Fisher information and Cramér-Rao lower bound for experimental design in parallel imaging.

PURPOSE: The Cramér-Rao lower bound (CRLB) is widely used in the design of magnetic resonance (MR) experiments for parameter estimation. Previous work has considered only Gaussian or Rician noise distributions in this calculation. However, the noise distribution for multi-coil acquisitions, such as in parallel imaging, obeys the noncentral χ-distribution under many circumstances. The purpose of this paper is to present the CRLB calculation for parameter estimation from multi-coil acquisitions. THEORY AND METHODS: We perform explicit calculations of Fisher matrix elements and the associated CRLB for noise distributions following the noncentral χ-distribution. The special case of diffusion kurtosis is examined as an important example. For comparison with analytic results, Monte Carlo (MC) simulations were conducted to evaluate experimental minimum standard deviations (SDs) in the estimation of diffusion kurtosis model parameters. Results were obtained for a range of signal-to-noise ratios (SNRs), and for both the conventional case of Gaussian noise distribution and noncentral χ-distribution with different numbers of coils, m.
RESULTS: At low-to-moderate SNR, the noncentral χ-distribution deviates substantially from the Gaussian distribution. Our results indicate that this departure is more pronounced for larger values of m. As expected, the minimum SDs (i.e., CRLB) in derived diffusion kurtosis model parameters assuming a noncentral χ-distribution provided a closer match to the MC simulations as compared to the Gaussian results.
CONCLUSION: Estimates of minimum variance for parameter estimation and experimental design provided by the CRLB must account for the noncentral χ-distribution of noise in multi-coil acquisitions, especially in the low-to-moderate SNR regime. Magn Reson Med 79:3249-3255, 2018.