Recently studies have demonstrated the utility of susceptibility tensor imaging (STI) in the brain, heart, and kidney (1-3). STI can characterize more subtle changes compared to DTI, for example, in white matter demyelination and nephron epithelial damage (4,5). However, STI remains a challenging protocol due to the requirement of physical reorientation with respect to the magnetic field. Advances in the field have reduced the number of orientations to three by assuming cylindrical symmetry (6,7). STI for the abdomen, even with three orientations, can be particularly challenging. Current studies of the kidney and heart are limited to ex-vivo imaging. In this study, we present a method to compute frequency tensor imaging (FTI) at a single image acquisition without rotating the object. FTI takes advantage of tissue structure already pointing in multiple directions with respect to the magnetic field in a single orientation dataset. Here, we use the image structure tensor as a prior to determine the directions of tissue structure. This technique offers the potential for susceptibility based tensor imaging of the abdomen in the clinic.

C57Bl/6 mouse (4 months) was perfusion fixed. Kidneys were excised and immersed in 2.5 mM ProHance. The specimen was placed in a sphere to facilitate multiple orientations for STI. A single orientation dataset was used for FTI. Images were acquired using 3D multi-echo GRE at 9.4T with these parameters: TR=50 ms, TE₁/ΔTE/TE_n=3.4/2.9/17.9 ms, FA=50°, resolution=55×55×55 μm³, STI directions=12, acquisition time=12 hours (STI) and 1 hour (FTI).

Each echo phase was unwrapped and background removed using a Laplacian and V-SHARP algorithm (8). The resultant frequency maps were echo summed to create an enhanced frequency map. The 12 frequency maps were used to calculate STI following (9).

The structure tensor (S) for each pixel was computed as the outer product of the image gradient vectors:

$$ S=\left( \begin{array}{ccc}f_x^2 & f_x\cdot f_y & f_x\cdot f_z \\f_x\cdot f_y & f_y^2 & f_y\cdot f_z \\f_x\cdot f_z & f_y\cdot f_z & f_z^2 \end{array} \right) $$

where $$$f_x=g_{x,\sigma}\star f(x,y,z)$$$, $$$f_y=g_{y,\sigma}\star f(x,y,z)$$$, and $$$f_z=g_{z,\sigma}\star f(x,y,z)$$$. The major eigenvectors from the structure tensor was used for vector projection. Candidate vectors were projected onto a target vector based on distance and angle (Fig. 1). The distance threshold was 3 pixels within the radial length from seed point to target point. The angle threshold was >25° between candidate vector and target vector. A maximum of 200 vectors were projected. Each pixel then has a matrix of frequency values and magnetic field unit vectors based on the vector transformation.

The frequency tensor (F) was calculated from frequency values at projected orientations (i = 1, 2, …, 200) following:

$$\bf f_i=\bf \hat{h}_i^T F \bf \hat{h}_i$$

where f is the frequency value and $$$\bf \hat{h}$$$ is the magnetic field unit vector. Since F is symmetric, it can be rewritten as six independent tensor elements (F₁₁ F₁₂ F₁₃ F₂₂ F₂₃ F₃₃). Tractography from structure tensor, STI, and FTI were performed using TrackVis.

The enhanced frequency map is shown in Fig. 2. The inset displays the structure tensor eigenvectors used for vector projections. The outer medulla was focused because of the wide range of vector orientations with respect to the magnetic field. Tractography results are shown in Fig. 3. The structure tensor tracks accurately span from the anteroposterior directions (red arrows) to the mediolateral direction (blue arrow). Similarly, FTI tracks cover the anteroposterior and mediolateral directions of kidney tubules. On the other hand, STI tracks are most coherent mediolaterally and less coherent anteroposteriorly.In this work, we demonstrate the feasibility of FTI at a single orientation. Structure tensor was used to estimate tubular directions. While the structure tensor produced the most coherent tracks, it is based on image intensities and does not provide the frequency information sensitive to local microstructure. For instance, one study demonstrated that renal tubules were visible in the magnitude images, however, STI tractography were virtually absent due to damages in the tubular walls (5). FTI assumes radial symmetry from a seed point and treats nephron segments from pole to pole as being homogenous in structure and composition. It treats nephron segments along the tubule as being distinct where tubular vectors are not projected or shared. Renal injuries can be more specific to nephron segments along the tubule than to the same segments in neighboring tubules (10). Most importantly, the resultant FTI shows sensitivity to the frequency information of tubules at varying orientations with respect to the magnetic field. FTI rids of the major challenge of performing multiple orientations and significantly reduces the acquisition time needed to calculate a susceptibility based tensor image.