Nuclear magnetic resonance: The contrast imaging problem

Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal problem and the Pontryagin Maximum Principle is used to select the optimal trajectories among the extremal solutions. Correlation between the contrast problem and the optimal transfer time problem is demonstrated. Further, we develop some analysis of the singular extremals and apply the results to examples of cerebrospinal fluid/water and grey/white matter of the cerebrum.