A physics-constrained Bayesian framework for QSM with uncertainty quantification and variational susceptibility...

Objective. Quantitative susceptibility mapping (QSM) requires inversion of a convolutional dipole operator whose null-space renders susceptibility reconstruction severely ill-posed, making the estimated susceptibility highly sensitive to measurement noise and modeling errors. Existing deep learning methods primarily produce deterministic reconstructions and typically perform susceptibility source separation as an independent post-processing step without explicitly modeling inversion uncertainty or incorporating probabilistic information into the decomposition process.Approach.QSM is formulated as a physics-constrained probabilistic inverse problem, and a Bayesian reconstruction framework is proposed for simultaneous susceptibility estimation and uncertainty quantification. A heteroscedastic Gaussian posterior, parameterized by a lightweight 3D encoder-decoder network, predicts voxel-wise posterior mean and variance. Physics consistency is enforced through the dipole forward model, encouraging physically consistent reconstruction while maintaining computational tractability. The estimated posterior mean and uncertainty are subsequently incorporated as confidence-weighted priors in a sign-constrained variational optimization to perform posterior-guided decomposition of total susceptibility into paramagnetic and diamagnetic components.Main Results.The proposed framework was evaluated on a publicly available multi-session head-and-neck QSM repeatability dataset. Experimental results demonstrate competitive reconstruction accuracy, reliable uncertainty estimation, and robust repeatability compared with existing reconstruction methods. A supplementary simulation study further verifies the numerical feasibility of the proposed posterior-guided susceptibility decomposition under controlled conditions.Significance.This work introduces a physics-constrained probabilistic reconstruction framework for QSM that combines Bayesian uncertainty estimation with posterior-guided susceptibility source separation. By explicitly modeling predictive uncertainty and propagating it to the decomposition stage, the proposed approach provides a more interpretable and uncertainty-aware reconstruction paradigm, offering a promising foundation for future probabilistic QSM methods and clinically reliable susceptibility analysis.