Accurate estimation of local susceptibility distributions using Quantitative Susceptibility Mapping (QSM) techniques requires removal of background field inhomogeneities. Background field removal in the abdomen, and particularly the liver, should be accurate near tissue/air interfaces due to air in the lungs and bowel. The purpose of this work is to introduce a background field removal method, Poisson Estimation of the Abdominal Local field (PEAL), and evaluate its accuracy, particularly near the edges of a tissue region, for application in abdominal QSM.Current background field methods include Laplacian Boundary Values (LBV [1]), Sophisticated Harmonic Artifact Reduction for Phase data (SHARP [2]), Regularization-Enabled SHARP (RESHARP [3]), and Projection onto Dipole Fields (PDF [4]). The location at which SHARP computes the value (the “compute point”) is at the center of a constant spherical kernel; therefore, voxels near the edge of the VOI must be eroded by the radius of the kernel. However, many important abdominal organs (liver, pancreas) are located closer to air-tissue interfaces: an off-center compute point may enable more accurate background field removal. The extended Poisson kernel [5] enables a compute point at any location within the sphere: $$ P_0(\vec{v},\vec{w}) = \frac{1-|\vec{v}|^2|\vec{w}|^2}{(1-2\vec{v}\cdot\vec{w}+|\vec{v}|^2|\vec{w}|^2)^\frac{3}{2}} $$ where $$$\vec{v}$$$ denotes a point within a sphere of radius=1, and $$$\vec{w}$$$ is the compute point. Similar to the construction of the SHARP kernel, the PEAL kernel is constructed by sampling the continouous extended Poisson kernel at discrete points given by $$$\vec{V}$$$ and $$$\vec{W}$$$: $$$\hat{P_0}[\vec{V},\vec{W}]=\frac{3}{4\pi R^3}P_0(\vec{v}R,\vec{w}R)$$$ where R is the radius (in voxels), then subtracting from a delta function: $$$P[\vec{V},\vec{W}]=\delta-\hat{P_0}[\vec{V},\vec{W}]$$$. We hypothesize that the PEAL kernel may enable improved background field removal near VOI edges.

Simulation: A cylinder of radius=12 voxels was placed in a 128x128x128 matrix, at a central location, and at various locations near the VOI edge (distance between the cylinder edge and the VOI edge=1,2,3…6 voxels). To model hepatic iron overload, the true magnetic susceptibility in voxels within the cylinder was set to +5 ppm, and it was set to 0 in all other voxels. (A liver with iron overload can have susceptibility difference from surrounding tissue up to +9 ppm [6].) Phase data were generated from this susceptibility distribution using the forward dipole calculation [7-9]. (Note that a field map in Hz can be converted to unwrapped phase data in radians by multiplying by 2πΔTE.) Kernels of radius=6 voxels [10] were generated for LBV, SHARP, RESHARP; PDF was also run (using code from [11]). PEAL kernels for radius=6 and compute points 1,2,3,4,5 voxels away from the kernel center were also generated. All computations were performed using Matlab (The Mathworks, Natick MA, USA).

Performance comparison: Comparison of the relative performance of each method was made by calculating the error (difference) in the estimated phase after background field removal and the simulated phase before background field removal. However, phase errors do not directly translate into QSM errors. In order to quantify the susceptibility error related to each background field removal method, dipole inversion was performed, with the cylindrical region constrained to one constant value, and the surrounding region constrained to 0. In this way, one estimated susceptibility value was produced for each 128x128x128 test case.

The phase map errors for the cylinder located 3 voxels from the edge of the VOI are shown in Fig. 1. The dotted line in the uncorrected phase shows the extent of the VOI, i.e., the uneroded mask. LBV, SHARP, RESHARP, and PDF all lead to focal errors near the edge. The PEAL methods result in small errors that have little variation across the VOI. Note the changing location of eroded masks with PEAL compute point. The estimated constant susceptibility errors within the cylinder are shown in Fig. 2. The first group of bars shows the error without any background field removal. LBV, RESHARP, and PDF have errors that increase as the cylinder approaches the VOI edge; SHARP also has high errors in that region. The PEAL methods have the smallest overall errors near the edge, while the error in the central location is comparable to LBV and SHARP. The high errors in RESHARP are due to the large cylinder size; a smaller cylinder or a larger matrix size leads to much smaller RESHARP errors (results not shown). Using the extended Poisson kernel for QSM background field removal shows promise for performance improvements at the edges of a VOI over current background removal methods. The optimal choice of compute point and kernel size remains to be investigated.