Daniela Morale

Stochastic modelling of tumour-induced angiogenesis

A major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations à la Langevin.

Asymptotic Behavior of a System of Stochastic Particles Subject to Nonlocal Interactions

Abstract In this article, we present a rigorous mathematical derivation of a macroscopic model of aggregation, scaling up from a microscopic description of a family of individuals subject to aggregation/repulsion, described by a system of Itô type stochastic differential equations. We analyze the asymptotics of the system for both a large number of particles on a bounded time interval, and its long time behavior, for a fixed number of particles. As far as this second part is concerned, we show that a suitable localizing potential is required, in order that the system may admit a nontrivial invariant distribution.

Rescaling Stochastic Processes: Asymptotics

In this chapter the authors investigate the links among different scales, from a probabilistic point of view. Particular attention is being paid to the mathematical modelling of the social behavior of interacting individuals in a biological population, on one hand because there is an intrinsic interest in dynamics of population herding, on the other hand since agent based models are being used in complex optimization problems. Among other interesting features, these systems lead to phenomena of self-organization, which exhibit interesting spatial patterns. Here we show how properties on the macroscopic level depend on interactions at the microscopic level; in particular suitable laws of large numbers are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction–diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large. As a working example, an interacting particle system modelling social behavior has been proposed, based on a system of stochastic differential equations, driven by both aggregating/repelling and external “forces”. In order to support a rigorous derivation of the asymptotic nonlinear integro-differential equation, compactness criteria for convergence in metric spaces of measures, and problems of existence of a weak/entropic solution have been analyzed. Further the temporal asymptotic behavior of the stochastic system of a fixed number of interacting particles has been discussed. This leads to the problem of the existence of nontrivial invariant probability measure.

Modeling and simulating animal grouping individual-based models

Abstract This paper contains a review of recent results about two interacting particle models describing aggregation of social individuals; a cellular automaton model and a system of of Ito type stochastic differential equations are described.

Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis.

In the modelling and statistical analysis of tumor-driven angiogenesis it is of great importance to handle random closed sets of different (though integer) Hausdorff dimensions, usually smaller than the full dimension of the relevant space. Here an original approach is reported, based on random generalized densities (distributions) á la Dirac-Schwartz, and corresponding mean generalized densities. The above approach also suggests methods for the statistical estimation of geometric densities of the stochastic fibre system that characterize the morphology of a real vascular system. A quantitative description of the evolution of tumor-driven angiogenesis requires the mathematical modelling of a strongly coupled system of a stochastic branching-and-growth process of fibres, modelling the network of blood vessels, and a family of underlying fields, modelling biochemical signals. Methods for reducing complexity include homogenization at mesoscales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales); in tumor-driven angiogenesis the two scales can be bridged by introducing a mesoscale at which one locally averages the microscopic branching-and-growth process, in presence of a sufficiently large number of vessels (fibers).

Polymer Crystallization Processes via Many Particle Systems

In this chapter we introduce a new approach that thanks to the multiple-scale structure, allows us to use mathematical techniques of averaging at the lower scale.

Randomness in self-organized phenomena. A case study: Retinal angiogenesis

This note presents a review of recent work by the authors on angiogenesis, as a case study for analyzing the role of randomness in the formation of biological patterns. The mathematical description of the formation of new vessels is presented, based on a system of stochastic differential equations, coupled with a branching process, both of them driven by a set of relevant chemotactic underlying fields. A discussion follows about the possible reduction of complexity of the above approach, by mean field approximations of the underlying fields. The crucial role of randomness at the microscale is observed in order to obtain nontrivial realistic vessel networks.

A stochastic model for simulation and forecasting of emergencies in the area of Milano

In this paper a method for estimating and forecasting the demand of ambulance service in the area of Milano is presented. We assume that time and location of an emergency service request are outcomes of a space-time marked point process. Thus we estimate the intensity of the process on the basis of the records of specific emergency call history over 3 years (2005–2007). The mean number of emergencies occurring daily is also related, via a linear regression model, to some exogenous variables, which can be measured and used for the forecasting procedure. Simulation results based on such estimates and real data observed during 2008 are presented.

MATHEMATICAL MORPHOLOGY AND UNCERTAINTY QUANTIFICATION APPLIED TO THE STUDY OF DUAL PHASE STEEL FORMATION

Dual Phase steels (DP steels) have shown high potential for automotive and other applications, due to their remarkable property combination between high strength and good formability. The mechanical properties of the material are strictly related with the spatial distribution of the two phases composing the steel, ferrite and martensite, and their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, thus one important industrial problem is to reduce the number of 2D sections needed to reconstruct or simulate in a realistic way the 3D geometry of the material. This reduction causes an increase of the uncertainty in the parameters estimates of suitable geometric models for the material. In this work we present an approach based on the definition of a germ-grain model which approximates the main geometric characteristics of the martensite. The parameters of the model are estimated on the basis of the morphological characteristics of the images of about 150 tomographic sections of a real sample, and plausibility regions for the estimated parameters are computed. The increase in uncertainty on the parameters estimates in presence of a reduced number of sections is then quantified in terms of increase in the volume of the corresponding plausibility regions. Even though the model still needs some improvements in the fitting with the real data, the overall procedure for uncertainty quantification that we have obtained can be generalized to other study cases and can be used by the industry to set up a suitable experimental plan to fit a model to the data with a desired accuracy.

Mathematical Morphology Applied to the Study of Dual Phase Steel Formation

Dual Phase steel (DP steel) has shown high potential for automotive and other applications, due to its remarkable combined properties of high strength and good formability. The mechanical properties of the material are strictly related to the spatial distribution of the two steel phases, ferrite and martensite, and with their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, so that one important industrial problem is to reduce the required number of 2D sections in order to either reconstruct the 3D geometry of the material, or to simulate realistic ones. In this work we will present a germ-grain statistical model which can be used for a best fitting of the main geometric characteristics of the martensite phase. The parameters of the model are estimated on the basis of morphological characteristics of the images of about 150 tomographic sections taken from a real sample. After optimization or tuning of the relevant parameters, the statistical model can then be used to identify the minimum number of sections of the sample which are needed to estimate the parameters in a reliable way.