Mutliscale theory for complex dynamical networks
Contrary to agent-based models, macroscopic models have a computational complexity independent of system size, making them more suited to the study of the systems at a large scale. As data are collected both on a histological scale addressing tissue microarchitecture and on a macroscale by non-invasive imaging modalities, developping multiscale models which account for different types of interactions may be an invaluable tool to understand real systems.
I. Micro-macro passage for complex interconnected fiber networks
Aims and tools:
In order to adress the question of the organization of adipose tissues at the whole tissue scale, we derived a macroscopic model from the agent-based formalism (described here) for dynamical interconnected fibrous networks. The agent-based model features fibers as segments of fixed length having the ability to form and break cross-links dynamically in time via Poisson processes. Fibers are subject to an alignment force at their junction, an external potential, random reposition and reorientation (Brownian motion), and a retraction force forcing them to maintain their attachment sites with their linked neighbors. Using scaling hypothesis on the model parameters and tools from kinetic theory, we first obtained a closed kinetic system describing the time evolution of the individual fiber distribution function and of the fiber links distrubtion function [1]. We then performed formally the hydrodynamic limit and obtained a system of two PDE describing the evolution of the fiber local density and local orientation. In the simplified case of a homogeneous fiber density, we were able to show existence of solution of the macroscopic model [2], and through numerical simulations, we obtained a very good correspondence between the agent-based and the continuum formulations, validating numerically the formal derivation of the macroscopic model [2].This study is the first step towards a deeper understanding of the mechanisms of fibrous tissue self-organization at different scales. These work have inspired a new project described further.
Collaborations:
P. Degond (Imperial College London), F. Delebecque (IMT, Toulouse)
Related publications:
P. Degond, F. Delebecque, D. Peurichard, Continuum model for linked fibers with alignment interactions, M3AS, 26 (2016) pp. 269–318, lien arxiv
D. Peurichard, Macroscopic model for linked fibers with alignment interactions: existence theory and numerical simulations, SIAM MMS 14-4 (2016), pp. 1175-1210
P. Degond, D. Peurichard, Modelling tissue self-organization: from micro to macro models, to appear as book chapter in Multiscale Models in Mechano and Tumor Biology: Modeling, Homogenization, and Applications, A. Gerisch, R. Penta, J. Lang (Editors)
II. Micro-macro passage for dynamical networks
Aims and tools:
Our derivation (cf part I) of a macroscopic model for interconnected networks inspired new works aiming at deriving a macroscopic model for a simplified dynamical network. The goal and challenge here was to obtain a macroscopic description of a system composed of microscopic interactions between individual particles, starting from a simplified view of the network. In [1], the authors successfully derived a macroscopic model for particles interacting via their close neighbors by creating/breaking dynamically in time connections view as springs. The simpler version of the macroscopic equation (compared to our model of part I) enabled a Fourier linear and non linear analysis to detect and fully characterize phase transitions in the equilibrium solutions of the continuum model (cf [1]). I joined the project to help investigating numerically the link between the agent-based and macroscopic formulations of this model [2] and we obtained a very good agreement between the two models. With this framework (simplified compared to our works of part I), we hope to investigate the micro-macro link getting rid of a scaling hypothesis we needed for the derivation, and which engenders a loss of information of the link forces at the macroscopic level. The hope is to obtain a macroscopic model for dynamical networks which exhibits the same rich behavior as the one offered by the agent-based models.
Collaborations:
P. Degond, J.A. Carrillo, E. Zatorska (Imperial College London), J. Barré (Nice Sophia Antipolis)
Related publications:
J. Barré, P. Degond, E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks. submitted, arXiv
J. Barré, J.A. Carillo, P. Degond, E. Zatorska, D. Peurichard, Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions, Journal of Nonlinear Science (2017), 1-34, doi:10.1007/s00332-017-9408-z lien arXiV
