Linear Acceleration of SADD Method for Three Compartments
Ana Karen Loya1 and Mariano Rivera1

1Computer Science, Centro de Investigacion en Matematicas AC, Guanajuato, Mexico

### Synopsis

The proposal attempts to model more properly the intra-voxel information from DW-MRI signals in order to obtain tissue diffusion properties in white matter. The method is based on Sparse and Adaptive Diffusion Dictionary (SADD) strategy that dynamically adapts a dictionary of diffusion functions by changing size and orientation of the diffusion tensors. In ISMRM2015, we demonstrated that our accelerated version (LASADD) reduces complexity and computational cost wrt SADD with similar quality results. This work extends the idea of LASADD to three compartments (intracellular, extracellular and cerebrospinal fluid) and presents experimental results depicting the computed properties about the diffusion structure.

### Purpose

We extend the Linearly Accelerated SADD multitensorial method for modelling the IntraCellular (IC), ExtraCellular (EC) and CerebroSpinal Fluid (CSF) tissue compartments (environments).

### Methods

The diffusion multifiber model used in Diffusion Basis Functions (DBF) [4] is a linear combination of Gaussians [6] or Diffusion Functions (DFs) under a Diffusion Tensor (DT) Model [2]. The SADD strategy [1] iteratively adapts the dictionary of DFs in three steps: prediction, correction and generation. Prediction uses a LASSO penalisation to find the best DFs to fit the signal diffusion. Correction rotates and change the size of each DT of the DFs selected. Generation consists in add DF combinations to the dictionary. LASADD [3] uses the DT representation $\mathbf{T}={\chi_1} \mathbf{v} \mathbf{v}^T + {\chi_2} \mathbf{I}$ that simplifies the correction step: the rotations are reduce to sum of vectors. Additionally, LASADD uses the first order Taylor series of exponential and cast the optimisations to simple least squares problems with boundary constraints.

In this work, we consider three types of environment, mentioned in [5], each one is represented with a particular dictionary. IC considers the diffusion inside the axons in the tract fiber, the DT of each $j$-tract is $\mathbf{T}_j^{IC}={\chi_1}_j \mathbf{v}_j \mathbf{v}_j^T$. EC represents the diffusion between the axons in the tract, the respective DT is an anisotropic ellipse $\mathbf{T}_j^{EC}={\chi_1} \mathbf{v} \mathbf{v}^T + {\chi_2} \mathbf{I}$. CSF is the diffusion isotropic outside the tracts and is modelled with a the spherical DT, $\mathbf{T}^{CSF}={\chi_2} \mathbf{I}$. Hence, the signal diffusion model is

$$S_i = \sum_{j} \left[\alpha_j^{IC} \psi_{i,j}^{IC} + \alpha_j^{EC} \psi_{i,j}^{EC} \right] + \alpha^{CSF} \psi_{i}^{CSF} + \eta_i,$$

where $\alpha$ and $\psi$ are the corresponding volume fraction and DF respectively, $i$ indicates the $i$-gradient measurement and $\eta$ is the Rician noise of acquisition. Our proposal consists of using the LASADD method for estimating: the number of fibers in a voxel, their orientations and their size. Then, we impose the computed orientations to the IC and EC compartments and, keeping fixed the orientations, we estimate the size profile of each DTs and the volume fractions according with the previous equation. The optimisations remain being least squares problems with boundary constraints.

### Experiments and Results

We compare the signal estimations of our proposal with DBF and LASADD. Fig. 1 presents the mean square error of the reconstruction for two experiments.

In the first experiment we use the data in Sparse Reconstruction Challenge for Diffusion MRI 2014 to observe the behaviour in crossing fibers at 45$^\circ$. Fig. 2a shows a simple DT model. Fig. 2b depicts the estimated DTs by DBF and Fig. 2c shows the respective results computed by LASADD.Fig. 2d displays the estimation of the IC compartments. Note that the detection of the IC compartments is confined to the crossing region, this can be explained by the packing of the "solid fibers" that reduces the space for transversal water diffusion. The trasversal space is increased in the regions outside the crossing. In the no-crossing regions, the EC compartment explains the diffusion phenomena, see Fig. 2e. Fig. 2f is the CSF compartments have limited presence in the borders of the tracts.

For the second experiment we use DW-MRI signals acquired from a healthy volunteer scanned on a Phillips Achieva TX 3.0 scanner with 16 channels. We have 4 B0 images and 64 multi-shell DW images with $b=\left\lbrace 2000,2500 \right\rbrace$ s/mm$^2$ with approximately SNR=30.In Fig. 3a we show a slide and select one zone with well-known specific crossing. Fig. 3b depicts the DT estimations by DBF. In Fig. 3c we see that LASADD seems to detect a more defined crossing. Fig. 3d, 3e and 3f shows the computed IC, EC and CSF compartments.

### Conclusions

We extended the linear approximation of SADD to three compartments. Our proposal shows the posible behaviour in each compartment using a dynamic dictionary by solving simple linear least squares problems. We evaluate the performance with benchmark dataset and demonstrated its capabilities using real data. Our future work will focus in validate our method with histological studies.

### Acknowledgements

No acknowledgement found.

### References

1. Aranda, R. et al. Sparse and Adaptive Diffusion Dictionary (SADD) for recovering intra-voxel white matter structure. Medical Image Analysis, 2015.26(1), 243-255.

2. Basser, P.J. et al. MR diffusion tensor spectroscopy and imaging. Biophysical journal, 66(1):259–267, 1994.

3. Loya-Olivas, A.K. et al. LASADD: Linear Acceleration Method for Adapting Diffusion Dictionaries. ISMRM 2015, Diffusion: Image Processing & Analysis Methods, 6438.

4. Ramirez-Manzanares, A. et al. Diffusion basis functions decomposition for estimating white matter intravoxelfiber geometry. IEEE-Transactions on Medical Imaging, 2007.26(8), 1091-1102.

5. Sherbondy, A.J. et al. Microtrack: An algorithm for concurrent projectome and microstructure estimation. Medical Image Computing and Computer-Assisted Intervention, 2010. 6361, 183-190.

6. Tuch, D.S. et al. High angular resolution diffusion imaging revealsintravoxel white matter fiber heterogeneity. Magnetic Resonance in Medicine, 48(4):577–582, 2002.

### Figures

Mean reconstruction error of the methods in the experiments.

Results in SPARC dMRI 2014.

Results in a brain.

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