Els Fieremans^{1}, Jelle Veraart^{1}, Benjamin Ades-Aron^{1}, Filip Szczepankiewicz^{2,3}, Markus Nilsson^{2}, and Dmitry S Novikov^{1}

The diffusion MRI signal, as measured with conventional linear tensor encoding (LTE), has been shown to have not enough features to fully model the white matter microstructure. Here we investigate whether adding spherical encoding (STE) to LTE makes microstructural parameter estimation more robust. On signal simulations and in in vivo MRI data, we demonstrate that the intra-axonal diffusivity and axonal water fraction are estimated with higher precision, thereby enabling a 20 minute whole brain protocol to extract brain microstructural parameters without imposing constraints or priors.

**Theory** The
Standard Model for the diffusion signal
$$S_{{\bf\,g}}(b)=\int_{|\mathbf{n}|=1}\!\!d\mathbf{n}\,\mathcal{P}(\mathbf{n})\,\mathcal{K}(b,{\bf\,g}\cdot\mathbf{n})\,\quad\quad(1)$$
is a convolution on a unit-sphere $$$|\mathbf{n}|=1$$$ between the fiber
orientation distribution function (ODF) $$$\mathcal{P}(\mathbf{n})$$$ and the
2-compartment elementary-fiber kernel
$$\mathcal{K}(b,\xi)=S_0\left[f\,e^{-bD_a\xi^2}+(1-f)\,e^{-bD_e^\perp-b(D_e^\parallel-D_e^\perp)\xi^{2}}\right]\,,\quad\quad(2)$$ parameterized by proton density $$$S_0$$$,
intra-axonal-axial $$$D_a$$$, extra-axonal-axial $$$D_e^\parallel$$$ and
extra-axonal-radial $$$D_e^\perp$$$ diffusivities, and axonal water fraction
$$$f$$$. Convolution (2) factorizes^{4,6-10} in the spherical-harmonic basis, $$S_{lm}(b)=p_{lm}\,K_l(b)\,.\quad\quad(3)$$
We employ the rotational-invariant (RotInv) formalism^{4,6} to factor out the ODF,
by introducing basis-independent rotational-invariants of the signal and ODF
$$$S_l\equiv\sqrt{\sum_{m=-l}^l\,S_{lm}^2}/\mathcal{N}_l$$$, $$$p_l\equiv\sqrt{\sum_{m=-l}^l\,p_{lm}^2}/\mathcal{N}_l$$$,
$$$\mathcal{N}_l=\sqrt{4\pi(2l+1)}$$$. In this way, we solve the system
$$S_l(b,x)=p_l\,K_l(b,x)\,,\quad\,l=0,2\,,\quad\,p_0\equiv1\quad\quad(4)$$ for
6 unknowns: fiber-compartment-parameters $$$x=\{S_0,f,D_a,D_e^\parallel,D_e^\perp\}$$$
and ODF-anisotropy-invariant $$$p_2$$$. Nonlinear fitting (4) is generally
unstable^{3,4,6}, characterized by multiple solutions and shallow
"trenches" near the fit minima. Here we evaluate the advantages of additionally
employing isotropic encoding
$$S_{\mathrm{iso}}(b)=S_0\left[f\,e^{-bD_a}+(1-f)\,e^{-b(D_e^\parallel+2D_e^\perp)}\right]\quad\quad(5)$$
that automatically factors-out the ODF, and senses the parameters $$$x$$$ in a
different combination, as an "orthogonal" measurement to improve
precision of the fit (4) for anisotropic compartments.

**Noise Propagation**
analysis was performed to investigate the accuracy and precision of the
joint RotInv fit (4)-(5), referred to as LTE+STE, versus the RotInv fit (4),
referred to as LTE. For the acquisition protocol described below, 2500 noise
realizations with SNR=100 at $$$b=0$$$, were simulated for one parameter vector
(Fig 1a) as well as for a range of biologically plausible parameter vectors
(Fig1b). Each fit was initiated using one random starting point sampled from:
$$$0≤f≤1$$$, $$$0\leq\,p_2\leq,1$$$, $$$0\leq\,D\leq\,4\mu$$$m^{2}/ms for all
diffusivities.

**Imaging** Seven
healthy volunteers (3 males, age range 26-35) were scanned on a 3T Siemens
Prisma scanner with a 32-channel head coil. A repeated scan within 30 days was
performed on 2 volunteers to evaluate scan-rescan precision. The LTE protocol
(10:53 min) used the product diffusion sequence and included 165 images
acquired using monopolar gradients along 30 directions for b=1000, 2000,
3000, and along 64 directions for b=5000 s/mm^{2}, in addition to 11 b=0-images.
The STE protocol (9:52min) included 70 images acquired using a prototype
spin-echo sequence that enables variable shapes of the b-tensor by numerically
optimized waveforms^{11}, for b-values ranging from b = 0 up to b = 3000 in steps
of 500s/mm^{2}. Other parameters were: acquisition matrix = 70 x 70, image
resolution = 3x3x3 mm^{2}, TE=102ms, TR=3800ms, grappa 2, multiband-2 (LTE), 38
oblique axial slices. Data was denoised^{12} and gibbs^{13}, eddy-current^{14},
Rician-bias^{15}, and B1-bias^{16} corrected prior to model fitting, similar as
described above.

Both simulations (Fig.1) and in vivo results (Fig 2-5) corroborate the finding of increase in precision of $$$D_a$$$ , and to a lesser extent $$$f$$$, when including STE in addition to LTE in the nonlinear model fitting (4)-(5), yet the precision of the extra-axonal diffusivity decreased in the in vivo data. While multiple solutions are still present (Fig.1,4), they are better identifiable as outliers compared using LTE only, which would imply re-doing the fit with different starting point to retrieve the other solution (not included here to provide fair presentation of the pitfalls of fitting).

The
obtained values over all WM regions of all subjects for
$$$D_a=2.2\pm0.66$$$ (LTE+STE), and
$$$2.17\pm0.67$$$ (LTE), are both in good agreement with reported values using other orthogonal
diffusion methods.^{17,18} Along with increased precision of $$$D_a$$$, the scan-rescan improved, as illustrated in Fig.4 for the PLIC. While the simulations kept the total number of measurements the same, the LTE+STE data had 70 additional weightings, which can explain some of the overall improvement. Future work will focus on incorporating a free water compartment (which may affect our corpus callosum results) and extending to other “orthogonal” acquisition methods by varying, e.g., the
echo^{17} or diffusion time.

1. Novikov, D. S., Jespersen, S. N., Kiselev, V. G. & Fieremans, E. Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation. arXiv preprint arXiv:1612.02059 (2016).

2. Stejskal, E. O. & Tanner, J. E. Spin Diffusion Measurements: Spin Echoes in the Presence of a Time‐Dependent Field Gradient. The Journal of Chemical Physics 42, 288-292, (1965).

3. Jelescu, I. O., Veraart, J., Fieremans, E. & Novikov, D. S. Degeneracy in model parameter estimation for multi-compartmental diffusion in neuronal tissue. NMR Biomed. 29, 33-47, (2016).

4. Novikov, D. S., Veraart, J., Jelescu, I. O. & Fieremans, E. Mapping orientational and microstructural metrics of neuronal integrity with in vivo diffusion MRI. arXiv preprint arXiv:1609.09144 (2016).

5. Mitra, P. P. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Physical Review B 51, 15074-15078 (1995).

6. Reisert, M., Kellner, E., Dhital, B., Hennig, J. & Kiselev, V. G. Disentangling micro from mesostructure by diffusion MRI: A Bayesian approach. NeuroImage 147, 964-975, (2017).

7. Tournier, J., Calamante, F., Gadian, D. & Connelly, A. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. Neuroimage 23, 1176-1185,(2004).

8. Anderson, A. W. Measurement of fiber orientation distributions using high angular resolution diffusion imaging. Magn. Reson. Med. 54, 1194-1206, (2005).

9. Jespersen, S. N., Kroenke, C. D., Østergaard, L., Ackerman, J. J. H. & Yablonskiy, D. A. Modeling dendrite density from magnetic resonance diffusion measurements. Neuroimage 34, 1473-1486, (2007).

10. Dell'Acqua, F. et al. A Model-Based Deconvolution Approach to Solve Fiber Crossing in Diffusion-Weighted MR Imaging. IEEE Transactions on Biomedical Engineering 54, 462-472, (2007).

11. Sjölund, J. et al. Constrained optimization of gradient waveforms for generalized diffusion encoding. J. Magn. Reson. 261, 157-168, (2015).

12. Veraart, J. et al. Denoising of diffusion MRI using random matrix theory. Neuroimage, doi:http://dx.doi.org/10.1016/j.neuroimage.2016.08.016.

13. Kellner, E., Dhital, B., Kiselev, V. G. & Reisert, M. Gibbs-ringing artifact removal based on local subvoxel-shifts. Magn. Reson. Med. 76, 1574-1581 (2016).

14. Andersson, J., Jenkinson, M. & Smith, S. in FMRIB Technial Report TR07JA2 (FMRIB, Oxford, 2007).

15. Koay, C. G. & Basser, P. J. Analytically exact correction scheme for signal extraction from noisy magnitude MR signals. J. Magn. Reson. 179, 317-322, (2006).

16. Zhang, Y., Brady, M. & Smith, S. Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Trans. Med. Imaging 20, 45-57, (2001).

17. Veraart, J., Novikov, D. S. & Fieremans, E. TE dependent Diffusion Imaging (TEdDI) distinguishes between compartmental T2 relaxation times. Neuroimage, (2017).

18. Veraart, J., Fieremans, E. & Novikov, D. S. Universal power-law scaling of water diffusion in human brain defines what we see with MRI. arXiv preprint arXiv:1609.09145 (2016).

Noise propagation: Left: scatter plots
between all estimated parameters using either LTE (red) or using both LTE+STE (blue). Cyan+ markers indicate the ground truth values. Spurious
correlations between different parameters decrease particularly for $$$D_a$$$ and $$$f$$$. Multiple solutions still exist, but can be identified more easily as outliers (also visible in
parametric maps, Fig. 3). Right: similar color encoding, but now
for scatter plots between ground truth and estimated values, and corresponding
histograms of the errors. $$$D_a$$$ and $$$f$$$-estimation improves
when including STE, resulting in sharper error distributions, while precision
of the other parameters appears unchanged.

Illustration of the "orthogonality" of LTE vs STE
on in vivo human brain for a voxel in the left anterior corona radiata: Left: the
rotational invariants $$$K_0$$$ and $$$K_2$$$ measured with LTE (circles)
along with two fits based on (4). The two sets of parameter solutions, A and B, listed in
the table, are markedly different; right: the measured STE signal (circles)
along with the corresponding two predicted models based on (5) for A and B.
While both A and B describe the LTE encoding equally well, the STE signal
clearly matches solution A, and not B.

Example
white matter parametric maps based
on LTE and LTE+STE. While the parametric maps of $$$D_e^{||}$$$, $$$D_e^\perp$$$ look similar between two approaches, the $$$D_a$$$, and to lesser extent $$$f$$$,
parametric maps are more homogeneous and yield more biologically plausible values
when including STE data in the nonlinear fitting than compared to
fitting to LTE data alone.

Scatter
plots and histograms of parameter values in the posterior limb of the internal
capsule (PLIC) of a 30 y/o female subject scanned two times, 30 days apart. Spurious peaks for $$$f$$$ in STE data vanish when including STE, and precision of $$$D_a$$$ increases, while precision of of $$$D_e^{||}$$$ decreases and $$$D_e^\perp$$$ and $$$p_2$$$ remained relatively unchanged, also reflected in the listed coefficients of variation.

Box
plots of all subjects for white matter ROIs: genu (GCC) and splenium corpus
callosum (SCC), posterior limb internal capsule (PLIC), anterior (ACR),
posterior (PCR) and superior corona radiate (SCR) using either LTE or LTE+STE
encoding. The decrease in range is observed in the PLIC, ACR, PCR and SCR (but not in the corpus callosum) for $$$D_a$$$ (and $$$f$$$) in parameters obtained
with LTE+STE versus LTE. Note also the increase in range for $$$D_{e,||}$$$ in LTE+STE versus LTE for most ROIs.