Mathematical Models for Analysis of Diffusion Weighted MRI
Speaker: Baba C. Vemuri , University of Florida, Gainesville, FL
Date: October 10 2008
Time: 2:00PM to 3:00PM
Location: 32-D507
Host: Polina Golland, CSAIL
Contact: Polina Golland, x38005, polina@csail.mit.edu
Relevant URL: Diffusion weighted MRI (DW-MRI) is a non-invasive imaging technique
that allows the measurement of water molecular diffusion through
tissue in vivo. The directional features of water diffusion allow one
to infer the connectivity patterns prevalent in tissue and possibly
track changes in this connectivity over time for various clinical
applications. There are two fundamental quantities that one is
normally interested in computing for analyzing DW-MRI data sets, they
are: (i) water molecule displacement probability function P(r) and
(ii) the diffusivity function. The former is needed in estimating the
fiber orientations required in tractography and the latter is needed
in computing clinically significant quantities such as mean
diffusivity, generalized anisotropy etc. I will first present a novel
statistical model for representing the sensed signal in the presence
of complex local geometries in neural tissue. This model is based on
representing the signal at each voxel as a continuous mixture model
and solving the resulting integral in closed-form leading to a
generalization of the now widely used Stejskal-Tanner diffusion tensor
model of signal decay. P(r) is then easily computed as the Fourier
transform of this novel signal representation and its maxima
correspond to the fiber orientations that can be used in tractography.
In the second part of the talk, I will present the mathematics of high
rank tensors, specifically, rank 4 tensors and present a novel and
full characterization of the space of symmetric positive definite rank
4 tensors along with an algorithm to estimate them from diffusion
weighted MRI data sets. This is achieved via the use of Hilbertu2019s
theorem on ternary quartics (that are equivalent to the rank 4
tensors) in conjunction with Iwasawa parametrization that leads to an
efficient estimation algorithm for these tensors. I will present
several experimental results on synthetic and real data sets
demonstrating the efficacy of these models and associated algorithms
via application to DW-MRI data from rat brains and spinal cords
respectively.
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