Jost M. Kollmeier^{1} and Jens Frahm^{1}

^{1}Biomed NMR, Max-Planck-Institut für biophysikalische Chemie, Göttingen, Germany

In undersampled radial phase-contrast imaging, Maxwell terms can vary for individual radial projections (spokes). Integrated into a model-based image reconstruction, a computationally expensive spoke-by-spoke correction has been proposed, for which the reconstruction times scale with the number of spokes per frame. To make this approach practical for large numbers of spokes, this work proposes to use k-means clustering of Maxwell terms in the spoke dimension and thereby reduce the computational costs of the model-based image reconstruction for phase-contrast MRI with radial Maxwell correction.

$$ F_{j,l}: x \mapsto \sum_s^k P_{l,s} \text{ FT} \left\{ \rho \cdot \exp\left(i \phi_{l.s} \right) \cdot \exp\left( i\sum_d E_{l.d} v_d \right) \cdot c_j \right\} $$

Here, multiple velocity maps $$$v_d$$$ are jointly estimated along with the complex image $$$\rho$$$ and coil profiles $$$c_j$$$. Individual velocity encodings ($$$l$$$) are assumed to only differ in phase ($$$E_{l.d} v_d$$$) or by the individual Maxwell phase error map $$$\phi_{l.s}$$$, where $$$s$$$ is the index along the clustered spoke dimension. The number of Fourier transforms, clustered sampling patterns ($$$P_{l,s}$$$) and clustered phase error maps ($$$\phi_{l.s}$$$) are set in advance by the parameter $$$k$$$ – the argument of the k-means algorithm. Therefore, $$$k$$$ decides whether all individual Maxwell phase errors are compensated for on a spoke-by-spoke basis ($$$k = N_S$$$ the number of spokes per frame) or a compromise is chosen between reconstruction time and accuracy of velocity quantification ($$$k <N_S$$$). Reconstructions of real-time three-directional velocity maps

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3. Bernstein et al. Concomitant gradient terms in phase contrast MR: analysis and correction. Magnetic Resonance in Medicine. 1998;39.

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Figure 1: Speed maps of the aortic arch with
different numbers of clusters (k) for radial Maxwell correction.

Figure 2:
Number of Fourier transforms in forward model (left), reconstruction time
(center) and root-mean-squared velocity error (right) as a function of number
of clusters (k) of concomitant phase errors. Here, k = 0 corresponds to image
reconstructions without Maxwell correction.

DOI: https://doi.org/10.58530/2022/3282