In this thesis, we wanted to understand the main mechanisms involved in the self-organization of adipose tissues. To this aim, we followed the methodology represented in the diagram of Figure 8. In close collaboration with a team of biologists and of image specialists, we first built an individual based model thanks to heuristic rules coming from the biological reality (chap I). The goal of this model was to show that the self-organization of cells and fibers into lobule-like structures of cells in an organized fiber network was mainly driven by mechanical rules and could be reproduced by a simple model engaging few agents and interactions.
Figure 8: A diagram of the use of mathematical modeling to answer a biological question. Based on heuristic rules coming from the biology, an individual based model is developed. A statistical analysis of the numerical structures obtained with the microscopic model together with a treatment of the experimental data allow a calibration and estimation of the model parameters for the microscopic model. A first validation of the IBM appears from the comparison model/experimental data.
By the use of kinetic theory, a kinetic model linked to the microscopic dynamics is developed, and its asymptotic limit leads to the derivation of a macroscopic equation, which can be analysed theoretically and numerically compared to its microscopic corresponding model.
The individual based model we proposed for adipose tissue morphogenesis was qualitatively and quantitatively compared to experimental data as a validation, and showed good agreement with the observations of real tissues. The confronta-tion of the model results to the experiments was in the frame of an interdisciplinary network with biologists, image processing specialists and computer scientists.
In order to model the tissue at a larger scale, we aimed to derive meso and macroscopic models from the IBM of chapter I using techniques of kinetic theory.
We first concentrate on the fibrous network and aim at obtaining a macroscopic model for fiber elements having the ability to cross-link or unlink each other and to align with each other at the cross links. In order to capture the correct effects of the microscopic dynamics on the large-scale structures, we aimed at linking the macroscopic model to its microscopic version as rigorously as possible. This is the aim of Chapter II, where we first derive a kinetic model from the underlying IBM and secondly, we perform a diffusion approximation of the latter to obtain the continuum model. The kinetic model provides a statistical mechanics description of the underlying IBM by investigating how the probability distribution of fibers in position and orientation space evolves in time. We obtain a close system of two equations describing the evolution of two distribution functions: the fiber distribution function and the cross-link distribution function. We then consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only and investigate its diffusion limit.
As the derivation of the macroscopic model is only formal, the correspondence between the obtained model and the underlying IBM needs to be confirmed by numerical simulations and mathematical proofs. This is the aim of Chapter III, where we perform a first study of the macroscopic model. In this chapter, we first show existence of stationary solutions to the macroscopic equation for fiber mean orientation, in the case of a homogeneous fiber density. We then numerically study the properties of the solutions, and show that the macroscopic model features a buckling phenomenon depending on the external force applied to the fibrous network and in a range of model parameters. This observation highlights physical properties of the fiber network featured by the macroscopic equation. As a first validation of the macroscopic model, we then compare numerically its solutions to the ones of its underlying IBM. In a range of parameters, we are able to show a good correspondence between the two models.
Chapter IV and V are beyond the scope of adipose tissues. Chapter IV is the result of a collaborative work with S. Motsch, where we are interested in under-standing how density constraints impact the propagation properties of a growing mass of cells. Cell-cell non overlapping constraints are ubiquitous in models for collective behaviors and, of particular interest, in modeling of tumor growth. It consists of considering that only a finite number of individuals/agents can occupy a
given space. This can be used for modeling incompressible fluids for instance. We first propose an agent-based model for cells -represented as 2D spheres- randomly moving and interacting through non overlapping interactions. The introduction of cell division and cell apoptosis in this model leads to special solutions such as propagation waves at the microscopic level. We then derive a macroscopic model from the underlying IBM, and show that if the particle dynamics features compact supported solutions, the macroscopic density keeps spreading. We therefore pro-pose a modified version of teh macroscopic model that we are able to link to the microscopic dynamics. We finally show that the two models are in good agreement.
Finally, chapter V is devoted to the extensions of our works. In a first part, we present the works (in progress) of B. Aymar and P. Degond for modeling the vasculogenesis. In the proposed model, four actors are considered: the capillaries, the blood flow, the oxygen and the tissue. In this model, we aim to build a hybrid model, in which the blood flow and oxygen are described by macroscopic variables (flow and density respectively), and the capillary network as a set of discrete elementary capillaries. The connection with our works lies in the way of modeling the capillaries, which bear analogies with the modeling of the fiber network in the model of chapter I. The second model is a work in progress with M. Ferreira, S.
Motsch and P. Degond and aims at modeling ballistic aggregation. The underlying idea of this work, common with our works on adipose tissue, consists in saying that complex geometrical structures can be modeled by means of connected elementary units of simple geometry: 2D spheres for instance. As a starting point of this work, we aim to develop a model for self-propelled 2D spheres that connect and remain together when they collide. We then study the type of geometrical structures that can be obtained at equilibrium with such a recruitment process.