In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects ...caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.
We give a practical tool to control the L∞-norm of the Steklov eigenfunctions in a Lipschitz domain in terms of the norm of the BV-trace operator. The norm of this operator has the advantage to be ...characterized by purely geometric quantities. As a consequence, we give a spectral stability result for the Steklov eigenproblem under geometric domain perturbations and several examples where stability occurs. In particular we deal with geometric domains which are not equi-Lipschitz, like vanishing holes, merging sets, approximations of inner peaks.
We consider optimal control problems associated to generally non-well posed Cauchy problems in a general framework. Firstly, we approximate the ill-posed problem with a family of well-posed one and ...show that solutions of the latter one converge to solutions of the former one. Secondly, we investigate the minimization problem associated with the approximated state equation. We prove the existence and uniqueness of minimizers that we characterize with the optimality systems. Finally, we show that minimizers of the approximated problems converge to the minimizers of the optimal control subjected to the ill-posed state equation that we characterize with a singular optimality system. This characterization is obtained as the limit of the optimality systems of the approximated minimization problem. We use the techniques of quasi-reversibility developed by Lattès and Lions in 1969. Our general framework includes classical elliptic second order operators with Dirichlet and Robin conditions, as well as the fractional Laplace operator with the Dirichlet exterior condition.
Abstract
Using a particle-based model, we examine the collective dynamics of skyrmions interacting with a funnel potential under dc driving as the skyrmion density and relative strength of the Magnus ...and damping terms are varied. For driving in the easy direction, we find that increasing the skyrmion density reduces the average skyrmion velocity due to jamming of skyrmions near the funnel opening, while the Magnus force causes skyrmions to accumulate on one side of the funnel array. For driving in the hard direction, there is a critical skyrmion density below which the skyrmions become trapped. Above this critical value, a clogging effect appears with multiple depinning and repinning states where the skyrmions can rearrange into different clogged configurations, while at higher drives, the velocity-force curves become continuous. When skyrmions pile up near the funnel opening, the effective size of the opening is reduced and the passage of other skyrmions is blocked by the repulsive skyrmion–skyrmion interactions. We observe a strong diode effect in which the critical depinning force is higher and the velocity response is smaller for hard direction driving. As the ratio of Magnus force to dissipative term is varied, the skyrmion velocity varies in a non-linear and non-monotonic way due to the pile up of skyrmions on one side of the funnels. At high Magnus forces, the clogging effect for hard direction driving is diminished.
Abstract
Using a particle-based model, we examine the collective dynamics of skyrmions interacting with a funnel potential under dc driving as the skyrmion density and relative strength of the Magnus ...and damping terms are varied. For driving in the easy direction, we find that increasing the skyrmion density reduces the average skyrmion velocity due to jamming of skyrmions near the funnel opening, while the Magnus force causes skyrmions to accumulate on one side of the funnel array. For driving in the hard direction, there is a critical skyrmion density below which the skyrmions become trapped. Above this critical value, a clogging effect appears with multiple depinning and repinning states where the skyrmions can rearrange into different clogged configurations, while at higher drives, the velocity-force curves become continuous. When skyrmions pile up near the funnel opening, the effective size of the opening is reduced and the passage of other skyrmions is blocked by the repulsive skyrmion-skyrmion interactions. We observe a strong diode effect in which the critical depinning force is higher and the velocity response is smaller for hard direction driving. As the ratio of Magnus force to dissipative term is varied, the skyrmion velocity varies in a non-linear and non-monotonic way due to the pile up of skyrmions on one side of the funnels. At high Magnus forces, the clogging effect for hard direction driving is diminished.
We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the ...fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint $\operatorname{div} B = 0$ to machine precision, both at the spatially and temporally discrete levels.
Intra-tumour heterogeneity (ITH) has a strong impact on the efficacy of the immune re-sponse against solid tumours. The number of sub-populations of cancer cells expressing different antigens and the ...percentage of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells) in a tumour are both expressions of ITH. Here, we present a spatially explicit stochastic individual-based model of the interaction dynamics between tumour cells and CD8+ T cells, which makes it possible to dissect out the specific impact of these two expressions of ITH on anti-tumour immune response. The set-up of numerical simulations of the model is defined so as to mimic scenarios considered in previous experimental studies. Moreover, the ability of the model to qualitatively reproduce experimental observations of successful and unsuccessful immune surveillance is demonstrated. First, the results of numerical simulations of this model indicate that the presence of a larger number of sub-populations of tumour cells that express different antigens is associated with a reduced ability of CD8+ T cells to mount an effective anti-tumour immune response. Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CD8+ T cells may reduce the effectiveness of anti-tumour immunity. Ultimately, the mathematical model presented in this paper may provide a framework to help biologists and clinicians to better understand the mechanisms that are responsible for the emergence of different outcomes of immunotherapy.
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