A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features

An adaptive methodology for risk classification of small homogeneous earthfill embankment dams integrating climate change projections

This paper presents the application of the advanced probabilistic slope stability model with precipitation effects developed to assess the performance of small homogeneous earthfill embankment dam slopes, when exposed to future seasonal precipitation scenarios. Here, the UK's latest probabilistic climate model known as UKCP09 is applied. To reflect the critical conditions conducive to slope failure, a benchmark has been developed to identify the change, if any, in the risk classification of the slope's performance level due to precipitation. Thus, enabling the reassessment of the dam's risk classification, as categorised by the Flood and Water Management Act 2010. Such an approach could therefore be well placed to support and enhance the decision-making process, its impact on the public, especially in relation to future climate effects

Embankment dam probabilistic assessment for climate scenarios

For small earthfill dams exposed to climate scenarios such as those defined in UKCP09, deterministic assessments are insufficient, and more sophisticated models are required. This paper presents a hybrid probabilistic methodology that quantitatively measures the notional reliability index against upstream and downstream slope failure for such dams exposed to variable precipitation. Upstream and downstream slope stability are selected here as representative significant limit states governing the dam's long-term performance. The governing equations for the limit states are defined using the sliding-block method incorporating the effects of infiltration through the embankment. Using standard and sloping Green–Ampt and closed-form van Genuchten methods, the rainfall effects on soils with variable saturation are considered, and the standard first-order second moment method is applied. The probabilistic model encompasses uncertainties associated with soil properties, dam geometry and rainfall parameters. The paper demonstrates notional reliability indices for the dam for selected precipitation scenarios. A benchmark is developed that reflects the critical conditions conducive to slope failure. The paper reflects on the implication of inclusion of probabilistic climate models for associated risks. Therefore the analysis is an effective new management tool for risk assessment of embankment dams as categorised by the Flood and Water Management Act 2010

A note on light velocity anisotropy

It is proved that in experiments on or near the Earth, no anisotropy in the one-way velocity of light may be detected. The very accurate experiments which have been performed to detect such an effect are to be considered significant tests of both special relativity and the equivalence principleComment: 8 pages, LaTex, Gen. Relat. Grav. accepte

Collective migration and patterning during early development of zebrafish posterior lateral line

The morphogenesis of zebrafish posterior lateral line (PLL) is a good predictive model largely used in biology to study cell coordinated reorganization and collective migration regulating pathologies and human embryonic processes. PLL development involves the formation of a placode formed by epithelial cells with mesenchymal characteristics which migrates within the animal myoseptum while cyclically assembling and depositing rosette-like clusters (progenitors of neuromast structures). The overall process mainly relies on the activity of specific diffusive chemicals, which trigger collective directional migration and patterning. Cell proliferation and cascade of phenotypic transitions play a fundamental role as well. The investigation on the mechanisms regulating such a complex morphogenesis has become a research topic, in the last decades, also for the mathematical community. In this respect, we present a multiscale hybrid model integrating a discrete approach for the cellular level and a continuous description for the molecular scale. The resulting numerical simulations are then able to reproduce both the evolution of wild-type (i.e. normal) embryos and the pathological behaviour resulting form experimental manipulations involving laser ablation. A qualitative analysis of the dependence of these model outcomes from cell-cell mutual interactions, cell chemical sensitivity and internalization rates is included. The aim is first to validate the model, as well as the estimated parameter values, and then to predict what happens in situations not tested yet experimentally. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'

Statistical mechanics model of angiogenic tumor growth

We examine a lattice model of tumor growth where survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the supply is correlated with distribution of tumor cells, which as we suggest might mimick the angiogenic growth, the extinction shows different, and most likely novel critical behaviour. Such a correlation affects also the morphology of the growing tumors and drastically raise tumor survival probability.Comment: 4 page

Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development

This paper presents a mathematical model of normal and abnormal tissue growth. The modelling focuses on the potential role that stress responsiveness may play in causing proliferative disorders which are at the basis of the development of avascular tumours. In particular, we study how an incorrect sensing of its compression state by a cell population can represent a clonal advantage and can generate hyperplasia and tumour growth with well known characteristics such as compression of the tissue, structural changes in the extracellular matrix, change in the percentage of cell type (normal or abnormal), extracellular matrix and extracellular liquid. A spatially independent description of the phenomenon is given initially by a system of nonlinear ordinary dierential equations which is explicitly solved in some cases of biological interest showing a rst phase in which some abnormal cells simply replace the normal ones, a second phase in which the hyper-proliferation of the abnormal cells causes a progressive compression within the tissue itself, and a third phase in which the tissue reaches a compressed state, which presses on the surrounding environment. A travelling wave analysis is also performed which gives an estimate of the velocity of the growing mass