Lattice Boltzmann Methods for Simulation of Diffusion-Weighted Magnetic Resonance Physics in Muscle or Brain
Diffusion tensor imaging allows in vivo measurement of the directional diffusion of water in muscle or brain. The correct interpretation of the results hinges on modeling microstructural information, water transport, and proton MR imaging physics. We have developed a class of Lattice Boltzmann methods to integrate the Bloch-Torrey partial differential equation, which governs this physics, in large domains containing cylindrical cells surrounded by complex extracellular matrix. Parallel implementation in a 8-core CPU computer system demonstrates linear speed-up, and plans are made to port this code to a multicore CPU/GPU platform.
Interface Problems and an Adaptive Time-Step Scheme
Many physical and biological problems involve interfaces separating different domains. To efficiently compute the dynamics of the interface, we develop an adaptive time-step scheme and test its performance in expanding/shrinking interface problems. The idea is to map the original time and space onto a new time and space such that the interface can evolve at an arbitrary speed in the new rescaled frame. In particular, for the expanding/shrinking interface problem, we choose (1) the space scaling function so that the expanding/shrinking interface is always mapped back to its initial size, i.e., the interface does not expand/shrink in the rescaled frame; (2) the time scaling function to speed up or slow down the motion of the interface, especially at later times when the interface expands slowly or shrinks extremely fast. We will also show some examples.