title: Harmonic analysis for the characterization and correction of geometric distortion in MRI.
abstract: OBJECTIVE
  Magnetic resonance imaging (MRI) is gaining widespread use in radiation therapy planning, patient setup verification, and real-time guidance of radiation delivery. Successful implementation of these technologies relies on the development of simple and efficient methods to characterize and monitor the geometric distortions arising due to system imperfections and gradient nonlinearities. To this end, the authors present the theory and validation of a novel harmonic approach to the quantification of system-related distortions in MRI.
  
  
  METHODS
  The theory of spatial encoding in MRI is applied to demonstrate that the 3D distortion vector field (DVF) is given by the solution of a second-order boundary value problem (BVP). This BVP is comprised of Laplace's equation and a limited measurement of the distortion on the boundary of a specified region of interest (ROI). An analytical series expansion solving this BVP within a spherical ROI is obtained, and a statistical uncertainty analysis is performed to determine how random errors in the boundary measurements propagate to the ROI interior. This series expansion is then evaluated to obtain volumetric DVF mappings that are compared to reference data obtained on a 3 T full-body scanner. This validation is performed within two spheres of 20 cm diameter (one centered at the scanner origin and the other offset +3 cm along each of the transverse directions). Initially, a high-order mapping requiring measurements at 5810 boundary points is used. Then, after exploring the impact of the boundary sampling density and the effect of series truncation, a reduced-order mapping requiring measurements at 302 boundary points is evaluated.
  
  
  RESULTS
  The volumetric DVF mappings obtained from the harmonic analysis are in good agreement with the reference data. Following distortion correction using the high-order mapping, the authors estimate a reduction in the mean distortion magnitude from 0.86 to 0.42 mm and from 0.93 to 0.39 mm within the central and offset ROIs, respectively. In addition, the fraction of points with a distortion magnitude greater than 1 mm is reduced from 35.6% to 2.8% and from 40.4% to 1.5%, respectively. Similarly, following correction using the reduced-order mapping, the mean distortion magnitude reduces to 0.45-0.42 mm within the central and offset ROIs, and the fraction of points with a distortion magnitude greater than 1 mm is reduced to 2.8% and 1.5%, respectively.
  
  
  CONCLUSIONS
  A novel harmonic approach to the characterization of system-related distortions in MRI is presented. This method permits a complete and accurate mapping of the DVF within a specified ROI using a limited measurement of the distortion on the ROI boundary. This technique eliminates the requirement to exhaustively sample the DVF at a dense 3D array of points, thereby permitting the design of simple, inexpensive phantoms that may incorporate additional modules for auxiliary quality assurance objectives.
authors.affiliation: Radiation Medicine Program, Princess Margaret Cancer Centre, University Health Network, Toronto M5G 2M9, Canada and Department of Radiation Oncology, University of Toronto, Toronto M5S 3E2, Canada.
authors.forename: Tony
authors.initials: T
authors.lastname: Tadic
authors.affiliation: Radiation Medicine Program, Princess Margaret Cancer Centre, University Health Network, Toronto M5G 2M9, Canada; Department of Radiation Oncology, University of Toronto, Toronto M5S 3E2, Canada; and Department of Medical Biophysics, University of Toronto, Toronto M5G 1L7, Canada.
authors.forename: David A
authors.initials: DA
authors.lastname: Jaffray
authors.affiliation: Radiation Medicine Program, Princess Margaret Cancer Centre, University Health Network, Toronto M5G 2M9, Canada and Department of Radiation Oncology, University of Toronto, Toronto M5S 3E2, Canada.
authors.forename: Teodor
authors.initials: T
authors.lastname: Stanescu
mesh_terms: D000465:Algorithms; D016477:Artifacts; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D021621:Imaging, Three-Dimensional; D060526:Magnetic Fields; D008279:Magnetic Resonance Imaging / Q000379:methods*; D008962:Models, Theoretical; D019047:Phantoms, Imaging; D011880:Radiotherapy Planning, Computer-Assisted / Q000379:methods*; D012016:Reference Values; D015203:Reproducibility of Results; D012984:Software