Nicolas Vauchelet

Bio: Nicolas Vauchelet is an academic researcher from University of Paris. The author has contributed to research in topics: Population & Uniqueness. The author has an hindex of 20, co-authored 111 publications receiving 1110 citations. Previous affiliations of Nicolas Vauchelet include French Institute for Research in Computer Science and Automation & École normale supérieure de Lyon.

Derivation of a Hele-Shaw type system from a cell model with active motion

Abstract: We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of ‘mushy region’ prepares the tumor invasion.

Traveling wave solution of the Hele–Shaw model of tumor growth with nutrient

Abstract: Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele–Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.

Chemotaxis: from kinetic equations to aggregate dynamics

Abstract: The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obtained through a precise definition of the nonlinear flux as well as the complete dynamics of aggregates, i.e. combinations of Dirac masses. Finally a particle method is used to build an adapted numerical scheme.

Numerical Methods for One-Dimensional Aggregation Equations

Abstract: We focus in this work on the numerical discretization of the one-dimensional aggregation equation $\partial_t\rho + \partial_x (v\rho)=0$, $v=a(W'*\rho)$, in the attractive case. Finite time blow up of smooth initial data occurs for potential $W$ having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in the hydrodynamic limit. Finally numerical simulations are provided to illustrate the results.

Incompressible limit of a mechanical model of tumour growth with viscosity

Abstract: Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Abstract: This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Some modified matrix eigenvalue problems.

Abstract: We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

Competing effects of attraction vs. repulsion in chemotaxis

Abstract: We consider the attraction–repulsion chemotaxis system under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ ℝn with smooth boundary, where χ ≥ 0, ξ ≥ 0, α > 0, β > 0, γ > 0, δ > 0 and τ = 0, 1. We study the global solvability, boundedness, blow-up, existence of non-trivial stationary solutions and asymptotic behavior of the system for various ranges of parameter values. Particularly, we prove that the system with τ = 0 is globally well-posed in high dimensions if repulsion prevails over attraction in the sense that ξγ - χα > 0, and that the system with τ = 1 is globally well-posed in two dimensions if repulsion dominates over attraction in the sense that ξγ - χα > 0 and β = δ. Hence our results confirm that the attraction–repulsion is a plausible mechanism to regularize the classical Keller–Segel model whose solution may blow up in higher dimensions.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

Abstract: This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...

The Hele-Shaw asymptotics for mechanical models of tumor growth

Abstract: Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. Our first goal here is to formulate a free boundary model of Hele–Shaw type, a variant including growth terms, starting from the description at the cell level and passing to the stiff limit in the pressure law of state. In contrast with the classical Hele–Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density. We then go on to consider a more complex model including the supply of nutrients through vasculature, and we study the stiff limit for the involved coupled system.