Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

We introduce an original approach for the cerebral white matter connectivity mapping from diffusion tensor imaging (DTI). Our method relies on a global modeling of the acquired magnetic resonance imaging volume as a Riemannian manifold whose metric directly derives from the diffusion tensor. These tensors will be used to measure physical three-dimensional distances between different locations of a brain diffusion tensor image. The key concept is the notion of geodesic distance that will allow us to find optimal paths in the white matter. We claim that such optimal paths are reasonable approximations of neural fiber bundles. The geodesic distance function can be seen as the solution of two theoretically equivalent but, in practice, significantly different problems in the partial differential equation framework: an initial value problem which is intrinsically dynamic, and a boundary value problem which is, on the contrary, intrinsically stationary. The two approaches have very different properties which make them more or less adequate for our problem and more or less computationally efficient. The dynamic formulation is quite easy to implement but has several practical drawbacks. On the contrary, the stationary formulation is much more tedious to implement; we will show, however, that it has many virtues which make it more suitable for our connectivity mapping problem. Finally, we will present different possible measures of connectivity, reflecting the degree of connectivity between different regions of the brain. We will illustrate these notions on synthetic and real DTI datasets.