Quantitative Magnetic Resonance Imaging: From Fingerprinting to Integrated Physics-Based Models

Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters, e.g., relaxation times T-1, T-2, or proton density p. Recently, in [Ma et al., Nature, 495 (2013), pp. 187-193], magnetic resonance fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two-step procedure: (1) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a precomputed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is nonconvex and that the accuracy of the MRF-type algorithms is limited by the "discretization size" of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two-step method, our model is dictionary-free and is rather governed by a single nonlinear equation, which is studied analytically. This nonlinear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples, for which improvement over MRF and its variants is also documented.