  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.
