A system of differential equations for coupled fluid and drug transport in vascularized (malignant) tissues is derived by a multiscale expansion. We start from mass and momentum balance equations, stated in the physical domain, geometrically characterized by the intercapillary distance (the microscale). The Kedem–Katchalsky equations are used to account for blood and drug exchange across the capillary walls. The multiscale technique (homogenization) is used to formulate continuum equations describing the coupling of fluid and drug transport on the tumor length scale (the macroscale), under the assumption of local periodicity; macroscale variations of the microstructure account for spatial heterogeneities of the angiogenic capillary network. A double porous medium model for the fluid dynamics in the tumor is obtained, where the drug dynamics is represented by a double advection–diffusion–reaction model. The homogenized equations are straightforward to approximate, as the role of the vascular geometry is recovered at an average level by solving standard cell differential problems. Fluid and drug fluxes now read as effective mass sources in the macroscale model, which upscale the interplay between blood and drug dynamics on the tissue scale. We aim to provide a theoretical setting for a better understanding of the design of effective anti-cancer therapies.
A permeation-diffusion-reaction model of gas transport in cellular tissue of plant materials