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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., Comment: 16 pages, 6 figures

Microstructure imaging combines tailored diffusion MRI acquisition protocols with a mathematical model to give insights into subvoxel tissue features. The model is typically fit voxel-by-voxel to the MRI image with least squares minimisation to give voxelwise maps of parameters relating to microstructural features, such as diffusivities and tissue compartment fractions. However, this fitting approach is susceptible to voxelwise noise, which can lead to erroneous values in parameter maps. Data-driven Bayesian hierarchical modelling defines prior distributions on parameters and learns them from the data, and can hence reduce such noise effects. Bayesian hierarchical modelling has been demonstrated for microstructure imaging with diffusion MRI, but only for a few, relatively simple, models. In this paper, we generalise hierarchical Bayesian modelling to a wide range of multi-compartment microstructural models, and fit the models with a Markov chain Monte Carlo (MCMC) algorithm. We implement our method by utilising Dmipy, a microstructure modelling software package for diffusion MRI data. Our code is available at github.com/PaddySlator/dmipy-bayesian.