Objective: Quantitative $T_1\rho$ imaging has potential for assessment of biochemical alterations of liver pathologies. Deep learning methods have been employed to accelerate quantitative $T_1\rho$ imaging. To employ artificial intelligence-based quantitative imaging methods in complicated clinical environment, it is valuable to estimate the uncertainty of the predicated $T_1\rho$ values to provide the confidence level of the quantification results. The uncertainty should also be utilized to aid the post-hoc quantitative analysis and model learning tasks. Approach: To address this need, we propose a parametric map refinement approach for learning-based $T_1\rho$ mapping and train the model in a probabilistic way to model the uncertainty. We also propose to utilize the uncertainty map to spatially weight the training of an improved $T_1\rho$ mapping network to further improve the mapping performance and to remove pixels with unreliable $T_1\rho$ values in the region of interest. The framework was tested on a dataset of 51 patients with different liver fibrosis stages. Main results: Our results indicate that the learning-based map refinement method leads to a relative mapping error of less than 3% and provides uncertainty estimation simultaneously. The estimated uncertainty reflects the actual error level, and it can be used to further reduce relative $T_1\rho$ mapping error to 2.60% as well as removing unreliable pixels in the region of interest effectively. Significance: Our studies demonstrate the proposed approach has potential to provide a learning-based quantitative MRI system for trustworthy $T_1\rho$ mapping of the liver.

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