A mathematical assessment of the precision of parameters in measuring resonance spectra

The accurate interpretation of in vivo magnetic resonance spectroscopy (MRS) spectra requires a complete understanding of the associated noise-induced errors. In this paper, we address the effect of complex correlated noise patterns on the measurement of a set of peak parameters. This is examined initially at the level of a single spectral analysis followed by addressing the noise-induced errors associated with determining the signal parameters from the peak parameters. We describe a relatively simple method for calculating these errors for any correlated noise pattern in terms of the noise standard deviation and correlation length. The results are presented in such a way that an estimate of the errors may be made from a single MRS spectrum. We also explore how, under certain circumstances, the lineshape of the signal may be determined. We then apply these results to reexamine a set of in vivo 31P MRS spectra obtained from rat brain prior to and following moderate fluid percussion injury. The approach outlined in this paper will demonstrate how meaningful results may be obtained from spectra where the signal-to-noise ratio (SNR) is quite small and where knowledge of the precise shape of the signal and the detail of the noise pattern is unknown. In essence, we show how to determine the expected errors in the spectral parameters from an estimate of the SNR from a single spectrum, thereby allowing a more discriminative interpretation of the data.