New Approaches to Estimation of White Matter Connectivity in Diffusion Tensor MRI: Elliptic PDEs and Geodesics in a Tensor-Warped Space

Here we investigate new approaches to quantifying the white matter connectivity in the brain, as discernible from Diffusion Tensor Magnetic Resonance Imaging data. Our first approach finds a steady-state concentration/heat distribution using the three-dimensional tensor field as diffusion/conductivity tensors. Our second approach casts the problem in a Riemannian framework, considering each tensor as a local warping of space, and finding geodesic paths in the space. Both approaches use the information from the whole tensor, and can provide numerical measures of connectivity as well as estimated tracts for visualization.

Previous work has employed an iterative technique to create time-of-arrival maps of a heat diffusion front. Instead, we solve directly for the steady state concentration, , which can also be thought of as a heat distribution in the tensor field:

In the figure above, the temperature (left) and the steady-state flow magnitude (center) demonstrate the flow from the source to the sink. Here, the source is bright in the temperature image and the sink is dark. In the center flow image, dark means high flow magnitude. The grayscale image on the right, a non-diffusion-weighted image, shows the corresponding anatomy.

Regularized Stochastic White Matter Tractography Using Diffusion Tensor MRI

The development of Diffusion Tensor MRI has raised hopes in the neuro-science community for in vivo methods to track fiber paths in the white matter. A number of approaches have been presented, but there are still several essential problems that need to be solved. In this paper a novel fiber propagation model is proposed, based on stochastics and regularization, allowing paths originating in one point to branch and return a probability distribution of possible paths. The proposed method utilizes the principles of a statistical Monte Carlo method called Sequential Importance Sampling and Resampling (SISR).