to calculate the mass flux at the pores (or the yield of reaction, in the case of reactive patches), denoted by the dimensionless Sherwood number S. The Sherwood number depends on the following dimensionless parameters: (1) the Damkohler number (k) which is the dimensionless reaction rate, (2) the Peclet number (P) which is the ratio of diffusion and convection time scales, (3) the area fraction (phi), and (4) the suction-Peclet number (P_Q). We obtain analytical closed form correlations for the Sherwood number for the case of transport of point particles using boundary element simulations and singular perturbation theory. The functional form of these correlations reveals the underlying physical mechanics of transport to a porous surface without the necessity to know the finer details. Then we develop a general Brownian dynamics algorithm to capture the effect of shape and size of the particle in the transport mechanics and support it with in vitro experiments. The details of our approach is describe below. Surface media with heterogeneity in the form of pores or reaction rates are typically modeled via an effective surface reaction rate or mass transfer coefficient employing the conventional ansatz of reaction-limited transport at the microscale. However, this assumption is not always valid, particularly when there is strong flow. To understand the physics at the length scale of the reactive patch size, we first analyze the flux to a single reactive patch. The shear flow induces a 3-D concentration wake structure downstream of the patch. When two patches are aligned in the shear direction, the wakes interact to reduce the per patch flux compared to the single patch case. Having determined the length scale of interaction between two patches, we study the transport to a periodic and disordered distribution of patches. We obtain an effective boundary condition for the transport to the patches that depends on local mass transfer coefficient (or reaction rate) and shear rate via the Sherwood number. We demonstrate that this boundary condition replaces the details of the heterogeneous surfaces at a wall-normal effective slip distance. The slip distance again depends on the shear rate, and weakly on the reaction rate and scales with the reactive patch size. These effective boundary conditions can be used directly in large scale physics simulations as long as the local shear rate, reaction rate and patch area fraction are known. We obtain various correlations for the Sherwood number as a function of (k, P, phi). In particular, we demonstrate that the 'method of additive resistances' provides a good approximation for the Sherwood number for a wide range of values of (k, P) for 0< phi< 0.78. When phi = 0.78, the reactive patches, which are modeled as circular disks, are in a closed packed configuration where the reactive regions are essentially connected to each other. To understand the physics when phi> 0.78, we examine the inverse problem of transport to a reactive, connected surface with inert disks. We show that the method of resistances approach to obtain the Sherwood number fails in the limit as phi approaches 1, i.e, in the dilute limit of periodic inert disks, due to the existence of a surface concentration boundary layer around each disk that scales with 1/k. This boundary layer controls the screening length between inert disks and allows us to introduce a new theory, thus providing new correlations for the Sherwood number that are highly accurate in the limit of phi approaching 1. After exploring the physics of transport to a heterogeneous surface in a shear flow, we study the original problem of transport to tumors which is inherently more complex due to presence of flow dynamics such as pressure driven suction flows. The Sherwood number is affected by the suction-Peclet number P_Q = PQ where the suction flow rate is characterized by the dimensionless parameter Q. We introduce a singular perturbation theory of advection-diffusion of point particles and express the extravasation rate, i.e., the Sherwood number, as an expansion in the limit of vanishing Peclet number and the suction-Peclet number. The expression for the Sherwood number is shown to be accurate in a wide regime of the dimensionless parameters by comparing against the theory of point particle transport to reactive disks from before (no-suction case). We then design Brownian Dynamics (BD) simulations for understanding the motion of general spheroids through the microvasculature. The BD simulations closely match the correlations for the point particles in the limit as the simulation particle size is made small. We study the effect of size and shape on the extravasation rates of each particle and show that particle geometry severely affects the flux. In general, for particles of same maximum dimension, the extreme aspect ratio particles are observed to provide a greater flux at the pore due to favorable alignment with streamlines, especially in the presence of suction. In the presence of strong suction flow, all particles are forced through the pore. These conclusions are supported by in vitro experiments where we measure the rate of diffusion of nanoparticles across a porous membrane using fluorometry analysis. We consider nanoparticles of spherical, discoid, and rod-like shapes and demonstrate that the BD simulations are able to predict the Sherwood number for transport across a membrane and relate it to the extravasation rates observed in in vivo experiments.