A Multiscale Modeling Approach to Transport of Nano-Constructs in Biological Tissues

Abstract

Nanomedicine is the emerging medical research branch which employs nanotechnological devices to improve clinical diagnosis and to design more effective therapeutic methodologies. In particular, functionalized nanoparticles have proved their clinical usefulness for cancer therapy, either as vectors for targeted drug delivery or for hyperthermia treatment. The effectiveness of such novel therapeutic strategies in nanomedicine exploits the capability of the nanoparticles to penetrate into the living tissue through the vascular network and to reach the targeted site. Accordingly, their success is tightly related to the control of the the multi-physics and multiscale phenomena governing the diffusion and transport properties of the nanoparticles, together with the geometrical and chemo-mechanical factors regulating the nanoparticles-tissue interactions. Indeed, the therapeutic effectiveness of earlier approaches was hindered by a limited ability in penetrating within the tumor tissue essentially due to microfluidic effects. Mathematical modeling is often employed in nanomedicine to analyze in silico the key biophysical mechanisms acting at different scales of investigations, providing useful guidelines to foresee and possibly optimize novel experimental techniques. Since these phenomena involve different characteristic time- and length-scales, a multiscale modeling approach is mandatory. In this work we outline how a multiscale analysis starts at the smallest scale, and its results are injected in large-scale models. At the microscale, the transport of nanoparticles is modeled either by the stochastic Langevin equation or by its continuous limit; in both cases short distance interaction forces between particles are considered, such as Coulomb and van der Waals interactions, and small disturbances of the fluid velocity field induced by the presence of nanoparticles are assumed. At the macroscopic scale, the living tissue is typically modeled as a homogeneous (homogenized) porous material of varying permeability, where the fluid flow is modeled by Darcy’s equation and nanoparticle transport is described by a continuum Diffusion-Reaction-Advection equation. One of the most significant features of the model is the ability to incorporate information on the microvascular network based on physiological data. The exploitation of the large aspect ratio between the diameter of a capillary and the intercapillary distance makes it possible to adopt an advanced computational scheme as the embedded multiscale method: with this approach the capillaries are represented as one-dimensional (1D) channels embedded and exchanging mass in a porous medium. Special mathematical operators are used to model the interaction of capillaries with the surrounding tissue. In this general context, we illustrate a bottom-up approach to study the transport and the diffusion of nanoparticles in living materials. We determine the permeability as well as the lumped parameters appearing in the nanoparticle transport equation at the tissue level by means of simulations at the microscale, while the macroscale tissue deposition rate is derived from the results of microscale simulations by means of a suitable upscaling technique.