Alexis Reymbaut^{1,2}, Paolo Mezzani^{1,3}, João Pedro de Almeida Martins^{1,2}, and Daniel Topgaard^{1,2}

^{1}Physical Chemistry, Lund University, Lund, Sweden, ^{2}Random Walk Imaging AB, Lund, Sweden, ^{3}Physics, Università degli Studi di Milano, Milan, Italy

In first approximation, the diffusion signal writes as the Laplace transform of an intra-voxel diffusion tensor distribution (DTD). Several algorithms have been introduced to estimate the DTD’s statistical descriptors (mean diffusivity, variance of isotropic diffusivities, mean squared diffusion anisotropy, etc.) by inverting data obtained from tensor-valued diffusion encoding schemes. However, the trueness and precision of these estimations have not been systematically assessed and compared across methods. Here, we compare such estimations in silico for a 1D Gamma fit, a generalized two-term cumulant approach, and 2D and 4D Monte-Carlo inversion techniques, using a common and clinically feasible tensor-valued acquisition scheme.

- the mean diffusivity $$$\mathrm{E}[D_\mathrm{iso}]$$$,
- the mean squared diffusion anisotropy $$$\tilde{\mathrm{E}}[D^2_\mathrm{aniso}]=\mathrm{E}[(D_\mathrm{iso}D_\Delta)^2]\,/\,\mathrm{E}[D_\mathrm{iso}]^2$$$,
- the variance of isotropic diffusivities $$$\mathrm{V}[D_\mathrm{iso}]$$$,

- 53 linearly encoded signals ($$$b_\Delta=1$$$) distributed as $$$6,10,16$$$ and $$$21$$$ points at $$$b=100,700,1400$$$ and $$$2000$$$ s/mm
^{2}, respectively. - 32 spherically encoded signals ($$$b_\Delta=0$$$) sampled as $$$6,6,10$$$ and $$$10$$$ points at $$$b=100,700,1400$$$ and $$$2000$$$ s/mm
^{2}, respectively. - one $$$b=0$$$ signal.

We compared the performances of four signal inversion techniques:

- for powder-averaged signals $$$\overline{\mathcal{S}}(b,b_\Delta)$$$, the Gamma fitting (Gamma)
^{7}and the 2D Monte-Carlo inversion (MC-2D).^{8} - for non powder-averaged signals $$$\mathcal{S}(\mathbf{b})=\mathcal{S}(b,b_\Delta,\Theta,\Phi)$$$, the covariance tensor approximation (Cov, a tensorial two-term cumulant expansion)
^{6}and the 4D Monte-Carlo inversion (MC-4D).^{9}

To ensure fair comparison of the different inversion techniques, all readily available as Matlab code on GitHub,

- A system is simulated by generating a set of ground-truth features $$$\{D_\mathrm{iso},D_\Delta,\theta,\phi,w\}$$$, from which the ground-truth descriptors and the ground-truth signals are computed using the general 4D kernel.
- Each signal inversion technique is run on an identical set of signals with added Rician noise at either signal-to-noise ratio (SNR) of 30 or at infinite SNR.
- Step 2 is repeated 100 times to build up statistics (medians and interquartile ranges) on estimation of the statistical descriptors.

We found that these methods all share similar precision (except for a lower precision of MC-2D) but differ in term of trueness. Gamma exhibits infinite-SNR biases when the signal strongly deviates from mono-exponentiality and is unaffected by orientational order. As for Cov, it shows infinite-SNR biases when this deviation originates either from the variance in isotropic diffusivities or from the low orientational order of anisotropic diffusion components, because signals from such distributions contain features that are not captured by a two-term cumulant expansion. MC-2D shows remarkable trueness in all studied systems. However, its trueness in systems that are anisotropic at the voxel scale drastically worsens if the acquisition scheme does not possess enough directions to yield a good rotational invariance of the powder-averaged signal. Finally, MC-4D presents no infinite-SNR bias but significantly suffers from noise in the data, while preserving contrast in finite-anisotropy systems.

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