Experimental Errors as Alternative to the Theoretical Cramér-Rao Minimum Variance Bounds in MRS: ER-ARSOS - Error Estimation by Multiple Quantification of Recombined ARSOS-filtered Output Signals

Introduction The Cramer-Rao minimum variance bound (CR-MVB) is widely used to estimate the theoretical minimal variances (errors) of estimated spectral parameters in localized single voxel magnetic resonance spectroscopy (SVS). The estimation of the CR-MVB requires only the spectral model function, the obtained set of optimal estimated spectrum model parameters, and the variance of the Gaussian noise. Further, the CR-MVB is valid under the condition that the assumed parametric model is complete. Due to the macromolecular baseline, unknown line-shapes, and other artefacts resulting from non-stationary signal acquisition conditions (e.g. patient motion, scannerinstabilities, etc.) or RF-pulse imperfections the parametric model is normally incomplete, and, strictly taken, the CR-MVBs estimates become invalid. Mistakenly, it is frequently believed that the CR-MVB accounts also for error sources like patient motion etc. mentioned above. In general the CR-MVB underestimates the variances in the estimated spectral parameters. This work presents a novel method to estimate the experimental errors of the parameters, as an alternative to the CR-MVBs for the case of single voxel spectroscopy (SVS). The only extra requirement for the proposed method is separate storage of the multiple signal acquisitions. Method Let Sm[n] (1 ≤ m ≤ M; 0 ≤ n ≤ N-1) be the m-th acquired single voxel time-domain signal during signal averaging having Gaussian noise and possibly contaminated by heavily tailed noise (e.g. outlier signals). Filtering the M low SNR signals Sm[n] by means of all rank selected order statistic filtering technique ARSOS [1] results in M high SNR output signals Om[n] (1 ≤ m ≤ M; 0 ≤ n ≤ N-1), which is a unique feature of ARSOS. Due to the fact that the noise of the output signals Om[n] have an asymmetric probability density function (pdf), straightforward non linear least squares NLLS-fitting of the M-output signals would result in biased parameter estimates. Fortunately, the M-output signals with asymmetric pdf can be recombined to 1⁄2M output signals with symmetric pdf (see ref.[1]). These 1⁄2M output signals can be fitted with one of the available fitting routines resulting in a 1⁄2M parameter vectors pm= (pm,1 , pm,2 , . , pm,K ). Separate statistical analysis of the set of 1⁄2M parameter vector elements {p1,k , p2,k. , , p1⁄2M,k } of the k-th fitted parameter enables the computation of the mean and variance var(pk) of each estimated model function parameter pk. The standard deviation stdev(pk) is defined as the √var(pk) and is regarded as experimental error estimate and can be used as alternative to the CR-MVB. Results For a SVS voxel localized in the parietal human white matter (PRESS/ TE=135/TR=1.5sec/N=2048/M=24/15x15x15mm, @3T) the signal amplitudes obtained with conventional processing with QUEST/CR-MVB (see ref. [2]) were compared to those obtained with ER-ARSOS-QUEST and ER-ARSOS errors. The MRS-signals were subjected to reliability testing and passed the tests (see ref. [1]). The peak areas obtained with both methods do not significantly differ statistically (TTEST; p>0.99). However, the errors obtained by conventional CR-MVB are on average a factor of 2.55 smaller than the average ER-ARSOS-errors (TTEST; p<0.03).