A probabilistic Bayesian approach to recover R2*$$ {R}_{2\ast } $$ map and phase images for quantitative susceptibility mapping.

PURPOSE: Undersampling is used to reduce the scan time for high-resolution three-dimensional magnetic resonance imaging. In order to achieve better image quality and avoid manual parameter tuning, we propose a probabilistic Bayesian approach to recover R2∗$$ {R}_2^{\ast } $$ map and phase images for quantitative susceptibility mapping (QSM), while allowing automatic parameter estimation from undersampled data. THEORY: Sparse prior on the wavelet coefficients of images is interpreted from a Bayesian perspective as sparsity-promoting distribution. A novel nonlinear approximate message passing (AMP) framework that incorporates a mono-exponential decay model is proposed. The parameters are treated as unknown variables and jointly estimated with image wavelet coefficients. METHODS: Undersampling takes place in the y-z plane of k-space according to the Poisson-disk pattern. Retrospective undersampling is performed to evaluate the performances of different reconstruction approaches, prospective undersampling is performed to demonstrate the feasibility of undersampling in practice. RESULTS: The proposed AMP with parameter estimation (AMP-PE) approach successfully recovers R2∗$$ {R}_2^{\ast } $$ maps and phase images for QSM across various undersampling rates. It is more computationally efficient, and performs better than the state-of-the-art l1$$ {l}_1 $$ -norm regularization (L1) approach in general, except a few cases where the L1 approach performs as well as AMP-PE. CONCLUSION: AMP-PE achieves better performance by drawing information from both the sparse prior and the mono-exponential decay model. It does not require parameter tuning, and works with a clinical, prospective undersampling scheme where parameter tuning is often impossible or difficult due to the lack of ground-truth image.