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.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

While drawing a link between the papers contained in this issue and those present in a previous one (Vol. 2, Issue 3), this introductory article aims at putting in evidence some trends and challenges on cancer modelling, especially related to the development of multiphase and multiscale models.

In this paper we explore a new model of field carcinogenesis, inspired by lungcancer precursor lesions, which includes dynamics of a spatially distributed population ofpre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cellproliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growthfactor molecules g(t, x) are produced by precancerous cells and may diffuse before they becomebound to other cells. The purpose of modelling is to investigate the existence of solutions,which correspond to formation of multiple spatially isolated lesions of pre-cancerous cellsor, mathematically, to stable spike solutions. These multiple lesions are consistent with thefield theory of carcinogenesis. In a previous model published by these authors, the influxof mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignantcolony of cells. In that model, stable patterns appeared only if some of the growth factorwas supplied from outside, arguably, a biologically tenuous hypothesis. In the present model,when μ > 0, that hypothesis is no more required, which makes this model more realistic.We present a range of results, both mathematical and computational, which taken togetherallow understanding the dynamics of this model. The equilibrium solutions in the currentmodel result from the balance between new premalignant colonies being initiated and theold ones dying out.