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

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.

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.

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.