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Study of the Bloch-Torrey equation associated to diffusion magnetic resonance imaging

Abstract : Diffusion magnetic resonance imaging (dMRI) is an experimental technique which aims at unraveling the microstructural properties of a sample well below the conventional spatial resolution of ``classic'' MRI. Although this technique has been proposed and applied in various contexts for several decades, many theoretical points remain to be clarified, even more with the permanent improvement of MRI scanners and experimental protocols. Notably, the understanding of the signal formation at high gradients is largely incomplete, in spite of the ``natural'' tendency to increase the gradient in order to probe finer and finer structural scales.We first revisit anisotropy effects. While micro- and macroscopic anisotropy have been largely studied over past years, the intermediate, ``mesocopic'' scale had not been investigated in a systematic way. We have obtained a generalized Mitra formula which improves significantly surface-to-volume ratio estimations for arbitrary domains and gradient waveforms.In a second chapter, we investigate permeability effects, that are crucial for biomedical applications. We critically revise three classical models of exchange for dMRI. Moreover, we obtain a general and flexible numerical and theoretical method to study diffusion trough several parallel permeable membranes.The last chapter is the heart of the thesis and contains a non-perturbative study of Bloch-Torrey equation, which governs the evolution of dMRI signal. At high gradient strength, we reveal theoretically, numerically, and experimentally the universality of the localization phenomenon, which opens promising perspectives to improve the sensitivity of the signal to the microstructure.
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Nicolas Moutal. Study of the Bloch-Torrey equation associated to diffusion magnetic resonance imaging. Mathematical Physics [math-ph]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX031⟩. ⟨tel-02926470⟩

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