Tract-based quantitative white matter fiber analysis: Mathematical frameworks

This is a standalone post. It is also part of the Quantitative white matter fiber analysis
series posted here.


There are three steps in tract-based data analysis: pre-processing, modeling and statistical analysis. The pre-processing may involve registration, tractography segmentation or clustering and the information contained in these results is often incorporated into the data analysis model. Fractional anisotropy (FA) is one such artifact and the analysis of FA profiles along tracts is usually included in most studies. The quantitative frameworks use both simple ideas such as the alignment of fibers for statistical analysis and more advanced mathematical concepts such as currents or techniques in Riemannian geometry.
These methods for tract-based analysis are listed below:

1) Fiber-tract oriented statistics
This model by Corouge et al.[1] is a prototype for computing statistics along a fiber bundle. The core idea was to give a compact description of the geometry and diffusion properties of a fiber bundle along its length profile. Corresponding points for a set of fibers are aligned and averages of DTI indices along the cross-section are computed. This can be done with linear or nonlinear Riemannian spaces.
This basic idea has also been extended to a population atlas where information contained in tractography results[2] or registration results[3] are evaluated along a tract-based coordinate system.

2) Statistical modeling and EM clustering of white matter fiber bundles
Individual fibers are parameterized and aligned with the goal of performing tract-oriented analysis over a population. A statistical model of pre-segmented fiber bundles is then calculated which serves as a prior for mixture model clustering. The Expectation-Maximization algorithm is used to infer membership probabilities and cluster parameters. Atlas guided clustering gives anatomically meaningful bundles. This is work by Maddah et al. [4].

3) Statistical model of white matter fiber bundles based on currents
This is a flexible framework where fiber bundles (a collection of curves) are modeled as currents. The space of currents is a vector space equipped with an inner product and norm which defines the distance between two bundles. This is a global distance that does not use point-wise correspondences [5]. Durrleman et al. give a statistical model for a fiber bundle atlas and it variability in a population.

4) Comprehensive Riemannian Framework
In this framework, Mani et al.[6] use various combinations of shape, scale, orientation and position, the physical features associated with white matter fibers, to define Riemannian feature spaces. This is useful since applications have different objectives and often need different sets of tools and metrics for optimal results. For each joint manifold, a (geodesic) distance metric that quantifies differences between fibers as well as tools for computing statistical summaries of samples are defined. Correspondences between fibers are implicitly established during these pairwise comparisons. The framework may be extended to accomodate features such as scalar diffusion indices.

5) Statistical mapping of medial models of white matter tracts
Yushkevich et al.[7] use a different geometric model by giving major fiber tracts a medial representation. While this is strictly not a tract-based model, it is included here because local statistics of tensor-based features can be computed.

Bibliography:

1) Isabelle Corouge, P.Thomas Fletcher, Sarang Joshi, Sylvain Gouttard, Guido Gerig, "Fiber Tract-Oriented Statistics for Quantitative Diffusion Tensor MRI Analysis," Medical Image Analysis 10 (2006), pp. 786-798.

2) Lauren J. O'Donnell, Carl-Fredrik Westin and Alexandra J. Golby. "Tract-Based Morphometry for White Matter Group Analysis," NeuroImage. Volume 45, Issue 3, (2009), pp. 832-844.

3) Casey B. Goodlett, P. Thomas Fletcher, John H. Gilmore, and Guido Gerig. "Group Analysis of DTI Fiber Tract Statistics with Application to Neurodevelopment," NeuroImage. (2009), 45(1 Suppl): S133–S142.

4) Maddah M., Grimson W.E.L., Warfield S.K. "Statistical Modeling and EM Clustering of White Matter Fiber Tracts", Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging. (2006);1, pp. 53-56.

5) M. Vaillant and J. Glaunès, "Surface matching via Currents", Proceedings of Information Processing in Medical Imaging, Lecture Notes in Computer Science vol. 3565, Springer (2005), pp. 381–392.

6)Meena Mani, Sebastian Kurtek, Christian Barillot, Anuj Srivastava. "A Comprehensive Riemannian Framework for the Analysis of White Matter Fiber Tracts," In ISBI'2010, pp. 1101-1104.

7) Yushkevich P.A., Zhang H., Simon T.J., Gee J.C., "Structure-specific statistical mapping of white matter tracts," (2008) NeuroImage, 41 (2), pp. 448-461.