Narrow Results

Permeation of small molecules across membrane channels can be measured by a multiscale computational protocol based on Brownian dynamics and the potential of mean force formalism. In this article we look at ways to compute the potential of mean force by reusing pre-existing molecular dynamics trajectories via a protocol centered on instantaneous forward/reverse transformations. We apply the method to the energetics of water across the narrow channel formed by Gramicidin A and reproduce several features of the energy barrier across the channel albeit at a coarse level of detail due to limits imposed by the exponential averages intrinsic to the method and the small size of the channel. The implications for ions and less dense systems are briefly discussed.

A multiscale model and numerical method for computing microstructures with large and inhomogeneous deformation is established, in which the microscopic and macroscopic information is recovered by coupling the finite order rank-one convex envelope and the finite element method. The method is capable of computing microstructures which are locally finite order laminates. Numerical experiments on a double-well problem show that plenty of stress free large deformations can be achieved by microstructures consisting of piecewise simple twin laminates.

An existing multiscale model is extended to study the response of a vascularised tumour to treatment with chemotherapeutic drugs which target proliferating cells. The underlying hybrid cellular automaton model couples tissue-level processes (e.g. blood flow, vascular adaptation, oxygen and drug transport) with cellular and subcellular phenomena (e.g. competition for space, progress through the cell cycle, natural cell death and drug-induced cell kill and the expression of angiogenic factors). New simulations suggest that, in the absence of therapy, vascular adaptation induced by angiogenic factors can stimulate spatio-temporal oscillations in the tumour's composition.

Numerical simulations are presented and show that, depending on the choice of model parameters, when a drug which kills proliferating cells is continuously infused through the vasculature, three cases may arise: the tumour is eliminated by the drug; the tumour continues to expand into the normal tissue; or, the tumour undergoes spatio-temporal oscillations, with regions of high vascular and tumour cell density alternating with regions of low vascular and tumour cell density. The implications of these results and possible directions for future research are also discussed.

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.

In this paper we consider the coupling between two diffusion-reaction problems, one taking place in a three-dimensional domain Ω, the other in a one-dimensional subdomain Λ. This coupled problem is the simplest model of fluid flow in a three-dimensional porous medium featuring fractures that can be described by one-dimensional manifolds. In particular this model can provide the basis for a multiscale analysis of blood flow through tissues, in which the capillary network is represented as a porous matrix, while the major blood vessels are described by thin tubular structures embedded into it: in this case, the model allows the computation of the 3D and 1D blood pressures respectively in the tissue and in the vessels.

The mathematical analysis of the problem requires non-standard tools, since the mass conservation condition at the interface between the porous medium and the one-dimensional manifold has to be taken into account by means of a measure term in the 3D equation. In particular, the 3D solution is singular on Λ. In this work, suitable weighted Sobolev spaces are introduced to handle this singularity: the well-posedness of the coupled problem is established in the proposed functional setting. An advantage of such an approach is that it provides a Hilbertian framework which may be used for the convergence analysis of finite element approximation schemes. The investigation of the numerical approximation will be the subject of a forthcoming work.

Cancer is a complex, multiscale process involving interactions at intracellular, intercellular and tissue scales that are in turn susceptible to microenvironmental changes. Each individual cancer cell within a cancer cell mass is unique, with its own internal cellular pathways and biochemical interactions. These interactions contribute to the functional changes at the cellular and tissue scale, creating a heterogenous cancer cell population. Anticancer drugs are effective in controlling cancer growth by inflicting damage to various target molecules and thereby triggering multiple cellular and intracellular pathways, leading to cell death or cell-cycle arrest. One of the major impediments in the chemotherapy treatment of cancer is drug resistance driven by multiple mechanisms, including multi-drug and cell-cycle mediated resistance to chemotherapy drugs. In this article, we discuss two hybrid multiscale modelling approaches, incorporating multiple interactions involved in the sub-cellular, cellular and microenvironmental levels to study the effects of cell-cycle, phase-specific chemotherapy on the growth and progression of cancer cells.