Dependence of the frequency distribution around a sphere on the voxel orientation

Abstract Microscopically small magnetic field inhomogeneities within an external static magnetic field cause a free induction decay in magnetic resonance imaging that generally exhibits two transverse components that are usually summarized to a complex entity. The Fourier transform of the complex-valued free induction decay is the purely real and positive-valued frequency distribution which allows an easy interpretation of the underlying dephasing mechanism. Typically, the frequency distribution inside a cubic voxel as caused by a spherical magnetic field inhomogeneity is determined by a histogram technique in terms of subdivision of the whole voxel into smaller subvoxels. A faster and more accurate computation is achieved by analytical expressions for the frequency distribution that are derived in this work. In contrast to the usually assumed simplified case of a spherical voxel, we also consider the tilt angles of the cubic voxel to the external magnetic field. The typical asymmetric form of the frequency distribution is reproduced and analyzed for the more realistic case of a cubic voxel. We observe a splitting of frequency distribution peaks for increasing tilt of the cubic voxel against the direction of the external magnetic field in analogy to the case for dephasing around cylindrical, vessel-like objects inside cubic voxels. These results are of value, e.g., for the analysis of susceptibility-weighted images or in quantitative susceptibility imaging since the reconstruction of these images is performed in cubic-shaped voxels.