Mathematical analysis and experimental investigation of noise in quantitative magnetic resonance imaging applied in polymer gel dosimetry

In polymer gel dosimetry, the spin-spin relaxation rate R2 = 1/T2 is related to the absorbed dose that is delivered to a gel phantom by high-energy radiation beams. In a two-points method, R2 is calculated from two differently T2-weighted images. In the many-points method, R2 is calculated by fitting the pixel intensities of a set of differently exponentially T2-weighted images. An analysis of the influence of noise on the resulting R2 image may contribute considerably to the enhancement of the accuracy of the dose map. The relation between the noise level in the differently T2-weighted images and the noise level in the R2 image is derived mathematically. This relation is dependent on the actual R2, on the choice of the echo times in the sequences used, and the fitting algorithm. Both a least square fit to the semi-logarithmic T2-relaxation plot and a maximum-likelihood estimation on the T2-relaxation plot were investigated. It was found that dose images obtained from R2 images through calibration will contain noise that originates from the noise in the R2 image and from the probability distribution of the coefficients of the calibration curve. We present a method for optimizing the echo times in order to maximize the signal-to-noise ratio (SNR) in the resulting R2 image for both methods and for both fitting algorithms. The mathematical considerations on the SNR as presented, can also be applied on other data sets which display a mono-exponential behavior (e.g. diffusion measurements, T1 relaxation).