The prevalence of sheet structure (the colloquial term for surfaces formed by two families of crossing fiber pathways) in the brain has been a debated issue since its proposal^1,2,3. This structure can be analyzed by means of the Lie bracket^1,4,5, which is a vector field that captures geometric properties of a pair of vector fields. In practice, such a pair of vector fields can be derived from diffusion MRI (dMRI) data by extracting peaks of the diffusion or fiber orientation distribution functions (FODs). In the case of sheet structure, the Lie bracket is constrained to lie in the plane spanned by the pair of vectors (or, equivalently, the “normal component of the Lie bracket” should be zero). This property can be used to distinguish “sheets” from “non-sheets” in dMRI data^4,5.

Due to the occurrence of noise, however, the sheet-constraint is never exactly fulfilled, and a single measurement does not provide information on its variability. This makes it difficult to quantify to what degree the local structure effectively resembles a sheet. In this work, we propose a new and robust local measure based on the Lie bracket that can be interpreted as the sheet probability index (SPI).

Lie bracket computation

We estimate the voxel-wise Lie bracket normal component $$$[X,Y]^\perp$$$ of pairs of vector fields $$$X$$$, $$$Y$$$ with a method described in Tax et al.^4,5. $$$[X,Y]^\perp$$$ is zero if there is local sheet structure, and deviates from zero if the local structure does not resemble a sheet. All vector fields were derived from peaks calculated using constrained spherical deconvolution^8,9 (CSD, $$$l_{max}=8$$$). For voxels containing three or more vectors all possible pairs are considered.

Sheet probability index (SPI)

To quantify the effect of noise on the Lie bracket, we generated $$$N_b$$$ residual bootstrap realizations¹⁰ for a single set of noisy measurements. Currently we rely on the bootstraps being normally distributed with data-derived mean $$$\mu$$$ and standard deviation $$$\sigma$$$, which is verified using a Shapiro-Wilks test. When normality holds, we can calculate the integral probability $$$P_\lambda$$$ bounded by $$$-\lambda$$$ and $$$\lambda$$$ (where we can tune the parameter $$$\lambda$$$) for the estimated distribution $$$N(\mu,\sigma)$$$. $$$P_\lambda$$$ produces a value that lies between $$$0$$$ and $$$1$$$ which we coin the sheet probability index (SPI) of the local sheet structure. Choosing a higher value for $$$\lambda$$$ means that the impact of deviations from zero on the estimated SPI will become smaller.

Data

Simulations: Analytical pairs of vector fields with known Lie bracket normal component (a “sheet” pair and “non-sheet” pair, Fig. 1) are used as the basis for simulated diffusion MRI data ($$$60$$$ directions, $$$b=3000s/mm^2$$$ using a ZeppelinStickDot model⁶). Rician noise was added with $$$SNR=20$$$;

MRI data: A single shell ($$$90$$$ directions, $$$b=3000\,s/mm^2$$$) of a Human Connectome Project⁷ dataset was extracted (voxel size $$$1.25\,mm$$$).

Simulations

Fig. 2 shows the range and mean of the estimated Lie bracket normal component $$$\widehat{[\cdot,\cdot]}^\perp$$$, where we have evaluated the pairs of vector fields (sheet in green and non-sheet in red) at different locations. The bootstraps (Fig. 2a) are well-defined representations of the “real” noise realizations (Fig. 2b). For the non-sheet case $$$P_\lambda$$$ decreases with increasing true $$$[\cdot,\cdot]^\perp$$$ ($$$\lambda=0.008$$$).

MRI data

Fig. 3a shows two examples of the $$$N_b=20$$$ bootstraps on real data. Fig. 3b shows histograms of $$$\widehat{[\cdot,\cdot]}^\perp$$$ at different locations (high-, medium-, and low-sheet probability area), visually confirming normality. Figs. 4 and 5 show SPI maps (a, $$$\lambda=0.008$$$) for two different slices, continuous areas of high (red) and low (blue) $$$P_\lambda$$$ (b), and pathways that run through these high SPI areas (c and d).

The proposed SPI measure is based on the previously investigated Lie bracket, and is designed as a robust and quantitative estimate of the local structure as illustrated in Fig. 2. Though the method ideally requires several independent samples of the normal component of the Lie bracket, Fig. 2 shows that bootstrapping can be a viable alternative in practice. Our first SPI maps of real data (Figs. 4 and 5) show that tracts can be explored in sheet and non-sheet areas in a guided and interactive way.