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(q,τ)=S(q,τ)/S0 at diffusion encoding position q and diffusion time τ. It uses an orthogonal basis that allows efficient representation of any multi-spherical diffusion signal with few parameters 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 δ→0 and separability in the dependence of the dMRI signal to q and τ [4]. We represent the fitted signal attenuation using the cross-product between the spatial Fourier basis Φi(q) [5] and temporal basis Tk(τ) [2] such that we fit ˆE(q,τ;c)=∑Nqi∑NτkcikΦi(q)Tk(τ) with basis coefficients c and Nq and Nτ the number of spatial or temporal basis functions, respectively. The voxel-wise optimization is as follows:
argminc∬
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.