1 Iterative Off-resonance and Signal Decay Correction for Improved Multi-echo Imaging in MRI Tobias Knopp, Holger Eggers, Hannes Dahnke, J¨ urgen Prestin, Julien S´ en´ egas Abstract— Local deviations of the main field and signal decay due to transverse relaxation lead to perturbations of the Fourier encoding commonly applied in magnetic resonance imaging. Hence, images acquired with trajectories having long readout times suffer from artefacts such as blurring, geometric distortion, and intensity inhomogeneity. These effects can be corrected by means of iterative reconstruction algorithms, provided a field map and a relaxation map are available. Recently, a fast gridding- based approach to field inhomogeneity correction was proposed. In this work, this algorithm is extended to also handle the signal decay due to relaxation. It is then embedded in a novel fixed- point iteration algorithm that allows for the joint estimation of the field map, relaxation map, and echo images from a single multi-echo acquisition. This joint estimation approach enables the application of fast acquisition trajectories in multi-echo imaging experiments, such as spiral and echo planar, while avoiding artefacts in the reconstruction of the echo images, the field map, and the relaxation map. Since the method dispenses with the acquisition of a separate calibration scan, an appreciable overall reduction in scan time can be achieved. The evaluation of the proposed algorithm in simulations and in-vivo experiments shows a significant improvement in the reconstruction of the echo images and the estimation of the relaxation map, as compared to the standard case, where no correction is applied. The demon- strated rapid convergence of the fixed-point iteration algorithm together with the computational efficiency of the gridding-based reconstruction keeps the overall computation time reasonable. Index Terms— Magnetic resonance imaging, image reconstruc- tion, gridding, field inhomogeneity, off-resonance correction, relaxation, signal decay correction, iterative reconstruction, spiral imaging, echo planar imaging I. I NTRODUCTION Measurement of relaxation rates in magnetic resonance imaging (MRI) is usually performed by means of multi-echo sequences [1], which allow for the acquisition of a series of images at increased echo times. The relaxation rate is then estimated by fitting voxel-wise an exponential function to the data of the echo time series. Such measurements are characterized by long repetition times (TR) and are, there- fore, associated with relatively long scan times. One way to shorten the acquisition time consists of applying fast imaging techniques that reduce the number of phase encoding steps while preserving the spatial resolution. For example, echo planar imaging [2] and spiral imaging [3] have been applied T. Knopp is with the Institute of Medical Engineering, University of L¨ ubeck, L¨ ubeck, Germany (e-mail: knopp@imt.uni-luebeck.de). H. Eggers, H. Dahnke, J. S´ en´ egas are with Philips Research Europe, Sector Medical Imaging Systems, Hamburg, Germany (e-mail: holger.eggers@philips.com, hannes.dahnke@philips.com, julien.senegas@philips.com). J. Prestin is with the Institute of Mathematics, University of L¨ ubeck, L¨ ubeck, Germany (e-mail: prestin@math.uni-luebeck.de). to collect data in multi-echo imaging experiments for T ⋆ 2 mapping. These trajectories acquire a larger portion of k- space at a given echo time, and have, therefore, long readout times. In that case however, additive dephasing due to off- resonance and relaxation-induced signal decay significantly perturb the Fourier encoding. Hence, images reconstructed by means of a Fourier transform, or by gridding in case of non- Cartesian trajectories, suffer from artefacts such as blurring, geometric distortion, and intensity inhomogeneity, depending on the chosen trajectory. Main field inhomogeneity may be due to susceptibility gradients occuring e.g. at the transition between air and tissue, as can be found in the vicinity of the sinus cavities in brain imaging or of the lung in liver imaging. Signal-decay artefacts are most pronounced in regions having short T ⋆ 2 values, such as some parts of the brain or the liver, and can be severe in the presence of iron-oxide contrast agents. This work proposes a new, time-efficient method to correct for off-resonance and signal decay artefacts in the reconstruction of multi-echo acquisitions and the estimation of the corresponding relaxation map. Field inhomogeneity and signal decay generally affect ac- quisition techniques with long readout times, not only in multi- echo acquisitions, and several approaches have been proposed to reduce their effects. While technical solutions, such as shimming, may limit main field inhomogeneity due to system imperfections to some extent, nonlinear spatial variations of the main field due to susceptibility changes can only be mitigated in very rare situations and only for a spatially limited field of view [4]. Alternatively, suitable corrections during the reconstruction process can be applied. In these techniques, the off-resonance term is explicitly taken into account in the modeling of the encoding matrix, and the obtained linear system is inverted [5]. This approach can be extended to include also the signal decay due to transverse relaxation [6], [7]. These correction schemes require, however, the knowledge of both a field map and a relaxation map, which usually need to be measured in a calibration scan. A field map can be measured by acquiring two images at different echo times and dividing their phase difference by the difference in echo times. The echo spacing should be kept small to avoid wrapping errors. Two practical issues may limit the applicability of an improved correction scheme based on pre-measured maps: the substantial scan time involved in acquiring additional data and the possible temporal variation of the field map and the relaxation map, e.g. due to physiological effects [8], between the calibration and the diagnostic scans. In the particular case of multi-echo imaging, however, data from different echo