Quantitative MR susceptibility mapping using piece-wise constant regularized inversion of the magnetic field

Magnetic properties characterized by susceptibility and chemical shift linearly modify the local magnetic field experienced by spins. A piece-wise constant solution using magnetic resonance imaging is found to the challenging inversion problem from field to magnetic properties. The magnetic field shifts were estimated from MR phase images. The MR magnitude images were segmented into many regions of uniform magnetic properties. Standard linear regression using the calculated magnetic field from each region allowed accurate susceptibility quantification. The technique was experimentally validated on a variety of samples including water, vegetable oil, air, Gadolinium, and superparamagnetic iron oxides. Susceptibility was measured with a precision better than 0.1 ppm, in a range of 10 ppm. In vivo feasibility was shown on the forearm for which soft-tissue, cortical bone, and bone marrow susceptibility, and chemical shift values in good agreement with literature data were obtained.