Sune Nørhøj Jespersen^{1,2}, Jonas Lynge Olesen^{1,2}, Andrada Ianus^{3,4}, and Noam Shemesh^{3}

Multidimensional diffusion weighting, such as magic angle spinning of the q-vector (q-MAS), relies on the assumption of multiple Gaussian compartments (MGC). Then the kurtosis measured with q-MAS can be fully ascribed to ensemble variance of isotropic diffusivity. However, in compartments with nongaussian diffusion, anisotropic time dependence of the diffusion tensor imparts orientation dependence on the q-MAS measured mean diffusivity, which in the presence of orientation dispersion leads to additional contributions to kurtosis. Yet another contribution arises from intracompartmental kurtosis. Using simulations and experiments we demonstrate that q-MAS derived diffusion kurtosis conflates variance in isotropic diffusivity with dispersion and intracompartmental kurtosis.

Under the MGC
assumption, the diffusion kurtosis $$${{K}_{I}}$$$ associated with q-MAS^{11}
can be related^{7} to the corresponding
mean (or isotropic) diffusivity $$${{D}_{I}}$$$
and the variance $$${{V}_{I}}$$$ of isotropic diffusivities, $$${{K}_{I}}=3{{V}_{I}}/D_{I}^2$$$.
However, in compartments with nongaussian diffusion, anisotropic time-dependence
of the diffusion tensor imparts orientation dependence on the isotropic diffusivity
measured with q-MAS, which in the presence of orientation dispersion leads to additional
contributions to the variance. Further, intracompartmental diffusion kurtosis
is another source of kurtosis beyond isotropic diffusion variance. These
additional contributions $$$\delta K_{I}^{(1)}$$$ and $$$\delta K_{I}^{(2)}$$$ are:

(i) For identical axially symmetric microdomains with orientations $$$\mathbf{\hat{u}}$$$ and diffusivities $$${{D}_{\parallel }}(t)$$$ and $$${{D}_{\bot }}(t)$$$, apparent isotropic diffusivity is $${{\tilde{D}}_{\text{I}}}={{\delta }_{ij}}B_{ij}^{\bot }+{{\langle {{u}_{i}}{{u}_{j}}\rangle }_{{\mathbf{\hat{u}}}}}B_{ij}^{\Delta }=\text{Tr}({{\text{B}}^{\bot }})+{{\langle {{\mathbf{\hat{u}}}^{\text{T}}}{{\text{B}}^{\Delta }}\mathbf{\hat{u}}\rangle }_{{\mathbf{\hat{u}}}}},$$ where $$${{\langle \cdot \rangle }_{{\mathbf{\hat{u}}}}}$$$ denotes an average over the orientations $$$\mathbf{\hat{u}}$$$ of the microdomains, $$B_{ij}^{\bot }=({{\gamma }^2}/2b)\int_{0}^{T}{d}{{t}_{1}}\int_{0}^{T}{d}{{t}_2}{{G}_{i}}({{t}_{1}}){{G}_{j}}({{t}_2}){{D}_{\bot }}(|{{t}_{1}}-{{t}_2}|))|{{t}_{1}}-{{t}_2}|$$

$$B_{ij}^{\Delta }=({{\gamma }^2}/2b)\int_{0}^{T}{d}{{t}_{1}}\int_{0}^{T}{d}{{t}_2}{{G}_{i}}({{t}_{1}}){{G}_{j}}({{t}_2})\left( {{D}_{\parallel }}(|{{t}_{1}}-{{t}_2}|)-{{D}_{\bot }}(|{{t}_{1}}-{{t}_2}|) \right)|{{t}_{1}}-{{t}_2}|,$$ and $$$\mathbf{G}$$$ is the diffusion gradient with associated b-value $$$b$$$.

(ii) With an intracompartmental diffusivity $$$\tilde{D}_{I}^{p}$$$ and kurtosis $$$K_{I}^{p}$$$ in pore $$$p$$$, this additional contribution becomes $$\delta K_{I}^{(2)}={{\langle {{(\tilde{D}_{I}^{p})}^2}K_{I}^{p}\rangle }_{p}}/\tilde{D}_{I}^2$$ where $$${{\langle \cdot \rangle }_{p}}$$$ denotes an average over the pore population. Thus, q-MAS derived diffusion kurtosis conflates variance in isotropic diffusivity with dispersion and intracompartmental kurtosis.

Figure 1 illustrates
that $$${{\tilde{D}}_{\text{I}}}$$$ depends on the orientation of a rectangular
box as function of the side length $$$a$$$, with a variability of up to varying up to more than 20% of the mean. The
corresponding maximum contribution $$$\delta K_{I}^{(1)}$$$ (blue) is
negligible for $$$a<5$$$µm but reaches 0.04 for larger pores. A powder
average limits $$$\delta K_{I}^{(1)}<0.012$$$. Figure 2 shows $$$\delta
K_{I}^{(1)}$$$ for other dimensions, and the red lines indicate regions for which
where $$$\delta
K_{I}^{(1)}$$$ is less than 0.01 (solid) and 0.05 (dashed). Figure 3
illustrates directional variability in pig spinal cord and compares to
$$${{\tilde{D}}_{\text{I}}}$$$ computed
from $$${{D}_{\parallel }}(t)$$$ and $$$D_\bot(t)$$$ obtained in ^{13}. Figure 4
evaluates the directional dependence of $$${{D}_{I}}$$$ measured in rat spinal cord, and estimates $$$\delta K_{I}^{(1)}$$$.
Finally, figure 5 illustrates the importance of intracompartmental kurtosis $$$\delta
K_{I}^{(2)}$$$ for the 4 model systems.

Orientational variability of the apparent isotropic
diffusivity for an infinitely long rectangular box as function of cross-sectional
side length $$$a$$$ using an isotropic diffusion encoding waveform of duration τ = 55 ms^{12}.
The upper x-axis allows translation to other durations by converting to $$$\tau
/{{t}_{D}}$$$ with $$${{t}_{D}}={{a}^2}/{{D}_{0}}$$$. Left: black curves show
the maximum and minimum possible apparent isotropic diffusivity for a single infinitely long rectangular box. The blue curve plots the corresponding range squared divided
by its mean squared - this corresponds to maximum $$$\delta K_{I}^{(1)}$$$ for
a system of two rectangular boxes. Right: $$$\delta K_{I}^{(1)}$$$ for a powder
sample.

$$$\delta
K_{I}^{(1)}$$$ caused by orientational variability of the apparent
isotropic diffusivity for a rectangular box as function of side lengths. The
solid/dashed red lines outline the region with $$$\delta K_{I}^{(1)}$$$ below 0.1/.05. Left: Maximum $$$\delta K_{I}^{(1)}$$$ for
a system of two rectangular boxes. Right: $$$\delta K_{I}^{(1)}$$$ for a powder sample. The pores must be
relatively isotropic in order not to confound $$${{K}_{I}}$$$, especially
for small dimensions.

Using the
inferred compartmental diffusivities from the plus branch of ^{13},
the apparent isotropic diffusivity $$${{\tilde{D}}_{I}}$$$ in intra- (left) and
extra-axonal (right) spaces was computed numerically as function of waveform
duration using the equations in Results part (i). Again, black curves give
maximum and minimum $$${{\tilde{D}}_{I}}$$$ over orientations, and the blue
curve the maximum $$$\delta {{K}_{I}}$$$ for a two component system.
The dashed line plots the actual isotropic diffusivity for the effective
diffusion time^{11} associated with τ.

Rat spinal cord T2-weighted image ($$${{S}_{0}}$$$),
FA, apparent isotropic diffusivity averaged over the 12 acquired directions, and
the corresponding standard deviation of $$${{\tilde{D}}_{I}}$$$ normalized by
its mean. The histogram shows the distribution of $$$\delta
K_{I}^{(1)}/{{K}_{I}}$$$(inset) in white matter, with a mean value close to 0.1.
Here $$$\delta K_{I}^{(1)}$$$was computed from the normalized variance of $$${{\tilde{D}}_{I}}$$$
over directions. Both $$${{\tilde{D}}_{I}}$$$and $$${{K}_{I}}$$$were estimated
from a DKI-fit for each direction separately, and$$${{K}_{I}}$$$was subsequently
averaged over directions.

Kurtosis $$${{K}_{I}}=\delta K_{I}^{(2)}$$$ as
function of dimension $$$a$$$ for the two model systems, infinitely long
rectangular box and permeable barriers, both for a single orientation and a
powder sample. In several of the systems, the values are at least as large as
reported values for $$$K_I$$$^{5,8}.
Perm. Comp. = permeable compartment.