Two Short Presentations related to Cancer Modeling

International audienceThis paper contains two short presentations related to the mathematical modeling of Cancer. The first part intends to introduce a tumour-immune system interaction, which describes the early dynamics of cancerous cells, competing with the immune system, potentially leading to either the elimination of tumoral cells or to the viability of a solid tumor. The second part of the paper addresses the case where a solid tumor has grown enough to initiate angiogenesis, a process which equips the tumor with its own blood network. Nash game theory is used to model the interaction between activators and inhibitors of the angiogenesis process

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays

Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker-D\"oring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.Comment: 28 page

Use of ultrasonography exams to determinate the parturition day by Yorkshire canine breed

The purpose of this study was verify the efficacy of ultrasonography to determinate the parturitionday by Yorkshire canine breed also to determinate a measures pattern embryonic vesicle diameter crown rumplenght biparietal diameter body diameter torax diameter abdomen diameter and femur length and also toestablish the linear regression formula to be used by veterinarians for this breed. The length of pregnancy fromthe date of the first mated to the first parturition signs resulted in 63,57 days. So to predict the date of parturitionwas used a multiple linear regression analyses. It was possible determinate a formula to predict the date ofparturition utilizing crown-rump length, biparietal diameter and femur length obtained a major correlation(R 0,998)., ,²Oobjetivo deste trabalho foi verificar a eficácia do método ultra-sonografico visando prever a data departo em cadelas da raça Yorkshire. Também foi objetivo determinar um padrão de mensurações de vesículagestacional, comprimento do feto, diâmetro biparietal, diâmetro do corpo, diâmetro do tórax, diâmetro deabdome e comprimento do fêmur, além de estabelecer uma fórmula de regressão linear para ser utilizada poroutros veterinários ultra-sonografistas nesta raça.Aduração da gestação a partir da data de primeira cópula atéos primeiros sinais de parto resultou numa média de 63,57 dias. Para poder predizer a data de gestação foirealizada uma análise de regressão linear multivariada. Foi possível determinar uma fórmula para prever a datade parturição em cadelas da raça Yorkshire utilizando o comprimento fetal diâmetro biparietal e comprimentodo fêmur obtendo-se significativa correlação (R =0,998)

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters

Boltzmann and Fokker-Planck equations modelling the Elo rating system with learning effects

In this paper we propose and study a new kinetic rating model for a large number of players, which is motivated by the well-known Elo rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated after each game. We state and analyse the respective Boltzmann type equation and derive the corresponding nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the Boltzmann and Fokker-Planck equation with various numerical experiments

Ethnic differences in DNA methyltransferases expression in patients with systemic lupus erythematosus

Systemic lupus erythematous (SLE) is a systemic autoimmune inflammatory disease with both genetic and epigenetic etiologies. Evidence suggests that deregulation of specific genes through epigenetic mechanisms may be a contributing factor to SLE pathology. There is increasing evidence that DNA methyltransferase activity may be involved. This study demonstrated modulation in expression of DNA methyltransferases (DNMTs) according to ethnicity in patients diagnosed with SLE. Furthermore, differential expression in one of the DNMTs was found in a subset of lupus patients on dehydroepiandrosterone (DHEA) therapy. Real-time PCR analyses of DNMT1, DNMT3A and DNMT3B in peripheral blood mononuclear cells from a cohort of African American and European American lupus and non-lupus women were conducted. Also, global DNA methylation was assessed using the MethylFlash.sup.TM methylated quantification colorimetric assay. These findings suggest that epigenetic changes may play a critical role in the manifestations of the disease observed among ethnic groups, particularly African American women who often have a higher incidence of lupus. DHEA therapy effects on DNMT3A expression in AA women warrant further investigation in a larger population

On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of nonlinear echoes; sharp scattering estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the nonlinear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications.Comment: News: (1) the main result now covers Coulomb and Newton potentials, and (2) some classes of Gevrey data; (3) as a corollary this implies new results of stability of homogeneous nonmonotone equilibria for the gravitational Vlasov-Poisson equatio