A multi-scale analysis of drug transport and response for a multi-phase tumour model

In this article, we consider the spatial homogenisation of a multi-phase model for avascular tumour growth and response to chemotherapeutic treatment. The key contribution of this work is the derivation of a system of homogenised partial differential equations describing macroscopic tumour growth, coupled to transport of drug and nutrient, that explicitly incorporates details of the structure and dynamics of the tumour at the microscale. In order to derive these equations, we employ an asymptotic homogenisation of a microscopic description under the assumption of strong interphase drag, periodic microstructure, and strong separation of scales. The resulting macroscale model comprises a Darcy flow coupled to a system of reaction-advection partial differential equations. The coupled growth, response, and transport dynamics on the tissue scale are investigated via numerical experiments for simple academic test cases of microstructural information and tissue geometry, in which we observe drug-and nutrient-regulated growth and response consistent with the anticipated dynamics of the macroscale system.