Fast nonlinear susceptibility inversion with variational regularization

Purpose Quantitative susceptibility mapping can be performed through the minimization of a function consisting of data fidelity and regularization terms. For data consistency, a Gaussian‐phase noise distribution is often assumed, which breaks down when the signal‐to‐noise ratio is low. A previously proposed alternative is to use a nonlinear data fidelity term, which reduces streaking artifacts, mitigates noise amplification, and results in more accurate susceptibility estimates. We hereby present a novel algorithm that solves the nonlinear functional while achieving computation speeds comparable to those for a linear formulation. Methods We developed a nonlinear quantitative susceptibility mapping algorithm (fast nonlinear susceptibility inversion) based on the variable splitting and alternating direction method of multipliers, in which the problem is split into simpler subproblems with closed‐form solutions and a decoupled nonlinear inversion hereby solved with a Newton‐Raphson iterative procedure. Fast nonlinear susceptibility inversion performance was assessed using numerical phantom and in vivo experiments, and was compared against the nonlinear morphology‐enabled dipole inversion method. Results Fast nonlinear susceptibility inversion achieves similar accuracy to nonlinear morphology‐enabled dipole inversion but with significantly improved computational efficiency. Conclusion The proposed method enables accurate reconstructions in a fraction of the time required by state‐of‐the‐art quantitative susceptibility mapping methods. Magn Reson Med 80:814–821, 2018. © 2018 International Society for Magnetic Resonance in Medicine.