Chantal MW Tax^{1}, Umesh S Rudrapatna^{1}, Thomas Witzel^{2}, and Derek K Jones^{1}

Combining multiple, complementary contrasts into one analysis will yield deeper understanding of white matter physiology than using diffusion MRI (dMRI) alone. Varying TE in a PGSE sequence would allow for the exploration of D-T2 spectra in tissue. However, typical hardware and time constraints render the acquisition of such diffusion/relaxation spectra in the living human impractical. In this work, we explore how 300 mT/m gradients of a Connectom scanner could help in further investigating 1) the reported TE dependency of DTI parameters and 2) D-T2 spectra in the living human brain.

An emerging zeitgeist in microstructural
imaging, as evidenced by discussion at the recent ISMRM diffusion workshop, is
that combining multiple, complementary contrasts (e.g. axon and myelin metrics)
into one analysis will yield deeper understanding of white matter physiology
than using diffusion MRI (dMRI) alone. Where typically only gradient direction
and strength are varied in microstructural dMRI experiments, resolving
different tissue components is likely facilitated by studying the signal
attenuation as a function of more experimental variables. Varying TE would
allow for the exploration of D-T2 spectra in tissue^{1,2,3}.

Recently, a TE-dependence of DTI parameters
was observed in monkey brain, invoking explanations of
compartment-specific T2^{4}. However, a similar pattern was not seen in the rat
brain^{5} – bringing into question whether this is a global phenomenon of neural
tissue, whether it would consistently be seen in the human brain^{6}, and whether other
mechanisms might be at play (e.g. SNR effects and water exchange). However, typical hardware and time
constraints render the acquisition of such diffusion/relaxation spectra in the
living human impractical: the long diffusion pulses limit the shortest TE and largest b-value that can be used while maintaining sufficient SNR. In this
work, we explore how 300 mT/m gradients of a Connectom scanner could help in
further investigating 1) the observed TE dependency of DTI parameters and 2)
D-T2 spectra in the living human brain.

In vivo human data:

A dataset was acquired with 30 directions
and one b0 image per shell (Delta=22ms, delta=8ms); b=[500,1000,2000,3000,4000,5000,6000,7000]s/mm^{2 }for the lowest TE possible (47ms with multiband=2) and b=[500,1000,3000,5000,7000]s/mm^{2} until
maxTE=127ms in steps of 16ms. Additional b0 images were acquired with
minTE=39ms (Fig.1).

Simulations:

1) Dependency of DTI on TE: To investigate
whether TE dependency could be caused by noise, WM signal was simulated from an
FA=0.7 tensor (b=[500,1000]s/mm^{2}) and T2 =70ms. Rician noise was added with SNR
defined on the b0 image with the shortest TE.

2) Feasibility of estimating D-T2 spectra:
subsets of the b-TE space were simulated for a configuration mimicking the
diffusion process perpendicular to fibers: non-exchanging compartments with T2=[80,80,15]ms, D=[0.08,0.9,0.1]$$$\mu$$$m^{2}/ms, volume
fraction=[0.35,0.45,0.2] representing intra-axonal, extra-axonal, and myelin
components respectively^{6,7}. Noise was added as in 1).

Processing:

Real data were corrected for subject motion and distortions using FSL eddy^{8}.

1) To investigate TE dependency, DTI was fitted for every TE using WLLS on the b=[500,1000]s/mm^{2} shells^{9}.

2) To investigate the feasibility of reconstructing D-T2 spectra, the mean signal in the plane spanned by the second and third eigenvector (estimated from the fit in 1)) was extracted for each b-value in an ROI containing the posterior limb of internal capsule (PLIC). This mean signal can be written as Fredholm integral of the first kind:

$$M(TE,b)=\int \int F(T2,D)K1(TE,T2)K2(b,D)dT2 dD$$

with kernels $$$K1(TE,T2)=exp(-TE/T2)$$$ and $$$K2(b,D)=exp(-bD)$$$,
discretized to $$$M=KF$$$ where vector $$$M$$$ contains the signal, $$$K$$$ is the dictionary with 50
diffusivities and 50 T2-values and $$$F$$$ is a vector representing the spectrum^{1,2,11}. Elastic net
regularization with positivity constraint was used to estimate F, i.e. $$$min_{F>=0} 0.5||M-KF||_2^2+ \lambda 1||F||_1+0.5 \lambda 2||F||_2^2$$$^{10}.

Fig. 1 shows an overview of the acquisitions, there is still signal present at the highest b-value and TE.

Fig. 2a shows a dependency of estimated DTI
features in the PLIC as function of TE in real data. A paired t-test was used to
compare FA, AD, RD, and MD differences at minTE and maxTE revealing a highly significant effect of TE
for the former three. (p<[10^{-23},10^{-12},10^{-13} respectively). For the simulations
looking solely at noise (Fig. 2b) the trend is not
evident and all differences are not significant
(p>0.77).

Fig. 3 shows the mean relative error of D-T2 estimates for each compartment as function of the regularization parameters (100 noise iterations). Based on this we choose l1 regularization with $$$\lambda 1$$$ = 0.003 for further analysis. While detecting the intra and extra axonal component seems feasible (spectra on the right), myelin is challenging.

Fig. 4 shows an estimated D-T2 spectrum in the PLIC perpendicular to the estimated
fiber direction, revealing one compartment with [T2,D]≈[80ms,0$$$\mu$$$m^{2}/ms] and one with
[T2,D]≈[65ms,1$$$\mu$$$m^{2}/ms], potentially corresponding to the two compartments also estimated
in Fig. 3a.

[1] Callaghan, P. T., Godefroy, S., & Ryland, B. N. (2003). Use of the second dimension in PGSE NMR studies of porous media. Magnetic resonance imaging, 21(3), 243-248.

[2] Venkataramanan, L., Song, Y. Q., & Hurlimann, M. D. (2002). Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions. IEEE Transactions on Signal Processing, 50(5), 1017-1026.

[3] Kim, D., Kim, J. H., & Haldar J. P. (2016) Diffusion-Relaxation Correlation Spectroscopic Imaging (DR-CSI): An Enhanced Approach to Imaging Microstructure. ISMRM, 0660.

[4] Qin, W., Shui Yu, C., Zhang, F., Du, X. Y., Jiang, H., Xia Yan, Y., & Cheng Li, K. (2009). Effects of echo time on diffusion quantification of brain white matter at 1.5 T and 3.0 T. Magnetic resonance in medicine, 61(4), 755-760.

[5] De Santis, S., Assaf, Y., and Jones, D.K. (2016) The influence of T2 relaxation in measuring the restricted volume fraction in diffusion MRI. ISMRM, 1998.

[6] Lin, M., Tong, Q., Ding, Q., Yan, X., Feiweier, T., He, H., & Zhong J. (2016). TE-Dependence of DTI-Derived Parameters in Human Brains and Elucidation with Monte Carlo Simulation. ISMRM diffusion workshop.

[7] Jelescu, I. O., Zurek, M., Winters, K. V., Veraart, J., Rajaratnam, A., Kim, N. S., ... & Fieremans, E. (2016). In vivo quantification of demyelination and recovery using compartment-specific diffusion MRI metrics validated by electron microscopy. Neuroimage, 132, 104-114.

[8] Andersson, J. L., & Sotiropoulos, S. N. (2016). An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. Neuroimage, 125, 1063-1078.

[9] Veraart, J., Sijbers, J., Sunaert, S., Leemans, A., & Jeurissen, B. (2013). Weighted linear least squares estimation of diffusion MRI parameters: strengths, limitations, and pitfalls. NeuroImage, 81, 335-346.

[10] Beck, A., & Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2(1), 183-202.

[11] Benjamini, D., & Basser, P.J. Use of marginal distributions constrained optimization (MADCO) for accelerated 2D MRI relaxometry and diffusometry

[12] Dietrich, O., Raya, J. G., Reeder, S. B., Reiser, M. F., & Schoenberg, S. O. (2007). Measurement of signal-to-noise ratios in MR images: Influence of multichannel coils, parallel imaging, and reconstruction filters. Journal of Magnetic Resonance Imaging, 26(2), 375-385.

Fig. 1: Overview of
acquisition. (a) Absolute signal intensities, gradient direction is varied for images of different b-values shown. (b) Normalized signal intensities per volume, there is
still structure visible at the highest TE (127ms) and b-value (7000s/mm^{2}). (c)
SNR of the b0 volumes as a function of TE^{12}.

Fig. 2: TE dependency
of DTI parameters. (a) Features in ROIs of the left and right PLIC. The b0
signal as a function of TE seems to be slightly non-monoexponential. A positive
correlation of FA/AD with TE and a negative correlation of RD can be observed^{4,6}.
(b) Simulation trying to mimic the situation in the PLIC with similar SNR as in real data (Fig. 1c). No significant trends can be observed.

Fig. 3: Connected
components that could be identified in the spectrum were matched to a ground truth component by
determining the minimum total relative difference in D-T2 space. The mean
relative error over 100 noise iterations was then computed and plotted for D
(upper row) and T2 (middle row). The lower row shows the mean number of
detected connected components, ideally 3. (a) shows l1
regularization ($$$\lambda 2$$$=0) and (b) shows l2 regularization ($$$\lambda 1$$$=0). Using both l1 and l2 regularization did not significantly improve estimates (results not shown). Example
spectra are shown for $$$\lambda 1$$$=0.003 and $$$\lambda 2$$$=0.001, respectively.

D-T2 spectrum of the radial
signal decay in a ROI in the PLIC, revealing two compartments with [T2,D]≈[80ms,0um2/ms] and [T2,D]≈[65ms,1um2/ms]. $$$\lambda 1$$$=0.003 and $$$\lambda 2$$$=0 was used.