20 research outputs found
Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities
This work focuses on an improved fractional Sobolev inequality with a
remainder term involving the Hardy-Littlewood-Sobolev inequality which has been
proved recently. By extending a recent result on the standard Laplacian to the
fractional case, we offer a new, simpler proof and provide new estimates on the
best constant involved. Using endpoint differentiation, we also obtain an
improved version of a Moser-Trudinger-Onofri type inequality on the sphere. As
an immediate consequence, we derive an improved version of the Onofri
inequality on the Euclidean space using the stereographic projection.Comment: 25 page
Modelling adhesion-independent cell migration
A two-dimensional mathematical model for cells migrating without adhesion
capabilities is presented and analyzed. Cells are represented by their cortex,
which is modelled as an elastic curve, subject to an internal pressure force.
Net polymerization or depolymerization in the cortex is modelled via local
addition or removal of material, driving a cortical flow. The model takes the
form of a fully nonlinear degenerate parabolic system. An existence analysis is
carried out by adapting ideas from the theory of gradient flows. Numerical
simulations show that these simple rules can account for the behavior observed
in experiments, suggesting a possible mechanical mechanism for
adhesion-independent motility.Comment: 22 pages and 9 figure
Micro- and Macroscopic Modeling of Crowding and Pushing in Corridors
Experiments with pedestrians revealed that the geometry of the domain, as
well as the incentive of pedestrians to reach a target as fast as possible have
a strong influence on the overall dynamics. In this paper, we propose and
validate different mathematical models at the micro- and macroscopic levels to
study the influence of both effects. We calibrate the models with experimental
data and compare the results at the micro -- as well as macroscopic levels. Our
numerical simulations reproduce qualitative experimental features on both
levels, and indicate how geometry and motivation level influence the observed
pedestrian density. Furthermore, we discuss the dynamics of solutions for
different modeling approaches and comment on the analysis of the respective
equations.Comment: 22 pages, 14 figure
The Influence of Nucleus Mechanics in Modelling Adhesion-independent Cell Migration in Structured and Confined Environments
: Recent biological experiments (Lämmermann et al. in Nature 453(7191):51-55, 2008; Reversat et al. in Nature 7813:582-585, 2020; Balzer et al. in ASEB J Off Publ Fed Am Soc Exp Biol 26(10):4045-4056, 2012) have shown that certain types of cells are able to move in structured and confined environments even without the activation of focal adhesion. Focusing on this particular phenomenon and based on previous works (Jankowiak et al. in Math Models Methods Appl Sci 30(03):513-537, 2020), we derive a novel two-dimensional mechanical model, which relies on the following physical ingredients: the asymmetrical renewal of the actin cortex supporting the membrane, resulting in a backward flow of material; the mechanical description of the nuclear membrane and the inner nuclear material; the microtubule network guiding nucleus location; the contact interactions between the cell and the external environment. The resulting fourth order system of partial differential equations is then solved numerically to conduct a study of the qualitative effects of the model parameters, mainly those governing the mechanical properties of the nucleus and the geometry of the confining structure. Coherently with biological observations, we find that cells characterized by a stiff nucleus are unable to migrate in channels that can be crossed by cells with a softer nucleus. Regarding the geometry, cell velocity and ability to migrate are influenced by the width of the channel and the wavelength of the external structure. Even though still preliminary, these results may be potentially useful in determining the physical limit of cell migration in confined environments and in designing scaffolds for tissue engineering
Conservation, convergence, and computation for evolving heterogeneous elastic wires
The elastic energy of a bending-resistant interface depends both on its
geometry and its material composition. We consider such a heterogeneous
interface in the plane, modeled by a curve equipped with an additional density
function. The resulting energy captures the complex interplay between curvature
and density effects, resembling the Canham-Helfrich functional. We describe the
curve by its inclination angle, so that the equilibrium equations reduce to an
elliptic system of second order. After a brief variational discussion, we
investigate the associated nonlocal -gradient flow evolution, a coupled
quasilinear parabolic problem. We analyze the (non)preservation of quantities
such as convexity, positivity, and symmetry, as well as the asymptotic behavior
of the system. The results are illustrated by numerical experiments.Comment: 34 pages, 13 figures. Comments are welcome
Canards in a bottleneck
In this paper we investigate the stationary profiles of a nonlinear
Fokker-Planck equation with small diffusion and nonlinear in- and outflow
boundary conditions. We consider corridors with a bottleneck whose width has a
global nondegenerate minimum in the interior. In the small diffusion limit the
profiles are obtained constructively by using methods from geometric singular
perturbation theory (GSPT). We identify three main types of profiles
corresponding to: (i) high density in the domain and a boundary layer at the
entrance, (ii) low density in the domain and a boundary layer at the exit, and
(iii) transitions from high density to low density inside the bottleneck with
boundary layers at the entrance and exit. Interestingly, solutions of the last
type involve canard solutions generated at the narrowest point of the
bottleneck. We obtain a detailed bifurcation diagram of these solutions in
terms of the in- and outflow rates. The analytic results based on GSPT are
further corroborated by computational experiments investigating corridors with
bottlenecks of variable width.Comment: arXiv admin note: text overlap with arXiv:2010.1442
Modelling adhesion-independent cell migration
International audienceA two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modelled as an elastic curve, subject to an internal pressure force. Net polymerization or depoly-merization in the cortex is modelled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility
Comparison and calibration of different electroporation models. Application to rabbit livers experiments
Electroporation is a complex phenomenon that occurs when biological tissues are subjected to electric pulses. The clinical interest for the phenomenon has constantly increased for the last decades. Indeed, electroporation makes it possible to either kill directly the cells in the target region (tumor) or to introduce molecules into living cells. However, one of the main limitation of using electroporation in the clinical routine comes from the technical difficulties raised by such therapies, in particular it is difficult to well determine the treated zone. Numerical modeling of the electric field magnitude could provide a powerful strategy to assess the treatment efficacy: thanks to well-designed models, the computation of the distribution of the electric field is achievable, providing a numerical evaluation of the treatment. The main objective of this work is to go further on the patient-adapted numerical modeling of the electric field magnitude by laying the ground of the possible electroporation models - which will be compared qualitatively - and their calibrations. This will be done in the framework of bioelectrical measurements on rabbit livers that come from the literature