# Director Field Analysis (DFA): Exploring Local White Matter Geometric Structure in diffusion MRI

## TL;DR

This paper introduces Director Field Analysis (DFA), a novel mathematical framework for quantifying local geometric structures and distortions in white matter using diffusion MRI data, enabling detailed analysis of white matter organization.

## Contribution

The paper presents new mathematical tools and scalar indices to analyze local orientational order, dispersion, and distortions in diffusion MRI data, a first in quantifying such features in spherical function fields.

## Key findings

- Quantitative description of orientational distortions (splay, bend, twist) in diffusion MRI data.
- Development of scalar indices for local geometric changes in white matter.
- Provision of tools applicable to both DTI and HARDI data for voxel-based analysis.

## Abstract

In Diffusion Tensor Imaging (DTI) or High Angular Resolution Diffusion Imaging (HARDI), a tensor field or a spherical function field (e.g., an orientation distribution function field), can be estimated from measured diffusion weighted images. In this paper, inspired by the microscopic theoretical treatment of phases in liquid crystals, we introduce a novel mathematical framework, called Director Field Analysis (DFA), to study local geometric structural information of white matter based on the reconstructed tensor field or spherical function field: 1) We propose a set of mathematical tools to process general director data, which consists of dyadic tensors that have orientations but no direction. 2) We propose Orientational Order (OO) and Orientational Dispersion (OD) indices to describe the degree of alignment and dispersion of a spherical function in a single voxel or in a region, respectively; 3) We also show how to construct a local orthogonal coordinate frame in each voxel exhibiting anisotropic diffusion; 4) Finally, we define three indices to describe three types of orientational distortion (splay, bend, and twist) in a local spatial neighborhood, and a total distortion index to describe distortions of all three types. To our knowledge, this is the first work to quantitatively describe orientational distortion (splay, bend, and twist) in general spherical function fields from DTI or HARDI data. The proposed DFA and its related mathematical tools can be used to process not only diffusion MRI data but also general director field data, and the proposed scalar indices are useful for detecting local geometric changes of white matter for voxel-based or tract-based analysis in both DTI and HARDI acquisitions. The related codes and a tutorial for DFA will be released in DMRITool.