.:: Stochastic Analysis Day - February 27, 2017 ::.

Tumors can be regarded as multicellular complex system of interacting cellular decision-makers. Decisions of interacting cells allow for the emergence of higher order organizational structures, such as tumors. At the same time, the new emergent structures exert selective forces on the cell phenotype decision dynamics. Here I investigate the impact on tumor development of a particular type of cancer cell decision-making the so-called migration/proliferation phenotypic plasticity (or Go-or-Grow) found in brain tumors. This mechanism implies a mutually exclusive switching between migratory and proliferative phenotypes. In this talk, I will discuss the identification of an emerging Allee effect, as a consequence of Go-or-Grow dependency on local tumor cell density, in terms of brain tumor initiation, persistency and potential therapeutic implications.

Currently, most of the basic mechanisms governing tumor-immune system interactions, in combination with modulations of tumor-associated vasculature, are far from being completely understood. In this talk I will present a mathematical model of vascularized tumor growth, where the main novelty is the modelling of the interplay between functional tumor vasculature and effector recruitment dynamics. Parameters were calibrated on the basis of different in vivo Rag1^-/- and wild-type (WT) BALB/c murine tumor growth experiments. The model analysis supports that vasculature normalization can be a plausible and effective strategy to treat cancer when combined with appropriate immuno-stimulation. We find that improved levels of functional vasculature, potentially mediated by vascular normalization or stress alleviation strategies, can provide beneficial outcomes in terms of tumor burden reduction and control. Normalization of tumor blood vessels opens a therapeutic window of opportunity to augment the anti-tumor immune responses, as well as to reduce the intratumoral immunosuppression and hypoxia due to vascular abnormalities. The potential success of normalizing tumor vasculature closely depends on the effector cell recruitment dynamics and tumor sizes. Furthermore, an arbitrary increase of initial effector cell concentration does not necessarily imply tumor control, and we evidence the existence of an optimal effector concentration range for tumor shrinkage. Based on these findings, we suggest a theory-driven therapeutic proposal that optimally combines immune- and vaso-modulatory interventions. Finally, I will also show an example of how the proposed mathematical model is used to investigate the therapeutic potential of bacterial infections against solid tumors.

We consider a mathematical model for the Rayleigh-Benard convection, the stochastic Swift-Hohenberg equation

∂_tu = - (1 + ∂_x²)²u + ε²νu - u³ + ε^³⁄₂ξ(t,x).

Near its change of stability, a solution of this equation can be described in a multiscale setting as the product of a slowly varying amplitude equation and a faster periodic wave.

In this talk, after a very brief introduction to the problem in its deterministic setting, I'll present some recent developments on this rigorous approximation in the unbounded space domain setting.

We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed (in a pathwise sense) under addition of noise. Most of the results in this direction are limited to SDEs and associated linear SPDEs.

In this talk, we show a regularization by noise result for a nonlinear SPDE, namely a stochastic scalar conservation law on R^d with a space-irregular flux:

∂_tv + b·∇[v²] + ∇v∘∂_tW = 0,

where b = b(x) is a given deterministic, possibly irregular vector field, W is a d-dimensional Brownian motion (∘ denotes Stratonovich integration) and v = v(t, x, ω) is the scalar solution. More precisely we prove that, under suitable Sobolev assumptions on b and integrability assumptions on its divergence, the equation admits a unique entropy solution. The result is false without noise.

The proof of uniqueness is based on a careful combination of arguments used in the linear case: first we show the renormalization property for the kinetic formulation of the equation, then we use second order PDE estimates and a duality argument to conclude.

If time permits, we will discuss also some open questions.

We will first give a brief overview of some recent developments in KPZ and coupled KPZ equations. Then, we will give a heuristic argument for Glauber-Kawasaki dynamics leading to a nonlinear SPDE under a certain scaling limit. This part is motivated by a suggestive talk of Franco Flandoli.

We study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s^th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties w.r.t. the fractional parameter s. We show that under certain conditions on the noise, optimality conditions for the control problem can be established. This is joint work with Enrico Valdinoci.

Stochastic Analysis Day

Monday February 27, 2017

PROGRAM

PARTICIPANTS

Venue

Aula Magna

Dipartimento di Matematica

Università di Pisa

Largo Bruno Pontecorvo 5

56127 Pisa

Italia