Déconvolution de Laplace : Estimation des temps de transit d'un agent de contraste en imagerie dynamique - Application au suivi de l'angiogénèse tumorale

In the context of cancer treatments, a major issue is to follow the effect of anti-angiogenesis drugs. If parametric models have been developed to achieve this goal, they suffer from being tissue-related and, moreover, if their pertinence is already questionable in heathy tissue, they are certainly wrong in tumors where the cell growth changes the nature of the tissue. In order to face these problems, nonparametric modeling of the blood flow exchanges has been imagined early in the 80's and started to be used in the second half of the 90's with the availability of high-frequency imaging techniques. Unfortunately, to date the estimation in such nonparametric models is highly unstable due to high level of ill-posedness.

After recalling the medical context which has motivated our study and describing the associated models, I will present a new nonparametric estimate in the Laplace deconvolution setting. This estimate is derived from the statistical analysis of Volterra equations of the first type, intimately linked to Laplace deconvolution. These estimate is shown to be adaptive in the sense that it achieves optimal rates of convergence up to the regularity of the unknown function even if this regularity is also unknown. This theoretical study is completed by simulations which show the proper behavior of these estimates.

Collaboration with Felix Abramovich (TAU), Charles-André Cuénod (UPD-HEGP), Marianna Pensky (UCF)