Are Cramér-Rao lower bounds an accurate estimate for standard deviations in in vivo magnetic resonance spectroscopy?

Due to inherent time constraints for in vivo experiments, it is infeasible to repeat multiple MRS scans to estimate standard deviations on the desired measured parameters. As such, the Cramér-Rao lower bounds (CRLBs) have become the routine method to approximate standard deviations for in vivo experiments. Cramér-Rao lower bounds, however, as the name suggests, are theoretically a lower bound on the standard deviation and it is not clear if and under what circumstances this approximation is valid. Realistic synthetic 3 T spectra were used to investigate the relationship between estimated CRLBs, true CRLBs and standard deviations. Here we demonstrate that, although the CRLBs are theoretically truly a lower bound on the standard deviation (not an equality) for the problem typically encountered in quantification, they are still an adequate approximation to standard deviation as long as the model perfectly characterizes the data. In the case when the macromolecule basis deviates from the measured macromolecules it was shown that the CRLBs can deviate from standard deviations by approximately 50% for N-acetylaspartic acid, creatine and glutamate and of the order of 100% or more for myo-inositol and γ-aminobutyric acid. In the case when the model perfectly reflects the data the CRLBs are within approximately 10% of standard deviations for all metabolites. The result of the CRLB being within 10% of standard deviations means that, for an accurate model, novel quantification methods such as machine learning or deep learning will not be able to obtain substantially more precise estimates for the desired parameters than traditional maximum-likelihood estimation.