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Nonparametric D(Ω)-distributions for model-free analysis of b(Ω)-encoded multidimensional diffusion MRI on ex vivo rat brain1A.I. Virtanen Institute for Molecular Sciences, University of Eastern Finland, Kuopio, Finland, 2CEMHTI, French National Centre for Scientific Research, Paris, France, 3Department of Chemistry, Lund University, Lund, Sweden
Synopsis
Nonparametric distributions of cell sizes or diffusion tensors have recently been applied to analyze clinically relevant data acquired with advanced diffusion encoding schemes building on oscillating gradients, targeting the frequency-dependence and cell size, or more general q-vector trajectories focusing on the tensorial aspects. We introduce nonparametric D(ω)-distributions as a joint analysis framework taking both frequency-dependence and tensorial properties into account, and demonstrate the approach with ex vivo rat brain data acquired with gradient waveforms exploring the relevant dimensions of the tensor-valued encoding spectrum b(ω).
Introduction
Distributions of scalar diffusivities (1) and compartment sizes (2), combined with data acquisition probing the diffusion time (3) and spectral content (4,5), have a long history in diffusion NMR spectroscopy (6) for investigations of multicomponent solutions and heterogeneous materials. While early studies relied on parametric distributions, such as bimodal (1) and log-normal (2), the development of efficient data inversion algorithms (7,8) enabled estimation of nonparametric distributions of diffusivities (9,10) and compartment sizes (11,12) for samples where the functional forms of the distributions are unknown, albeit at the expense of increasing the sensitivity to noise and the risks of overfitting.Taking the anisotropy and sub-voxel heterogeneity of brain tissues into account, distributions of diffusion tensors D have been proposed for analysis of diffusion MRI data (13,14). Similarly to the development path in NMR, the early parametric distributions (13,14) were followed by nonparametric ones (15), here however only after translating isotropic-anisotropic correlation strategies from solid-state NMR spectroscopy (16) into a diffusion context (17-19).
Motivated by the recent surge of papers with human in vivo studies focusing on either the encoding frequency ω (20-25) or the anisotropy (“shape”) of the encoding tensor b (26-35), as well as pre-clinical studies indicating the potential value of a combined approach (36,37), we here introduce nonparametric D(ω)-distributions for model-free analysis of b(ω)-encoded data (38,39) and demonstrate the method on ex vivo rat brain.
Methods
A healthy adult rat was intracardially perfused with 0.9% saline followed by 4% paraformaldehyde. After extraction, the brain was sagittally sectioned and placed in a solution of phosphate buffer saline (PBS) 0.1 M and gadoteric acid (Dotarem 279.3 mg/ml; Guerbet) 24 h before scanning. During the scan, the brain was immersed in perfluoropolyether (Galden; TMC Industries).MRI was performed on a Bruker Avance-III HD 500 MHz spectrometer equipped with an 11.7 T magnet and a MIC-5 gradient probe giving 3 T/m gradients on-axis. Images were acquired at 90 µm3 isometric resolution and 111×111×10 matrix size using Bruker’s multi-slice multi-echo (MSME) sequence customized for diffusion encoding with general gradient waveforms according to the scheme in Fig 1. After image reconstruction in Bruker’s ParaVision, the data was exported to MRtrix (40) for denoising and Matlab (41) for analysis.
The previous D-distributions (13,14,18,42) are here modified to include a simple Lorentzian ω-dependence of the tensor eigenvalues, corresponding to exponential velocity autocorrelation function with decay rate Γ(43). By constraining the tensor shapes to axisymmetric ones (44), each component of the distribution is described by its weight w, the ω-dependent axial and radial eigenvalues, DA(ω) and DR(ω), and the polar and azimuthal angles, θ and Φ. Within the Lorentzian approximation, the ω-dependence is given by DA/R(ω) = D∞ – [D∞ – D0,A/R]/(1 + ω2/ ΓA/R2), where D∞ = DA/R(ω → ∞) and D0,A/R = DA/R(ω= 0). The b(ω)-encoded signal for a single component is proportional to wexp(–β) where β is obtained by numerical integration of the generalized scalar product b(ω):D(ω) over ω (39).
Following our earlier works (42,45-47), Monte Carlo inversion (48) is used to estimate ensembles of distributions consistent with the measured data. For generating parameters maps, the distributions in the primary analysis space [D0,A,D0,R,θ,Φ,D∞,ΓA,ΓR] are evaluated for selected values of ω, giving [DA(ω),DR(ω),θ,Φ], and transformed to the dimensions of isotropic diffusivity Diso(ω) = (DA(ω) + 2DR(ω))/3 and squared normalized anisotropy DΔ(ω)2 = (DA(ω) – DR(ω))2/(DA(ω) + 2DR(ω))2 (49), as well as the lab-frame diagonal values Dxx(ω), Dyy(ω), and Dzz(ω). Parameter maps are then calculated as means E[x], variances Var[x], and covariances Cov[x,y] over relevant dimensions and sub-divisions (“bins”) of the distribution space (15).
Results & Discussion
Figure 2 shows D(ω)-distributions for selected voxels in ex vivo rat brain. The results for pure white matter (WM), gray matter (GM), and PBS voxels (WM: low Diso and high DΔ2, GM: low Diso and low DΔ2, and PBS: high Diso and low DΔ2) are used to define three bins in the Diso-DΔ2 plane and generate maps of nominally tissue type-specific per-bin signal fractions and diffusion metrics (see Figure 3) consistent with earlier in vivo results (42). The ω-dependence of the metrics is reported in Figure 4, reproducing earlier observations obtained with oscillating gradients (50) – particularly striking for the pyramidal cell layer in hippocampus. In comparison to state-of-the-art oscillating gradient protocols (5), the herein investigated range of ω is however rather limited, which could be remedied by inclusion of additional waveforms optimized to move the spectral content of b(ω) to the lowest and highest values of ω within the constraints of gradient strength and echo time.Conclusions & Outlook
The nonparametric D(ω)-distributions are here introduced as a general framework for analysis of b(ω)-encoded diffusion MRI data including aspects of both oscillating gradients and tensor-valued encoding. The per-voxel distributions for ex vivo rat brain give parameter maps consistent with literature data. In light of recent impressive results with either oscillating gradients or b-tensor encoding in clinically relevant cases, such as breast tumors (24,25), ischemic stroke (21), multiple sclerosis (33), epilepsy (34), and brain tumors (35), using essentially identical pulse sequences with only minor differences in diffusion gradient waveforms, we foresee that the two approaches will eventually be merged and, for instance, analyzed with our novel nonparametric distributions.Acknowledgements
This work was financially supported the Swedish Foundation for Strategic Research (ITM17-0267) and Swedish Research Council (2018-03697), Academy of Finland (#323385) and Erkko Foundation. DT owns shares in Random Walk Imaging AB (Lund, Sweden, http://www.rwi.se/), holding patents related to the described methods.References
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