Advances in Sensitivity Encoding With Arbitrary k-Space Trajectories Klaas P. Pruessmann, 1 Markus Weiger, 1 Peter Bo ¨ rnert, 2 and Peter Boesiger 1 * New, efficient reconstruction procedures are proposed for sensi- tivity encoding (SENSE) with arbitrary k-space trajectories. The presented methods combine gridding principles with so-called conjugate-gradient iteration. In this fashion, the bulk of the work of reconstruction can be performed by fast Fourier transform (FFT), reducing the complexity of data processing to the same order of magnitude as in conventional gridding reconstruction. Using the proposed method, SENSE becomes practical with non- standard k-space trajectories, enabling considerable scan time reduction with respect to mere gradient encoding. This is illus- trated by imaging simulations with spiral, radial, and random k- space patterns. Simulations were also used for investigating the convergence behavior of the proposed algorithm and its depen- dence on the factor by which gradient encoding is reduced. The in vivo feasibility of non-Cartesian SENSE imaging with iterative re- construction is demonstrated by examples of brain and cardiac imaging using spiral trajectories. In brain imaging with six receiver coils, the number of spiral interleaves was reduced by factors ranging from 2 to 6. In cardiac real-time imaging with four coils, spiral SENSE permitted reducing the scan time per image from 112 ms to 56 ms, thus doubling the frame-rate. Magn Reson Med 46:638 – 651, 2001. © 2001 Wiley-Liss, Inc. Key words: sensitivity encoding; SENSE; k-space trajectories; spiral imaging; iterative reconstruction Arrays of simultaneously operated receiver coils have re- cently attracted increasing attention as a means of enhanc- ing scan speed in MRI. Several authors have proposed imaging schemes designed to utilize parallel signal acqui- sition with multiple coils for the purpose of reducing scan time (1– 8). Exploiting knowledge of spatial coil sensitiv- ity, these techniques enable the reduction of the number of gradient-encoding steps without compromising spatial resolution or the field of view (FOV). Due to the specific role of coil sensitivity, array imaging with reduced k-space sampling requires particular consid- erations for image generation. As coil sensitivity is used as a means of signal encoding, the net encoding functions are no longer harmonics of the FOV as in conventional Fourier imaging. Consequently, image reconstruction from under- sampled multiple-coil data cannot be performed by mere Fourier transform (FT) but requires specialized processing. A general approach to the reconstruction problem was described in Ref. 6, introducing the SENSitivity Encoding (SENSE) method. In this work, reconstruction formulae were derived for array imaging with arbitrary coil config- urations and k-space trajectories. However, in their general form these formulae are numerically rather demanding. An efficient implementation was previously described for the special case of sampling k-space along a regular Cartesian grid (6). In this case, SENSE processing may be performed in a fashion similar to earlier Cartesian ap- proaches (2,4). In a first step, conventional Fourier recon- struction is performed for each coil element, yielding a set of aliased images with reduced FOV. In a second step, the aliasing is undone using knowledge of the individual coil sensitivities. This latter step requires little computation because the aliasing effect superimposes the signal of only a small number of volume elements in each pixel of the reduced FOV. In other words, the point spread function (PSF), which corresponds to Cartesian undersampling, is well behaved in that it exhibits sharply localized, equidis- tant peaks. With general k-space trajectories, such as spirals (9,10) or radial (11,12) and stochastic (13) schemes, the situation is more complicated. Similarly to the Cartesian case, SENSE permits reducing the density of general sampling patterns while maintaining the covered k-space area. How- ever, arbitrary k-space patterns entail highly variable PSFs. With spiral acquisition, for example, the PSF exhibits con- tinuous, ring-shaped structures rather than evenly spaced, isolated peaks (14). As a consequence, the aliasing that results from reducing k-space density is considerably more complex and prevents straightforward unfolding in the image domain. This observation illustrates why general SENSE reconstruction has so far been seriously hampered by its numerical complexity (15). In this work we propose a novel, highly efficient imple- mentation of non-Cartesian SENSE processing. The new approach is based on the idea of performing reconstruction iteratively, as recently proposed independently by the au- thors and by Kannengieer et al. (16,17). It is shown that so-called weak reconstruction with SNR optimization can be accomplished using the conjugate-gradient (CG) itera- tion method (18). The key improvement in reconstruction speed is then achieved by combining fast Fourier trans- form (FFT) with forward and reverse gridding operations for efficient execution of the CG iteration loops. Further speed benefits are accomplished by several measures of preconditioning and numerical optimization. Using the proposed procedure, the computation times required for non-Cartesian SENSE reconstruction are con- siderably diminished, making sensitivity-encoded imag- ing practical with arbitrary acquisition patterns. The gen- eral applicability of the method has been verified by im- aging simulations with spiral, radial, and random trajectories, as well as Cartesian sampling for comparison. The in vivo feasibility of the new approach is demon- 1 Institute of Biomedical Engineering, University of Zu ¨ rich and Swiss Federal Institute of Technology Zu ¨ rich, Switzerland. 2 Philips Research Laboratory Hamburg, Division Technical Systems, Ham- burg, Germany. Grant sponsor: EUREKA; Grant number: EU1353; Grant sponsor: KTI; Grant number: 3030.2. *Correspondence to: Dr. P. Boesiger, Institute of Biomedical Engineering, University of Zu ¨ rich and ETH Zu ¨ rich, Gloriastrasse 35, CH-8092 Zu ¨ rich, Swit- zerland. Received 22 November 2000; revised 17 January 2001; accepted 5 February 2001. 2001 ISMRM Young Investigator I.I. Rabi Award Finalist. Magnetic Resonance in Medicine 46:638 – 651 (2001) © 2001 Wiley-Liss, Inc. 638