Single-shell diffusion MRI data with b=1000: which non-tensor models are feasible?
Andrew D Davis1, Geoffrey B Hall2, Benicio N Frey2, Stephen C Strother3, Glenda M MacQueen4, Stefanie Hassel4, Jacqueline K Harris4, Mojdeh Zamyadi5, Stephen R Arnott5, Jonathan Downar6, and Sidney H Kennedy3

1Psychology, Neuroscience, and Behaviour, McMaster University, Hamilton, ON, Canada, 2McMaster University, Hamilton, ON, Canada, 3University of Toronto, Toronto, ON, Canada, 4University of Calgary, Calgary, ON, Canada, 5Rotman Research Institute, Toronto, ON, Canada, 6Krembil Research Institute, Toronto, ON, Canada


Single-shell diffusion data with b=1000 is commonly acquired in clinically focused studies, and fit with a tensor to calculate microstructure parameters, despite that model's limitations. This study fit several multi-compartment models to such data. The most promising models were: (i) two-fiber ball-stick with a single common diffusivity parameter, which provided the most stable results, and (ii) two-fiber ball-zeppelin, which provided low-noise and stable parameter maps and dispersion values with burn-in settings above 20,000 jumps. The default model in bedpostx yielded lower quality results in this data, even with extremely long processing times.


Diffusion MRI (DMRI) acquisitions increasingly incorporate high b-values, multiple shells, and high angular resolution.1 However, acquisitions using b=1000 s/mm2 and 30 directions or less (hereafter, sparse data) are common. As they are often part of information-rich data sets, it is imperative to gain the most information possible from them. The limitations of tensor-derived scalar metrics are well known (e.g. crossing fibers).2,3 Prior work has incorporated CSF into a tensor model,4,5 and extracted microstructural parameters from CSF-free regions.6 However, the application of compartmental models7 to sparse data has not been widely adopted. The exception is tractography, where ball-stick models are fit with Markov chain Monte Carlo (MCMC), but scalar outputs are not reported, nor are burn-in and other parameter settings. Default values are presumably used, which may be unsuitable.

This study addresses the following questions: (i) Which non-tensor models can be fit to sparse DMRI data? (ii) Which MCMC model settings are appropriate for such fits?


DMRI data was acquired on a 3T scanner (Discovery MR750, 8-channel coil; GE Healthcare, USA) as part of a larger IRB-approved study.8 Single-shot SE-EPI images were obtained over 31 directions (b=1000s/mm^2), with 6 b=0 scans (isotropic 2.5mm voxels, TR=8s, TE=94ms, ASSET R=2). Data was preprocessed for motion and eddy currents, and a tensor was fit using weighted-least-squares.9 One typical control subject was selected (SNR=27 in b=0 images), and analysis was performed on a thin axial slab in subject native image space (at approximately z=+8mm in MNI co-ordinates). A modest Linux-based computer was used (four 2.67GHz cores).

This work attempts to fit single- and two-fiber ball-stick10 (BS/BSS, respectively) and ball-zeppelin7 (BZ/BZZ) models to DMRI data using the Levenberg–Marquardt (LM) algorithm (Camino's modelfit11) and the MCMC method (FSL's bedpostx9). The BS1/BSS1 models use a common diffusivity parameter for both compartments, while BS2/BSS2 use a distribution of diffusivities. The second fiber population was subject to automatic relevance determination.12

In testing with LM, only the BS1 model fit was possible. Other models, including zeppelin-stick, ball-cylinder, and dual-cylinder either failed to run or to converge. All six models were fit using MCMC, with the default burn-in (1000), and a larger range in two-fiber models (0–50,000).


Fitting the BS1 model using the LM and MCMC methods yielded very similar results (Fig. 1), though the fitting failed in a minority voxels with LM. Consequently, only the MCMC fitting of the BS1 model was considered further.

The six models fit using MCMC yielded plausible tissue-fraction maps in this data (Fig. 2). However, the noisy diffusivity maps from the distributed-d models motivated further trials with higher burn-in (Fig. 3). Increased burn-in eliminated the visible noise in the BZZ model, however the BSS2 map retained noise even at very high burn-in.

Model comparison plots (Fig. 4) show that two-fiber models (especially BZZ) identify higher tissue fraction, though with a corresponding increase in variance. The diffusivity values differ considerably among the models, with the BS1 and BSS1 models having lower variance. The high variance of BZZ improves at long burn-in (Fig. 5), but that of BSS2 does not.


The nature of the sparse data used here limited the choice of compartment models (e.g. two compartments, two fiber populations). However, several models were fit to this data using MCMC, and a single-fiber ball-stick model was fit with LM.

All six models included an isotropic component, offering an advantage over the tensor in areas with CSF or GM partial-volume effects. The single-fiber models and tensor were similarly limited in crossing fiber areas, reporting artificially low tissue-fraction. The maps from two-fiber models were improved in this regard, and BZZ reported the highest WM tissue-fractions, probably due to the incorporation of dispersion in the model. The diffusivity values were more variable among models, and the more complex models required long burn-in times. A key observation from the plots is that the BSS2 and BZZ diffusion SD values tended toward zero with increased burn-in, meaning the BSS2 model approached BSS1. Therefore, BSS2 is not a good model choice for this data.

Recommended model settings based on these results would be the BSS1 model with burn-in of 10,000 jumps, or the BZZ model with 20,000 (17.5h or 42h for whole brain, respectively). The next step in this research is testing data from different scanners, then assessing model outputs for their predictive power as biomarkers.13


The bedpostx default model and parameter settings were not well suited to this single-shell, b=1000 DMRI data, producing noisy diffusivity maps with high variance. However, with appropriate parameter choices, two-fiber ball-stick and ball-zeppelin models were successfully used. The fits produced physiologically plausible parameter maps that bear further study for their value in differentiating subject populations.


The authors wish to acknowledge the contributions of the imaging technologists and study co-ordinators associated with the CAN-BIND project.


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Figure 1. Comparison of fitting BS1, (ball-stick model with a single value for diffusivity), using the Levenberg–Marquardt (LM) algorithm (in Camino) or the Markov chain Monte Carlo (MCMC) method (in bedpostx, with default burn-in=1000). The voxel-wise mean-difference plot for white matter voxels indicates good agreement between the two methods, showing roughly 99% agreement of voxels within the confidence intervals (z-score=1.96) for both tissue fraction and diffusivity, and mean differences close to 0. The distinctive pattern above the agreement interval represents voxels in which MCMC returned a low, but non-zero, tissue-fraction, but LM returned 0.

Figure 2. Images from tensor and MCMC models with default burn-in setting. Top row: FA and tissue fraction maps (intensity window: 0–1). In general, the two-fiber models yield higher tissue fractions, especially in the crossing-fiber region indicated by the arrow. Bottom row: mean diffusivity maps (intensity window: 0–3.5um^2/ms). The contrast of the ball-stick diffusivity maps is different than tensor MD, especially in WM where the values are higher, while the ball-zeppelin maps are similar to MD. The distributed-model diffusivity maps are very noisy, motivating further study at higher burn-in values.

Figure 3. Comparison of diffusivity maps for two-fiber MCMC models with increasing burn-in (BI) values. Top row: BSS1 model; diffusivity maps have little obvious noise and appear stable with increasing burn-in. Middle row: BSS2 model. Bottom row: BZZ model. In these distributed-diffusivity models, the speckled noise artifact generally decreases with increasing burn-in. However, even with very long burn-in, the diffusivity map for BSS2 remains noisy.

Figure 4. Comparison of tensor and MCMC models with default burn-in of 1000 jumps. (A) Processing times; 0.2 s/voxel implies ~ 5h for a whole brain, and times scale linearly with burn-in. (B) WM tissue-fraction; two-fiber models assign greater tissue fractions than single-fiber. (C) Dispersion (a measure of uncertainty) about first fiber in WM; the two-fiber models are higher, as the second fiber may be absorbing some dispersion. (D) Mean diffusivity over the whole slab, with BSS2 having the highest values and variability. Note: there is no value for dispersion in the tensor, but the time value was 0.00005 s/voxel.

Figure 5. Comparison of 2-fiber models with increased MCMC burn-in values. (A) Tissue-fraction in WM; all curves decrease with burn-in, as more weight is assigned to the isotropic compartment. (B) Mean diffusivity, with very low variability in BSS1; BZZ reaches a steady state after 10,000 jumps, but BSS2 retains high standard deviation and a decreasing trend even after 50,000 jumps. (C) Dispersion about fiber 1 in WM; all models show a rapid initial decrease. (D) Standard deviation of diffusivity estimated within the model; Both models tend towards 0 with increasing burn-in. Note this parameter is always 0 for BSS1.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)