Rafael Neto Henriques^{1}, Jonas Lynge Olesen ^{2,3}, Sune Nørhøj Jespersen^{2,3}, and Noam Shemesh^{1}

^{1}Champalimaud Research, Champalimaud Centre for the Unknown, Lisbon, Portugal, ^{2}Center of Functionally Integrative Neuroscience (CFIN) and MINDLab, Clinical Institute, Aarhus University, Aarhus, Denmark, ^{3}Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark

Diffusional kurtosis imaging (DKI) quantifies the non-Gaussian degree of diffusion using the kurtosis tensor. However, kurtosis can depend on conflicting sources of non-Gaussian diffusion such as Gaussian diffusion variance (inter-compartmental kurtosis) or the presence of restricted and hindered effects inside compartments (intra-compartmental kurtosis). Here, we develop and apply a novel double diffusion encoding method that is capable of providing the full directional dependence of both inter- and intra-compartmental kurtosis which can be summarized into two distinct kurtosis tensors and thereby improving kurtosis specificity and potentially providing information for diffusion model validation.

$$\log{S(b_1,b_2,\mathbf{n})}=\log\left\langle\exp\left[-\left(b_{1}+b_{2}\right)D_c(\mathbf{n})+\frac{1}{6}\left(b_{1}^2+b_{2}^2\right)D_c^2(\mathbf{n})K_c(\mathbf{n})+O(b^3)) \right] \right\rangle$$(Eq.1)

where $$$b_1$$$ and $$$b_2$$$ are the b-values of the two diffusion gradients applied and $$$\left\langle...\right\rangle$$$ represents the average across different compartments. Eq.1 can be approximated up to the second order in $$$b$$$ by:

$$\log{S(b_1,b_2,\mathbf{n})}=-\left(b_{1}+b_{2}\right)D(\mathbf{n})+\frac{1}{6}\left(b_{1}+b_{2}\right)^2D^2(\mathbf{n})K^{inter}(\mathbf{n})$$

$$+\frac{1}{6}\left(b_{1}^2+b_{2}^2\right)D^2(\mathbf{n})K^{intra}(\mathbf{n})$$(Eq.2)

where $$$D(\mathbf{n})$$$, $$$K^{inter}(\mathbf{n})$$$ and $$$K^{intra}(\mathbf{n})$$$ are the total directional diffusivity, and total inter- and intra-compartmental kurtosis:

$$D(\mathbf{n})=\left\langle D_c(\mathbf{n})\right\rangle,$$

$$K^{inter}(\mathbf{n})=3\frac{\left ( \left\langle D_c(\mathbf{n})\right\rangle^2-\left\langle D_c(\mathbf{n})\right\rangle^2\right )}{D^2(\mathbf{n})},$$

$$K^{intra}(\mathbf{n})=\frac{\left\langle D_c^2(\mathbf{n})K_c(\mathbf{n})\right\rangle}{D^2(\mathbf{n})}$$(Eq.3)

From Eq.2, $$$K^{intra}(\mathbf{n})$$$ can be extracted using two DDE experiments with constant total b-value=$$$b=b_1+b_2$$$ (Fig.1B):

$$K^{intra}(\mathbf{n})=\frac{12}{D^2(\mathbf{n})b^2}\left(\log{S(b,0,\mathbf{n})}-\log{S(b/2,b/2,\mathbf{n})}\right)$$(Eq.4)

$$\log{S(b_1,b_2,\mathbf{n})}=-\left(b_{1}+b_{2}\right)D(\mathbf{n})+\frac{1}{6}\left(b_{1}+b_{2}\right)^2\overline{D}^2n_in_jn_kn_lW^{inter}_{ijkl}+$$

$$\frac{1}{6}\left(b_{1}^2+b_{2}^2\right)\overline{D}^2n_in_jn_kn_lW^{intra}_{ijkl}$$(Eq.5)

**Simulations.**Synthetic signals are produced for two restricted infinite cylinders of different radius (i.e., different RD and RK values), Fig.2A. For these simulations $$$K^{intra}(\mathbf{n})$$$ is estimated for axial and radial directions and for powder-averaged signals at different b-values=b (from 0 to 2.5ms/μm^{2}). $$$\mathbf{W}^{inter}$$$ and $$$\mathbf{W}^{intra}$$$ are then fitted to all these signals simultaneously.**Tissue extraction.**Animal experiments were preapproved by the competent institutional and national authorities (European Directive 2010/63). A C57BL/6J mice brain (13 weeks) was transcardially perfused, immersed in 4% Paraformaldehyde (PFA) solution (24h), washed in Phosphate-Buffered Saline (PBS) solution (48h) and placed in a 10mm NMR tube with Fluorinert.**MRI Experiments.**MRI scans were performed at 37^{o}C using a 16.4 T Aeon Bruker scanner equipped with a Micro5 probe (which produces gradients up to 3000 mT/m). DDE data were acquired for $$$S(b, 0,\mathbf{n})$$$ and $$$S(b/2,b/2,\mathbf{n})$$$ experiments (Fig1.B) for b=[0,1.00,1.25,1.50,2.00,2.5ms/μm2] and for the 45 directions (Fig1.C). Other acquisition parameters are set to:Δ=τ_{m}=13ms, δ=1.5ms; TR/TE=2200/52ms, resolution=130×130x900μm^{3}, partial Fourier acceleration=1.42. $$$K_{intra}$$$ is first analysed separately for each b-value repetition, then $$$\mathbf{W}^{inter}$$$ and $$$\mathbf{W}^{intra}$$$ and respective invarianted are extracted from all acquired data.

**Simulations:**Radial $$$K^{intra}$$$ and $$$\overline{K}^{intra}$$$ estimates detect the expected negative intra-compartmental kurtosis effects of restricted cylinders (Fig.2C-D). The directional dependence of $$$\mathbf{W}$$$, $$$\mathbf{W}^{inter}$$$ and $$$\mathbf{W}^{intra}$$$ is presented in Fig.2E-G. The $$$\mathbf{W}^{inter}$$$ and $$$\mathbf{W}^{intra}$$$ decouples the positive non-Gaussian effects of diffusion variance across compartments (Fig.2F) from negative non-Gaussian effects of restricted diffusion (Fig.2G).**MRI experiments:**WM and GM ROIs placed at b=0 images revealed SNRs=70±20 and 110±10 (Fig.3A). For a representative slice and b=2.5ms/μm^{2}, the $$$S(b, 0,\mathbf{n})$$$ and $$$S(b/2,b/2,\mathbf{n})$$$ estimates for $$$\mathbf{n}=[1,0,0]$$$ and $$$\mathbf{n}=[0,1,0]$$$ are displayed in Fig.3B-C, while powder-averaged $$$\overline{S}(b,0)$$$ and $$$\overline{S}(b/2,b/2)$$$ signal are shown in Fig.3D. The differences between these maps reveals that $$$S(b, 0,\mathbf{n})$$$ is generally higher than $$$S(b/2, b/2,\mathbf{n})$$$ (Fig.3B3, Fig.3C3, and Fig.3D3)), which indicates positive intra-compartmental kurtosis values for both GM and WM (Fig.3B4, Fig.3C4, and Fig.3D4). Visually the $$$K^{intra}(\mathbf{n})$$$ estimates for individual directions present low precision; however, results from powder-averaged $$$\overline{K}^{intra}$$$ estimates indicates that intra-compartmental kurtosis effects are lower in WM (red arrow in Fig. 3D4).

Fig.4 shows metrics extracted from different tensors. As expected the total RK is higher than the total AK in WM (Fig.4B) which is revealed to be mainly caused by high RK^{inter}values (Fig.4C2). Both AK and RK maps reveal that intra-compartmental kurtosis is the highest source in GM.

1. Jensen JH, Helpern JA, Ramani A, et al. Diffusional Kurtosis Imaging: The Quantification of Non-Gaussian Water Diffusion by Means of Magnetic Resonance Imaging, Magn. Reson. Med. 2005; 53: 1432–1440

2.Stejskal EO, Tanner JE. Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient. The Journal of Chemical Physics 1965; 42(1): 288-292. doi:10.1063/1.1695690

3. Shemesh N, Jespersen SN, Alexander DC, et al.. Conventions and nomenclature for double diffusion encoding NMR and MRI. Magn. Reson. Med. 2016: 75: 82–87. doi:10.1002/mrm.25901

4. Cheung JS, Wang E, Lo EH, Sun PZ. Stratification of heterogeneous diffusion MRI ischemic lesion with kurtosis imaging: evaluation of mean diffusion and kurtosis MRI mismatch in an animal model of transient focal ischemia. Stroke 2012; 43: 2252–4. doi: 10.1161/STROKEAHA.112.661926

5. Falangola MF, Jensen JH, Babb JS, et al. Age-related non-Gaussian diffusion patterns in the prefrontal brain. J. Magn. Reson. Imaging 2008; 28: 1345–1350. doi:10.1002/jmri.21604

6. Jensen JH, Helpern JA, Tabesh A. Leading non-Gaussian corrections for diffusion orientation distribution function. NMR Biomed. 2014; 27: 202-211. doi: 10.1002/nbm.3053

7. Henriques RN, Correia MM, Nunes RG, Ferreira HA. Exploring the 3D Geometry of the Diffusion Kurtosis Tensor - Impacts on the Development of Robust Tractography Procedures and Novel Biomarkers. Neuroimage 2015; 111:85-99. doi.: 10.1016/j.neuroimage. 2015.02.004

8. Assaf Y, Cohen Y. Non‐mono‐exponential attenuation of water and N‐acetyl aspartate signals due to diffusion in brain tissue. J Magn Reson. 1998;131:69–85.

9. Yablonskiy DA, Bretthorst GL, Ackerman JJ. Statistical model for diffusion attenuated MR signal. Magn Reson Med. 2003;50:664–669.

10. Jensen JH, Helpern JA. MRI quantification of non‐Gaussian water diffusion by kurtosis analysis. NMR Biomed. 2010;23:698–710.

11. Jespersen, White Matter biomarkers from diffusion MRI. J. Magn. Reson. Imaging 2018; 291: 127–140. doi: 10.1016/j.jmr.2018.03.001

12. Callaghan PT, Coy A, MacGowan D, et al. Diffraction‐like effects in NMR diffusion studies of fluids in porous solids. Nature. 1991;351:467–469.

13. van Gelderen P, DesPres D, van Zijl PC, Moonen CT. Evaluation of restricted diffusion in cylinders: Phosphocreatine in rabbit leg muscle. J Magn Reson B. 1994;103:255–260.

14. Paulsen JL, Özarslan E, Komlosh ME et al. Detecting compartmental non-Gaussian diffusion with symmetrized double-PFG MRI. NMR Biomed. 2015; 28: 1550-6. doi: 10.1002/nbm.3363

15. Dhital B, Kellner E, Kiselev VG, Reisert M. The absence of restricted water pool in brain white matter. Neuroimage 2018; 182: 398-406. doi:10.1016/j.neuroimage.2017.10.051

16. Henriques RN, Jespersen SN, Shemesh N. Microscopic anisotropy misestimation in spherical‐mean single diffusion encoding MRI. Magn Reson Med. 2019; 81(5):3245-3261. doi: 10.1002/mrm.27606.

17. Lampinen B, Szczepankiewicz F, Mårtensson J, et al. Neurite density imaging versus imaging of microscopic anisotropy in diffu- sion MRI: A model comparison using spherical tensor encoding. Neuroimage. 2017;147:517–531.

18. Mitra PP. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Phys. Rev. B 1995; 51: 15074– 15078. doi: 10.1103/PhysRevB.51.15074

19. Jespersen SN. Equivalence of double and single wave vector diffusion contrast at low diffusion weighting. NMR in Biomed. 2011; 25(6): 813-818. doi: 10.1002/nbm.1808.

20. Jespersen SN, Lundell H, Sønderby CK, Dyrby TB. Orientationally invariant metrics of apparent compartment eccentricity from double pulsed field gradient diffusion experiments. NMR Biomed. 2013; 26(12):1647-62. doi: 10.1002/nbm.2999.

21. Jespersen SN, Buhl N. The displacement correlation tensor: Microstructure, ensemble anisotropy and curving fibers. J. Magn. Reson. 2011; 208: 34–43.doi: 10.1016/j.jmr.2010.10.003.

22. Koch MA, Finsterbush J. Compartment size estimation with double wave vector diffusion-weighted imaging. Magn. Reson. Med. 2008; 60(1); 90- 101. doi:10.1002/mrm.21514

23. Özarslan E. Compartment shape anisotropy (CSA) revealed by double pulsed field gradient MR. J Magn Reson. 2009;199: 56–67.

24. Jensen JH, Hui ES, Helpern JA. Double-pulsed diffusional kurtosis imaging. NMR Biomed. 2014; 27:363-370. doi: 10.1002/nbm.3094

25. Kaden E, Kruggel F, Alexander DC. Quantitative mapping of the per‐axon diffusion coefficients in brain white matter. Magn Reson Med. 2016;75:1752–1763.