Tuesday, 17 April 2018, 3 p.m. (sharp),
Dr. David Hewett, University College London
at the conference room of IMATI-CNR in Pavia, will give a lecture titled:

Scattering by fractal screens - functional analysis and computation

and at 4.15 p.m (sharp),
Dr. Abramo Agosti, Politecnico di Milano
at the conference room of IMATI-CNR in Pavia, will give a lecture titled:

A diffuse interface model for tumour growth with applications to neuro-oncology

as part of the Applied Mathematics Seminar (IMATI-CNR e Dipartimento di Matematica, Pavia).

At the end a refreshment will be organized.


Abstract Dr. David Hewett:

The mathematical analysis and numerical simulation of acoustic and
electromagnetic wave scattering by planar screens is a classical
topic. The standard technique involves reformulating the problem as a
boundary integral equation on the screen, which can be solved
numerically using a boundary element method. Theory and computation
are both well-developed for the case where the screen is an open
subset of the plane with smooth (e.g. Lipschitz or smoother) boundary.
In this talk I will explore the case where the screen is an arbitrary
subset of the plane; in particular, the screen could have fractal
boundary, or itself be a fractal. Such problems are of interest in the
study of fractal antennas in electrical engineering, light scattering
by snowflakes/ice crystals in atmospheric physics, and in certain
diffraction problems in laser optics. The roughness of the screen
presents challenging questions concerning how boundary conditions
should be enforced, and the appropriate function space setting. But
progress is possible and there is interesting behaviour to be
discovered: for example, a sound-soft screen with zero area (planar
measure zero) can scatter waves provided the fractal dimension of the
set is large enough. Accurate computations are also challenging
because of the need to adapt the mesh to the fine structure of the
fractal. As well as presenting numerical results, I will outline some
of the outstanding open questions from the point of view of numerical
analysis. This is joint work with Simon Chandler-Wilde (Reading) and
Andrea Moiola (Pavia).

Abstract :

In this talk I will describe a diffuse-interface model based on mixture theory recently used to model tumor growth, and in particular to model the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM). Using thermodynamic principles, a Cahn-Hilliard type equation with degenerate mobility is obtained, in which the spreading dynamics of the multiphase tumour is coupled through a growth term with a parabolic equation describing a nutrient species dynamics.

A single-well potential of Lennard-Jones type is used in the model to describe the cell-cell mechanical interactions. As a consequence, the degeneracy set of the mobility and the singularity set of the potential do not coincide. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equations with a double-well potential analyzed in the literature.

I will show some analytical results for simplified versions of the model, together with different finite element approximations of the problem which preserve the analytical properties of the continuous solutions.

By comparing the observed in-vitro evolution of a culture of glioblastoma cells to the growth and coarsening dynamics described by the model, I will prove the effectiveness of using a single-well potential to describe the GBM cells interaction.

The tumor growth model is finally fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of a patient brain. I will show the predictions of numerical simulations and the comparison with the clinical data for a case test study, observing a good accordance with the data and highlighting the ground-breaking potential of the model for delivering accurate patient-specific predictions.

This work has been conducted in collaboration with Pasquale Ciarletta, Maurizio Grasselli, Paola F. Antonietti, Marco Verani, Chiara Giverso, Elena Faggiano and Aymeric Stamm.