Diffusion Tensor Magnetic Resonance Tomography and Cerebral White Matter Tractography

The technique of diffusion tensor imaging provides two unique insights into tissue microstructure: It quantifies non- invasively diffusion anisotropy of proton motion, which is a useful signum of cerebral white matter integrity, and provides an estimate of the principal direction of axon fib ers, which enables white matter tractography. The paper presents a novel approach to the magnetic resonance modality of diffusion tensor imaging which is based on the Lévy-Khintchin integral representation of the expectation value of Lévy processes thought of as random walks in continuous time, that is they are stochastic processes with independent and stationary increments. The proof is based on nilpotent harmonic analysis, contact geometry and the symbolic calculus of quantum field theory. Diffusion tensor imaging provides indirect insights into the brain microstructural characteristics of patients suffering of different forms of dementias, improving the comprehension of the underlying pathophysiological processes that result in macroscopic brain tissue loss over time.