Constrained spherical deconvolution (CSD) is a robust approach to resolve the fibre orientation distribution (FOD) from diffusion MRI (dMRI) data^[1]. The FOD from single-shell single-tissue (SSST)-CSD only models white matter (WM); it will be distorted/inappropriate when other tissue types are (partially) present; i.e., grey matter (GM) and cerebrospinal fluid (CSF). Multi-shell multi-tissue (MSMT)-CSD was proposed to solve this issue^[2], but requires multi-shell data. We aim to achieve similar results/benefits, by using only single-shell (+b=0) data.

Single subject dMRI data were acquired on a Siemens 3T scanner, with voxel size 2.5×2.5×2.5mm³, and a multi-shell scheme (b=0,1000,2000,3000s/mm² respectively for 5,17,31,50 directions + additional b=0 volume with reversed-phase encoding). The data were corrected for susceptibility-induced distortions^[3], eddy-current-induced distortions and motion^[4], and bias-fields^[5].

We use these terms to refer to subsets of the data:

MS-data (multi-shell data): all images over all 3 dMRI shells + b=0 data.

SS-data (single-shell data): the 50 directions at b=3000s/mm².

B0-data: b=0 images.

SS+B0-data: combination of the latter 2. (often informally called "single-shell" data)

Conservative regions or individual voxels deemed to contain "pure" samples of single-fibre-WM,GM,CSF were selected to estimate the tissue response functions (guided by FA and ADC maps). MSMT-CSD results are shown in Fig.1, first column. The WM-tissue outcome is presented using FOD-based directionally-encoded colour (DEC), weighted by the WM-FOD integral^[6]. SSST-CSD results are shown in the second column: WM is overestimated, because GM/CSF parts are not estimated. Both results match the findings of [2].

First naive approach: applying MSMT-CSD directly to SS+B0-data. Even under non-negativity constraints, given isotropy of GM and only two b-values, GM can be (and is) fitted by a WM+CSF mixture (Fig.1, third column). The WM is still grossly overestimated; most fundamental problems of SSST-CSD results remain.

Second naive approach: applying MSMT-CSD, but using only WM+GM. This yields a more aggressive "cleanup" of WM (Fig.1, fourth column). CSF gets fitted as "hyper-GM" (far beyond the 0-to-1 range): the only/best means to fit its high B0-data. But this also (partially) happens in WM+CSF mixtures, resulting in an overly aggressive cleanup; e.g., enlarged ventricles, eroded nearby WM... even at the cost of GM not being able to represent the non-B0 WM anisotropy!

The naive approaches' results provide important insights. GM sits "in the range between WM and CSF". Fitting only WM+GM provides an underestimate of WM. A similar property holds for fitting only CSF+GM: this yields an underestimate of CSF.

This inspired us to design a specialised optimiser to tease out the WM-GM-CSF parts from SS+B0-data. Without going into details, the overall strategy is:

1.Initialise WM to 0.

2.Fit only CSF+GM, given WM as prior constraint. This yields an underestimate of CSF.

3.Fit only WM+GM, given CSF as prior constraint. Since CSF is an underestimate, the resulting WM will be as well.

This marks the end of an iteration, yielding underestimates of CSF/WM, and consequently an overestimate of GM. The next iteration is initialised with the current (under)estimate of WM.

In the theory of MSMT-CSD^[2], the B0-data are regarded like any other b-value/shell; so "formally", SS+B0-data is a case of 2-shell data (even though often informally called "single-shell" data). Retaining consistency, we refer to our specialised strategy as a 2-shell 3-tissue CSD approach; 2S3T-CSD for short.

We performed 2S3T-CSD on the SS+B0-data for 4 iterations (this took 13 minutes for the whole volume, on a standard desktop computer). The final result is shown in Fig.1, fifth column. Note how closely the outcome resembles the MSMT-CSD (on MS-data) result. Fig.2 shows the WM-GM-CSF estimates after each iteration. Note how, even after iteration #1, the WM estimate is already informed by the initial (under)estimate of CSF; e.g., the fornix starts to reappear. Over iterations, the WM/CSF are recovered. This is most apparent at, e.g., the ventricle borders, where excess GM is swiftly eliminated; but also happens in other regions. Figs.3-4 present tractography results, to further support the benefits of 2S3T-CSD for SS+B0-data. Fig.5 shows further 2S3T-CSD results, offering all typical outputs previously only offered by MSMT-CSD^[2].

Informed CSD^[7] attempts this as well, but requires acquisition of a high-resolution T1-image, subvoxel-accuracy registration and intricate spatial segmentation. 2S3T-CSD leverages the full potential of dMRI.

While it was initially believed that multiple tissue types could not be distinguished using single-shell data^[2], we hereby provide the first proof of the contrary. Leveraging (relative) properties of the 3 common brain tissue types (WM,GM,CSF), we obtain close to the same results/benefits as MSMT-CSD, yet from single-shell (+b=0) data and without any external spatial/anatomical priors, using a novel iterative method: 2-shell 3-tissue CSD (2S3T-CSD).