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

Quantitative Susceptibility Mapping (QSM) requires inversion of a convolutional dipole operator whose null-space renders the reconstruction problem severely ill-posed, such that small perturbations in the measured field can induce large variations in the estimated susceptibility distribution. While recent deep learning approaches improve empirical reconstruction quality, they typically produce deterministic estimates and treat susceptibility source separation as an independent post-processing step, without modeling inversion uncertainty or enforcing structured physical consistency. In this work, we formulate QSM as a physics-constrained probabilistic inverse problem and propose a Bayesian reconstruction framework that performs susceptibility estimation and uncertainty quantification. The susceptibility distribution is modeled using a heteroscedastic Gaussian posterior parameterized by a neural network, yielding voxel-wise posterior mean and variance. A dipole-domain consistency term enforces fidelity to the physical forward model, acting as an implicit null-space regularizer that stabilizes the learned inversion operator. The estimated posterior is further incorporated into a sign-constrained variational formulation to decompose total susceptibility into paramagnetic and diamagnetic components in a post-processing step, enabling uncertainty-aware source separation. This coupling between probabilistic reconstruction and variational decomposition mitigates error accumulation inherent in sequential pipelines. Experiments on a multi-session Head and Neck QSM repeatability dataset demonstrate competitive reconstruction accuracy, meaningful uncertainty calibration, and stable susceptibility decomposition, further validated on a simulated dataset. The proposed approach establishes a physics-consistent probabilistic inversion paradigm for QSM, providing an interpretable and uncertainty-aware solution to dipole inversion.