Bloch Equation Based Algebraic Reconstruction for MRI using Frequency-Modulated Pulses

Introduction Magnetic resonance imaging of ultrashort T2 spins generally requires the use of a short radiofrequency pulse delivered with high peak power. In contrast, frequency-modulated (FM) pulses enable uniform spin excitation with a relatively low peak power, because they deliver a flat excitation profile and energy is distributed in time due to sequential excitation of spins by the frequency sweep. However, the spin excitation with an FM pulse induces a quadratic phase distribution that can not be retrieved by application of linear magnetic field gradients (1). The recently described FM-pulse-based ultrashort T2 imaging method known as SWIFT (2) overcomes this quadratic phase problem by the correlation method (3). More recently, an algebraic reconstruction method based on an approximation of instantaneous excitation along with the frequency sweep has also been introduced and used to reconstruct SWIFT image (4). However, while these two methods regard spin dynamics as a linear system, the actual spin system has non-linearity in its time evolution that is described by the Bloch equation. Hence, incorporation of the non-linearity into the reconstruction method should improve the image quality. Here, we introduce a Bloch equation based algebraic reconstruction method applied to COncurrent Dephasing and Excitation (CODE) with FM pulse excitation (Fig.1). Theory In CODE, k-space sampling is performed in a radial manner, as the orientation of the magnetic field gradient changes in a stepwise manner (5). The experimentally acquired signal vector composed of N complex sampled points, S(ti), is expressed as