Nonlinear magnitude and linear phase behaviors of T2* imaging: theoretical approximation and Monte Carlo simulation

The underlying source of brain imaging by T2*-weighted magnetic resonance imaging (T2*MRI) is the intracranial inhomogeneous tissue magnetic susceptibility (denoted by χ) that causes an inhomogeneous field map (via magnetization) in a main field. By decomposing T2*MRI into two steps, we understand that the 1st step from a χ source to a field map is a linear but non-isomorphic spatial mapping, and the 2nd step from the field map to a T2* image is a nonlinear mapping due to the trigonometric behavior of spin precession signals. The magnitude and phase calculations from a complex T2* image introduce additional nonlinearities. In this report, we look into the magnitude and phase behaviors of a T2* image (signal) by theoretical approximation and Monte Carlo simulation. We perform the 1st-order Taylor expansion on intravoxel dephasing formula of a T2* signal and show that the T2* magnitude is a quadratic mapping of the field map and T2* phase is a linear isomorphic mapping. By Monte Carlo simulation of T2*MRI for a span of echo times (with B0 = 3 T and TE = [0,120] ms), we first confirm the quadratic magnitude and linear phase behaviors in small phase angle regime (via TE < 30 ms), and then provide more general magnitude and phase nonlinear behaviors in large phase angle scenarios (via TE > 30 ms). By solving the inverse problem of T2*MRI, we demonstrate χ tomography and conclude that the χ source can be reliably reconstructed from a T2* phase image in a small phase angle regime.