Alexandru Vlad Avram^{1}, Elizabeth Hutchinson^{2}, and Peter Basser^{2}

We compute the higher-order statistics of the 3D spin displacement
probability distributions measured with mean apparent propagator (MAP) MRI and quantify
microstructural tissue parameters such as the mean kurtosis (MK), axial kurtosis
(K_{∥}),
radial kurtosis (K_{⊥}) and kurtosis fractional anisotropy (FA_{K}). This extension
of the family of MAP tissue parameters provides a direct link between the
frameworks of MAP MRI and other advanced diffusion techniques facilitating
interpretation of findings in clinical MAP MRI studies in the context of
existing literature on advanced diffusion MRI applications.

We derive analytical expressions for computing the higher-order
statistics of 3D probability distributions of spin displacements (i.e., diffusion propagators) measured with mean apparent propagator (MAP) MRI^{1}, and for quantifying microstructural
tissue parameters^{2}
conventionally derived with methods that rely on the cumulant expansion of the
MR signal phase, such as generalized diffusion tensor imaging (GDTI)^{3,4},
and, in particular, diffusional kurtosis imaging (DKI)^{5}. This extension of MAP MRI provides
the ability to make a direct quantitative comparison of MAP-derived parameters
and parameters obtained from other diffusion methods, and to compare and
harmonize findings in the scientific and clinical diffusion MRI literature.

We acquired high-quality diffusion MRI dataset (250x250x250µm^{3},
TE/TR=36/700ms) in fixed ferret brain with diffusion gradients applied along
orientations uniformly sampled on the unit sphere at five b-values (b_{max}=13500s/mm^{2}).
We measured the diffusion propagators using MAP MRI (up to order 6) and derived
microstructural parameter maps of return-to-origin, -axis, and -plane
probabilities (RTOP, RTAP, RTPP); total, axial, and radial non-gaussianity (NG,
NG_{∥}, NG_{⊥}); and propagator anisotropy
(PA).

From the 3D diffusion propagators measured in the MAP MRI functional
basis $$$\Psi_{m_{1}m_{2}m_{3}}(\bf{u},\bf{r})$$$ (up to order $$$M_{max}=6$$$) defined by scaling
vector **u**^{1}: $$P({\bf{r}})=\sum_{M=0}^{M_{max}}\sum_{m_{1}+m_{2}+m_{3}=M}^{}a_{m_{1}m_{2}m_{3}}\Psi_{m_{1}m_{2}m_{3}}(\bf{u},\bf{r})$$ we quantify the
higher-order statistical moment tensors of order N $$${\bf{\mu_{N}}}=\mu_{n_{1}}\mu_{n_{2}}\mu_{n_{3}}=\int_{-\infty}^{\infty}P({\bf{r}})x^{n_{1}}y^{n_{2}}z^{n_{3}}d{\bf{r}}$$$ (denoted
here using the so-called “occupation number” notation^{6}) both numerically, by integrating $$$P({\bf{r}})$$$, and
analytically, by using
a series of linear transformations $$\bf{\mu_{N}=aY_{N}U_{N}}$$, where $$${\bf{a}}=a_{m_{1}m_{2}m_{3}}$$$ is the
row vector of MAP MRI coefficients that describes $$$P({\bf{r}})$$$, $$${\bf{Y_{N}}}=Y_{m_{1}n_{1}}Y_{m_{2}n_{2}}Y_{m_{3}n_{3}}$$$ is a constant matrix with $$Y_{mn}=K_{m+n}\sqrt{m!}\sum_{r=0,2,...}^m\frac{(-1)^{\frac{r}{2}}2^{m-r}}{r!!(m-r)!}\Gamma(\frac{m+n-r+1}{2})$$, $$$K_{m+n}=1$$$ if m and n are even, and 0 otherwise, $$$\Gamma(x)$$$ is the Gamma function, and $$${\bf{U_{N}}}=u_x^{n_{1}}u_y^{n_{2}}u_z^{n_{3}}\sqrt{\frac{2^{N}}{\pi^{3}}}$$$ is a diagonal scaling matrix. From $$$\bf{\mu_{N}}$$$ we compute the cumulant tensors^{7} and higher-order diffusion tensors (HOTs) as
described in^{3}, along with parameters such as
mean kurtosis (MK), axial kurtosis (K_{∥}), radial kurtosis (K_{⊥})
and kurtosis fractional anisotropy (FA_{K})^{2,5,8}.

In addition, we directly analyze the DWI data with GDTI (order 6) by first re-orienting the **q**-vectors
in the DTI reference frame (which is the same as in MAP MRI) before fitting the
data^{1,9}.
We measure the kurtosis tensors^{5} along with the DKI parameters MK, K_{∥}, K_{⊥},
and FA_{K}, to compare with the MAP-derived quantities.

The moment tensors of MAP propagators derived
analytically (**Eq.2**) were verified by numerical integration. **Fig.1** shows how GDTI (or DKI) analysis can be performed directly in the DTI
reference frame (just like MAP MRI^{1}) by re-orienting the **q**-vectors (or b-matrices respectively)
before fitting the data^{1,9}, without affecting the measurement of rotation-invariant parameters^{10}
such as the MK, K⊥, K∥,
and FA_{K}^{2}. Analyzing the
data in the DTI reference frame^{9} “diagonalizes” the HOTs^{11} and may provide opportunities for regularization and sparse encoding.

Microstructural parameters MK, K_{⊥},
K_{∥}, and FA_{K} derived from the higher-order
statistics of MAP propagators showed good agreement with corresponding parameters
derived with GDTI/DKI (**Fig.2**) and complement the MAP MRI microstructural assessment
(**Fig. 3**). Differences between the parameters in **Fig.2** may be attributed to
truncations of MAP and GDTI series approximations.

Computation of DKI parameters within the analytical MAP framework does not require numerical integration, and is therefore faster and more
accurate than methods that measure the propagators numerically, such
as diffusion spectrum imaging (DSI)^{12}.

Due to the limited number of DWI measurements the truncation of the MAP (or GDTI) series approximation inherently leads to inaccuracies
(small oscillations) in the measured propagators for very large displacement
values. While **Eq.2** allows the exact computation of tensor
moments (and HOTs) with arbitrarily high order form MAP propagators, in
practice, very high-order statistics may amplify these spurious oscillations
in the propagator approximations leading to physically
inaccurate results.

Both MAP MRI and GDTI reconstruct the diffusion propagators
analytically. The Gram-Charlier Series (GCS) approximates the GDTI propagator from
its cumulants using Gauss-Hermite functions^{3,13 }similar to the MAP basis functions. However, the use of the
physicists’ Hermite polynomials in MAP – compared to the statisticians’ Hermite
polynomials in GCS – may allow a more robust approximation of probability
distributions^{14} with orthogonal functions
that have the desired asymptotic physical behavior^{1}.

This study serve as cross-validation of MAP MRI and diffusion
methods that rely on the cumulant expansion of the MR signal phase to quantify
features of the diffusion propagators. It extends the family of MAP microstructural
parameters to include HOT-derived metrics, and shows that MAP MRI subsumes not
only DTI but also GDTI/DKI. More importantly, it provides a direct link between
the frameworks of MAP MRI and other advanced diffusion techniques facilitating
interpretation of findings in clinical MAP MRI^{15} studies in the context of
existing literature on advanced diffusion MRI applications.

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2 Hui, E. S., Cheung, M. M., Qi, L. Q. & Wu, E. X. Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis. Neuroimage 42, 122-134, (2008).

3 Liu, C., Bammer, R. & Moseley, M. E. Generalized Diffusion Tensor Imaging (GDTI): A Method for Characterizing and Imaging Diffusion Anisotropy Caused by Non-Gaussian Diffusion. Israel Journal of Chemistry 43, 145-154 (2003).

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8 Jensen, J. H. & Helpern, J. A. MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR Biomed 23, 698-710 (2010).

9 Basser, P. J., Mattiello, J. & LeBihan, D. MR diffusion tensor spectroscopy and imaging. Biophys J 66, 259-267 (1994).

10 Qi, L. Q., Wang, Y. J. & Wu, E. X. D-eigenvalues of diffusion kurtosis tensors. J Comput Appl Math 221, 150-157 (2008).

11 Comon, P. in 10th IFAC Symposium on System Identification (IFAC-SYSID). 77-82 (IEEE).

12 Wedeen, V. J., Hagmann, P., Tseng, W.-Y. I., Reese, T. G. & Weisskoff, R. M. Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn Reson Med 54, 1377-1386 (2005).

13 Liu, C., Bammer, R., Acar, B. & Moseley, M. E. Characterizing non-gaussian diffusion by using generalized diffusion tensors. Magn Reson Med 51, 924-937 (2004).

14 Wellig, M. Robust series expansions for probability density estimation. (California Institute of Technology, Computer Vision Lab, 1999).

15 Avram, A. V. et al. Clinical feasibility of using mean apparent propagator (MAP) MRI to characterize brain tissue microstructure. Neuroimage 127, 422-434 (2016).