Exploring the Physical Limits of Saturation Contrast in Magnetic Resonance Imaging

TLDR

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 engineering, biology and economics.Solving an optimal control problem leads to the determination of a particular trajectory, that is a solution of an associated Hamiltonian system constructed from the PMP and satisfying given boundary conditions.This approach has found remarkable applications in quantum computing and NMR spectroscopy, but its application to MRI has been limited to the numerical design of slice-selective 90u and 180u pulses 12 .Despite the efficiency of MRI techniques currently used in clinics, some aspects still pose fundamental problems of both theoretical and practical interest.The enhancement of contrast remains one of the crucial questions for improving image quality and the corresponding medical diagnosis.The use of particular pulse sequences to generate image contrast based on relaxation rates is not new in MRI, since this question was raised at the beginning of the development of MRI in the 1970s.Different strategies have been proposed, such as the Inversion Recovery sequence 13,14 for T 1 contrast and pulses for ultra short echo time experiments for T 2 contrast 15 (See Eq. ( 1) for the definition of T 1 and T 2 parameters).Here, we go beyond such intuitive methods by using the powerful machinery of optimal control, which provides in this case not just an improved performance but an