Luigi Preziosi

This book is an overview of recent mathematical models developed to examine stages of cancer growth and development, from single mutant cell to metastatic spread of the disease.

La finalità del libro è quella di presentare i concetti di base della Meccanica dei Continui a studenti che frequentano sia corsi di laurea che corsi di dottorato in Matematica, Fisica e Ingegneria. Questo obiettivo è perseguito da una parte limitando per quanto possibile i prerequisiti culturali necessari per la comprensione della materia e dall’altra mantenendo un linguaggio sì rigoroso, ma anche semplice e colloquiale. In questo modo il testo risulta adatto anche agli studenti che per la prima volta si interessano alla Meccanica dei Continui e che non hanno conoscenze pregresse specifiche. Nella presentazione di ogni argomento gli aspetti teorici sono corredati di esempi ed esercizi, tutti svolti. Inoltre, pur mantenendo la tradizionale e irrinunciabile struttura ipotetico-deduttiva, il libro presenta, soprattutto nei capitoli finali dedicati ai solidi e ai fluidi, una attenzione particolare alle applicazioni che gli studenti potrebbero incontrare in altri insegnamenti.

The aim of these lecture notes is to give an introduction to several mathematical models and methods that can be used to describe the behaviour of living systems. This emerging field of application intrinsically requires the handling of phenomena occurring at different spatial scales and hence the use of multiscale methods. Modelling and simulating the mechanisms that cells use to move, self-organise and develop in tissues is not only fundamental to an understanding of embryonic development, but is also relevant in tissue engineering and in other environmental and industrial processes involving the growth and homeostasis of biological systems. Growth and organization processes are also important in many tissue degeneration and regeneration processes, such as tumour growth, tissue vascularization, heart and muscle functionality, and cardio-vascular diseases.

A flexible, cell-level, and lattice-based technique, the cellular Potts model accurately describes the phenomenological mechanisms involved in many biological processes. Cellular Potts Models: Multiscale Extensions and Biological Applications gives an interdisciplinary, accessible treatment of these models, from the original methodologies to the latest developments. The book first explains the biophysical bases, main merits, and limitations of the cellular Potts model. It then proposes several innovative extensions, focusing on ways to integrate and interface the basic cellular Potts model at the mesoscopic scale with approaches that accurately model microscopic dynamics. These extensions are designed to create a nested and hybrid environment, where the evolution of a biological system is realistically driven by the constant interplay and flux of information between the different levels of description. Through several biological examples, the authors demonstrate a qualitative and quantitative agreement with the relative experimental data. The cellular Potts model is increasingly being used for the mathematical modeling of a wide range of biological phenomena, including wound healing, tumor growth, and cancer cell migration. This book shows how the cellular Potts model can be used as a framework for model building and how extended models can achieve even better biological practicality, accuracy, and predictive power.

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa­ tions is a relevant, maybe the fundamental, aspect of the interactions be­ tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula­ tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re­ lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be­ havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa­ tions is a relevant, maybe the fundamental, aspect of the interactions be­ tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula­ tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re­ lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be­ havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

This book presents applications to several fluid dynamics problems in both the bounded and unbounded domains in the framework of the discrete velocity models of kinetic theory. The proposition of new models for dense gases, gases with multi-components, and gases with chemical reactions are also included. This is an up-to-date book on the applications of the discrete Boltzmann equation.