Mathematical methods for magnetic resonance based electric properties tomography

Magnetic resonance-based electric properties tomography (MREPT) is a recent quantitative imaging technique that could provide useful additional information to the results of magnetic resonance imaging (MRI) examinations. Precisely, MREPT is a collective name that gathers all the techniques that elaborate the radiofrequency (RF) magnetic field B1 generated and measured by a MRI scanner in order to map the electric properties inside a human body. The range of uses of MREPT in clinical oncology, patient-specific treatment planning and MRI safety motivates the increasing scientific interest in its development. The main advantage of MREPT with respect to other techniques for electric properties imaging is the knowledge of the input field inside the examined body, which guarantees the possibility of achieving high-resolution. On the other hand, MREPT techniques rely on just the incomplete information that MRI scanners can measure of the RF magnetic field, typically limited to the transmit sensitivity B1+. In this thesis, the state of art is described in detail by analysing the whole bibliography of MREPT, started few years ago but already rich of contents. With reference to the advantages and drawbacks of each technique proposed for MREPT, the particular implementation based on the contrast source inversion method is selected as the most promising approach for MRI safety applications and is denoted by the symbol csiEPT. Motivated by this observation, a substantial part of the thesis is devoted to a thoroughly study of csiEPT. Precisely, a generalised framework based on a functional point of view is proposed for its implementation. In this way, it is possible to adapt csiEPT to various physical situations. In particular, an original formulation, specifically developed to take into account the effects of the conductive shield always employed in RF coils, shows how an accurate modelling of the measurement system leads to more precise estimations of the electric properties. In addition, a preliminary study for the uncertainty assessment of csiEPT, an imperative requirement in order to make the method reliable for in vivo applications, is performed. The uncertainty propagation through csiEPT is studied using the Monte Carlo method as prescribed by the Supplement 1 to GUM (Guide to the expression of Uncertainty in Measurement). The robustness of the method when measurements are performed by multi-channel TEM coils for parallel transmission confirms the eligibility of csiEPT for MRI safety applications.