A mathematical model of the tumour growth along a blood vessel is proposed. The model employs the mixture theory approach to describe a tissue which consists of cells, extracellular matrix and liquid. The growing tumour tissue is supposed to be surrounded by the host tissue. Tumours where complete oxydation of glucose prevails are considered. Special attention is paid to consistent description of oxygen consumption and growth processes based on the energy balance. A finite difference numerical method is proposed. The level set method is used to track an interface between the tissues.

This work reports on vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable of reproducing interesting traffic phenomena without using experimental information as modeling assumptions. To this end, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions.

This paper deals with models of living complex systems, chiefly human crowds, by methods of conservation laws and measure theory. We introduce a modeling framework which enables one to address both discrete and continuous dynamical systems in a unified manner using common phenomenological ideas and mathematical tools as well as to couple these two descriptions in a multiscale perspective. Furthermore, we present a basic theory of well-posedness and numerical approximation of initial-value problems and we discuss its implications on mathematical modeling.

We present a multiphase mathematical model for tumor growth which incorporates the remodeling of the extracellular matrix and describes the formation of fibrotic tissue by tumor cells. We also detail a full qualitative analysis of the spatially homogeneous problem, and study the equilibria of the system in order to characterize the conditions under which fibrosis may occur. © 2010 Elsevier Ltd.

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. © EDP Sciences, 2009.

The realization of innovative transport services requires greater flexibility and inexpensive service. In many cases the solution is to realize demand responsive transportation system. A Demand Responsive Transport System (DRTS) requires the planning of travel paths (routing) and customer pick-up and drop-off times (scheduling) according to received requests. In particular, the problem has to deal with multiple vehicles, limited capacity of the fleet vehicles and temporal constraints (time windows). A DRTS may operate according to static or dynamic mode.

Resorting to a multiphase modelling framework, tumours are described here as a mixture of tumour and host cells within a porous structure constituted by a remodelling extracellular matrix (ECM), which is wet by a physiological extracellular fluid. The model presented in this article focuses mainly on the description of mechanical interactions of the growing tumour with the host tissue, their influence on tumour growth, and the attachment/detachment mechanisms between cells and ECM.

In this paper we propose a classification of crowd models in built environments based on the assumed pedestrian ability to foresee the movements of other walkers. At the same time, we introduce a new family of macroscopic models, which make it possible to tune the degree of predictiveness of the individuals.