Optimal Experiment Design for Magnetic Resonance Fingerprinting: Cramér-Rao Bound Meets Spin Dynamics

Abstract

Magnetic resonance (MR) fingerprinting is a new quantitative imaging paradigm, which simultaneously acquires multiple MR tissue parameter maps in a single experiment. In this paper, we present an estimation-theoretic framework to perform experiment design for MR fingerprinting. Specifically, we describe a discrete-time dynamic system to model spin dynamics, and derive an estimation-theoretic bound, i.e., the Cramér-Rao bound (CRB), to characterize the signal-to-noise ratio (SNR) efficiency of an MR fingerprinting experiment. We then formulate an optimal experiment design problem, which determines a sequence of acquisition parameters to encode MR tissue parameters with the maximal SNR efficiency, while respecting the physical constraints and other constraints from the image decoding/reconstruction process. We evaluate the performance of the proposed approach with numerical simulations, phantom experiments, and in vivo experiments. We demonstrate that the optimized experiments substantially reduce data acquisition time and/or improve parameter estimation. For example, the optimized experiments achieve about a factor of two improvement in the accuracy of T2 maps, while keeping similar or slightly better accuracy of T1 maps. Finally, as a remarkable observation, we find that the sequence of optimized acquisition parameters appears to be highly structured rather than randomly/pseudorandomly varying as is prescribed in the conventional MR fingerprinting experiments.

Flexible and efficient optimization of quantitative sequences using automatic differentiation of Bloch simulations

“…It provides a lower bound on the variance of a parameter estimate for an unbiased maximum likelihood estimator. When an analytical expression for the magnetization is available, the bound can be calculated and optimized as has been shown in multiple works . An analytical expression for the magnetization is available for a wide range of parameter mapping sequences, including but not limited to: DESPO (Variable Flip Angle Mapping) and DESPO, Inversion‐Recovery Mapping, multi‐echo spin echo mapping, Diffusion Weighted Imaging and quantitative double‐echo in steady‐state (DESS) .…”

Parameter map error due to normal noise and aliasing artifacts in MR fingerprinting

“…However, Sommer et al found inconsistency in this method in the presence of undersampling artifacts. With a rich history in statistical estimation theory, several groups have used the Cramer‐Rao lower bound to assess MRF parameter estimation in the presence of Gaussian white noise . However, Gaussian white noise does not accurately characterize the undersampling noise that is prevalent in MRF and dependent on signal amplitude.…”

Magnetic resonance fingerprinting review part 2: Technique and directions

“…. In contrast to the FA patterns found in, there are no constraints placed on consecutive flip angle changes, resulting in an FA pattern with large variations and rapid changes. However, in both cases, the point is made that by rigorously optimizing the sequence structure for MRF‐FISP, shorter sequences with improved efficiency may be possible than have been previously demonstrated.…”

“…The FA and TR patterns in (a) are from Cohen et al using the interior‐point optimization, applied to a cost function that emphasizes the orthogonality of the dictionary matrix. In (b) are FA and TR patterns, which are based on the Cramér–Rao lower bound for unbiased estimators. "Optimized I" and "Optimized II" refer to the constraints put on the FA pattern.…”

“…The Cramér–Rao bound is a statistical tool that places a lower bound on the variance of an unbiased estimator, and has recently been applied to derive optimal sequence parameters for T 1 mapping, and separately to increase the precision for the relaxation values estimated with DESPOT . A rigorous derivation of a cost function to characterize the signal to noise ratio (SNR) efficiency of the MRF sequence is presented in Zhao et al, using the Cramér–Rao lower bound. SNR efficiency is defined in terms of the variance in the estimated tissue properties from the MRF sequence.…”