Adam Rich^{1}, Ning Jin^{2}, Yingmin Liu^{3}, Lee C. Potter^{4}, Orlando P. Simonetti^{3,5,6}, and Rizwan Ahmad^{1,3}

We present a variable density Cartesian sampling method that allows retrospective adjustment of temporal resolution, providing added flexibility for real-time applications where optimal temporal resolution may not be known in advance. This method, called CArtesian sampling with Variable density and Adjustable temporal resolution (CAVA), is validated using real-time, free-breathing phase-contrast MRI data from four volunteers. Diagnostic quality images were successfully recovered at different temporal resolutions. Also, flow quantification based on CAVA was in good agreement with the breath-held segmented acquisition. In summary, CAVA provides a Cartesian alternative to Golden Angle-based radial sampling and can benefit a wide range of 2D real-time applications.

Generating CAVA - To distribute $$$M$$$ samples on a k_{y}-t Cartesian grid with $$$N_s$$$ phase-encoding (PE) lines, we first generate sampling on a smaller grid with $$$N_s$$$ PE lines $$$(N_s =\frac{N}{s},s>1)$$$.
Starting from a randomly selected first PE index, $$$p_s(1)$$$, the index of the subsequent PE lines is sequentially advanced by $$$ gN_s$$$, yielding $$ p_s(i+1) =\left<p_s(i) + gN_s\right>_{N_s} \,\,\,\,\, \text{(Eq. 1)},$$
where $$$ p_s(i) $$$ is the PE index of the $$$i^{\text{th}}$$$ $$$ (i = 1,2,...,M)$$$, (sample on the grid with $$$N_s$$$ PE lines, $$$g$$$ is the golden ratio
(0.618), and $$$\left<\cdot\right>_{N_s}$$$ represents modulo operation. Now the samples $$$ p_s(i) $$$ can be projected to a larger grid using a nonlinear stretching operation, yielding $$ p(i) = \left[ p_s(i) - k\,\text{sign}\left(\frac{N_s}{2}-p_s(i)\right) \left| \frac{N_s}{2}-p_s(i)\right|^{\alpha}+\frac{1}{2}(N-N_s)\right]\,\,\,\,\, \text{(Eq. 2)},$$ where $$$p(i)$$$ is the PE index of the $$$i^{\text{th}}$$$ sample on the grid with $$$N$$$ PE lines, $$$s$$$ controls the relative acceleration rate at the center of k-space compared to the overall acceleration rate, with $$$s>1$$$ ensuring that the sampling density is higher at the center of k-space, $$$\alpha\geq$$$ controls the transition from high-density central region to low-density outer region, $$$\left[\cdot\right]$$$ represents the rounding operation, and constant $$$k>0$$$ is selected such samples $$$p_s=1$$$ and $$$p_s=N_s$$$ are mapped to $$$p=1$$$ and $$$p=N$$$, respectively. In this work, we selected $$$s=3$$$ and $$$\alpha=3$$$.

Figure 1 depicts the stretching process from smaller $$$N_s$$$-grid to the final $$$N$$$-grid. Figure 2 shows CAVA when the same sequence of $$$p$$$ values is binned with different lines per frame (LPF) values. Figure 3 shows two interleaved CAVA samplings for PC-MRI, where the prestretching offset between the two samplings was set at $$$gN_s/2$$$

Data Acquisition—For
validation,
data
was
collected
from
four
volunteers
on
a
3T
(Prima,
Siemens
Healthcare,
Erlangen,
Germany) using a gradient echo, through-plane flow quantification pulse sequence with TE = 2.12ms, TR = 4.1ms, flip angle = 15^{o}, spatial resolution 2.9x2.4 mm, acquisition matrix 84x128, and 10s of free breathing acquisition time.

Image Recovery—Images
were
reconstructed
using
ReVEAL^{2 }at
six
different
temporal
resolutions:
4 LPF (33.0 ms), 5 LPF
(41.2ms), 6 LPF (49.4ms), 8 LPF (65.9ms), 10 LPF (82.4ms) and 15 LPF (123.6ms) with net acceleration rate R = 21, 16.8, 14, 10.5, 8.4, 5.6 respectively. To obtain reference values, segmented acquisition was performed using a gradient echo, through-plane flow quantification sequence with TE = 2.53ms, TR = 4.63ms, temporal resolution 37.1ms, flip angle =15^{o}, and spatial resolution 2.1x2.1mm.

Image Analysis—All contours were drawn using freely available Segment software (Medviso, Lund Sweden). Peak velocity (PV) and stroke volume (SV) were computed for each heartbeat in the real time data and compared against values from the reference segmented acquisition.

[1] Guo L, Derbyshire JA, Herzka DA. Magnetic Resonance in Medicine 2015;76:417-429

[2] Rich A, Potter LC, Jin N, Ash J, Simonetti OP, Ahmad R. Magnetic Resonance in Medicine 2015;76:689-701

Figure 1: The stretching process employed in CAVA. First, a smaller grid is sampled with Golden Ratio jumps between consecutive samples (Eq. 1). Only eight samples and their acquisition order is shown. The samples are then projected to a larger (final) grid (Eq. 2), enabling variable density. Here, $$$N$$$=120 and $$$N_s$$$=40.

Figure 2: Retrospective adjustment of temporal resolution with CAVA. All sampling patterns were generated from the same sequence of phase coding index. The resolution was retrospectively changed by assigning different number of lines per frame (LPF). Here, $$$M=360$$$, $$$N=96$$$.

Figure 3: An example of CAVA for 2D PC-MRI. The composite sampling pattern is a union of two independent CAVA, with one used to collect compensated data and the other used to collect velocity encoded data. The two CAVA patterns consistently interleave each other because they are shifted with respect to each other before stretching. Here, LPF=8 and $$$N=96$$$.

FIgure 4: Tabulated values for PV and SV from four volunteers. The mean and standard deviation for all the heartbeats for a given LPF value are shown as a percentage of the reference value.

Figure 5: Data reconstructed using ReVEAL. (a) Selected frames from a single volunteer for LPF=4, 5, 6, 8, 10, 15. (b) Peak velocity profiles from different LPF values. (c) Volumetric flow rate from different LPF values.