Exploring the Physical Limits of Saturation Contrast in Magnetic Resonance Imaging

Abstract

Magnetic Resonance Imaging has become nowadays an indispensable tool with applications ranging from medicine to material science. However, so far the physical limits of the maximum achievable experimental contrast were unknown. We introduce an approach based on principles of optimal control theory to explore these physical limits, providing a benchmark for numerically optimized robust pulse sequences which can take into account experimental imperfections. This approach is demonstrated experimentally using a model system of two spatially separated liquids corresponding to blood in its oxygenated and deoxygenated forms. Since its discovery in the forties, Nuclear Magnetic Resonance (NMR) has become a powerful tool 1,2 to study the state of matter in a variety of domains extending from biology and chemistry 3 to solid-state physics and quantum computing 4,5 . The power of NMR techniques is maybe best illustrated by medical imaging 6 , where it is possible e.g. to produce a three-dimensional picture of the human brain. NMR spectroscopy and Magnetic Resonance Imaging (MRI) involve the manipulation of nuclear spins via their interaction with magnetic fields. All experiments in liquid phase can be described in a first approach as follows. A sample is held in a strong and uniform longitudinal magnetic field denoted B 0 . The magnetization of the sample is then manipulated by a particular sequence of transverse radio-frequency magnetic pulses B 1 in order to prepare the system in a particular state. The analysis of the radio-frequency signal that is subsequently emitted by the nuclear spins leads to information about the structure of the molecule and its spatial position. One deduces from this simple description that the crucial point of this process is the initial preparation of the sample, i.e. to design a corresponding pulse sequence to reach this particular state with maximum efficiency. The maximum achievable efficiency can be determined for the transfer between well defined initial and target states 7 if relaxation effects can be neglected. In imaging applications, where relaxation forms the basis for contrast, a very large number of different strategies have been proposed and implemented so far with the rapid improvement of NMR and MRI technology 2,6 . However, there was no general approach to provide the maximum possible performance and the majority of these pulse sequences have been built on the basis of intuitive and qualitative reasonings or on inversion methods such as the Shinnar-Le Roux algorithm 8 . Note that this latter can be applied only in the case where there is no relaxation effect and radio-frequency inhomogeneity.A completely different point of view emerges if this problem is approached from an optimal control perspective. Optimal control theory was created in its modern version at the end of the 1950s with the Pontryagin Maximum Principle (PMP) [9][10][11] . Developed originally for problems in space mechanics, optimal control has become a key tool in a large spectrum of applications including eng...

Training Schrödinger’s cat: quantum optimal control

“…Singular optimal arcs have been used to derive control fields in some example quantum systems [77,[82][83][84][85][86][87]. This approach can also be advantageously combined with numerical optimization techniques in order to manipulate more complicated systems [88].…”

“…for Δt sim → 0, the gain of the optimal-control schemes approaches infinity [71,314]. Examples of geometric optimal control applications involving singular arcs [68] in NMR are relaxation-optimized polarization transfer experiments [78,79,82,315], minimal-time controls for the saturation of spins [77], for maximizing contrast in MRI [88] and for maximizing the achievable signal-to-noise ratio per unit time [316,317].…”

“…A current challenge is to improve the computational speed and accuracy of the algorithms. A promising approach to this end is the combined use of numerical and geometric optimal control methods [88]. Analogously to open quantum systems, improving the computational speed will be an important prerequisite to attack control problems of increasing complexity, representing experimental settings in a realistic way.…”

Fully efficient time-parallelized quantum optimal control algorithm

“…In this framework, different numerical optimal control algorithms [7][8][9] have been developed and applied to a large variety of quantum systems. Optimal control was used in physical chemistry in order to steer chemical reactions [3], but also for spin systems [10,11] with applications in Nuclear Magnetic Resonance [7,[12][13][14][15][16] and Magnetic Resonance Imaging [17][18][19]. Recently, optimal control has attracted attention in view of applications to quantum information processing, for example as a tool to implement high-fidelity quantum gates in minimum time [4,20,21].…”

Discrete-valued-pulse optimal control algorithms: Application to spin systems

“…The design of control sequences accounting for experimental constraints is a central task in a variety of domains in quantum dynamics extending from photochemistry, Nuclear Magnetic Resonance (NMR) and quantum information science [1][2][3][4][5][6][7][8][9]. Nowadays, Optimal Control Theory (OCT) reveals to be a highly efficient and versatile tool to bring answers to the different issues raised by the experimental setups [2,[10][11][12][13][14][15][16][17][18][19][20][21][22].…”