Gradient Optimization for arbitrary k-space trajectories using Active Contour (GO-Active)
Pavan Poojar1, Bikkemane Jayadev Nutandev2, Ramesh Venkatesan3, and Sairam Geethanath1

1Medical Imaging Research Centre, Dayananda Sagar College of Engineering, Bangalore, India, 2Bangalore, India, 3Wipro-GE Healthcare, Bangalore, India

### Synopsis

K-space trajectories such as cartesian, radial, spiral are not optimal for traversing arbitrary k-space shapes. GO-Active is a novel acquisition technique which is a combination of active contour and convex optimization where active contour was used to obtain arbitrary k-space trajectory and convex optimization was employed to optimize the gradients based on hardware constraints. Reconstruction was performed using Non Uniform Fast Fourier Transform and compressed sensing. Retrospective study was performed on six brain datasets and phantom, where as prospective study was carried out on the phantom respectively. Current and Future work involves application of GO-Active on in vivo data prospectively.

### Purpose

In dynamic MRI methods such as DCE MRI, DWI, the shape of the significant values of k-space depends on the structure of the organ and is typically arbitrary. The conventional k-space trajectories such as Radial, Spiral, Cartesian, etc. are inadequate in terms of covering such arbitrary k-space shapes efficiently. Here, we demonstrate one such method that combines the usage of active contours and convex optimization (cvx) to obtain the desired gradient waveforms.

### Methods

The Active Contour (AC) [1] technique is typically used for segmentation of images based on their texture. It is an energy minimization technique which, under the influence of internal and external forces moves likes a snake and is given by $E_{snake}=\int_{0}^{1} E_{int}(V(s))+E_{image}(V(s))+E_{con}(V(s))ds$ (1), $E_{int}(V(s))$ where $E_{image}(V(s))$ is the internal energy of the spline due to bending, $E_{con}(V(s))$ is the image forces and is the external constraint force. An undersampled mask of a given k-space can be used as an image and AC can be used to obtain spirals of arbitrary shapes to traverse the k-space. The relationship between the k-space trajectory and gradients is given by $k(t)=\frac{\gamma}{2\pi}\int_{0}^{T}g(t)dt$ (2) [2]. The cvx [3] can be used to obtain optimal gradient waveforms for the by solving for (3), subjected to maximum gradient amplitude (Gmax), maximum slew rate (SRmax) and total time duration, where is the k-space trajectory from AC, is the integration matrix and is the gradient waveform. The integration matrix is formulated based on the trapezoidal rule. Figure 1 summarizes the workflow described above. Studies: i) Retrospective studies: Phantom: The k-space data was acquired on a 1.5T scanner (Optima, GE) for a circular phantom as shown in figure 2(a) with TR/TE=34/5.3ms, matrix size 256x256, slice thickness 5mm, Total acquisition time=70s. The original k-space mask was undersampled (20%) and morphological operations of erosion and dilation were performed. Tweaked spiral like arbitrary k-space trajectory was obtained from the AC. The number of points on the trajectory was subsampled to match the memory requirements of the computer. The gradient waveforms were obtained by solving equation $\parallel(k-Axg(t))\parallel$ (3) using cvx subjected to the constraints SRmax = 100T/m/s, Gmax = 33mT/m and a total time duration of T = 40ms. The gradient waveforms were verified by integrating them analytically. The images were reconstructed using Fourier transform with density compensation. In-vivo: The k-space data was acquired for brain from six subjects with TR/TE=3000/150ms, matrix size 256x128, slice thickness 5mm. Similar method was followed for acquisition and reconstruction as described previously. ii) Performance evaluation: The Normalized Root Mean Squared Error (NMRSE) was obtained by evaluating the Euclidean distance between the input k-space trajectory and the k-space trajectory obtained from the designed gradients. For each evaluation one among the three variables (Gmax, SRmax, time, and resolution) was varied while keeping the other variables constant. iii) Prospective studies: The gradient waveforms designed for the phantom were played on a 1.5T (Optima, GE). The gradient waveforms were designed for 20% undersampling and were interpolated to match the scanner requirements (2048 points, ∆t = 20µs, total acquisition time 40.94ms). k-space points were acquired and the resulting image was reconstructed using non uniform fast fourier transform [4] followed by compressed sensing for reconstruction. A baseline image obtained from the complimentary mask was used to provide high frequency components and was scaled accordingly.

### Results

Figure 2(a) represents the image of the phantom and brain respectively which were acquired for the prospective studies. Figure 2(b) and 2(d) represent the respective k-space masks along with the k-space trajectory and verification in colour codes. Figure 2(e) and 2(f), shows the image of retrospectively reconstructed phantom while 2(f) represents the image obtained from prospective studies. The image difference as represented in figure 2(h) is significantly low. The gradient waveforms, which were played on, the scanner is as shown in figure 3. The performance evaluation curves are as represented in figure 4. The percentage coverage after the morphological operations were performed was observed to have an error difference of ± 2%. The NRMSE obtained shows a decreasing trend for the change in time duration as expected since, when the time duration increases, the gradients have more time for extending to farther regions of k-space. In figure 4(d) the image resolution is changed, as the resolution increases the extent of k-space reduces therefore requiring less gradient strength and slew rate to reach the reduced k-space region. Further the gradient waveforms can be split to traverse the k-space in multiple shots similar to a multi-shot spiral trajectory.

### Acknowledgements

No acknowledgement found.

### References

[1] M. Kass et. al. IJCV, 1988. [2] Hand book of MRI Pulse Sequences, M. A. Bernstein [3] M. Grant et. al. DCP, 2014.[4] M. Lusting et. al. MRM, 2007.

### Figures

Figure 1: Flow chart of the propose method

Figure 2: (a) Phantom Image (b) The undersampled (20% of original k-space) mask along with morphological bound (red), subsampled k-space trajectory (blue) and verified result (green). (c) in vivo brain image, (d) undersampled mask of brain (e) and (g) Recostructed images after active contour technique, (f) Prospective reconstruction using NUFFT, (h) difference image .

Figure 3: Gradient waveforms designed for acquisition of phantom data, each comprising 2048 points and total time duration of 40.94 ms.

Figure 4: Performance evaluation (a) Maximum gradient amplitude constraint, (b) Maximum slew rate constraint, (c) Time duration constraint. (d) Resolution constraint

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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