Mapping spatiotemporal diffusion inside the human brain using a numerical solution of the diffusion equation

Diffusion is an important mechanism for molecular transport in living biological tissues. Diffusion magnetic resonance imaging (dMRI) provides a unique probe to examine microscopic structures of the tissues in vivo, but current dMRI techniques usually ignore the spatiotemporal evolution process of the diffusive medium. In the present study, we demonstrate the feasibility to reveal the spatiotemporal diffusion process inside the human brain based on a numerical solution of the diffusion equation. Normal human subjects were scanned with a diffusion tensor imaging (DTI) technique on a 3-T MRI scanner, and the diffusion tensor in each voxel was calculated from the DTI data. The diffusion equation, a partial-derivative description of Fickʹs law for the diffusion process, was discretized into equivalent algebraic equations. A finite-difference method was employed to obtain the numerical solution of the diffusion equation with a Crank–Nicholson iteration scheme to enhance the numerical stability. By specifying boundary and initial conditions, the spatiotemporal evolution of the diffusion process inside the brain can be virtually reconstructed. Our results exhibit similar medium profiles and diffusion coefficients as those of light fluorescence dextrans measured in integrative optical imaging experiments. The proposed method highlights the feasibility to noninvasively estimate the macroscopic diffusive transport time for a molecule in a given region of the brain.