Pavan Poojar^{1}, Bikkemane Jayadev Nutandev^{2}, Ramesh Venkatesan^{3}, and Sairam Geethanath^{1}

^{1}Medical Imaging Research Centre, Dayananda Sagar College of Engineering, Bangalore, India, ^{2}Bangalore, India, ^{3}Wipro-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.