Background field removal by solving the Laplacian boundary value problem

Abstract

The removal of the background magnetic field is a critical step in generating phase images and quantitative susceptibility 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 boundary 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.

An illustrated comparison of processing methods for phase MRI and QSM: removal of background field contributions from sources outside the region of interest

“…Wen et al [33] determined the background field by solving the equation S ′* H bkg = 0 in the ROI using the Jacobi iterative method [17, 9], $H_{\text{bkg}}^{(n+1)}=S\ast H_{\text{bkg}}^{(n)}$, and using the total field, H , for initialization of the algorithm. In this method, dubbed the Iterative Spherical Mean Value (iSMV) method, the boundary constraint is enforced by replacing in each iteration step the background field values in the boundary region with the total field.…”

Background field removal using a region adaptive kernel for quantitative susceptibility mapping of human brain

“…The unit dipole is defined as d ( r ) = (3 cos 2 θ − 1)/| r | 3 , with θ denoting the angle between B 0 and r . The perturbation field B Δ in the VOI can be decomposed into the contributions of the local field B loc and the background field B bkg [18,20] as $$B^{\mathrm{\Delta }}=B_{\mathit{loc}}+B_{\mathit{bkg}}$$…”

“…Δ B bkg = 0, with the 3D Laplacian operator $\mathrm{normal\Delta }={\partial }_x^2+{\partial }_y^2+{\partial }_z^2$ . Using this assumption, B loc can be obtained by solving the partial differential equation Δ B Δ = Δ B loc with appropri ate boundary conditions [20]. In addition, using the harmonic mean value theorem from the SHARP method [7], we can introduce an interim variable B inter as $$B_{\mathit{inter}}=B^{\mathrm{\Delta }}-\rho false\otimes B^{\mathrm{\Delta }}=B_{\mathit{loc}}-\rho false\otimes B_{\mathit{loc}}$$where ρ is a nonnegative, radially symmetric, normalized convolution kernel and the symbol ⊗ denotes the 3D convolution operator.…”

“…We explored the energy functional in the level set formulation to partition the phase data into different subdomains according to the field inhomogeneity. In the regions of large susceptibility variation, such as tissues around the skull and paranasal sinuses, the background field is commonly one or two orders larger than the local field [20]. Our proposed scalable kernel is capable to adjust the kernel radius into a small size, while the kernel weights focus on the sphere center with adjacent weights approaching zero rapidly.…”

“…Yet an SMV with equal weights does not perform well in estimating the local field for voxels with high field gradients [19,25]. The Laplacian boundary value (LBV) method was proposed to remove the background field by solving the boundary value problems of Laplace’s or Poisson’s equation to retain data near the boundary [20]. However, in most studies, brain regions around the paranasal sinuses are usually excluded due to insufficient background field removal [2,16,18].…”