Weak‐harmonic regularization for quantitative susceptibility mapping

Abstract

Purpose: Background-field removal is a crucial preprocessing step for quantitative susceptibility mapping (QSM). Remnants from this step often contaminate the estimated local field, which in turn lead to erroneous tissue-susceptibility reconstructions. The present work aimed to mitigate this undesirable behavior with the development of a new approach that simultaneously decouples background contributions and local susceptibility sources on QSM inversion.Methods: Input phase data for QSM can be seen as a composite scalar field of local effects and residual background components. We developed a new Weak-Harmonic (WH) regularizer to constrain the latter and to separate the two components. The resulting optimization problem was solved with the Alternating Directions of Multipliers Method (ADMM) framework to achieve fast convergence. In addition, for convenience a new ADMM-based preconditioned nonlinear Projection onto Dipole Fields (nPDF) solver was developed to enable initializations with wrapped-phase distributions. WH-QSM, with and without nPDF preconditioning, was compared to the original (ADMM-based) Total Variation QSM algorithm in phantom and in vivo experiments.Results: WH-QSM returned improved reconstructions irrespective of the method used for background- field removal, though the proposed nPDF method often obtained better results. Streaking and shadowing artifacts were substantially suppressed, and residual background components were effectively removed.Conclusion: WH-QSM with field preconditioning is a robust dipole inversion technique and has the potential to be extended as a single-step formulation for initialization with uncombined multi-echo data.