Diffusion tensor imaging and tractography

(c) 2020 Moo K. Chung
University of Wisconsin-Madison


 July 17, 2020

Diffusion tensor imaging (DTI) provides the directional information of water molecule diffusion in the white matter fibers of the human brain. It is usually represented as the collection of 6 multivariate images that represent a 3 by 3 symmetric positive definite matrix called diffusion coefficients or diffusion tensors. The diffusion coefficients can be used to understand the pattern of white fibers in the brain.

Kernel smoothing in DTI

DTI often requires spatial smoothing to reduce noise. Most previous regularization works on diffusion tensor magnetic resonance images have been based on either anisotropic heat equations or streamline approaches. We developed a novel regularization method using iterated anisotropic kernels that avoids solving diffusion equations or streamline equations while improves upon numerical stability (Chung et al. 2003, Lee et al. 2005). The bandwidth of kernel is proportionally matched to the diffusion tensor to smooth out more along the tensor fields (Figure 1). The proportionally constant is chosen to minimize the sum of the squared intensity difference between before and after smoothing  while maximizing the smoothness of the image. The kernel method can be shown to increases the signal-to-noise ratio while preserving the anisotropy of the tensor fields. This formulation can be shown to be equivalent to the edge enhancing diffusion equation approaches.

Figure 1. Probabilistic connectivity obtained by iterated heat kernel smoothing

Structural connectivity

Streamline based tractography is often used in determining the connectivity between brain regions. In Chung et al. 2019, 10 million tracts were extracted then biologically informed 1 million tracts were extracted. The DTI tractography was done by Li Shen's group at Univ. of Penn. Figures 2 and 4 show the average connectivity of males and females (reproduced from Figure 3 in  Chung et al. 2019). The average connectivity matrices and t-stat map used in producing Figure 2 are stored as MAT file: chung.2019.CNI.mat. The MAT file contains the sample mean of female (mean(x)), the sample mean of males(mean(y)), the sample variance of females (var(x), the sample variance of males (var(y)). It also contains the t-stat of sex difference computed as 


The MAT file also contains the t-stat pooling the two groups (Figure 3) computed as


Here m and n are the sample size in females and males respectively.

Notice: If you use MAT file for whatever purpose, please reference
Chung et al. 2019 [1].

Figure 2. Average connectivity matrices of females (left) and males (right) between 116 AAL parcellations. The two-sample t-statistic result (female − male). Females have more structural connections between brain regions than males.

Figure 3. t-stat on combined sample of female and male. This is the average connectivity of whole population  (female and male combined) that account for group variability difference. This connectivity map can be used as a prior structural information, where functional connectivity can be overlaid somehow.

Figure 4. The t-stat map in Figure 2 was used to construct 3D graph representation of network differences.  Only the connections that are statistically significant (thresholded at −4.05 and 3.96) after multiple comparisons correction at 0.05 are shown. Females have more connections in most parts of the brain while males are more connected in the frontal regions of the brain. Rapid Acceleration of the Permutation Test via Transpositions


[1] Chung, M.K., Xie, L, Huang, S.-G., Wang, Y., Yan, J., Shen, L. 2019 Rapid acceleration of the permutation test via transpositions, International Workshop on Connectomics in NeuroImaging, Lecture Notes in Computer Science (LNCS) 11848:42-53. An earlier version with different application arXiv:1812.06696


This project is funded by NIH R01 EB022856 and R01 EB028753.