Rutger Fick^{1}, Alexandra Petiet^{2}, Mathieu Santin^{2}, Anne-Charlotte Philippe^{2}, Stephane Lehericy^{2}, Rachid Deriche^{1}, and Demian Wassermann^{1}

Effective representation of the diffusion signal's dependence on diffusion time is a sought-after challenge in diffusion MRI (dMRI). As a solution, we recently proposed Multi-Spherical Diffusion MRI (MS-dMRI) to represent the dMRI signal in this four-dimensional space - varying over gradient strength, direction and diffusion time. Our representation allows for the estimation of time-dependent q-space features, providing unique insights on the tissue microstructure. In the study, we assess test-retest reproducibility of these indices in three C57Bl6 wild-type mice. We find that due to our effective regularization methodology during signal fitting, these time-dependent features can be estimated reliably without overfitting the data.

MS-dMRI represents the diffusion signal attenuation $$$E(\textbf{q},\tau)=S(\textbf{q},\tau)/S_0$$$ at diffusion encoding position $$$\textbf{q}$$$ and diffusion time $$$\tau$$$. It uses an orthogonal basis that allows efficient representation of any multi-spherical diffusion signal with few parameters $$$\textbf{c}$$$, using effective regularization approaches imposing signal smoothness and sparsity. We follow Callaghan et al.'s description of time-dependent diffusion in pores and assume an infinitely short gradient pulse $$$\delta\rightarrow0$$$ and separability in the dependence of the dMRI signal to $$$\textbf{q}$$$ and $$$\tau$$$ [4]. We represent the fitted signal attenuation using the cross-product between the spatial *Fourier* basis $$$\Phi_i(\textbf{q})$$$ [5] and temporal basis $$$T_k(\tau)$$$ [2] such that we fit $$$\hat{E}(\textbf{q},\tau;\textbf{c}) = \sum_i^{N_\textbf{q}}\sum_k^{N_\tau} \textbf{c}_{ik}\,\Phi_i(\textbf{q})\,T_k(\tau)$$$ with basis coefficients $$$\textbf{c}$$$ and $$$N_\textbf{q}$$$ and $$$N_{\tau}$$$ the number of spatial or temporal basis functions, respectively. The voxel-wise optimization is as follows:

$$ \textrm{argmin}_{\textbf{c}} \overbrace{\iint\left[E(\textbf{q},\tau) - \hat{E}(\textbf{q}, \tau ; \textbf{c})\right]^2d\textbf{q}d\tau}^{\textrm{Data Fidelity}}+ \overbrace{\iint\left[\nabla^2 \hat{E}(\textbf{q}, \tau; \textbf{c})\right]^2d\textbf{q}d\tau}^{\textrm{Smoothness}} + \overbrace{\|\textbf{c}\|_1}^{\textrm{Sparsity}},\textrm{ subject to}\overbrace{\hat{E}(0, \tau;\textbf{c})=1}^{\textrm{Attenuation Boundary Condition}}$$

Once the coefficients $$$\textbf{c}$$$ are known, we can directly estimate features of the multi-spherical *diffusion propagator* $$$\hat{P}(\textbf{R},\tau;\textbf{c})$$$ through our spatial basis' Fourier properties [5]. We illustrate this by computing the *time-dependent *indices for the Mean Squared Displacement (MSD) [6]; and Return-To-Origin, Return-To-Axis and Return-To-Plane Probability (RTOP, RTAP and RTPP) [5], whose values are inversely related to the mean, perpendicular and parallel diffusivity, respectively. We apply our method in-vivo to three C57Bl6 wild-type mice. For each mouse we acquire test-retest spin-echo multi-spherical images in an 11.7 Tesla Bruker scanner. We show our scheme on the left of Figure 1. We acquire 35 different ``shells'' with one b0 each and a total of 480 DWIs using $$$\delta=5ms$$$. We measure five equispaced "$$$\tau$$$-shells" $$$\Delta=\{10.8, 13.1, 15.4, 17.7, 20 \}$$$ms and seven approximately equispaced "gradient shells" between $$$\{50-490\}$$$mT/m. The voxels are of size $$$100\times 100\times 500\,\mu \textrm{m}$$$. We corrected each dataset from eddy currents and motion artifacts using FSL's eddy_correct, and drew a region of interest in the middle slice of the corpus callosum, taking voxels with Fractional Anisotropy $$$>0.45$$$. We show a sagittal slice with the ROIs on the right of Figure 1.

[1] Fick, Rutger, et al. "Multi-Spherical Diffusion MRI: Exploring Diffusion Time Using Signal Sparsity." MICCAI 2016 Workshop on Computational Diffusion MRI (CDMRI'16). 2016.

[2] Fick, Rutger, et al. "A unifying framework for spatial and temporal diffusion in diffusion MRI." International Conference on Information Processing in Medical Imaging. Springer International Publishing, 2015.

[3] Ozarslan, Evren, et al. "Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal." Journal of Magnetic Resonance 183.2 (2006): 315-323.

[4] Callaghan, Paul T. "Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation." Journal of Magnetic Resonance, Series A 113.1 (1995): 53-59.

[5] Ozarslan, Evren, et al. "Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure." NeuroImage 78 (2013): 16-32.

[6] Fick, Rutger HJ, et al. "MAPL: Tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data." NeuroImage 134 (2016): 365-385.

left: Multi-Spherical diffusion acquisition using a spin echo sequence. Every point represents a shell with uniformly spread DWIs on the sphere. right: FA illustrations of the test-retest mice with in red the region of interest (ROI) voxels in the corpus callosum.

Regularized fitting error of our model while randomly subsampling the data from 400 to 100 fitted DWIs. We broke the y-axis into two parts. The top part uses log-scaling to show the much higher fitting error of unregularized MS-dMRI. The bottom part uses regular scaling and shows our much lower regularized fitting error. For the regularized result, we see that the fitting error is robust to subsampling for all data sets except for Test Subject 3, whose error is nearly twice as high as in all other cases.

Mean and $$$0.75\times$$$Standard deviation of the MSD (top) and RTOP, RTAP and RTPP (bottom) in the corpus callosum for the test and retest data (red and green) of every subject. We used a $$$0.75$$$ multiplier to better separate index groups. For comparison, the gray tones show MSD isolines for different free diffusion coefficients. In subject 1 the test-retest indices overlap closely for every metric, indicating excellent reproducibility. Subject 2 shows similar overlap for q-space indices, but the MSD is slightly off. Subject 3 does not show the same consistency, which corresponds to the higher fitting error found in Figure 2.