Department of Biomedical Engineering, Columbia University Fu Foundation School of Engineering and Applied Science, New York, New York, USA.
Department of Radiology, Columbia University College of Physicians and Surgeons, New York, New York, USA.
NMR Biomed. 2021 Jul;34(7):e4521. doi: 10.1002/nbm.4521. Epub 2021 Apr 19.
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.
由于体内实验的固有时间限制，重复多次 MRS 扫描以估计所需测量参数的标准差是不可行的。因此，克拉美-罗下限 (CRLB) 已成为估计体内实验标准差的常规方法。然而，正如其名，克拉美-罗下限只是标准差的理论下界，并且不清楚这种近似在什么情况下是有效的。使用逼真的合成 3T 谱来研究估计的 CRLB、真实 CRLB 和标准差之间的关系。在这里，我们证明尽管 CRLB 在通常遇到的定量问题中理论上确实是标准差的真正下界（不是等式），但只要模型能完美地描述数据，它们仍然是标准差的一个充分近似。在大分子基础与所测量的大分子偏离的情况下，CRLB 对于 N-乙酰天门冬氨酸、肌酸和谷氨酸的标准差的偏差可以达到 50%左右，而对于肌醇和 γ-氨基丁酸的偏差则达到 100%或更高。在模型完美反映数据的情况下，对于所有代谢物，CRLB 与标准差的偏差在 10%以内。CRLB 与标准差的偏差在 10%以内的结果意味着，对于准确的模型，像机器学习或深度学习这样的新定量方法将无法比传统的最大似然估计更精确地估计所需参数。
