title: Primal-dual and forward gradient implementation for quantitative susceptibility mapping.
abstract: OBJECTIVE
  To investigate the computational aspects of the prior term in quantitative susceptibility mapping (QSM) by (i) comparing the Gauss-Newton conjugate gradient (GNCG) algorithm that uses numerical conditioning (ie, modifies the prior term) with a primal-dual (PD) formulation that avoids this, and (ii) carrying out a comparison between a central and forward difference scheme for the discretization of the prior term.
  
  
  METHODS
  A spatially continuous formulation of the regularized QSM inversion problem and its PD formulation were derived. The Chambolle-Pock algorithm for PD was implemented and its convergence behavior was compared with that of GNCG for the original QSM. Forward and central difference schemes were compared in terms of the presence of checkerboard artifacts. All methods were tested and validated on a gadolinium phantom, ex vivo brain blocks, and in vivo brain MRI data with respect to COSMOS.
  
  
  RESULTS
  The PD approach provided a faster convergence rate than GNCG. The GNCG convergence rate slowed considerably with smaller (more accurate) values of the conditioning parameter. Using a forward difference suppressed the checkerboard artifacts in QSM, as compared with the central difference. The accuracy of PD and GNCG were validated based on excellent correlation with COSMOS.
  
  
  CONCLUSIONS
  The PD approach with forward difference for the gradient showed improved convergence and accuracy over the GNCG method using central difference. Magn Reson Med 78:2416-2427, 2017. © 2017 International Society for Magnetic Resonance in Medicine.
authors.affiliation: Department of Radiology, Weill Cornell Medical College, New York, New York, USA.
authors.forename: Youngwook
authors.initials: Y
authors.lastname: Kee
authors.affiliation: Department of Physiology, Systems Biology and Biophysics, Weill Cornell Medical College, New York, New York, USA.
authors.forename: Kofi
authors.initials: K
authors.lastname: Deh
authors.affiliation: Department of Biomedical Engineering, Cornell University, Ithaca, New York, USA.
authors.forename: Alexey
authors.initials: A
authors.lastname: Dimov
authors.affiliation: Department of Radiology, Weill Cornell Medical College, New York, New York, USA.
authors.forename: Pascal
authors.initials: P
authors.lastname: Spincemaille
authors.affiliation: Department of Radiology, Weill Cornell Medical College, New York, New York, USA.|Department of Biomedical Engineering, Cornell University, Ithaca, New York, USA.
authors.forename: Yi
authors.initials: Y
authors.lastname: Wang
mesh_terms: D000465:Algorithms; D016880:Anisotropy; D001921:Brain / Q000000981:diagnostic imaging*; D005682:Gadolinium / Q000737:chemistry; D064368:Healthy Volunteers; D006801:Humans; D007091:Image Processing, Computer-Assisted; D008279:Magnetic Resonance Imaging; D015233:Models, Statistical; D016011:Normal Distribution; D019047:Phantoms, Imaging; D011786:Quality Control; D015203:Reproducibility of Results; D012984:Software
chemical_list: D005682:Gadolinium
keywords: Gauss-Newton conjugate gradient; discretization methods; morphology enabled dipole inversion (MEDI); primal-dual formulation; quantitative susceptibility mapping (QSM)