Homogenised active fluid flow in partially tumourous vascular tissues

Abstract We derive a new multiscale model describing fluid flow actively responding to the application of an external load in a vascularised, partially cancerous tissue. We start from a mesoscale where the difference between the interstitial and vessels compartment is geometrically smoothed out, but the healthy and tumourous regions can be clearly identified. As such, our starting point is a double Darcy's system of partial differential equations (PDEs) which effectively accounts for both the microscale mass exchange between interstitial and vessels' compartments, and the influence of external volume loads. We then apply the asymptotic homogenisation technique to the resulting system of four PDEs (for the vessels and interstitial pressure in both the healthy and tumourous regions) by assuming that the mesoscale and the macroscale (where differences between tumours and healthy regions can no longer be appreciated) are well separated. We then obtain a new macroscale double‐Darcy's system of two PDEs at the macroscale equipped with suitable sources encoding (a) the role of the overall blood exchange between interstitial and vessels' compartments, (b) the influence of the external loads properly mediated by the mesoscale properties of the system, and (c) the role of geometry and volume fraction of the tumourous and healthy regions (which is an indication of the tumour stage) through the coefficients of the model. These are the effective permeabilities and hydraulic conductivities. These latter, as well as the overall body forces, are to be computed by solving appropriate periodic cell problems. We present a sample numerical solution of the model by assuming long cylindrical tumourous inclusions at the mesoscale, a spherical geometry at the macroscale, and by considering a uniform volume load related to the application of a magnetic field, which is relevant in the context of cancer hyperthermia.