Yu Yulee Li^{1,2}, Shams Rashid^{1}, Yang Cheng^{1}, William Schapiro^{1}, Kathleen Gliganic^{1}, Ann-Marie Yamashita^{1}, Marie Grgas^{1}, Michelle Maragh^{1}, and Jie Jane Cao^{1,3}

Radial imaging is k-space variant, but mostly uses k-space invariant methods in image reconstruction. This permits reconstructing images with lower computation complexity at a cost of performance. Here a k-space variant parallel imaging reconstruction technique is developed to reconstruct Cartesian data directly from multi-channel radial samples with affordable computation. It is demonstrated that this technique offers the ability to collect real-time images with a temporal resolution of 40ms and a spatial resolution of 1.7mm. The new technique outperforms those gridding-based methods with k-space invariant algorithms in a stress cardiac test.

A previously-developed framework
of correlation imaging^{1-3} is used to convert radial parallel imaging
reconstruction into the estimation of correlation functions. As shown in Figure
1, every Cartesian datum, *d _{m}*(

To implement k-space variant
reconstruction with commonly-available computer hardware, the correlation
imaging model in Figure 1 is modified by introducing an additional filter *w*(* k_{c}*)
in Figure 3. This k-space filter suppresses MRI signals from the outside of
heart anatomy, making it possible to reconstruct images within a small FOV. By
reducing the FOV, Cartesian k-space data may be undersampled without any loss. This
allows for the calculation of only a subset of the full Cartesian k-space data,
making it possible to reconstruct images with an affordable computation cost.

For feasibility demonstration, cardiac imaging experiments were performed with a balanced steady-state free-precession sequence. To investigate reconstruction in simulation studies, a full k-space single-phase radial dataset was collected with breath-holding (segmented, FOV 300mm, 128 views, 1.2mm spatial resolution, TR/TE 4.6/2.3ms, flip angle 65°). The new technique was demonstrated in a stress (biking) cardiac test. In this test, the subject was first scanned when resting with a heart rate of ~60bpm and then when biking with a heart rate of ~100bpm. Real-time cardiac images were collected with free-breathing (FOV 300mm, 220 time frames, 16 views per time frame, 1.7 mm spatial resolution, TR/TE 2.5/1.3ms, flip angle 50°). The results were compared with online reconstruction provided by the MRI manufacturer.

1. Li, Y et al., MRM 2012, 68:2005-2017.

2. Li, Y et al., MRM 2015, 74(6): 1574-1586.

3. Li, Y, Proc Intl Soc Mag Reson 2017, 24: 5156.

4. Shannon, C et al., Proc Institute of Radio Engineers 1949, 37(1): 10-21.

5. Whittaker, T., Proc Royal Soc Edinburgh 1915, 35: 181-194.

Figure 1.
A correlation imaging framework^{1-3} is used to convert radial
parallel imaging reconstruction into the estimate of multi-channel correlation
functions. This framework minimizes the least square error between every
Cartesian datum at **k**_{c} and the
linear combination of its multi-channel radial neighbors at **k**_{r}'s. The linear weights for
reconstruction can be resolved point-wisely from linear equations with the
coefficients equal to correlation functions on the non-Cartesian grid (**k**_{r'} -**k**_{r} and **k**_{c}-**k**_{r'}). If these correlation
functions be estimated, Cartesian data can be reconstructed directly from
multi-channel radial data in a k-space variant fashion.

Figure
2. Based on Nyquist theorem, radial samples are written as the
Whittaker-Shannon interpolation from Cartesian data in the central k-space.
This yields a set of linear equations for resolving low-resolution images which
are in turn used to estimate correlation functions in Cartesian space. The
estimated Cartesian-space correlation functions are further used to estimate
correlation functions at an arbitrary k-space position using Whittaker-Shannon interpolation. It should be noted that correlation imaging (Figure 1) requires
correlation functions only at low |**k**|
values and an estimate of central k-space correlation functions
(low-resolution) is thus sufficient for image reconstruction.

Figure
3. Reduced-FOV reconstruction may be implemented by introducing an additional
k-space filter *w*(**k**_{c}) in the correlation imaging model illustrated by Figure
1. Correspondingly, the right side of linear equations should be modified with reduced FOV correlation functions generated by passing the original correlation functions through the filter *w*(**k**_{c}). The resolved linear
weights will lead to the reconstruction of reduced-FOV images (bottom-right).

Figure 4. Comparison
of full-FOV and reduced-FOV k-space variant reconstruction methods in reference
to gridding. In all the experiments with different view numbers, both two methods perform better than
gridding while the reduced-FOV reconstruction needs less computation time. This
permits implementing reduced-FOV reconstruction with affordable computation
hardware for clinical use.

Figure 5. Real-time radial imaging
with reduced-FOV k-space variant reconstruction in reference to that with
online gridding-based reconstruction provided by the MRI manufacturer. The
real-time data include 16 views in each time frame, providing a spatial
resolution of 1.7mm and a temporal resolution of 40ms. The k-space variant
reconstruction gives less artifacts and better spatial/temporal resolutions. This
gain is more significant in images collected when biking with both rapid heart-beats/respiration
and body movements.