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# The tensor distribution function.

^{1}, Zhu S, Zhan L, McMahon K, de Zubicaray GI, Meredith M, Wright MJ, Toga AW, Thompson PM.

### Author information

- 1
- Neuropsychiatric Hospital and LONI (Laboratory of NeuroImaging), University of California, Los Angeles, California 90095, USA. feuillet@ucla.edu

### Abstract

Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues.

- PMID:
- 19097208
- PMCID:
- PMC2770429
- DOI:
- 10.1002/mrm.21852

- [Indexed for MEDLINE]

### Publication types, MeSH terms, Grant support

#### Publication types

#### MeSH terms

- Algorithms*
- Brain/anatomy & histology*
- Computer Simulation
- Data Interpretation, Statistical
- Diffusion Magnetic Resonance Imaging/methods*
- Humans
- Image Enhancement/methods*
- Image Interpretation, Computer-Assisted/methods*
- Imaging, Three-Dimensional/methods*
- Models, Neurological
- Models, Statistical
- Nerve Fibers, Myelinated/ultrastructure*
- Reproducibility of Results
- Sensitivity and Specificity
- Statistical Distributions

#### Grant support

- R01 EB007813/EB/NIBIB NIH HHS/United States
- R01 HD050735-02/HD/NICHD NIH HHS/United States
- R01 EB008281-12/EB/NIBIB NIH HHS/United States
- R01 EB007813-02/EB/NIBIB NIH HHS/United States
- U54 RR02183/RR/NCRR NIH HHS/United States
- R01 HD050735/HD/NICHD NIH HHS/United States
- U54 RR021813-05/RR/NCRR NIH HHS/United States
- U54 RR021813/RR/NCRR NIH HHS/United States
- R01 EB008281/EB/NIBIB NIH HHS/United States

### LinkOut - more resources

#### Full Text Sources

- Wiley
- Europe PubMed Central - Author Manuscript
- Ovid Technologies, Inc.
- PubMed Central - Author Manuscript