Inverse problem in quantitative susceptibility mapping

Quantitative susceptibility mapping (QSM) is a new medical imaging technique that can visualize magnetic susceptibility, changes of which in tissue indicate various disease processes involving iron transport. The inverse problem of QSM is to recover the susceptibility distribution of the human body from the measured local field that is expressed by the convolution of the susceptibility distribution with the magnetic field generated by a unit dipole. The inverse problem is ill-posed due to the presence of zeros at a cone in the Fourier representation of the unit dipole kernel. Reconstruction methods have been greatly improved to give better recovery of tissue susceptibility data for QSM, and various clinical applications have been pursued. However, rigorous mathematical analyses for the inverse problem, such as demonstrations of the existence and uniqueness of solutions and error characterizations, have not yet been presented. This paper provides for the first time not only a theoretical ground for QSM but also the underlying cause of streaking artifacts.