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Correlation Tensor Imaging - resolving non-Gaussian diffusion sources of in vivo tissues
Rafael Neto Henriques1, Sune Nørhøj Jespersen2,3, and Noam Shemesh1
1Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal, 2Center of Functionally Integrative Neuroscience (CFIN) and MINDLab, Clinical Institute, Aarhus University, Aarhus, Denmark, 3Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Synopsis

Resolving the different sources of diffusional kurtosis can increase DKI’s specificity towards different microstructural features. Such sources can be resolved using the correlation tensor imaging (CTI) – a novel double diffusion encoding technique that does not rely on common assumptions of time-independent diffusion. Here, a minimal acquisition protocol for CTI is designed and used to characterize the diffusional kurtosis of living rat brains for the first time. We here develop an approach to acquire CTI data in vivo and show that it can robustly decouple inter-compartmental kurtosis sources (anisotropic and isotropic diffusivity variances) from intra-compartmental kurtosis sources.

Introduction

Diffusion Kurtosis Imaging (DKI)1is a sensitive biomarker of microstructural alterations in health and disease2,3. However, several sources of non-Gaussian diffusion contribute to DKI’s contrast (Fig.1A), thereby diminishing its specificity. Techniques based on q-space trajectory encoding (QTE) schemes can currently resolve two inter-compartmental kurtosis sources: microscopic anisotropic (anisotropic kurtosis, Fig.1B) and the variance of compartment’s mean diffusivities (isotropic kurtosis, Fig. 1C)4,5, which has been shown to be useful for distinguishing different tumour types6-8. However, QTE assumes vanishing intra-compartmental kurtosis and diffusion time independence (Fig. 1D) – which may be a strong assumption in some biological tissues9-10. Recently, different kurtosis contributions were resolved in an assumption-free fashion using correlation tensor imaging (CTI)11– a novel approach based on the correlations tensors of double diffusion encoding (DDE) signals12-20. However, CTI acquisitions were very long and incompatible with in vivo conditions. Here, we designed a minimal protocol for CTI, which can faithfully report on all its metrics, and performed the first CTI experiments in vivo aiming to characterize different intra- and inter-compartment kurtosis sources.

Methods

  • Theory: CTI relies on the cumulant expansion of the DDE signal14-20. DDE imparts two independent q-vectors ($$$\mathbf{q}_1$$$, $$$\mathbf{q}_2$$$), spanning diffusion times $$$\Delta_{1}=\Delta_{2}=\Delta$$$ separated by a mixing time $$$\tau_{m}$$$ (Fig.2A). At long $$$\tau_{m}$$$, the fourth order cumulant expansion of these signals in multiple compartment non-Gaussian systems is given by11,14-16:
    $$\log{E(\mathbf{q}_1,\mathbf{q}_2)}=-\left(q_{1i}q_{1j}+q_{2i}q_{2j}\right )\Delta D_{ij}+\frac{1}{6}\left(q_{1i}q_{1j}q_{1k}q_{1l}+q_{2i}q_{2j}q_{2k}q_{2l}\right )\Delta^2\overline{D}^2W_{ijkl} +\left(q_{1i}q_{1j}q_{2k}q_{2l}\right )\Delta^2C_{ijkl}$$
    (Eq.1)
    where $$$E(\mathbf{q}_1,\mathbf{q}_2)$$$ is the normalized signal attetuation, $$$D_{ij}$$$ is the diffusion tensor, $$$W_{ijkl}$$$ is the kurtosis tensor, $$$C_{ijkl}$$$ is the covariance tensor, and $$$\overline{D}$$$ is the mean diffusivity. From Eq.1, the sources of kurtosis can be extracted as follows11:
  1. the total kurtosis $$$K_{T}$$$ can be computed from $$$W_{ijkl}$$$;
  2. the two inter-compartmental kurtosis sources ($$$K_{aniso}$$$ and $$$K_{iso}$$$) can be extracted from $$$C_{ijkl}$$$;
  3. an intra-compartmental kurtosis source ($$$K_{intra}$$$) can be estimated as $$$K_{T}-K_{aniso}-K_{iso}$$$, since $$$W_{ijkl}$$$ inclides all sources of kurtosis, while $$$C_{ijkl}$$$ includes only $$$K_{aniso}$$$ and $$$K_{iso}$$$.
  • Proposed protocol for CTI: Analogously to DKI, CTI requires at least three measurements with different diffusion-gradient intensities. To decouple $$$W_{ijkl}$$$from $$$C_{ijkl}$$$ CTI also requires acquisitions with unequal wave vectors (i.e., $$$|\mathbf{q}_2|\neq|\mathbf{q}_2|$$$). The combination of q-vector magnitude pairs used herein is summarized in Fig.2B. To resolve the individual elements of $$$C_{ijkl}$$$, , each gradient intensity combination has to be repeated for different directions of $$$\mathbf{q}_1$$$ and$$$\mathbf{q}_2$$$. Here, we used the directions of the 5-design (12 parallel + 60 perpendicular $$$\mathbf{q}_1$$$-$$$\mathbf{q}_2$$$ direction combinations)15 in addition to 45 parallel $$$\mathbf{q}_1$$$-$$$\mathbf{q}_2$$$ directions.
  • MRI experiments: Animal experiments were preapproved by the competent institutional and national authorities (European Directive 2010/63).Data was acquired from N=2 female long Evans rats (14/15 weeks old) under anaesthesia (~2.5% Isoflurane, 28% oxygen) on a 9.4 T Bruker Biospec scanner equipped with an 86 mm quadrature transmission coil and a 4-element array reception cryocoil (Bruker).
    DDE experiments were performed using the following diffusion parameters: bmax=2ms/μm2 (corresponding to a maximum total b-value of 4ms/μm2, $$$\Delta=\tau_m=$$$=12ms, and $$$\delta$$$=3ms). Additionally, anti-parallel DDE signal $$$E(\mathbf{q}_1,-\mathbf{q}_1)$$$ were acquired for checking the long mixing time approximation16-18.
    Other acquisition parameters included: TR/TE=3000/48.5 ms, in-plane resolution = 200×200 μm2, slice thickness = 1mm, 2 averages, partial fourier acceleration = 1.40, total acquisition time ~2h.

Results

  • Raw data is shown in Fig.3. ROIs placed in superior white matter (WM) and grey matter (GM) presented SNRb=0 of 41±.5 and 41±4, respectively (green and red ROIs, Fig.3A). Inferior brain regions had lower SNRb=0 (26.5±0.5 for the blue ROI) due to the cryocoil profile. Powder-averaged signal decays for different total b-values of 2 and 4ms/mm2 and for parallel, anti-parallel and perpendicular DDE experiments are displayed in Fig.3B and Fig.3C.
  • Maps resolving different kurtosis sources using CTI are displayed in Fig. 4 for all slices of both animals scanned. As expected, $$$K_T$$$ is higher than any of its sources (Fig.4A). $$$K_{aniso}$$$dominates in white matter (Fig.4B), while $$$K_{iso}$$$ is high in regions near the ventricle (Fig.4C, red arrows). Higher $$$K_{iso}$$$ in inferior brain regions are due to noise biases (Fig.4C, white arrows). $$$K_{intra}$$$ is consistently positive for both animals (Fig. 4D).

Discussion

CTI can resolve different kurtosis sources in a model-free fashion and without relying on the Gaussian regime approximation. The long mixing time requirement was experimentally verified by a near unity ratio of parallel and anti-parallel DDE signals (Fig.3B4 and Fig.3C4). In this regime, CTI’s $$$K_{aniso}$$$estimates are sensitive to the high microscopic anisotropy in WM. These measures can be used to study WM without confounding fiber dispersion effects. Interestingly, $$$K_{iso}$$$ highlights regions with increased free water partial volume effects, suggesting that $$$K_{iso}$$$ may be useful for detecting edema. We also presented the first in vivo CTI’s $$$K_{intra}$$$ estimates, whose positive values might reflect extra-cellular non-Gaussian diffusion effect or axial intra-cellular short-range disorder21. Future studies accelerating CTI further and investigating the effects of higher-order terms and parameters noise robustness can increase CTI’s potential value even further.

Conclusion

CTI is compatible with in vivo acquisitions, and its contrasts reflect the different kurtosis sources, including intra-compartmental kurtosis, thereby offering a more complete characterization of kurtosis in vivo. Moreover, the more general CTI framework might be used to validate and calibrate faster q-trajectory encoding acquisitions.

Acknowledgements

This study was funded by the European Research Council (ERC) (agreement No. 679058).

References

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Figures

Fig.1 - Schematic representation of the different sources of diffusional kurtosis estimates. The total diffusional kurtosis (A) depends on the: 1) ensemble microscopic anisotropy (anisotropic kurtosis (B)); variance of the mean diffusivities across different compartments (isotropic kurtosis (C)); and 3) restricted, hindered and time-dependent diffusion due to the interaction of water molecules with the boundaries of compartments or its macromolecules (intra-compartmental kurtosis (D)).

Fig.2 – Proposed acquisition protocol for CTI: (A) Parameters of a standard DDE pulse sequence (Δ1 and Δ2 are the diffusion gradients’ separation time, δ their duration, and τm is the mixing time between the two diffusion encoding modules marked by the blue and red lines); (B) The six q-value magnitude combinations of the proposed acquisition protocol – note that q-value magnitudes and are fully determined for a given bmax value. Each q-value magnitude combinations are acquired for 117 pairs of q1 and q2 directions.

Fig.3 – Raw DDE data of a representative rat: (A) b0 images of three coronal slices, where a superior GM (red), superior WM (green) and inferior GM (blue) ROIs are defined; (B) Powder-averaged data for b-value=2ms/μm2 and for: (B1) parallel, (B2) anti-parallel, (B3) perpendicular, (B4) ratio parallel/anti-parallel, and (B5) ratio parallel/perpendicular experiments; (C) Powder-averaged data of slice #2 for b-value=4ms/μm2 and for: (C1) parallel, (C2) anti-parallel, (C3) perpendicular, (C4) ratio parallel/anti-parallel, and (C5) ratio parallel/perpendicular experiments.

Fig.4 – CTI kurtosis measures extracted from all acquired slices and for both in vivo rat brains: (A) maps of the total kurtosis KT; (B) maps of the anisotropic kurtosis Kaniso; (C) maps of the isotropic kurtosis Kiso; and (D) maps of the intra-compartmental kurtosis Kintra. Note that Kanisois higher in WM, Kiso shows higher values in regions near cerebral ventricles (red arrows) and inferior brain regions (white arrows), while Kintra is consistently positive for both animals.

Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)
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