Mathematical models of cell migration and self-organization in embryogenesis

In this thesis we deal with mathematical models and numerical simulations for cell migration and self-organization in embryogenesis. The part of biology which studies the formation and development of the embryo from fertilization until birth is called embryology. Morphogenesis is then the part of embryology which is concerned with the development of patterns and forms. It is well known that although morphogenesis processes are controlled at the genetic scale, genes themselves cannot create the pattern. In general a series of biological mechanisms of self-organization intervene during the early development and the formation of particular biological structures can not be anticipated solely by genetic information. This needs to be taken into account in the choice of a suitable mathematical formulation of such phenomena. Two main main topics will be investigated: we will analyze and mathematically model the self-organizing cell migration in the morphogenesis of the lateral line in the zebrafish (Danio rerio); in a second part, starting from this model, we will propose, and will study both from the analytical and the numerical point of view, a mathematical model of collective motion under only alignment and chemotaxis effects. The present thesis is organized in four chapters. In Chapter 1 we will introduce biological elements about the morphogenetic process occurring in the development of the lateral line in a zebrafish. After a first discussion on the lateral line system and on its fundamental relevance in the current scientific research, we will focus on the main mechanisms of chemical signaling and collective cell migration that will be taken into account later in our mathematical formulation of the phenomenon. In Chapter 2 we will provide a mathematical-modelling background that, starting from the morphogenesis on the chemical scale, will gradually lead us to discuss the existing mathematical models, proposed in the last years to describe collective motion in living system and in particular in the biological field. Example of numerical simulations, and their comparison with experimental evidences will be briefly shown, taken from the recent modelling literature. In Chapter 3 we will introduce a mathematical model describing the self-organizing cell migration in the zebrafish lateral line primordium. We will discuss the derivation of the model, justifying our modelling choices and comparing them with the existing literature. The proposed model will adopt a hybrid “discrete in continuous” description, where cells are treated as discrete entities moving in a continuous space, and chemical signals at molecular level are described by continuous variables. On the chemical scale we will employ diffusion and chemotaxis equations, while on the cellular scale a Newtonian second order equation for each cell will take into account typical effects arising from collective dynamics models. Cell dimension will be recovered introducing suitable detection radii and nonlocal effects. Particular steady states, corresponding to emerging structures, said neuromasts, will then be investigated and their stability will be numerically assessed. Moreover, after a description of the designed numerical approximation scheme, some dynamical simulations will be proposed to show the powerful and the limit of our approach. Finally, we will discuss the estimate of the parameters of the model, derived in part by the biological and the modelling literature, in part by the stationary model or by a numerical data fitting. In Chapter 4 we will propose a Cucker and Smale-like mathematical model of collective motion. Our hybrid model will describe a system of interacting particles under an alignment and chemotaxis effect. From an analytical point of view local and global existence and uniqueness of the solution will be proved. Furthermore, the asymptotic behaviour of the model will be investigated on a linearized form of the system. From a numerical point of view, through an approximation scheme based on finite differences, the full nonlinear system will be simulated and some significant dynamical tests will be shown. Numerical results will be compared with those analytical, and new perspectives will be proposed

Mathematical models of cell migration and self-organization in embryogenesis

In this thesis we deal with mathematical models and numerical simulations for cell migration and self-organization in embryogenesis. The part of biology which studies the formation and development of the embryo from fertilization until birth is called embryology. Morphogenesis is then the part of embryology which is concerned with the development of patterns and forms. It is well known that although morphogenesis processes are controlled at the genetic scale, genes themselves cannot create the pattern. In general a series of biological mechanisms of self-organization intervene during the early development and the formation of particular biological structures can not be anticipated solely by genetic information. This needs to be taken into account in the choice of a suitable mathematical formulation of such phenomena. Two main main topics will be investigated: we will analyze and mathematically model the self-organizing cell migration in the morphogenesis of the lateral line in the zebrafish (Danio rerio); in a second part, starting from this model, we will propose, and will study both from the analytical and the numerical point of view, a mathematical model of collective motion under only alignment and chemotaxis effects. The present thesis is organized in four chapters. In Chapter 1 we will introduce biological elements about the morphogenetic process occurring in the development of the lateral line in a zebrafish. After a first discussion on the lateral line system and on its fundamental relevance in the current scientific research, we will focus on the main mechanisms of chemical signaling and collective cell migration that will be taken into account later in our mathematical formulation of the phenomenon. In Chapter 2 we will provide a mathematical-modelling background that, starting from the morphogenesis on the chemical scale, will gradually lead us to discuss the existing mathematical models, proposed in the last years to describe collective motion in living system and in particular in the biological field. Example of numerical simulations, and their comparison with experimental evidences will be briefly shown, taken from the recent modelling literature. In Chapter 3 we will introduce a mathematical model describing the self-organizing cell migration in the zebrafish lateral line primordium. We will discuss the derivation of the model, justifying our modelling choices and comparing them with the existing literature. The proposed model will adopt a hybrid “discrete in continuous” description, where cells are treated as discrete entities moving in a continuous space, and chemical signals at molecular level are described by continuous variables. On the chemical scale we will employ diffusion and chemotaxis equations, while on the cellular scale a Newtonian second order equation for each cell will take into account typical effects arising from collective dynamics models. Cell dimension will be recovered introducing suitable detection radii and nonlocal effects. Particular steady states, corresponding to emerging structures, said neuromasts, will then be investigated and their stability will be numerically assessed. Moreover, after a description of the designed numerical approximation scheme, some dynamical simulations will be proposed to show the powerful and the limit of our approach. Finally, we will discuss the estimate of the parameters of the model, derived in part by the biological and the modelling literature, in part by the stationary model or by a numerical data fitting. In Chapter 4 we will propose a Cucker and Smale-like mathematical model of collective motion. Our hybrid model will describe a system of interacting particles under an alignment and chemotaxis effect. From an analytical point of view local and global existence and uniqueness of the solution will be proved. Furthermore, the asymptotic behaviour of the model will be investigated on a linearized form of the system. From a numerical point of view, through an approximation scheme based on finite differences, the full nonlinear system will be simulated and some significant dynamical tests will be shown. Numerical results will be compared with those analytical, and new perspectives will be proposed

A Macroscopic Mathematical Model For Cell Migration Assays Using A Real-Time Cell Analysis

Experiments of cell migration and chemotaxis assays have been classically performed in the so-called Boyden Chambers. A recent technology, xCELLigence Real Time Cell Analysis, is now allowing to monitor the cell migration in real time. This technology measures impedance changes caused by the gradual increase of electrode surface occupation by cells during the course of time and provide a Cell Index which is proportional to cellular morphology, spreading, ruffling and adhesion quality as well as cell number. In this paper we propose a macroscopic mathematical model, based on \emph{advection-reaction-diffusion} partial differential equations, describing the cell migration assay using the real-time technology. We carried out numerical simulations to compare simulated model dynamics with data of observed biological experiments on three different cell lines and in two experimental settings: absence of chemotactic signals (basal migration) and presence of a chemoattractant. Overall we conclude that our minimal mathematical model is able to describe the phenomenon in the real time scale and numerical results show a good agreement with the experimental evidences

A hybrid model of cell migration in zebrafish embryogenesis

Starting from the results of recent biological experiments, we propose a discrete in continuous mathematical model for the morphogenesis of the posterior lateral line system in zebrafish. Our hybrid description is discrete on the cellular level and continuous on the molecular level. We prove the existence of steady solutions corresponding to the formation of particular biological structures, the neuromasts. Numerical simulations are performed to show the dynamics of the model and its accuracy to describe the evolution of the cell aggregate by a qualitative and quantitative point of view. Some related models, applied to the collective motion of cells, and to the behaviour of cardiac stem cells, are indicated

A hybrid model of cell migration in zebrafish embryogenesis

Starting from the results of recent biological experiments, we propose a discrete in continuous mathematical model for the morphogenesis of the posterior lateral line system in zebrafish. Our hybrid description is discrete on the cellular level and continuous on the molecular level. We prove the existence of steady solutions corresponding to the formation of particular biological structures, the neuromasts. Numerical simulations are performed to show the dynamics of the model and its accuracy to describe the evolution of the cell aggregate by a qualitative and quantitative point of view. Some related models, applied to the collective motion of cells, and to the behaviour of cardiac stem cells, are indicated

Virtual endoscopy in odontogenic s sinus disease. Study technique and main pathological findings

The use of CT scans in dental pathology is an established technique. The potential applications of Dentascan are further enhanced by the use of virtual navigation software, resulting in endoscopy-like imaging of the maxillary sinus, thus optimising both the diagnostic and therapeutic approach to sinus pathology of dental origin. The aim of this paper is to illustrate the technical-methodological aspects of maxillary sinus virtual esdoscopy with Dentascan software and to document the most important and frequent diseases

A discrete in continuous mathematical model of cardiac progenitor cells formation and growth as spheroid clusters (Cardiospheres)

We propose a discrete in continuous mathematical model describing the in vitro growth process of biophsy-derived mammalian cardiac progenitor cells growing as clusters in the form of spheres (Cardiospheres). The approach is hybrid: discrete at cellular scale and continuous at molecular level. In the present model, cells are subject to the self-organizing collective dynamics mechanism and, additionally, they can proliferate and differentiate, also depending on stochastic processes The two latter processes are triggered and regulated by chemical signals present in the environment. Numerical simulations show the structure and the development of the clustered progenitors and are in a good agreement with the results obtained from in vitro experiments