Modélisation stochastique de processus pharmaco-cinétiques, application à la reconstruction tomographique par émission de positrons (TEP) spatio-temporelle. (Stochastic modeling of pharmaco-kinetic processes, applied to PET space-time reconstruction)

L'objectif de ce travail est de developper de nouvelles methodes statistiques de reconstruction d'image spatiale (3D) et spatio-temporelle (3D+t) en Tomographie par Emission de Positons (TEP). Le but est de proposer des methodes efficaces, capables de reconstruire des images dans un contexte de faibles doses injectees tout en preservant la qualite de l'interpretation. Ainsi, nous avons aborde la reconstruction sous la forme d'un probleme inverse spatial et spatio-temporel (a observations ponctuelles) dans un cadre bayesien non parametrique. La modelisation bayesienne fournit un cadre pour la regularisation du probleme inverse mal pose au travers de l'introduction d'une information dite a priori. De plus, elle caracterise les grandeurs a estimer par leur distribution a posteriori, ce qui rend accessible la distribution de l'incertitude associee a la reconstruction. L'approche non parametrique quant a elle pourvoit la modelisation d'une grande robustesse et d'une grande flexibilite. Notre methodologie consiste a considerer l'image comme une densite de probabilite dans (pour une reconstruction en k dimensions) et a chercher la solution parmi l'ensemble des densites de probabilite de . La grande dimensionalite des donnees a manipuler conduit a des estimateurs n'ayant pas de forme explicite. Cela implique l'utilisation de techniques d'approximation pour l'inference. La plupart de ces techniques sont basees sur les methodes de Monte-Carlo par chaines de Markov (MCMC). Dans l'approche bayesienne non parametrique, nous sommes confrontes a la difficulte majeure de generer aleatoirement des objets de dimension infinie sur un calculateur. Nous avons donc developpe une nouvelle methode d'echantillonnage qui allie a la fois bonnes capacites de melange et possibilite d'etre parallelise afin de traiter de gros volumes de donnees. L'approche adoptee nous a permis d'obtenir des reconstructions spatiales 3D sans necessiter de voxellisation de l'espace, et des reconstructions spatio-temporelles 4D sans discretisation en amont ni dans l'espace ni dans le temps. De plus, on peut quantifier l'erreur associee a l'estimation statistique au travers des intervalles de credibilite.

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