The Dot…wherefore art thou? Search for the isotropic restricted diffusion compartment in the brain with spherical tensor encoding and strong gradients
Chantal M.W. Tax1, Filip Szczepankiewicz 2,3, Markus Nilsson2, and Derek K Jones1

1CUBRIC, School of Psychology, Cardiff University, Cardiff, United Kingdom, 2Clinical sciences, Lund, Lund University, Lund, Sweden, 3Random Walk Imaging AB, Lund, Sweden


The accuracy of biophysical models requires that all relevant tissue compartments are modelled. The so-called “dot compartment” is a conjectured compartment that represents small cells with apparent diffusivity approaching zero. We establish an upper limit of the “dot-fraction” across the whole brain in vivo, by using ultra-high gradients and optimized gradient waveforms for spherical tensor encoding. We report a notable signal above the noise floor in the cerebellar gray matter even for an extremely high b-value of 15000 s/mm2. For cerebral tissue, the dot-fraction seems negligible, and we consider how exchange may have affected this result.


Biophysical modelling of the diffusion MRI (dMRI) signal can be used for tissue microstructure characterisation by carefully selecting model compartments with a relevant impact on the signal1. The inclusion of a “dot-compartment” is motivated by the ubiquity of small cells, wherein water may be trapped and its diffusion highly restricted2. Previous work investigating the minimum model requirements for brain white matter (WM) was based on “linear tensor encoding” (e.g., Stejskal-Tanner) and showed that including a dot-compartment better explained the dMRI signal3,4. However, its inclusion is not generally adopted in vivo5,6. Probing the dot-compartment in anisotropic tissue is challenging with linear encoding, due to the strong relation between encoding direction and orientation distribution of anisotropic tissue microenvironments. Here, we instead use “spherical tensor encoding” (STE) to render signals insensitive to orientation and anisotropy7,8. By so doing, the signal becomes the Laplace transform of the distribution of isotropic diffusivities8,9, where a signal plateau indicates a dot-compartment. Previous STE-based results suggest a negligible dot-fraction in WM10, but the use of gradient amplitudes of below 80mT/m limited the maximal b-value and SNR needed for accurate assessment of small dot fractions. In this work, we: 1) leverage asymmetric STE waveforms and ultra-strong gradients to significantly reduce the TE and increase SNR; 2) study the signal decay for b-values up to 15000 s/mm2; 3) extend the search to both the cerebellum and cerebrum. This facilitates a more accurate estimation of the dot signal fraction across the whole brain, in vivo.


The signal arising from each compartment represented by diffusion tensor $$$\mathbf{D}$$$ probed by b-tensor $$$\mathbf{B}$$$ can be described by $$$f\cdot\exp(\mathbf{B:D})$$$, which in the specific case of STE and two non-exchanging compartments (one dot-compartment) simplifies to


For infinite SNR, the $$$S(0)$$$-normalised signal at the plateau equals the dot signal fraction $$$(f_{\text{dot}}=S_{\text{plateau}}/S(0))$$$. Fig.1 shows the simulated signal in the case of two non-exchanging compartments with isotropic diffusivities of 1 and 0 µm2/ms. In the absence of a plateau, the normalised signal at $$$b_{\text{max}}$$$ can serve as an upper limit of $$$f_{\text{dot}}$$$, because $$$f_{\text{dot}}≤S_{b_{\text{max}}}/S(0)$$$. The accuracy is limited by the presence of the noise floor but still yields an upper-limit of $$$f_{\text{dot}}$$$, at least in the absence of exchange.


Data: A healthy volunteer was scanned on a 3T, 300mT/m Siemens Connectom using a prototype spin-echo sequence that enables arbitrary b-tensor encoding. We used b-values=[0,250,1500,3000,...,15000]s/mm2 repeated [1,6,9,12,...,36] times, respectively. No in-plane acceleration was used, voxel size=4.4×4.4×6 mm3, matrix=64×64, 26 slices, TR/TE=5000/90ms, partial-Fourier=6/8, bandwidth=1594Hz/pix. Maxwell-compensated waveforms11 were optimized numerically12, and yielded a diffusion time of approximately 20ms. This approach renders superior encoding efficiency compared to standard 1-scan-trace imaging (requires TE=270ms for b=15000s/mm2). An MPRAGE was acquired for brain segmentation.

Processing: To investigate whether a potential plateau arising in the signal decay was not solely an effect of the noise floor, the data was corrected for Rician bias13-15. Masks for different tissue types were created in Freesurfer based on the MPRAGE registered to the susceptibility-distortion corrected16 dMRI data, and used to guide the definition four regions for further analysis: cerebral white and gray matter (WM and GM), and cerebellar white and gray matter (cWM and cGM).


Fig.2 shows that the signal in most of the cerebral tissue was reduced to the noise floor at $$$b$$$>9000 s/mm2. However, cGM was observed to have a remarkably high signal at high b-values, remaining well above the noise floor even at b=15000 s/mm2, unlike any other tissue.

Fig.3 shows that the signal-verus-b curve in cGM was markedly non-monoexponential, with signal values above the noise floor for all sampled b-values. However, no plateau is found at b=15000s/mm2. In the absence of exchange, the estimated upper-bound of $$$f_{\text{dot}}$$$ in WM, GM, cWM and cGM is approximately 0.5%,0.5%,1% and 2%, respectively.

Discussion & Conclusion

In the cerebrum, we did not find evidence of a large dot-fraction. In the cerebellum, however, the non-negligible signal at very high b-values points to the existence of a dot-like compartment. The absence of a signal plateau could be caused by compartments with low but non-zero isotropic diffusivity, or a true dot-compartment with zero diffusivity but non-negligible exchange. The effect of exchange17,18 is simulated in Fig.4, and can result in an underestimation of $$$f_{\text{dot}}$$$.

Regardless of model assumptions, the high signal retention at high b-values in the cerebellum demonstrates that its microstructure is remarkably different compared to the cerebrum. We speculate that this may originate from granule and/or Purkinje cells, that have a morphology consistent with this finding (Fig.5). If so, STE at extremely high b-values can become a very specific biomarker for better understanding of diseases such as autism spectrum disorders19, spinocerebellar ataxia20, and Alzheimer disease21,22, where such cells are affected.


We thank Siemens Healthcare for access to the pulse sequence programming environment, and Fabrizio Fasano from Siemens Healthcare for support. We thank Umesh Rudrapatna for technical support and feedback and Samuel St-Jean for useful discussions. The work was supported by a Wellcome Trust Investigator Award (096646/Z/11/Z) and a Wellcome Trust Strategic Award (104943/Z/14/Z). The data were acquired at the UK National Facility for In Vivo MR Imaging of Human Tissue Microstructure funded by the EPSRC (grant EP/M029778/1), and The Wolfson Foundation. CMWT is supported by a Rubicon grant (680-50-1527) from the Netherlands Organisation for Scientific Research (NWO) and Wellcome Trust grant (096646/Z/11/Z).


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Fig. 1: Simulations of two non-exchanging compartments with diffusivities of $$$D_{\text{iso}}$$$ = 1 and $$$D_{\text{dot}}$$$ = 0 µm2/ms for different SNR, and different signal fractions $$$f_{\text{dot}}$$$ (represented by the different colours). At low SNR, the rectified noise floor inflates the estimated upper limit of $$$f_{\text{dot}}$$$. As SNR increases, smaller signal fractions can be resolved, where the relative signal approaches $$$f_{\text{dot}}$$$.

Fig. 2: Magnitude DWIs at central (top row) and inferior (bottom row) slice-positions at variable b values. Coloured outlines represent tissue types (red = white matter, yellow = gray matter, green = cerebellar white matter, blue = cerebellar gray matter), and were generated from the intersection of the mask-isosurface of each type with the slice.

Fig. 3: shows the sample median and 1-99th percentile of the S(0) normalised measured signals (a) and Rician-bias corrected signals (b) across voxels in each ROI (location visualised in figure a, right). The horizontal line in (a) represents the estimated mean noise floor across voxels. Of the regions investigated, cGM and wGM show the most significant signal above the noise floor, indicating that the tissue is not comprised of a compartments with a single isotropic diffusivity.

Fig. 4: Simulated signal at variable SNR, $$$f_{\text{dot}}$$$ and exchange times assuming a two compartment Kärger-model17,18. Here, diffusivities were set to 1 and 0 µm2/ms. At infinite exchange times, the relative signal approaches the true signal fraction of the dot compartment, as in Fig. 1. However, at exchange times as long as 500 ms there is a relevant loss of signal at high b-values caused by exchanging particles. This suggests that the estimated upper limit of $$$f_{\text{dot}}$$$ is negatively biased in the presence of exchange.

Fig. 5: Coronal DWI that covers the cerebrum and cerebellum at b = 15 000 s/mm2 (left), nissl stained coronal slice (right) showing presence of cells in a rhesus monkey brain (http://www.brain-map.org/, access date: 2017-11-07). The figure showcases the agreement between signal retention and high cell density that is expected in the cerebellum.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)