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The spectral tilt plot (STP) – new microstructure signatures from spectrally anisotropic b-tensor encoding

Samo Lasic^1,2, Filip Szczepankiewicz³, Markus Nilsson³, Tim B. Dyrby^1,4, and Henrik Lundell¹
¹Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Hvidovre, Copenhagen, Denmark, ²Random Walk Imaging, Lund, Sweden, ³Clinical Sciences, Lund University, Lund, Sweden, ⁴Department of Applied Mathematics and Computer Science, Technical University of Denmark, Copenhagen, Denmark

Synopsis

Tensor-valued diffusion encoding can probe diffusion tensor distribution unconfounded by orientation and heterogeneity. Since different cell morphologies may yield similar apparent diffusion tensors, inferring specific microstructural features remains challenging. Further information can be accessed by considering time-dependent diffusion. We show that rotational dependence of spherical tensor encoding, caused by spectral anisotropy, can be prominent on a preclinical scanner. The presented analysis may reveal different microstructural signatures depending on cell shape, which could be relevant for tissue modelling. Our results on fixed monkey brain suggest that brain cells exhibit anisotropic restricted diffusion along all directions, unlike in the cylindrical diffusion model.

Introduction

Tensor-valued diffusion encoding employs b-tensors of varying shapes such as in linear tensor encoding (LTE) or spherical tensor encoding (STE) to probe diffusion tensor distribution unconfounded by orientation and heterogeneity^1-10. However, inferring specific cell morphologies from apparent diffusion tensors remains challenging.

Time-dependent diffusion (TDD) can confound microstructure assessments when different b-tensors are not tuned, i.e. have different sensitivities to TDD^11,12. We have shown how STE combined with tuned/detuned LTE can simultaneously asses size and anisotropy¹¹.

TDD may cause STE to not be rotationally invariant^12-14, which may add to the intra-compartmental kurtosis in the second-order attenuation^14,15. However, rotational dependence of STE, detectable also at low b, is a consequence of orientation dependent sensitivity to TDD or spectral anisotropy^12,16.

In clinical setting, a preliminary study has shown that STE is rotationally invariant in healthy brain¹⁷. However, our pre-clinical experiments enabling much shorter encoding times reveal strong TDD on a fixed monkey brain¹¹.

Here we account for the spectral principle axes system (SPAS), identify the low frequency axis ($$$\mathbf{u}_\mathrm{LF}$$$) and sort apparent diffusivities ($$$D$$$) based on the angles $$$\sigma$$$ between $$$\bf{u}_\mathrm{LF}$$$ and the main diffusion tensor axes ($$$\boldsymbol{\mu}$$$). With simplistic restricted diffusion models we illustrate how the "spectral tilt plot" (STP), i.e. $$$D$$$ vs. $$$\sigma$$$, could reveal different microstructural signatures. The microstructures sensed in the monkey brain can be inferred from comparison with our models.

Theory

For a given diffusion spectrum $$$D_{ij}(\omega)$$$, the apparent diffusivity ($$$D$$$) can be calculated based on the encoding spectra $$$s_{ij}(\omega)$$$ defined in terms of the Fourier transforms of dephasing waveforms $$$\mathbf{q}(t)$$$ (see Fig. 1), $$ s_{ij}(\omega) \equiv q_i(\omega)\bar{q}_j(\omega). $$

In calculations (Fig. 3) we used axisymmetric $$$D_{ij}(\omega)$$$ for restricted diffusion with known eigen-spectra^12,18 $$$\lambda_1(\omega) = \lambda_2(\omega)\neq \lambda_3(\omega)$$$. $$$D$$$ can in this case be expressed with polar and azimuthal angles $$$\theta$$$ and $$$\phi$$$ as \begin{equation} \begin{split} D(\theta,\phi) = (\sin^2\phi+\cos^2\phi \cos^2\theta) \Lambda_{111} + \cos^2{\phi} \sin^2\theta \Lambda_{113} + \\ (\cos^2{\phi} + \sin^2\phi \cos^2\theta) \Lambda_{221} + \sin^2\phi \sin^2\theta \Lambda_{223} + \\ \sin^2\theta \Lambda_{331} + \cos^2\theta \Lambda_{333} + \\ 2 \, \cos{\phi} \, \cos{\theta} \, \sin{\theta} \, \Delta_{13} + 2 \, \cos{\phi} \, \sin{\phi} \, \sin^2\theta \, \Delta_{12} + 2 \, \sin{\phi} \, \cos{\theta} \, \sin{\theta} \, \Delta_{23}, \end{split} \end{equation} where $$ \Lambda_{ijk} \equiv \frac{1}{b} \frac{1}{2\pi} \int_{-\infty}^{\infty} s_{ij}(\omega) \lambda_k(\omega) d\omega,$$ $$ \Delta_{ij} \equiv \Lambda_{ij3} - \Lambda_{ij1}$$ and the b-value is given by the spectral trace $$$s(\omega)$$$¹² as $$ b=\frac{1}{2\pi}\int_{-\infty}^{\infty}s(\omega)d\omega.$$

Calculations

Spectral anisotropy was color coded based on weighting given by projections of encoding spectra¹². The low frequency band was adjusted based on $$$s(\omega)$$$ to account for the cumulative power of $$$b/3$$$. The SPAS was determined by the eigenvectors of the low band filtered b-tensors (Fig. 2), where $$$\bf{u}_\mathrm{LF}$$$ corresponds to the largest eigenvalue. The angle relative to the main diffusion axis $$$\boldsymbol{\mu}$$$ is given by $$$\sigma = \arccos{(\boldsymbol{\mu} \cdot \bf{u}_\mathrm{LF})}$$$.

$$$\lambda(\omega)$$$ were calculated for compartments with varying degrees of normalized apparent diffusion anisotropy ($$$D_\Delta$$$)^5,6, i.e. thin sticks with longitudinal restricted diffusion, prolate and oblate ellipsoids, and cylinders (see details in the Fig. 3 caption).

Watson orientation distribution function (ODF) was used with order parameters (OP)^5,6 of 0, 1/3, 2/3 and 1. To cover the full range of $$$\sigma$$$, $$$\boldsymbol{\mu}$$$ was rotated along 15 uniform directions.

Experiment

Data from an excised brain of a 3.5 year old vervet monkey were acquired on a 4.7 T preclinical system at 23.0 $$$\pm$$$ 0.2 $$$^\circ$$$C as previously described¹¹. The same 12 b-values between 240 and 4800 $$$\mathrm{s/mm^2}$$$ were used for STE and tuned LTE. $$$\bf{u}_\mathrm{LF}$$$ was rotated along 15 uniform directions (Fig. 1). STE employed q-MAS¹⁹ and the LTE was given by a single channel from STE (projection 2 in Fig. 2). Apparent diffusivities were obtained from low b-values.

Results and Discussion

Spectral anisotropy can be assessed by determining the SPAS, i.e. by finding the eigen representation of the filtered b-tensor. The low frequency band is most sensitive to restricted diffusion. Projections along the SPAS show that our STE is spectrally nearly axisymmetric with significantly more low frequency power along $$$\bf{u}_\mathrm{LF}$$$ (projection 1, Fig. 2).

Fig. 3 illustrates different predicted signatures for $$$D$$$ vs. $$$\sigma$$$. As expected, the modulation increases with increasing OP. Importantly, TDD leads to modulation also for STE. At larger $$$\sigma$$$, STE yields increasingly wider range of $$$D$$$ due to small differences in spectral content between the tangential axes (projections 2 and 3 in Fig. 2).

Three distinct modulation patterns are visible in Fig. 3 and summarized in Table 1: (1) For sticks and prolate ellipsoids, $$$D$$$ decrease/increase with $$$\sigma$$$ for LTE/STE. (2) For cylinders, $$$D$$$ decrease with $$$\sigma$$$ for LTE and STE. (3) For oblate ellipsoids $$$D$$$ increase/decrease for LTE/STE. We can see that in all cases STE and LTE yield similar diffusivities at approximately $$$\sigma \approx 54.7^{\circ}$$$, which corresponds to the projection where LTE and STE have equivalent power spectra.

Conclusion

Considering spectral anisotropy in tensor-valued encoding reveals different microstructural signatures depending on the shape of restricted compartments. The suggested analysis could be useful in identifying relevant tissue models. Our results on fixed monkey brain suggest that brain cells exhibit anisotropic restricted diffusion along all directions, unlike in the cylindrical diffusion model. Future work will focus on more quantitative analysis and possibilities to further experimentally exploit spectral anisotropy.

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 804746). SL is also supported by Random Walk Imaging.

References

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Figures

Figure 1: Gradient and dephasing waveforms with the associated spectral trace for STE. The waveforms are shown for the x, y and z channels (in red, green, blue) corresponding to the encoding rotation shown in Fig. 2.

Figure 2: Spectral content for the STE from Fig. 1 color coded based on 3 frequency bands with equal encoding power determined from $$$s(\omega)$$$ and its cumulative sum (inset). Projections 1-3 along the SPAS (dotted lines), relative to the laboratory frame (XYZ), have the most (along $$$\bf{u}_\mathrm{LF}$$$) to the least power in the low frequency band. The orange lines illustrate an ODF and its main direction $$$\boldsymbol{\mu}$$$ for which the angle $$$\sigma$$$ can be calculated.

Figure 3: STP differentiates sticks from cylinders. Noiseless calculations for Watson ODF with varying OP (columns) of axisymmetric restrictions with varying $$$D_\Delta$$$: sticks ($$$d = 5\mathrm{\mu m}, D_\Delta = 1$$$), prolate ellipsoids ($$$R_3/R_1 = 5\mathrm{\mu m}/1\mathrm{\mu m}, D_\Delta = 0.99$$$), cylinders ($$$R = 5\mathrm{\mu m}, D_\Delta = 0.38$$$), oblate ellipsoids ($$$R_3/R_1 = 1\mathrm{\mu m}/5\mathrm{\mu m}, D_\Delta = -0.5$$$). In all cases, $$$D_0 = 2\times 10^{-9} m^2/s$$$. Compare the marked examples (dotted) with experimental results (Fig. 4)

Figure 4: Experimental STP - $$$D$$$ vs $$$\sigma$$$ from the fixed monkey brain: $$$D$$$ values for all voxels (markers) and fits (lines) for tuned LTE (red) and STE (blue) in the labeled regions of white matter (left, 12315 voxels) and grey matter (right, 4545 voxels). Compare to the marked calculations in Fig. 3.

Table 1: Summary of $$$D$$$ vs. $$$\sigma$$$ patterns for structures in Fig. 3. Up/down arrows indicate increasing/decreasing values with $$$\sigma$$$. Here $$$D_\mathrm{STE}$$$ and $$$D_\mathrm{LTE}$$$ are values for STE and tuned LTE.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)

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