Background field removal by solving the Laplacian boundary value problem Dong Zhou a , Tian Liu b , Pascal Spincemaille a and Yi Wang a,c,d * The removal of the background magnetic field is a critical step in generating phase images and quantitative suscep- tibility maps, which have recently been receiving increasing attention. Although it is known that the background field satisfies Laplace’s equation, the boundary values of the background field for the region of interest have not been explicitly addressed in the existing methods, and they are not directly available from MRI measurements. In this paper, we assume simple boundary conditions and remove the background field by explicitly solving the bound- ary value problems of Laplace’s or Poisson’s equation. The proposed Laplacian boundary value (LBV) method for background field removal retains data near the boundary and is computationally efficient. Tests on a numerical phantom and an experimental phantom showed that LBV was more accurate than two existing methods. Copyright © 2014 John Wiley & Sons, Ltd. Keywords: background field removal; boundary value problem of partial differential equation (PDE); Laplace’ s equation; Poisson’ s equation; full multigrid (FMG) algorithm; susceptibility weighted imaging (SWI); phase imaging; quantitative susceptibility mapping (QSM) INTRODUCTION In recent years, quantitative susceptibility mapping (QSM) has emerged as a promising imaging technique in the field of MRI (1–14). As a biomarker that specifically reflects tissue magnetic properties, QSM provides valuable, clinically relevant information about iron metabolism, oxygen metabolism, calcification, white matter tract (myelin lipids) anisotropy, and contrast agent biodistribution (15–24). QSM solves the field-to-source inverse problem, obtaining the magnetic susceptibility map from the magnetic field estimated from MRI data. Since only the magnetic field induced by the tissue is of interest, the elimination of the magnetic field induced by sources outside the region of interest (ROI) is required (2,13,25). This procedure, known as background field removal, directly affects the calculated susceptibility values (11). It is a re- quired pre-processing step for susceptibility weighted imaging (SWI) as well (26). The physical origin of the background field includes main field inhomogeneity (imperfect shimming) and susceptibility sources outside the ROI. In brain imaging, for ex- ample, besides the main field, the magnetic field measured on frontal lobe is a summation of the frontal lobe-induced field as well as the field induced by the skull and nasal cavity. Ideally, the background field can be directly measured by performing a separate reference scan (4,5,7). In the reference scan, the sample within the ROI is replaced by uniform materials with known susceptibility while the shimming is kept the same. For a clinical data acquisition, however, such a reference scan is impractical or impossible to perform. Several methods have been proposed to solve this problem via data post-processing, making the reference scan unnecessary. All post-processing methods directly or indirectly exploit the fact that the back- ground field is a harmonic function (1,27), i.e., it is the solution of Laplace’ s equation (28). Their underlying strategies fall into two types. The first type of method fits the total magnetic field with a set of basis functions, ideally harmonic functions, and the part that can be fitted is considered to be background field. In practice, the basis functions in use include dipole field func- tions (10,11,25), low order polynomials (4,6,26,29,30) and low fre- quency Fourier basis (6,7,10,31–33). The second type of method utilizes the spherical mean value (SMV) property of harmonic functions to eliminate the background field (1,7,9,13,34,35). Since the background field is the solution of Laplace’ s equa- tion, all post-processing methods are effectively numerical solvers for this particular partial differential equation (PDE). How- ever, in contrast to the usual way in which PDEs are solved, they generally do not explicitly incorporate conditions on boundary values. Without boundary value matching, the solution of a PDE is not uniquely determined. In the case of Laplace’ s equation and Poisson’ s equation, the solution is undetermined up to any * Correspondence to: Y. Wang, Department of Radiology, Weill Cornell Medical College, New York, NY, USA. E-mail: yiwang@med.cornell.edu a D. Zhou, P. Spincemaille, Y. Wang Department of Radiology, Weill Cornell Medical College, New York, NY, USA b T. Liu Medimagemetric, LLC, New York, NY, USA c Y. Wang Department of Biomedical Engineering, Cornell University, Ithaca, New York, NY, USA d Y. Wang Biomedical Engineering, Kyung Hee University, Republic of Korea Abbreviations used: LBV, Laplacian boundary value; FMG, full multigrid; PDF, projection onto dipole fields; SHARP, sophisticated harmonic artifact re- duction for phase data; PDE, partial differential equation; SMV, spherical mean value; QSM, quantitative susceptibility mapping; SWI, susceptibility weighted imaging; ROI, region of interest; GRE, gradient echo. Research article Received: 13 June 2013, Revised: 20 November 2013, Accepted: 25 November 2013, Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/nbm.3064 NMR Biomed. 2014 Copyright © 2014 John Wiley & Sons, Ltd.