Modelling tissue self-organization: from micro to macro models

In this chapter, we present recent works concerned with the derivation of a macroscopic model for complex interconnected fiber networks from an agent-based model, with applications to, but not limited to, adipose tissue self-organization. Starting from an agent-based model for interconnected fibers interacting through alignment interactions and having the ability to create and suppress cross-links, the formal limit of large number of individuals is first investigated. It leads to a kinetic system of two equations: one for the individual fiber distribution function and one for the distribution function of connected fiber pairs. The hydrodynamic limit, in a regime of instantaneous fiber linking/unlinking then leads to a macroscopic model describing the evolution of the fiber local density and mean orientation. These works are the first attempt to derive a macroscopic model for interconnected fibers from an agent-based formulation and represent a first step towards the formulation of a large scale synthetic tissue model which will serve for the investigation of large scale effects in tissue homeostasis

Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation

We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of nonlinear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive

Large-scale dynamics of self-propelled particles moving through obstacles: model derivation and pattern formation

We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of non linear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive

Modelling tissue self-organization: from micro to macro models

In this chapter, we present recent works concerned with the derivation of a macroscopic model for complex interconnected fiber networks from an agent-based model, with applications to, but not limited to, adipose tissue self-organization. Starting from an agent-based model for interconnected fibers interacting through alignment interactions and having the ability to create and suppress cross-links, the formal limit of large number of individuals is first investigated. It leads to a kinetic system of two equations: one for the individual fiber distribution function and one for the distribution function of connected fiber pairs. The hydrodynamic limit, in a regime of instantaneous fiber linking/unlinking then leads to a macroscopic model describing the evolution of the fiber local density and mean orientation. These works are the first attempt to derive a macroscopic model for interconnected fibers from an agent-based formulation and represent a first step towards the formulation of a large scale synthetic tissue model which will serve for the investigation of large scale effects in tissue homeostasis

Modelling pattern formation through differential repulsion

Motivated by experiments on cell segregation, we present a twospecies model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments

Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions

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Particle interactions mediated by dynamical networks: Assessment of macroscopic descriptions

We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling process is very fast, the macroscopic model takes the form of a single aggregation–diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the one-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modeling of dynamical networks at a large scale

A new model for the emergence of blood capillary networks

We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and intersticial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper