Quantitative susceptibility mapping by regulating the field to source inverse problem with a sparse prior derived from the Maxwell Equation: validation and application to brain

INTRODUCTION The field inhomogeneity mapped using MRI offers the opportunity to characterize tissue magnetic susceptibility. The field measured in a voxel is a dipole kernel convolution of all tissue susceptibility sources surrounding the voxel [1]. The zero cone surfaces of the dipole kernel in Fourier domain undersamples the measured field, causing illposedness for the inverse problem of deriving susceptibility source maps from the magnetic field [2,3]. However, if a proper prior could be found to sparsify the signal, even highly undersampled data may allow an exact reconstruction [4]. Here, we propose an approach for achieving quantitative susceptibility mapping (QSM) by exploiting the sparsity derived from Maxwell’s equation and the T2* weighted image. Phantom experiments demonstrate artifact free and accurate susceptibility reconstruction. High quality QSM of human brain indicates the encouraging advances of QSM.