Theoretical Investigation of Random Noise-Limited Signal-to-Noise Ratio in MR-Based Electrical Properties Tomography

In magnetic resonance imaging-based electrical properties tomography (MREPT), tissue electrical properties (EPs) are derived from the spatial variation of the transmit RF field $(B_{1}^{+})$ . Here we derive theoretically the relationship between the signal-to-noise ratio (SNR) of the electrical properties obtained by MREPT and the SNR of the input $B_{1}^{+}$ data, under the assumption that the latter is much greater than unity, and the noise in $B_{1}^{+}$ at different voxels is statistically independent. It is shown that for a given $B_{1}^{+}$ data, the SNR of both electrical conductivity and relative permittivity is proportional to the square of the linear dimension of the region of interest (ROI) over which the EPs are determined, and to the square root of the number of voxels in the ROI. The relationship also shows how the SNR varies with the main magnetic field $(B_{0})$ strength. The predicted SNR is verified through numerical simulations on a cylindrical phantom with an analytically calculated $B_{1}^{+}$ map, and is found to provide explanation of certain aspects of previous experimental results in the literature. Our SNR formula can be used to estimate minimum input data SNR and ROI size required to obtain tissue EP maps of desired quality.