Skip to main content
  • Workshop presentation
  • Open access
  • Published:

Velocity spectrum imaging using radial k-t SPIRiT


Fourier velocity encoding (FVE) [P.R.Moran,MRI(1),1982] assesses the distribution of velocities within a voxel by acquiring a range of velocity encodes (kv) points. The ability to measure intra-voxel phase dispersion, however, comes at the expense of clinically infeasible scan times. We have recently extended [C.Santelli,ESMRMB(345),2011] the auto-calibrating parallel imaging technique SPIRiT [M.Lustig,MRM(64),2010] to exploit temporal correlations in dynamic k-t signal space and successfully applied it to radially undersampled FVE data. Prior assumption of Gaussian velocity spectra additionally allows undersampling along the velocity encoding dimensions [P.Dyverfeldt,MRM(56),2006]. In this work, a scheme is proposed to non-uniformly undersample the kv-axes in addition to undersampling k-t space for reconstructing mean and standard deviation (SD) of the velocity spectra for each voxel in aortic flow measurements.



2D radial (FOV=250mmx250mm) fully sampled cine FVE data of the aortic arch for 3 orthogonal velocity components was obtained from 5 healthy volunteers on a 3T Philips Achieva scanner (Philips Healthcare, Best, The Netherlands) using a 6 element receive array. Three different first gradient moments corresponding to encoding velocities of 25cm/s, 50cm/s and 200cm/s were applied along with a reference point (kv=0). Undersampled radial data sets were obtained by separately re-gridding these 4-point measurements onto Golden-angle profiles (Fig.1a).

Figure 1
figure 1

a) Dynamic Golden-angle acquisition providing an optimal distritbution of radial profiles [S.Winkelmann,IEEETransMedIm(26),2007]. D relates the reconstructed Cartesian k-t data, x, to the measured k-t points, y. b) Every Cartesian k-t sample point is expressed as linear combination of neighboring points in dynamic k-t space across all coils. c) The shift-invariant interpolation kernel weights (indicated by the arrows in the green neighborhood-mask) in G are obtained by fitting them to the fully sampled k-t calibration area in a Tikhonov-regularized least-squares sense. d) Unconstrained Lagrangian, where I denotes identity and λ a regularization parameter. The latter was set to 0.125.


The interpolation operator G, enforcing consistency between calibration data from a fully sampled centre of k-space and reconstructed Cartesian k-space points, x, is extended for dynamic MRI by including temporal correlations between adjacent data frames (Fig.1b). Data consistency is imposed using gridding-operator D (Fig.1a). Then, x is recovered by solving the minimization problem in Fig.1d). Reconstruction was performed for every kv-point separately using dedicated software implemented in Matlab (Natick,MA,USA). A 7x7x3 neighborhood in kx-ky-t space was chosen for the k-t space interpolation kernel. The weights were calculated from a 30x30x(nr cardiac phases) calibration area (Fig.1c). Mean and SD of velocity distributions were calculated for the resulting coil-combined images.


Fig.2a) compares the mean root-mean-square error (RMSE) of the reconstructed mean velocities and SDs in the aortic arch for different undersampling factors and for each flow direction (M-P-S). Fig.2b) shows in-plane streamlines reconstructed from the acquired mean velocities and turbulence intensity maps calculated from SD values.

Figure 2
figure 2

a) Coil-combined magnitude frame from fully sampled reference data with plots of mean velocities and SDs along phase encoding direction P. RMSE of reconstructed mean velocities (top) and SDs (bottom) from a ROI placed over the aortic arch and averaged over all time frames and volunteers. For each component, means were derived with a 3-point method [A.T.Lee, MRM(33), 1995] whereas SDs were fitted to all 4-point measurements in a least-squares sense. The mean velocity RMSE of the in-plane flow directions (M-P) is significantly smaller compared to the S-component. b) Top row: Stream lines derived from reference and undersampled systolic data plotted over velocity magnitude map. Bottom row: Corresponding turbulence intensity maps [J/m3] calculated according to [P.Dyverfeldt, JMRI(28), 2008]. The streamlines are congruent with the turbulence and phase dispersion maps, respectively.


A novel auto-calibrating reconstruction technique for dynamic radial imaging was successfully applied to undersampled 4-point FVE data from five healthy volunteers. Results show that up to 12-fold radial undersampling provides accurate quantification of mean velocities and turbulence intensities derived from velocity spectra.

Author information

Authors and Affiliations


Rights and permissions

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Santelli, C., Kozerke, S. & Schaeffter, T. Velocity spectrum imaging using radial k-t SPIRiT. J Cardiovasc Magn Reson 14 (Suppl 1), W59 (2012).

Download citation

  • Published:

  • DOI: