MATLIS (Mathematics in Life and Social Sciences)
The scientific Programme to be described below intends to explore a number of relevant problems in Biology, Medicine and Sociology where the use of advanced mathematical methods is expected to provide significant help to solve the issues therein addressed . On the other hand, the precise formulation of the questions to be considered is likely to be a source of new and challenging mathematical problems . It is therefore expected that a successful completion of the studies described below will translate into significant advances in all fields involved.
We next proceed to describe the basic research units which will constitute the core of this Programme.
The term vasculogenesis is used to denote the formation of blood vessels starting from their cell precursors (angioblasts) to produce vascular nets with characteristic geometrical forms. These include vessels with different sizes, ranging from capillaries to arteries and veins. This process is characteristic of the embryonic state in vertebrates, while angiogenesis (the formation of vessels from a preexisting vasculature) always leads to the formation of small vessels, and continues to happen during adulthood. We are interested in deriving models for the formation of early vasculature that could be checked against experimental data obtained from chick embryos. Such models should in particular include aspects as signalling from the endoderm, angioblasts differentiation and the onset of the first vascular networks. We are also interested in the formulation and subsequent analysis of mathematical models for advanced stages of vascular networking. In particular, such models should account for existing data on characteristic sizes and critical cell densities corresponding to a stable vasculature
Modelling and Simulation of Blood Coagulation.
Blood coagulation is a robust security mechanism of human organisms, which prevents bleeding from minor injuries to occur . Any disruption in such a system may have significant consequences. For instance, an impaired ability of blood to coagulate is cause of haemophilia, a serious hereditary disorder. On the other hand, an inordinate increase in the activation of the blood coagulation system may lead to abnormal thrombi formation, and consequently to a number of thrombotic pathologies .The process of blood coagulation makes use of a complex array of interdependent, and finely tuned , biochemical reactions (the so-called biochemical cascade, BC), of which many details are known by now . However, much remains to be done in terms of ascertaining which points in such cascade are more suitable as targets for external control , and how such control has to be implemented, in order to obtain significant therapeutic improvements with respect to current protocols.
Optimization in Radiotherapy.
Radiotherapy consists in the delivery of ionizing radiation with curative goals. It represents a major tool for treating solid tumours in any anatomical site . While the first cures of cancer by radiation were reported in 1899, shortly after the discovery of the X-rays, it was soon learnt that radiotherapy requires of precise dosimetry and delivery to hit at the lesion while sparing surrounding normal tissue. Nowadays, radiotherapy is a fully interdisciplinary technique which provides major challenges in Biology, Engineering, Medicine and Physics, and where Mathematics has already gained solid ground. Indeed , mathematical methods are paramount in Image Treatment, a key aspect in the choice of the so-called Planned Target Volume (PTV), the region that has to be irradiated . Moreover, Mathematics has also shown its strength to develop a quantitative radiobiology, that is , a detailed description of the interaction between radiation and living matter . In particular, the question of predicting the biological response to incoming radiation has become a central issue in the field during the last thirty years, once the first mathematical models for that purpose were derived.
In this Programme we plan to tackle the question of deriving an integrated model for optimal radiation delivery, starting from a sharp identification of the PTV by means of Imaging Techniques , to the design of an optimal irradiation strategy based on the study of suitable variational problems . These will often involve non-standard constraints or penalty terms, which will require of advanced variational methods to identify the corresponding minimizers, as well as to implement efficient numerical approximations thereof.
Mathematical Models of Criminality and Emergent Social Behaviours.
Crime, and in general unacceptable social behaviour, has always been a serious concern in human societies. It is to be noted that the concept of criminal behaviour includes, together with some widely accepted facts (e.g. murder), others which largely differ among various cultures and historical periods. From a modelling point of view, a given society can be considered as a playground for a number of interacting populations , whose joint evolution leads to emergent behaviours (sometimes termed as social phase transitions) while at the same time has to cope with one or several subpopulations which refuse to play by the rules - the criminals . In recent years, mathematical methods have proved their strength in dealing with the dynamical aspects of such process, a step forward with respect to the classical, static picture provide by statistical techniques.
In this Programme, we aim at deriving and analysing agent-based models of social behaviour based on population dynamics methods on the one hand, as well as on game theory on the other . Such techniques will be instrumental to gain insight into the evolution of complex populations subject to resource competition and under the action of global constraints. Among these, we should mention the total wealth variation , this last been influenced by aspects as changes in environment and main resources.