Ting Gong^{1}, Hongjian He^{1}, Zhichao Lin^{2}, Zhiwei Li^{2}, Qiqi Tong^{1}, Yi Sun^{3}, Feng Yu^{2}, and Jianhui Zhong^{1,4}

Multi-shell, multi-tissue, constrained spherical deconvolution is an appealing method for the reconstruction of fiber orientation distribution function (fODF), which is of great importance for solving complex fiber configurations to achieve reliable tractography. However, many diffusion measurements and multiple reconstruction steps are required. In this study, the deep neural network were employed to form a multi-output regression problem for establishing a fast and direct estimation of fODF. The proposed method offers a new streamlined reconstruction procedure which exhibits great potential for accelerating the reconstruction of fODF with whole-brain coverage, with satisfactory accuracy in two minutes.

We use the DNN to form a multi-output regression problem, with a spherical harmonics (SH)-based representation of fODFs as the target output, which lowers the requirement for network structure and provides a smooth representation of the data distributed on a sphere, compared to amplitude-represented fODFs. In our DNN method, down-sampled DWIs are used as input and SH coefficients as the output, directly avoiding any intermediate steps. Several options of depth and width were tested for the DNN architecture, and the 2000-1500-1000-800-500-200 setting was chosen for a balance between performance and cost of time and memory, as shown in Figure 1 and Table I.

The ethics committee of the local institute approved this
human study. The DWI data were collected from three healthy subjects on a MAGNETOM Prisma 3T MR scanner (Siemens Healthcare, Erlangen, Germany) equipped with a 64-channel head-neck coil using
a single-shot EPI sequence: TR/TE= 7000/67 ms, FOV= 210×210 mm^{2}, number
of slices= 50, and resolution= 2.5×2.5×2.5 mm^{3}. Diffusion weightings
of b = 1000, 2000, and 3000 s/mm^{2} were applied in 23, 45, and 68
directions ^{4} with 14 b = 0 images equally entered, resulting in a total
of 150 DWIs. In addition, b = 0 images with an opposite phase-encoding direction were also acquired.

After DWI preprocessing for motion and
distortion correction,^{ 5} MSMT-CSD reconstruction was performed in
MRtrix (http://www.mrtrix.org/) using an unsupervised method.^{6} fODF was represented by the SH series of the order lmax,
amounting to (lmax +1)(lmax +2)/2 SH coefficients. Based on the highest
recommended order of 8, ^{7} the training labels were set as the 45
fODF SH coefficients reconstructed from all 150 DWIs. One subject’s data were
reserved for testing whereas 90% of the other two subjects’ data were used for
training, and 10% constituted a validation set. Keras ^{8}
was used for training and testing with Tensorflow running backend. ^{9}
All the codes were run on a platform with an NVIDIA
GTX-1080 graphics card and CUDA Deep Neural Network library^{10}
installed.

When decreasing the DWIs, the overall root-mean-squared errors of the DNN on whole brain white matter rose gently and were lower than the MSMT-CSD reconstruction after its sharp jump from 105 to 90 DWIs, as shown in Figure 2. The first SH coefficient showed that detailed structures are still maintained for the DNN method at 15 DWIs, whereas it is mostly lost for model reconstruction, indicating its unreliable fODF estimates.

The fODFs of three typical ROIs corresponding to k=1, 2, and 3 fiber directions are depicted in Figure 3, and its statistics of angular errors are shown in Figure 4. The reducing DWIs cause a decreasing lmax in the model reconstruction results, leading to less discriminating fODF lobes and a failure of recovery of less obvious directions when the DWIs are fewer than 45 in two fiber regions and 75 in three fiber regions. In the DNN method, the morphology of fODFs remains intact, even when the DWIs decrease to 15. The steady trend of solid curves in Figure 4 suggests that the DNN method preserves the information contained in the original data very well. The smaller angular errors suggest that the DNN outperforms the model reconstruction when the DWIs are fewer than 105.

Finally, fiber tractography using extracted peaks from fODFs was conducted to validate the effectiveness of the learned fODFs. The model-reconstructed results miss several large branches of fibers, causing severe track volume loss and track length truncation when n is less than 60.

1. Jeurissen B, Leemans A, Jones D K, et al. Probabilistic fiber tracking using the residual bootstrap with constrained spherical deconvolution. Human Brain Mapping, 2011, 32(3): 461-479.

2. Jeurissen B, Tournier J, Dhollander T, et al. Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data. NeuroImage, 2014: 411-426.

3. Golkov V, Dosovitskiy A, Sperl J I, et al. q-Space Deep Learning: Twelve-Fold Shorter and Model-Free Diffusion MRI Scans. IEEE Transactions on Medical Imaging, 2016, 35(5): 1344-1351.

4. Caruyer E, Lenglet C, Sapiro G, et al. Design of multishell sampling schemes with uniform coverage in diffusion MRI. Magnetic Resonance in Medicine, 2013, 69(6): 1534-1540.

5. Andersson J L, Sotiropoulos S N. An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. NeuroImage, 2016: 1063-1078.

6. Dhollander T, Raffelt D, Connelly A. Unsupervised 3-tissue response function estimation from single-shell or multi-shell diffusion MR data without a co-registered T1 image. Proc. ISMRM Workshop on Breaking the Barriers of Diffusion MRI, vol. 5, 2016.

7. Tournier J, Calamante F, Connelly A, et al. Determination of the appropriate b value and number of gradient directions for high‐angular‐resolution diffusion‐weighted imaging[J]. NMR in Biomedicine, 2013, 26(12): 1775-1786.

8. Chollet F et al., “Keras,” https://github.com/fchollet/keras, 2015.

9. Abadi M, Barham P, Chen J, et al. TensorFlow: a system for large-scale machine learning. operating systems design and implementation, 2016: 265-283.

10. Chetlur S, Woolley C, Vandermersch P, et al. cuDNN: Efficient Primitives for Deep Learning. arXiv: Neural and Evolutionary Computing, 2014.

11. Koppers S, Merhof D. “Direct estimation of fiber orientations using deep learning in diffusion imaging.” in Proc. MLMI. Springer, 2016, pp. 53–60.

Figure 1. Network
architecture used in this work. The network depth and width are determined by
conducting an error rate test on an independent validation set, as shown in
Table I. When decreasing the number of DWIs, the maximum order of SH also
decreases for the model reconstruction method, causing a loss of angular
resolution. However, for the DNN based method, a maximum order of 8 can always
be achieved with 45 output SH coefficients.

Figure 2. (A) The
overall root-mean-squared error of all 45 coefficients of the DNN method (red)
and model reconstruction method (blue) in brain white matter, estimated from
different numbers of DWIs. The reference standard is a model reconstruction
result using 150 DWIs. Model reconstruction could not generate responses of
lmax 8 when the DWIs were fewer than 60. (B) The first SH coefficient, which is
the only interpretable coefficient, represents the contrast of total apparent
fiber density per voxel, thus conveying structural information. The blurred
structure from model fitting with 15 DWIs indicates its unreliable fODF
estimates.

Figure 3. fODFs
and extracted peak directions of three typical ROIs of (A) single, (B) two, and
(C) three fiber regions (k=1, 2, 3), where n is the number of DWIs and lmax is
the SH order. (Top) Reference results from the model reconstruction with all the
DWIs, (middle) model-reconstructed results for reduced n and lmax, and (bottom)
the DNN results for reduced n and lmax of 8. The reduced DWIs cause decreased
order in the model reconstruction method, whereas a highest recommended order
of 8 can always be achieved for the DNN.

Figure 4. The angular error (◦) for the three ROIs
containing a (a) single fiber, (b) two fibers, and (c) three fibers. The solid
line represents the DNN, and the dashed line represents the model
reconstruction. The DNN exhibits a steady performance in these conditions, particularly
when the number of DWIs is limited. In such a case, the model reconstruction
method cannot infer the correct number of fibers and the angular error is unacceptable.
The mean angular errors obtained by taking the mean of the whole k directions
are listed in Table II.

Figure 5. Fiber tractography comparison. A whole brain
seeding deterministic tractography [46] was conducted with 1000 tracks
extracted for each case using the same ROI on the body of the corpus callosum.
(a-d) Results from 150, 60, 30, and 15 DWIs with model reconstruction; (e-f)
results from 150, 60, 30, and 15 DWIs with the DNN. The track volume (TV) and
mean track length (mTL) are listed for each case. The yellow circles mark the
missing branches.