interactions: existence theory and numerical simulations
This chapter is about to be submitted.
Abstract:
In this paper, we study the macroscopic model of [2] for fibers interacting through linking/unlinking and alignment interactions. As the continuous model has been derived from the particle dynamics, we here aim to (i) gain insight into the properties of the solutions to the macroscopic model and (ii) propose a vali-dation of the macroscopic model via numerical comparison between its solutions and the ones obtained with the particle model. We provide an existence result and perform numerical simulations of the stationary solutions of the macroscopic equation. The numerical simulations of the macroscopic model lead to interest-ing features such as the emergence of a bucklinterest-ing phenomenon which highlights the physical properties of the macroscopic fiber network. We finally propose a first numerical comparison between the microscopic and macroscopic models. The numerical simulations reveal that the microscopic and macroscopic models are in good agreement, providing we adapt the regime of study.
keywords: cross-linked fibers, alignment, individual based model, mean-field limit, macroscopic model, non linear elliptic equation, numerical simulations
AMS classification: 35J60, 35Q92, 82C40, 82C22, 82C31, 82C70, 92C10,
92C17
Acknowledgments: DP gratefully acknowledges the hospitality of Imperial College London, where part of this research was conducted. DP wishes to grate-fully thank Pierre Degond (Imperial College London) for his support and helpful suggestions. DP also thank Fanny Delebecque (IMT, Toulouse), Louis Casteilla (StromaLab, Toulouse) and Anne Lorsignol (StromaLab, Toulouse) for helpful discussions.
Sommaire
1 Introduction . . . 155 2 Microscopic model and mean-field limit . . . 157 3 Macroscopic derived model . . . 164 4 Micro-Macro comparison . . . 175 5 Conclusion . . . 181 6 Appendix A . . . 185 7 Appendix B . . . 190 8 Appendix C . . . 193 9 Appendix D . . . 197
1 Introduction
Biological fiber networks such as the Extra-cellular Matrix of adipose tissues are complex cross-linked structures providing internal and external mechanical sup-port to cells, playing major roles in cell and tissue functions [3, 4]. The plasticity of these dynamical networks results from their ability to break and reform cross-links, giving to the tissue the ability to change shape and adapt in response to biological and mechanical stimuli [5, 6]. Because of their vital role in the mechan-ical behavior of cells and tissues, there is a great deal of interest in understanding such structures. However, they are challenging to model due to the large number of components and the complexity of the interactions within the structure. Sev-eral Individual Based Model (IBM) have been proposed to understand the elastic properties of fibrous networks [14, 7, 19, 16], but the computational cost of an IBM tremendously increases with the size of the system. Therefore, IBM for fiber net-works remain spatially limited. To model larger systems, some authors treat the network as a continuous medium [21, 20, 22] or couple both particle and continuous visions by homogenization methods to create a less detailed regularly-patterned discrete model [17]. The challenge for these last models is to construct accurate constitutive laws and homogenization techniques to encompass the dynamics of
the fiber network, even if it implies a loss of information at the individual level.
To overcome this weakness of the macroscopic models, a solution is to derive a macroscopic model from a microscopic one. The macroscopic model will gain in predictability compared to models based on phenomenological considerations [11].
The derivation of macroscopic models from microscopic ones has been intensively studied by many authors, and consists in the large scale limit of a kinetic version of the microscopic model [23]. The kinetic equation of the microscopic model is obtained by considering the limit of a large number of individuals. It describes the evolution of a particle distribution function which is the probability density for a particle to be in a given configuration at a given time. The hydrodynamic limit of this kinetic model is then obtained using conservation properties of the system.
In [2], the authors performed the formal derivation of a kinetic model starting from an individual based model for a dynamical cross-linked network interacting through alignment interactions. The hydrodynamic limit of the kinetic model then leads to a macroscopic model. We here want to provide a first study of the proposed macroscopic model, and compare it to its microscopic version.
The 2D IBM consists of fibers represented as straight lines of fixed length which have the ability to connect and disconnect to their crossing neighbors. Fiber mo-tion is driven by a reacmo-tion to an external mechanical potential modeling the acmo-tion of the cells in the medium (the cells are not modeled as proper agents). Cross-linked fibers are supposed to continuously align together, and we consider small random fiber motion and random fiber reorientation. The maintain of fiber links is modeled through the use of a distance-based potential which cancels when the fiber links are maintained. In [2], it has been shown that such a system leads, at the limit of large number of particles, to a system of two kinetic equations: (i) one for the one-particle distribution function - probability for a fiber to have a given orientation and position-, and (b) one for the two-particle distribution function - probability for two fibers to be in a given configuration in the domain-. The authors show that the system has an intrinsic closure relation and that it can be entirely described by these two distribution functions. Then, in a regime of fast linking/unlinking of fibers, it is shown that the two-particle distribution function reduces to the product of two one-particle distribution functions. This is due to the fact that this scaling hypothesis reduces the action of fiber links and the memory effect of the links is therefore lost. Through the use of the concept of generalized collision invariant introduced in [23], the one-particle distribution function con-verges towards the equilibria of a collision operator composed of the cross-linked fiber alignment interactions and of the noise intensity. In the hydrodynamic limit, two evolution equations are obtained: (a) for the density function and (b) for the mean local orientation of fibers.
In this paper, we aim to provide a first analysis of the macroscopic model of [2]
and to compare the macroscopic model to its microscopic version. In a first part (section 2), we present the microscopic model and show in appendix 6 that the IBM for fiber motion can be seen as a gradient descent for a quadratic penalization of a minimization problem that was proposed in [1]. We then introduce the kinetic equation associated to the IBM and the scaling for the derivation of the macro-scopic model. In a second part (section 3), we present the macromacro-scopic model and provide an existence result for the stationary solutions in the case of homoge-neous fiber density. Then, we perform numerical simulations for the macroscopic model, and show that interesting physical features appear such as a buckling phe-nomenon depending upon the model parameters. Finally (section 4), we provide a first numerical comparison between the macroscopic model and the underlying IBM.