Samo Lasič^{1}, Henrik Lundell^{2}, and Daniel Topgaard^{3}

Unprecedented microstructural details of heterogeneous materials such as tissue are non-invasively accessible by multidimensional diffusion encoding (MDE) MRI using varying shapes of b-tensors. MDE can probe multidimensional distributions of diffusion tensors in terms of their size, shape and orientation. For robust conclusions, time-dependent diffusion effects need to be considered in MDE. A convenient recipe for generating b-tensors of varying shape was implemented including a single adjustable tuning parameter. The resulting b-tensors can thus be tuned for sensitivity to time-dependent diffusion.

Introduction

Microstructural features of heterogeneous materials, such as average
cell eccentricity or variance of cell density, which are inferred from optical
microscopy images, can be quantified non-invasively by multidimensional diffusion encoding (MDE) MRI using varying shapes of b-tensors^{1-5}. With MDE, distributions of
diffusion tensors can be characterized in terms of size, shape and orientation^{3,4}. This information is directly encoded in the measured signal by MDE,
which is not possible by conventional diffusion MRI. In contrast, due to the
inability of providing more specific data, conventional approaches heavily rely
on data modeling assumptions, rendering conclusions unreliable^{6,7}. If not accounted for, time-dependent
diffusion effects may still skew MDE results. This can happen if different
b-tensors are probing diffusion at different time scales. As we have
demonstrated previously using spherical and linear b-tenors, spectral analysis of
encoding and diffusion allows designing tuned MDE waveforms, which yield equal
first-order signal attenuation. The tuning dimension also provides an additional
orthogonal measurement suited to probe correlations between anisotropy and size
of restrictions^{8}.
A convenient recipe can be
used to yield b-tensors with arbitrary anisotropy and asymmetry^{3,4}. Here
we have implemented spectral tuning by modifying the previous recipe with a single
adjustable tuning parameter.

Frequency analysis of encoding

To the first order, signal attenuation is given byDesigning tuned b-tensors of varying shape

Various b-tensor shapes and the associated q-trajectories can be constructed using two key parameters: $$$\zeta$$$ and $$$\Delta\Psi$$$, yielding anisotropy and asymmetry of b-tensors, $$$\it{b}_{\mathrm{\Delta}}$$$ and $$$\it{b}_{\mathrm{\eta}}$$$, respectively. Varying anisotropy and asymmetry of b-tensors allows detecting anisotropy and asymmetry of diffusion tensors. Gradient waveforms are generated following the steps outlined in refs. 3,4 and illustrated in Fig. 1A. First, the azimuth angle $$$\Psi(t)$$$ is calculated from gradient $$$G_{\mathrm{a}}(t)$$$ and $$$\Delta\Psi$$$. Then, the polar angle $$$\zeta$$$ is used to yield $$$\mathbf{G}(t)$$$ and the corresponding dephasing $$$\mathbf{q}(t)$$$ (Fig 1B). Introducing the tuning parameter (TP) as an amplitude of oscillation inserted between the bipolar lobes of the $$$G_{\mathrm{a}}(t)$$$ (red patch in Fig. 1A) allows tuning the average dephasing power spectrum shown in Fig. 1C. The TP was adjusted for each b-tensor shape to yield approximately equal apparent isotropic diffusivities for diffusion restricted in a sphere of arbitrary size. The colors for b-tensor shapes and average power spectra shown in Figs. 2 and 3 correspond to the average encoding frequency mapped to the range of colors red-green-blue.Results and discussion

The original recipe^{3,4} to generate b-tensors of varying shapes (Fig. 2A) can be easily
modified to achieve spectral tuning (Fig. 2B) using a single adjustable tuning
parameter (TP). An optimal TP can be found considering restricted diffusion.
For a spherical restriction, tuning yields approximately equal apparent
isotropic diffusivities in the entire range of relative encoding times $$$\sqrt{D_0
\tau}/R$$$ (not shown). Fig. 2 demonstrates how a wide range of average
encoding frequencies (color-coded in b-tensor shapes) is homogenized by tuning.
The average color-coded power spectra can be seen in Fig. 3.
A few caveats need be
address with our simplistic tuning implementation. First, our tuning was
performed for the case of restricted diffusion with its characteristic
diffusion spectrum $$$D(\omega)$$$^{9}. Considering
a radically different case of e.g. incoherent flow would present a different
tuning problem. Achieving perfect tuning for arbitrary $$$D(\omega)$$$ remains
a difficult if not impossible task. Second, in our tuning implementation, we
ignored the effects caused by spectral anisotropy^{10}. This
effect has been first described by de Swiet and Mitra^{11} for the case of spherical b-tensors. We have suggested the potential of using
spectral anisotropy as yet another encoding dimension exclusively sensitive to
anisotropic restrictions^{10}.
Third, additional tuning parameters would be necessary to further improve the
tuning, which is particularly important when probing anisotropic restricted
diffusion. This task is a subject future work. Forth, the suggested waveform
design is conceptually neat but not optimal in terms of hardware demands.
Although our waveforms can be applied on high performance gradient systems^{1-5},
optimizations considering gradient amplitude, slew and coil heating limitations^{12}
will be needed for efficient clinical applications.

HL is kindly supported by the European Research Council under the European Union's Horizon 2020 research and innovation program (grant agreement No. 804647 – C-MORPH)

DT is supported by the Swedish Research Council (VR) 2018-03697 and by the Swedish Foundation for Strategic Research (SSF) ITM17-0267

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2. 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.

3. Topgaard D, Director orientations in lyotropic liquid crystals : diffusion MRI mapping of the Saupe order tensor, Phys. Chem. Chem. Phys., 2016, 18, 8545–8553.

4. Topgaard D, Multidimensional diffusion MRI, J. Magn. Reson., 2017, 275, 98–113.

5. Topgaard D, NMR methods for studying microscopic diffusion anisotropy, in Diffusion NMR of confined systems: fluid transport in porous solids and heterogeneous materials, New Developments in NMR no. 9, R. Valiullin, Ed. Cambridge, UK: Royal Society of Chemistry, 2017.

6. Jones DK, Knösche TR, Turner R, White matter integrity, fiber count, and other fallacies: the do’s and don’ts of diffusion MRI, Neuroimage, 2013, 73, 239–254.

7. Novikov DS, Kiselev VG, Jespersen SN, On modeling, Magn. Reson. Med., 2018, 79, 3172–3193.

8. Lundell H, Nilsson M, Dyrby T, Parker G, Cristinacce P, Zhou F, Topgaard D, Lasič S, Microscopic anisotropy with spectrally modulated q-space trajectory encoding., in Proc. Intl. Soc. Mag. Reson. Med. 25, 2017, 1086.

9. Stepišnik J, Time-dependent self-diffusion by NMR spin-echo, Phys. B, 1993, 183, 343–350.

10. Lundell H, Nilsson M, Westin C-F, Topgaard D, Lasič S, Spectral anisotropy in multidimensional diffusion encoding, in Proc. Intl. Soc. Mag. Reson. Med. 26, 2018, 0887.

11. De Swiet TM, Mitra PP, Possible Systematic Errors in Single-Shot Measurements of the Trace of the Diffusion Tensor, J. Magn. Reson. Ser. B, 1996, 111, 15–22.

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.

Figure 1: Recipe for generating tuned
b-tensors of varying shapes. A. The tuning parameter (red patch under the
curve) is introduced in the 1D gradient waveform $$$G_{\mathrm{a}}(t)$$$. The azimuth
angle $$$\Psi(t)$$$ is calculated based on a given maximum excursion of the
q-trajectory $$$\Delta\Psi$$$. The $$$\Delta\Psi$$$ together with the polar
angle $$$\zeta$$$ are used to determine the shape of b-tensors and the corresponding
3D waveforms $$$\mathbf{G}(t)$$$. The dephasing waveforms $$$\mathbf{q}(t)$$$
(B) and their average power spectrum (C) are calculated as described in the
analysis section. The X, Y, Z gradient/dephasing waveforms are color-coded in
red, green, blue, respectively.

Figure 2: Different b-tensor shapes (non-tuned
and tuned) and the corresponding gradient waveforms for the XYZ channels. The
tensors are color-coded as an average encoding frequency weighted by the
average power spectra and mapped to the range of colors red-green-blue. The X,
Y, Z gradient waveforms are color-coded in red, green, blue, respectively.

Figure 3: Mean power spectra
(non-tuned and tuned) corresponding to the b-tensors shown in Figure 2. Patches
under the curves are color-coded the same as the b-tensors in Figure 2 (based
on the average encoding frequency).