  Purpose: Previous quantitative MR imaging studies using self-supervised deep
learning have reported biased parameter estimates at low SNR. Such systematic
errors arise from the choice of Mean Squared Error (MSE) loss function for
network training, which is incompatible with Rician-distributed MR magnitude
signals. To address this issue, we introduce the negative log Rician likelihood
(NLR) loss. Methods: A numerically stable and accurate implementation of the
NLR loss was developed to estimate quantitative parameters of the apparent
diffusion coefficient (ADC) model and intra-voxel incoherent motion (IVIM)
model. Parameter estimation accuracy, precision and overall error were
evaluated in terms of bias, variance and root mean squared error and compared
against the MSE loss over a range of SNRs (5 - 30). Results: Networks trained
with NLR loss show higher estimation accuracy than MSE for the ADC and IVIM
diffusion coefficients as SNR decreases, with minimal loss of precision or
total error. At high effective SNR (high SNR and small diffusion coefficients),
both losses show comparable accuracy and precision for all parameters of both
models. Conclusion: The proposed NLR loss is numerically stable and accurate
across the full range of tested SNRs and improves parameter estimation accuracy
of diffusion coefficients using self-supervised deep learning. We expect the
development to benefit quantitative MR imaging techniques broadly, enabling
more accurate parameter estimation from noisy data.
