Etude de quelques modèles en imagerie photoacoustique

This thesis work is related to  photo-acoustic imaging techniques which are new multiwave modalities in medical imaging that combine both high resolution of ultrasounds and contrast of optical methods. We precisely studied  the inverse problem  that consists of determining the optical coefficients of biological tissues from measurement of acoustic waves generated by the photo-acoustic effect. The photo-acoustic inverse problem proceeds in two steps.We first retrieve the initial pressure  from the measurement of the pressure wave on a part of the boundary of the sample. The first inversion takes then the form of a linear inverse source problem and provides internal data for the optical waves that are more sensitive to the contrast of the absorption and diffusion coefficients. In a  second step we recover the optical coefficients from the acquired internal data. The aim of this work is to study the two inversions in  different contexts.  In the first part, we develop a model that takes into account the variations of the acoustic speed in the medium. Indeed, most of the inversion methods suppose that the acoustic speed is constant, and this assumption can lead to errors in the reconstruction of the optical coefficients. The second part of this work is the derivation of stability estimates for the photo-acoustic inverse problem in a layered medium. We prove that the reconstruction is getting worse with depth. This is one of the main drawbacks of the photo-acoustic method, the imaging depth is limited to a few centimeters. The last part is about photo-acoustic generation with plasmonic nanoparticles. They enhance the photo-acoustic signal around them, so that we can investigate the tissue more deeply. We derive the mathematical model of the photo-acoustic generation by heating nanoparticles, and we solve the photo-acoustic inverse problem in this context.

Directors:

• Mr Laurent Desbat (Professeur - Université Grenoble Alpes )
• Mr Faouzi Triki (Maître de Conférénces - Université Grenoble Alpes )

Raporteurs:

• Mr Kais Ammari (Professeur - Université de Monastir )
• Mr Abdellatif El Badia (Professeur - Université de Compiègne )

Examinators:

• Mr Eric Soccorsi (Maître de conférénces - Université Aix Marseille )
• Mr Eric Bonnetier (Professeur - Université Grenoble Alpes )