Sheet Probability Index (SPI): Characterizing the geometrical organization of the white matter with diffusion MRI

Abstract

The question whether our brain pathways adhere to a geometric grid structure has been a popular topic of debate in the diffusion imaging and neuroscience society. Wedeen et al. (2012a b) proposed that the brain’s white matter is organized like parallel sheets of interwoven pathways. Catani et al. (2012) concluded that this grid pattern is most likely an artifact, resulting from methodological biases that cause the tractography pathways to cross in orthogonal angles. To date, ambiguities in the mathematical conditions for a sheet structure to exist (e.g. its relation to orthogonal angles) combined with the lack of extensive quantitative evidence have prevented wide acceptance of the hypothesis. In this work, we formalize the relevant terminology and recapitulate the condition for a sheet structure to exist. Note that this condition is not related to the presence or absence of orthogonal crossing fibers, and that sheet structure is defined formally as a surface formed by two sets of interwoven pathways intersecting at arbitrary angles within the surface. To quantify the existence of sheet structure, we present a novel framework to compute the sheet probability index (SPI), which reflects the presence of sheet structure in discrete orientation data (e.g. fiber peaks derived from diffusion MRI). With simulation experiments we investigate the effect of spatial resolution, curvature of the fiber pathways, and measurement noise on the ability to detect sheet structure. In real diffusion MRI data experiments we can identify various regions where the data supports sheet structure (high SPI values), but also areas where the data does not support sheet structure (low SPI values) or where no reliable conclusion can be drawn. Several areas with high SPI values were found to be consistent across subjects, across multiple data sets obtained with different scanners, resolutions, and degrees of diffusion weighting, and across various modeling techniques. Under the strong assumption that the diffusion MRI peaks reflect true axons, our results would therefore indicate that pathways do not form sheet structures at every crossing fiber region but instead at well-defined locations in the brain. With this framework, sheet structure location, extent, and orientation could potentially serve as new structural features of brain tissue. The proposed method can be extended to quantify sheet structure in directional data obtained with techniques other than diffusion MRI, which is essential for further validation.

Quantifying the brain's sheet structure with normalized convolution

“…Previous work has shown that the Lie bracket, a mathematical concept from differential geometry, captures information on the existence of sheet structure (Tax et al, 2015; Tax et al, 2014a; Tax et al, 2016b; Wedeen et al, 2012b; Wedeen et al, 2014). Specifically, the Lie bracket can be computed from two vector fields, and is a vector field itself.…”

“…In addition, dMRI peaks are not clustered into distinct vector fields a priori. In previous work we have proposed to compute the Lie bracket by reconstructing an extensive amount of path neighborhoods (Tax et al, 2015; Tax et al, 2014a; Tax et al, 2016b), inspired by the qualitative reconstructions in (Wedeen et al, 2012b). More specifically, by taking small ‘loops’ along the tractography pathways of two vector fields, we could obtain an estimate of the Lie bracket.…”

“…In this so-called flows-and-limits approach, dMRI peaks were clustered into distinct vector fields during tractography. By computing the Lie bracket for repeated dMRI measurements or for different bootstraps of a single measurement, we were able to derive a sheet probability index (SPI) (Tax et al, 2016a; Tax et al, 2016b). SPI values close to 1 indicate that the underlying vector fields are highly supportive of sheet structure.…”

“…In this work we use the pipeline to compute the SPI developed in previous work (Tax et al, 2016b), but propose a new approach to compute the Lie bracket based on a conceptually different but equivalent definition. Instead of using the flows-and-limits approach, we here use the fact that the vector fields are defined on subsets of the 3D Euclidean space.…”

“…In Section 2 we will first briefly recapitulate the mathematical condition for the existence of sheet structure, involving the Lie bracket of vector fields (Tax et al, 2015; Tax et al, 2014a; Tax et al, 2016b; Wedeen et al, 2012b; Wedeen et al, 2014). In addition, we will present the flows-and-limits and coordinate definitions in more detail, and discuss their implementations.…”