Generalized Lorentzian Tensor Approach (GLTA) as a biophysical background for quantitative susceptibility mapping

Purpose: Quantitative susceptibility mapping (QSM) is a potentially powerful technique for mapping tissue magnetic susceptibility from gradient recalled echo (GRE) MRI. Herein we aim to derive the relationships between GRE signal phase and the underlying tissue microstructure and magnetic susceptibility at the cellular level. Methods: We use Maxwell's equations and a statistical approach to derive the expression for the magnetic-susceptibility-induced MR signal frequency shift of the GRE signal in single- and multicompartment systems, in which inhomogeneous magnetic field is induced by the cellular constituents (proteins, lipids, iron, etc.) distributed in intra- and extracellular spaces. Results: We introduce the Generalized Lorentzian Tensor Approach (GLTA) that accounts for both types of anisotropy: the anisotropy of magnetic susceptibility and the structural tissue anisotropy. In the GLTA the frequency shift due to the local environment is characterized by the Lorentzian tensor L which has a substantially different structure than the susceptibility tensor χ. While components of χ are simply compartmental susceptibilities "weighted" by their relative volumes, the components of L are weighted by specific numerical factors depending on tissue micro-symmetry and parameters related to the MR pulse sequence. We also provide equations bridging phenomenological and microscopic considerations. Conclusion: The GLTA provides a consistent background for deciphering phase data.