Ivan I Maximov^{1}, Sebastian Vellmer^{2}, Rüdiger Stirnberg^{3}, and Tony Stöcker^{3}

The
diffusion MRI represents a signal obtained from the relatively large
voxel size consisting of complex tissue microstructure. Modern
diffusion MRI strategies typically work with one parametric dimension
associated with either *b*-value or
diffusion time. In turn, spatial anisotropy of biological tissue
demands to take into account a high angular resolution. In order to
simplify the interpretation of the diffusion signal, we introduce
isotropic diffusion weightings. Essentially, we sample the diffusion signal by 3D oscillating
gradient method. Novel biomarkers such as surface-to-volume ratio and
mean neurite diameter are presented.

In
order to provide an accurate IDW sequence example we used the
FAMEDcos^{3} sequence with equal diffusion times along all
Cartesian axes. This sequence is based on the application of harmonic
functions with phase jumps in order to satisfy the condition of
mutual gradient orthogonality (see Fig. 1). The chosen diffusion
times were 72, 76, 80, 84, 88, and 92 ms with b-values equal to 0,
200, and 400 ms/mm^{2}. Acquired diffusion data were
corrected for eddy-current and susceptibility-based distortions using
the *eddy* utility from FSL^{6}, averaged over 4
acquisitions, and smoothed with the Gaussian kernel of 1.5mm^{3}
in order to decrease the Gibbs ringing artefacts. The acquired spatial resolution was isotropic 1.8mm^{3}. The short diffusion time
limit is described by Mitra's formula^{1}:

$$D(t) = D_0(1- \frac{4}{3d \sqrt{π}} \frac{S}{V} \sqrt{D_0 t})$$

where
D_{0} is the diffusion coefficient of a free water, *d* = 3 is the spatial dimension. In order
to estimate diffusion coefficients for each diffusion time we used a
two step algorithm: linear robust fit and constrained optimisation.
Next, the estimated diffusion coefficients were used for S2V
evaluation by fitting of Eq. (1) with and without D_{0}
variations. Non-varying D_{0} was fixed and equal to 3 µm^{2}/ms.
We performed measurements on a healthy volunteer (29 years old) at a
Siemens MAGNETOM 7T scanner. The volunteer gave written informed
consent prior to participation. The study was approved by the local
ethical committee.

- Mitra et al., Diffusion propagator as a probe of the structure of porous media. Phys Rev Lett 68 (1992) 3555.
- Novikov et al., Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation. arXiv: 1612.02059v1 (2016).
- Vellmer et al., Comparative analysis of isotropic diffusion weighted imaging sequences. JMR 275 (2017) 137.
- Westin et al., Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. Neuroimage 135 (2016) 345.
- Maximov et al., Fast isotropically weighted intravoxel incoherent motion brain imaging at 7T. NMR Biomed (submitted).
- Andersson and Sotiropoulos, An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. Neuroimage 125 (2016) 106.
- Sjolund et al., Constrained optimization of gradient waveforms for generalized diffusion encoding. JMR 261 (2015) 157-168.
- Vellmer et al., Anisotropic diffusion phantoms based on microcapillaries. JMR 279 (2017) 1-10.
- Novikov et al., Mapping orientational and microstructural metrics of neuronal integrity with in vivo diffusion MRI. arXiv:1609.09144 (2016).

Scheme
of the used diffusion gradient shapes.** a) **Gradient shapes; **b)
***q*-shapes; **c)** Fourier transformation of *q*-values.

Resulting
surface-to-volume ratio obtained by evaluation of Eq. (1) without **(a**)
and with (**b,c**) D_{0} variations. (**d**) manual segmentation of the corpus
callosum for histogram analysis . The first row of histograms is S2V
values, the second row of histograms is the estimated mean axon
diameters.