Rafael Neto Henriques^{1}, Sune N Jespersen^{2,3}, and Noam Shemesh^{1}

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.

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 by^{6}:$$\log{E}(\boldsymbol{q}_1,\boldsymbol{q}_2)=-(q_{1i}q_{1j}+q_{2i}q_{2j})\Delta
D_{ij}+\frac{1}{16}(q_{1i}q_{1j}q_{1k}q_{1l}+
q_{2i}q_{2j}q_{2k}q_{2l})\Delta^2\overline{D}^2 W_{ijkl}+$$
$$\frac{1}{14}q_{1i}
q_{1j}q_{2k}q_{2l}Z_{ijkl}+O(q^6)\,\,\,\,(2)$$
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 $$$W_{ijkl}$$$.

Simulations: Synthetic data
were generated using the MISST package^{12} 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 directions^{7} 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/μm^{2}.
Other acquisition parameters were as follows: TR/TE=2500/52.1ms, voxel
resolution 0.175×0.175×0.9mm.

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/μm^{2} (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.

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