We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This equation can be transformed into an equivalent singular equation of Cauchy type which allows us to give the explicit formula for the solution. The numerical method proposed in this paper consists of applying the Lagrange interpolation to the inner Cauchy type singular integral in the latter formula after subtracting the singularity. For the error of this method weighted norm estimates as well as estimates on discrete subsets of knots are given.

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.

In this paper, we systematically apply the mathematical structures by time-evolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discrete-time Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon-positive measures generated by a push-forward recursive relation.

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.

In this paper a macroscopic model of tumor cord growth is developed, relying on the mathematical theory of deformable porous media. Tumor is modeled as a saturated mixture of proliferating cells, extracellular fluid and extracellular matrix, that occupies a spatial region close to a blood vessel whence cells get the nutrient needed for their vital functions. Growth of tumor cells takes place within a healthy host tissue, which is in turn modeled as a saturated mixture of non-proliferating cells.

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.

In this work the phenomenon of contact inhibition of growth is studied by applying an individual based model and a continuum multiphase model to describe cell colony growth in vitro. The impact of different cell behavior in response to mechanical cues is investigated. The work aims at comparing the results from both models from the qualitative and, whenever possible, also the quantitative point of view. Crown Copyright © 2009.

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.