Synopsis

The cumulant expansion of Double Diffusion Encoding (DDE) involving correlation tensors has been previously theoretically presented but never exploited beyond microscopic anisotropy detection. Here, we propose the correlation tensor imaging (CTI) as novel approach capable of mapping correlation tensor features using DDE acquisitions. The correlation tensor can provide unique information: as a first step, we theoretically and experimentally demonstrate that CTI can be used to resolve the different underlying sources of diffusion kurtosis vis-à-vis isotropic and anisotropic variance of tensors and restricted diffusion (µK). The ensuing estimates bode well for many future applications.

Introduction

Theory

To assess the different non-Gaussian sources, it is useful to express the powder-average signal kurtosis $$${K_p}$$$ for a generic system compromised of multiple non-Gaussian diffusion compartments:$$\overline{D }^2K_p=\frac{12}{5}\langle V_\lambda(\mathbf{D_i})\rangle+3V(\overline{D_i})+\overline{D }^2\mu K\,\,\,\,(1)$$where $$$\overline{D}$$$ is the mean diffusivity, $$$\mathbf{D_i}$$$ models the apparent diffusivity of individual compartments, $$$\overline{D_i}$$$ is the mean diffusivity of individual compartments, $$$\langle V_\lambda(\mathbf{D_i})\rangle$$$ is the averaged variance of $$$\mathbf{D_i}$$$ eigenvalue’s (microscopic anisotropy term), $$$V(\overline{D_i})$$$ is the variance of $$$\overline{D_i}$$$ across individual compartments (inter-compartment kurtosis), and $$$\mu K$$$ is a term modelling the effects of intra-compartmental kurtosis.

In the long timing time regime, DDE signals $$$E(\boldsymbol{q}_1,\boldsymbol{q}_2)$$$ can be expressed in terms of the following cumulant expansion by⁶:$$\log{E}(\boldsymbol{q}_1,\boldsymbol{q}_2)=-(q_{1i}q_{1j}+q_{2i}q_{2j})\Delta D_{ij}+\frac{1}{16}(q_{1i}q_{1j}q_{1k}q_{1l}+ q_{2i}q_{2j}q_{2k}q_{2l})\Delta^2\overline{D}^2 W_{ijkl}+$$ $$\frac{1}{14}q_{1i} q_{1j}q_{2k}q_{2l}Z_{ijkl}+O(q^6)\,\,\,\,(2)$$ where $$$\boldsymbol{q}_1$$$ and $$$\boldsymbol{q}_2$$$ are the q-vectors of the two diffusion encoding pulses, $$$D_{ij}$$$ and $$$W_{ijkl}$$$ are the elements of the diffusion and kurtosis tensor, and $$$Z_{ijkl}$$$ are the elements of the 4nd order correlation tensor.

In the long timing time regime, $$$Z_{ijkl}$$$ can also be converted to the covariance tensor^10,11 - $$$C_{ijkl}= Z_{ijkl}/(4(\Delta-\delta/3)^2)$$$. From this, the terms of Eq.1 can be estimated as following:

1.Microscopic anisotropy:^7,11$$W_{aniso}= W_{aniso}=\frac{12}{5}\langle V_\lambda(\mathbf{D_i})\rangle=$$ $$\frac{8}{15}[ C_{1111}+D_{11}^2+C_{2222}+D_{22}^2+C_{3333}+D_{33}^2-C_{1122}-D_{11}D_{22}-$$ $$C_{1133}-D_{11}D_{33}-C_{2233}-D_{22}D_{33}+3(C_{1212}+D_{12}^2+C_{1313}+D_{13}^2 +C_{2323}+D_{23}^2)]\,\,\,\,(3)$$2.Inter-component kurtosis:¹¹$$W_{iso}=3V(\overline{D_i})=\frac{1}{2}(C_{1111}+C_{2222}+C_{3333}+2C_{1122}+2C_{1133}+2C_{2233})\,\,\,\,(4)$$3.Intra-compartmental kurtosis:$$\mu W=\overline{D }^2\mu K=\overline{D}^2K_p–W_{aniso}–W_{iso}\,\,\,\,(5)$$where $$$\overline{D}$$$ and $$$K_p$$$ can be estimated from $$$D_{ij}$$$ and $$$W_{ijkl}$$$.

Methods

Simulations: Synthetic data were generated using the MISST package¹² for two scenarios: 1) two Gaussian diffusion components representing intra- and extra-cellular spaces; 2) restricted infinite cylinders with radius=2.5μm and a Gaussian diffusion representing restricted intra-cellular space and hindered extra-cellular space. To assess the CTI acquisition parameters requirements, simulations were also produced for two protocols of $$$\boldsymbol{q}_1$$$ and $$$\boldsymbol{q}_2$$$ pairs of vectors: 1) with equal magnitude (16 b-value combinations used, Fig.1a); and 2) with asymmetric magnitudes (16 b-value combinations used, Fig.1b). For each b-value combination, 5-design directions⁷ were taken in addition to 45 parallel $$$\boldsymbol{q}_1$$$ and $$$\boldsymbol{q}_2$$$ directions. To assess higher order term effects, simulations were repeated using different $$$b_{max}$$$ values. Other protocol parameters were as follows: Δ=τ/δ=15/1.5ms. Simulations were produced noise-free to assess the technique’s full potential.

MRI experiments: Animal experiments were pre-approved by the institutional and national authorities (according to European Directive 2010/63). An adult mouse brain (N=1) was transcardially perfused and immersed in 4% PFA for 24h, and then washed in PBS for at least 48h prior to scanning. Data was acquired using a 16.4 T Aeon Ascend Bruker scanner equipped with a Micro5 probe with gradient coils capable of producing up to 3000 mT/m in all directions, and for the two protocols with $$$b_{max}=1.5$$$ms/μm². Other acquisition parameters were as follows: TR/TE=2500/52.1ms, voxel resolution 0.175×0.175×0.9mm.

Results and Discussion

CTI simulations: CTI produced accurate $$$W_{aniso}$$$ estimates for both encoding protocols (Fig.2a1/Fig.2b1). However, CTI produced inflated $$$W_{iso}$$$ estimates for protocol 1 (Fig. 2a2). $$$W_{iso}$$$ estimates were stabilised when asymmetric magnitudes of diffusion encoding pairs were used (Fig. 2b2). However, these only approached the ground-truth for small $$$b_{max}$$$ values due to higher order effects (Fig.2b2). m$$$W$$$ estimates were only close to the zero ground truth for protocol 2 (Fig.2a3/Fig.2b3). Although $$$W_{iso}$$$ is biased by high order terms, CTI $$$W_{iso}$$$ estimates are still sensitive to their increases and decreases(Fig.3).

Sensitivity to μ$$$W$$$: Simulations considering intra-comparmtal kurtosis effects are shown in Fig.4. $$$W_{aniso}$$$ and $$$W_{iso}$$$ are similar to those considering the Gaussian assumption (Figs.4a/Fig.4b). μ$$$W$$$ reveals the expected negative kurtosis in simulations for $$$b_{max}$$$ lower than 0.7ms/μm² (Fig.4c).

MRI experiments: As predicted by the simulations, $$$W_{aniso}$$$ maps are robust and independent of acquisitions protocol (Fig.5a1/Fig.5b1/Fig.5c1). The overestimation $$$W_{iso}$$$ for protocol 1 (Fig.5c2) is consistent to the prediction in from Fig.2a2. Consistent with simulations, negative m$$$W$$$ values are present in white matter regions. This is in contrast with two recent studies that attempted to measure intra-compartmental kurtosis^13,14. We ascribe the discrepancy to conflating effects of orientation dispersion.

Conclusion

Acknowledgements

References

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