Since cancer is a complex phenomenon that incorporates events occurring on differentlength and time scales, therefore multiscale models are needed if we hope to adequately addresscancer specific questions. In this paper we present three different multiscale individual-cell-basedmodels, each motivated by cancer-related problems emerging from each of the spatial scales: extracellular,cellular or subcellular, but also incorporating relevant information from other levels.We apply these hybrid models to investigate the influence of the microenvironement on tumourinvasion, cell-cell collaboration and competition leading to the initiation and growth of epithelialtumours, and to evolution of cell phenotypes/genotypes arising in tumours growing in differentoxygen concentrations. We also discuss how these models relate to one another and can be used tobridge biological scales relevant to cancer.

Mathematical models of tumour spheroids, proposed since the early seventies, have beengenerally formulated in terms of a single diffusive nutrient which is critical for cell replicationand cell viability. Only recently, attempts have been made to incorporate in the models the cellenergy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).By assuming glucose and lactate as the only fuel substrates, we propose a simple model for thecell ATP production which takes into account the main reactions that occur in the glycolytic andthe oxidative pathway. Under the assumption that cell death occurs when ATP production fallsbelow a critical level, we have studied the free boundary problem for the concentration of glucose,lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core isguaranteed for a sufficiently large size of the spheroid.

Despite recent advances, treatment of patients with aggressive Non-Hodgkin'slymphoma (NHL2) has yet to be optimally designed. Notwithstanding the contribution ofmolecular treatments, intensification of chemotherapeutic regimens may still be beneficial.Hoping to aid in the design of intensified chemotherapy, we put forward a mathematicaland computational model that analyses the effect of Doxorubicin on NHL over a widerange of patho-physiological conditions. The model represents tumour growth both indiffusion-limited settings, that is, in small avascular tumours and tumour cords, and inperfusion-limited settings, e.g. in well-vascularized tumours. Model simulations indicatedthe presence of a critical regimen intensity below which treatment will fall short of tumourelimination. Taking this critical intensity into account, we compared two regimenintensification strategies: Dose escalation and regimen densification, i.e. reducing theinter-dosing interval. In the diffusion-limited setting, dose escalation was somewhat moreefficient than regimen densification. In the perfusion-limited setting, both intensificationstrategies yielded similar results. The present study coupled with a realistic myelotoxicitymodel may add insight on the optimisation of NHL intensified chemotherapy design.

A model of tumor growth in a spatial environment is analyzed. The model includesproliferating and quiescent compartments of tumor cells indexed by successively mutated cell phenotypesof increasingly proliferative aggressiveness. The model incorporates spatial dependencedue to both random motility and directed movement haptotaxis. The model structures tumor cellsby both cell age and cell size. The model consists of a system of nonlinear partial differentialequations for the compartments of tumor cells, extracellular matrix, matrix degradative enzyme,and oxygen. The existence, uniqueness, positivity, regularity, and growth characteristics of thesolutions are investigated.

In this paper we propose a mathematical model to describe the evolution of leukemiain the bone marrow. The model is based on a reaction-diffusion system of equations in a porousmedium. We show the existence of two stationary solutions, one of them corresponds to the normalcase and another one to the pathological case. The leukemic state appears as a result of a bifurcationwhen the normal state loses its stability. The critical conditions of leukemia developmentare determined by the proliferation rate of leukemic cells and by their capacity to diffuse. Theanalytical results are confirmed and illustrated by numerical simulations.

We study the mathematical properties of a general model of cell division structuredwith several internal variables. We begin with a simpler and specific model with two variables, wesolve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-timeconvergence. The main difficulty comes from natural degeneracy of birth terms that we overcomewith a regularization technique. We then extend the results to the case with several parameters andrecall the link between this simplified model and the one presented in [6]; an application to thenon-linear problem is also given, leading to robust subpolynomial growth of the total population.

A mathematical model of the tumour growth along a blood vessel is proposed. Themodel employs the mixture theory approach to describe a tissue which consists of cells, extracellularmatrix and liquid. The growing tumour tissue is supposed to be surrounded by the hosttissue. Tumours where complete oxydation of glucose prevails are considered. Special attention ispaid to consistent description of oxygen consumption and growth processes based on the energybalance. A finite difference numerical method is proposed. The level set method is used to trackan interface between the tissues. The simulations show localization of the tumour within a limiteddistance from the vessels and constant expansion velocity along the vessels.