Chantal M.W. Tax^{1}, João P. de Almeida Martins^{2,3}, Filip Szczepankiewicz^{3,4}, Carl-Fredrik Westin^{5}, Maxime Chamberland^{1}, Daniel Topgaard^{2}, and Derek K Jones^{1}

Studying the diffusion MRI signal as a function of more experimental parameters allows to establish correlations between different chemical and physical properties and to disentangle different compartments. Such measurements are common in the field of physical chemistry to characterise heterogeneous media, but are rendered impractical on human scanners due to hardware limitations. Here, we leverage ultra-strong gradients to acquire a 5-dimensional in-vivo human brain correlation dataset, which allows the characterisation of microstructural features through unconstrained inversion.

The concept of studying the signal attenuation as a function of more
than one experimental variable has been used extensively in NMR to characterise
heterogeneous materials. Motivated by the possibility to distinguish between environments
with different physical and chemical properties and to establish correlations
between these properties, diffusion-T2-relaxometry varies both b-value and echo
time (TE) in repeated spin-echo experiments^{1-4}. Subsequently, inversion techniques can
transform these 2-dimensional signals into a joint diffusivity-T2 distribution^{5}.

In the case of anisotropic diffusion, however, a single
diffusion-dimension is insufficient to describe the process in each
microenvironment. By adding a shape dimension, the medium can now be described
by a distribution of diffusion tensors (DTD) with varying shape $$$D_{\Delta}$$$ and
size $$$D_{iso}$$$ (i.e. average diffusivity)^{6}. Previous work has obtained
the full shape-size-DTD in liquid crystals from unconstrained inversion based
on data averaged over directions^{7}. Fundamentally, this inversion
relies on the availability of data with not only different diffusion encoding directions and b-values, but also different encoding
“shapes” $$$b_{\Delta}$$$, as characterised by the b-tensor $$$\mathbf{B}$$$^{8,9}. This approach was only recently further extended to obtain
the full shape-size-orientation-DTD – adding another two dimensions $$$\theta$$$ and
$$$\phi$$$ – in orientationally ordered liquid crystals^{10}. The
5-dimensional T2-DTD $$$P$$$ provides a comprehensive description of the chemical
composition, density, size, shape, and orientation of a heterogeneous medium:

$$S(TE,b,b_{\Delta},\Theta,\Phi)=S(0)\int^\inf_0\int^\inf_0\int^1_{-1/2}\int^\pi_0\int^{2\pi}_0K(TE,b,b_{\Delta},\Theta,\Phi,T2,D_{iso},D_{\Delta},\theta,\phi)\times P(T2,D_{iso},D_{\Delta},\theta,\phi)d\phi d\theta dD_{\Delta}dD_{iso}dT2.$$

Critically, such measurements have been made on small bore systems with field gradient strengths that outstrip those typically available for human brain imaging. In this work, we translate the multi-dimensional approach to the human brain, leveraging the ultra-strong gradients of a Connectom scanner, and obtain the full T2-DTD without imposing priors on the number of compartments and without fixing parameters. We perform tractography on the DTD, allowing for the visualisation of along-tract properties of the tensor that was used for tract-propagation.

Data: A healthy volunteer was scanned on a 3T 300mT/m-gradient Siemens Connectom
system with a prototype spin-echo
sequence that enables diffusion encoding with arbitrary gradient waveforms^{11}.
Images with different b-tensors and
TE were acquired, yielding a 5-dimensional parameter space (Fig.1). The waveforms were numerically optimized^{12}, compensated for Maxwell terms^{13} and matched
to have similar diffusion times^{14}. Remaining settings were: no
in-plane acceleration, voxel size = 3x3x3 mm3, matrix=70x70, 15 slices, TR=2800ms,
partial-Fourier=6/8,
bandwidth=2100 Hz/pix

Processing: The data were corrected for misalignment due to subject
motion and eddy currents^{15,16}. The T2-DTD was estimated using a
non-negative least squares algorithm that doesn’t assume a fixed number of
compartments^{7}. Briefly, the inversion
is performed via a directed iterative approach wherein randomly generated $$$[T2,\mathbf{D}]$$$ sets are successively fitted to the
measured signal in order to find the ten most probable $$$[T2,\mathbf{D}]$$$ solutions. While this approach allows us to obtain a
T2-DTD distribution that agrees well with the acquired data, such solutions are
by no means unique. Tractography on DTD components with an FA>0.2 was
performed using the FiberNavigator^{17}.

The 5-dimensional acquisition protocol presented here results in the most comprehensive in-vivo human brain correlation dataset to-date. The richness of the data allows the characterisation of microstructural features through an unconstrained inversion. Because of the few assumptions, the approach could be invaluable in explorative neuroscience studies in health and disease where it is unclear what type of tissue constituents are present, and could be used together with biophysical or statistical modelling to find more suitable sets of constraints.

We have performed fibre tractography on the full DTD which resulted in bundles consistent with anatomy. The combination of b-tensor shape and T2 makes this protocol highly useful for free water mapping and the investigation of tract- and compartment-specific diffusion properties and T2. The method is flexible and can be scaled down to lower dimensional correlation protocols.

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[2] Bernin D, Topgaard D. NMR diffusion and relaxation correlation methods: New insights in heterogeneous materials. Curr Opin Colloid Interface Sci 2013;18:166-172.

[3] Song Y-Q, Venkataramanan L, Kausik R, Heaton N. Two-dimensional NMR of diffusion and relaxation. In: Valiullin R, editor. Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials. Cambridge, UK: Royal Society of Chemistry; 2016. p 111-155.

[4] van Dusschoten D, de Jager PA, Van As H. Extracting diffusion constants from echo-time-dependent PFG NMR data using relaxation-time information. J Magn Reson A 1995;116:22-28.

[5] Hürlimann MD, Venkataramanan L, Flaum C. The diffusion–spin relaxation time distribution function as an experimental probe to characterize fluid mixtures in porous media. J Chem Phys 2002;117:10223-10232.

[6] Topgaard D. Multidimensional diffusion MRI. J Magn Reson 2017;275:98-113.

[7] de Almeida Martins JP, Topgaard D. Two-dimensional correlation of isotropic and directional diffusion using NMR. Phys Rev Lett 2016;116:087601.

[8] Eriksson S, Lasič S, Nilsson M, Westin C-F, Topgaard D. NMR diffusion encoding with axial symmetry and variable anisotropy: Distinguishing between prolate and oblate microscopic diffusion tensors with unknown orientation distribution. J Chem Phys 2015;142:104201.

[9] Westin, C. F., Knutsson, H., Pasternak, O., Szczepankiewicz, F., Özarslan, E., Van Westen, D., Mattisson, C., Bogren, M., O'donnell, L. J., Kubicki, M., Topgaard, D. & Nilsson, M. 2016. Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. Neuroimage, 135, 345-62.

[10] Topgaard D. Diffusion tensor distribution imaging. Proc Intl Soc Mag Reson Med 2017;25:0279.

[11] Szczepankiewicz, F., Sjölund, J., Ståhlberg, F., Lätt, J. & Nilsson, M. 2016. Whole-brain diffusional variance decomposition (DIVIDE): Demonstration of technical feasibility at clinical MRI systems. arXiv:1612.06741.

[12] Sjölund J, Szczepankiewicz F, Nilsson M, Topgaard D, Westin C-F, Knutsson H. Constrained optimization of gradient waveforms for generalized diffusion encoding. J Magn Reson 2015;261:157-168.

[13] Szczepankiewicz, F. and Nilsson, M. 2018. Maxwell-compensated waveform design for asymmetric diffusion encoding. Submitted to Int. Soc. Magn. Reson. Med. Paris, France.

[14] Lundell, H., Nilsson, M., Dyrby, T. B., Parker, G. J., Hubbard, P., Zhou, F., Topgaard, D. & Lasic, S. 2017. Microscopic anisotropy with spectrally modulated q-space trajectory encoding. Int. Soc. Magn. Reson. Med. Honolulu, Hawaii.

[15] Nilsson, M., Szczepankiewicz, F., van Westen, D., & Hansson, O. 2015. Extrapolation-based references improve motion and eddy-current correction of high b-value DWI data: application in Parkinson’s disease dementia. PloS one, 10(11), e0141825.

[16] Klein et al., 2010. IEEE TMI 29(1),196-205

[17] Chamberland, M. et al. "Real-time multi-peak tractography for instantaneous connectivity display." Frontiers in neuroinformatics 8 (2014).

Fig.1: 5-dimensional parameter space: a) Images with
varying b-value (or b-tensor size; 1 dimension) and direction (or b-tensor
first eigenvector; 2 dimensions). The images shown were obtained with a “linear
b-tensor” shape, as used in conventional diffusion MRI acquisitions. 1 b=0
image and 78 directions across 4 shells with b-values [100,1450,2750,4000]s/mm^{2}
were acquired per set. b) Images with varying b-tensor shape $$$b_{\Delta}=(b_{\parallel}-b_{\bot})/(b_{\parallel}+2b_{\bot})$$$ (1 dimension; linear $$$b_{\Delta}$$$=1, cigar $$$b_{\Delta}$$$=0.5,
spherical $$$b_{\Delta}$$$=0, and planar $$$b_{\Delta}$$$=-0.5) and TE (1 dimension;
TE=[77,100,125,150]ms for the linear-,cigar-, and spherical encoding, and an
additional acquisition at TE=50ms for linear encoding). For the spherical
encoding, the b-values acquired were [100,700,1300,1800,2400,3000]s/mm^{2}.

Fig.2:
a) Spatially resolved DTDs overlaid
on a 2D parametric map of the mean transverse relaxation rate $$$R2=1/T2$$$. The DTDs are visualized as scatter plots with x and y axes representing
the size and shape dimensions, respectively. The circle area and colour give the
probability and orientation of each component. CSF partial volume region
(middle top), and crossing fibre region (middle bottom) are enlarged. b) each tensor of
the DTD plotted as a glyph coloured according to orientation. Zooms are shown of the crossing of the corpus callosum (CC) and Cingulum
(CG), and of partial volume with CSF around the fornix.

Fig.3: Tractography on the
full DTD (resolution: 3 x 3 x 3 mm³)

Fig.4: Conventional colouring
of tracts according to the underlying scalar image vs colouring according to
properties of the tensor that was used for tract propagation. a) voxel-wise FA
vs tensor-specific FA, where the latter is consistently higher for each
individual tract in the WM. The bottom image shows a reconstruction of the
fornix, where the voxel-wise FA is low due to partial volumeing with CSF. b) voxel-wise
T2 vs tensor-specific T2.