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Directed flows of cells in vivo are essential in morphogenesis. They shape living matter in phenomena involving cell mechanics and regulations of the acto-myosin cytoskeleton. However the onset of coherent motion during collective cell migration is still poorly understood. Here we show that coherence is set by spontaneous alignments of cell polarity by designing cellular rings of controlled dimensions. A tug-of-war between opposite polarities dictates the onset of coherence, as assessed by tracking live cellular shapes and motions in various experimental conditions. In addition, we identify an internally driven constraint by cellular acto-myosin cables at boundaries as essential to ensure coherence and active force is generated as evaluated by the high RhoA activity. Its contribution is required to trigger coherence as shown by our numerical simulations based on a novel Vicsek-type model including free active boundaries. Altogether, spontaneous coherent motion results from basic interplay between cell orientations and active cables at boundaries.

This manuscript gathers my contributions focused on developing new mathematical and computational methods for analyzing biological flows as complex multiphysics and multiscale phenomena. The description of the underlying mechanisms stems from the basic principles of fluid dynamics and is translated into systems of partial or ordinary differential equations. The overall objective of this work is the study of these equations atthe continuous and discrete levels, their coupling and the development of a reliable and efficient computational framework to implement various numerical methods to approximate the solutions to these problems. The numerical simulations incorporate realistic geometries, are thoroughly validated against experimental data and target specific biomedical applications. The first chapter focuses on three-dimensional models,in which the motion of a biofluid in a complex, realistic geometry is governed by the incompressible Navier-Stokes equations in a domain with inlet and outlet boundaries, the main application in view being the study of the cerebral venous network. The purpose of Chapter 2 is two-fold: (i) first, we present contributions towards the elaboration of several reduced 0d models describing the coupled dynamics of different biofluids in the eye-cerebral system; (ii) second, we describe a new splitting strategy for the numerical solving of coupled systems of partial and ordinary differential equations for fluid flows. In Chapter 3, the fluid dynamics description from the previous chapters is enriched to take into account the combined effects of flow and different structures, from a multiphysics perspective. The resulting framework is subsequently utilized for simulating blood flow inthe aorta, in view of a specific biomedical application and for the numerical simulation of particulate flows, with an emphasis on the issue of how to handle contacts.

International audience; We are developing a Mathematical Virtual Simulator (MVS)[1] that combines multiscale mathematical modeling, advanced numerical techniques and effcient computational tools motivated by biomedical applications in the context of ophthalmology. The design of MVS is constituted by a three-dimensional poro-elastic system, representing the biomechanics and hemodynamics of the ocular posterior tissues, and a lumped-parameter circuit that describes the blood circulation in the retina and the blood flow in the central retinal vessels. Driven by the physics of the problem, which requires a blood systemic view, we do not impose artificial boundary condition for the hemodynamics; rather, we couple the description of the flow by means of a system of partial differential equations with a system of Ordinary Differential Equations (ODEs).We have elaborated a novel splitting approach to numerically solve the multiscale problem in an efficient, accurate and affordable manner, which involves the coupling between Darcy equations and ODE systems.The algorithm is inspired by the work of Carichino et al.[2] and Glowinski[3] and it is based on a semi-discretization in time based on operator splitting in order to guarantee the physical energy balance at the discrete level. As a result, unconditional stability with respect to the time step choice is ensured by the implicit treatment of interface conditions within the Darcy sub-steps, whereas the coupling between the 3d and 0d sub-steps is enforced via appropriate initial conditions for each sub-step. Notably, unconditional stability is attained without the need of sub-iterating between Darcy and ODE sub-steps. In our case, the Darcy problem is solved using HDG (Hybridizable Discontinuous Galerkin) method to achieve high accuracy on fluxes, which have a fundamental role from a biological viewpoint.In conclusion, we have formulated and implemented a numerical algorithm combining HDG discretization with an unconditionally stable operator splitting method for multiscale problems in order to simulate the physiological behavior of blood flow in the ocular posterior segment. More generally, the algorithm might be utilized for other problems arising in engineering and life sciences, where a robust and efficient couplingbetween a 3d poro-elastic model and a 0d fluid flow circuit described by a system of ODEs is required.References[1] L. Sala, C. Prud'homme, G. Guidoboni, M. Szopos., Towards a full model for ocular biomechanics, fluid dynamics and hemodynamics.. J. for Modeling in Ophthalmology, to appear.[2] L. Carichino, G. Guidoboni, M. Szopos., Energy-based operator splitting approach for the time discretization of coupled systems of partial and ordinary differential equations for fluid flows: the Stokes case.Journal of Computational Physics, 364, 235-256.[3] R. Glowinski, Finite Element methods for incompressible viscous flow. In Handbook of Numerical Analysis, Vol. IX, North-Holland, Amsterdam, 2003, pp. 3-1176.

International audience; In the last 20 years there have been lots of progress in 3D medical imaging (such as Magnetic Resonance Imaging, MRI, and X-ray Computed Tomography, CT) and in particular in modalities to visualise vascular structures. The resulting images have been successfully used in various clinical applications, in particular for cerebrovascular pathologies (e.g., neurosurgery planning; stenoses, aneurysm or thrombosis quantification; arteriovenous malformation detection and follow-up, etc.). The complexity of the processing and analysis of these images (size, information vs noise, artifacts, etc) led to the development of imaging tools such as vessel filtering, segmentation and quantification. There is however, until now, no database of synthetic images and associated ground-truths (segmented data) available in cerebrovascular images contrary to morphological brain image analysis (e.g. brainweb).In the ANR Vivabrain project, we combine the skills of several communities: computer science, applied mathematics, biophysics, and medicine to remedy the aforementioned observation. In particular we focus on complex multi-disciplinary problems such as (i) the handling of inter-individual cerebrovascular variability, (ii) the generation of computational meshes, (iii) the simulation of blood flows in the complete cerebrovascular system 3D+time (3D+t) including calibration and validation and (iv) the accurate simulation of the physical processes involved in MRA acquisition sequences in order to finally obtain realistic virtual angiographic images.

International audience; Elevated intraocular pressure (IOP), clinically referred to as ocular hypertension (OHT), is a recognized risk factor for irreversible vision loss. Moreover, clinical studies showed that alterations in blood pressure (BP) are often associated with many ocular diseases. Thus, the combined effect of IOP and BP should be taken into account when assessing disease risks and evaluating the outcomes of different treatments on a specific patient. For example, the outcomes of IOP-lowering medications used for the treatment of OHT are observed to vary significantly among individuals, even though the factors influencing such variability are not yet fully understood. Here we propose a mathematical model to theoretically investigate the relationship between IOP and BP, with the aim of evaluating their potential impact on the individual response to IOP-lowering medications.

International audience; Over the past few decades, the computational fluid dynamics has evolved to become an important tool for the description of the complex multi-physics, multi-scale phenomena characterizing blood flow. Its reliability depends both on the verification of the numerical methods and on the validation of the mathematical models. The aim of the first part of the talk is to present a preconditioning framework for the linear system arising from the finite element discretizations and time advancing finite difference schemes of the 3D steady and unsteady Navier-Stokes equations. In particular, we are interested in preconditioners based on an algebraic factorization of the system's matrix which exploit its block structure, such as the Pressure Convection-Diffusion (PCD) preconditioner , the SIMPLE preconditioner or the LSC preconditioner, see [Elman et al. (2014)]. A comparison between the efficiency of these preconditioners is ascertain by testing them over the 3D backward facing step benchmark. The iteration counts using the PCD preconditioner are independent of mesh size and high order finite elements and mildly dependent on Reynolds numbers for steady flow problems which is not the case for the other preconditioners. In the second part of the talk we describe a framework for the solution of flow problems relevant to biomechanics strongly supported by the aforementioned solving strategies. We assess the efficiency of this framework through experimental data for fluid flow in a nozzle model with rigid boundaries, a device designed to reproduce acceleration, deceleration and recirculation, features commonly encountered in medical devices [Stewart et al. (2012)]. The flow rates were chosen to cover laminar (Re = 500), transient (Re = 2000) and turbulent (Re = 3500) regimes. The numerical results displayed during the presentation are obtained using the open-source library Feel++ (Finite Element method Embedded Language in C++, www.feelpp.org). Figure 1: Computational Fluid Dynamics FDA Benchmark at Re = 2000.

The aim of this thesis is to study several questions which arise in the theory of elasticity, by using methods of mathematical analysis and differential geometry. In the one-dimensional case, related to the study of elastic wires, we prove some existence, uniqueness and stability results for a curve in Sobolev spaces. Then, we treat the general case of an immersion of arbitrary dimension and co-dimension of a submanifold in an Euclidean space. We show that the classical existence and uniqueness result for such an immersion can be extended up to the boundary of the submanifold, under a specific, but mild, regularity assumption on this set. Moreover, we show that the mapping constructed in this fashion is locally Lipschitz-continuous with respect to suitable topologies. Finally, we reconsider the study of elastic wires, to obtain some linear and nonlinear Korn inequalities for curves in dimension 3., Le but de cette thèse est d'étudier des questions issues de la théorie de l'élasticité en utilisant des méthodes d'analyse mathématique et de géométrie différentielle. Dans le cas mono-dimensionnel, qui est lié à l'étude des fils élastiques, nous prouvons des résultats d'existence, d'unicité et de stabilité d'une courbe dans des espaces de Sobolev. Nous traitons ensuite le cas général d'une immersion de dimension et de co-dimension quelconques d'une sous-variété dans l'espace euclidien. Nous montrons ainsi que le résultat classique d'existence et d'unicité d'une telle immersion peut être étendu jusqu'au bord de la sous-variété, sous une hypothèse de régularité peu restrictive sur celui-ci. En outre, nous montrons que l'application ainsi construite est localement lipschitzienne pour les topologies appropriées. Enfin, nous revenons à l'étude des fils élastiques, pour obtenir des inégalités de Korn linéaires et non linéaires pour les courbes en dimension 3.

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