Simulations of diffusion in neural tissues have traditionally been limited to analytical solutions, to grid-based solvers, or to small-scale Monte Carlo simulations. None of these approaches has had the capability to simulate realistic complex neural tissues on the scale of even a single voxel of reasonable (i.e. clinical) size. An approach is described that combines a Monte Carlo Brownian dynamics simulator capable of simulating diffusion in arbitrarily complex polygonal geometries with a signal integrator flexible enough to handle a variety of pulse sequences. Taken together, this package provides a complete and general simulation environment for diffusion MRI experiments. The simulator is validated against analytical solutions for unbounded diffusion and diffusion between parallel plates. Further results are shown for aligned fibers, varying packing density and permeability, and for crossing straight fibers.

We present a second-order accurate algorithm for solving the free-space Poisson's equation on a locally-refined nested grid hierarchy in three dimensions. Our approach is based on linear superposition of local convolutions of localized charge distributions, with the nonlocal coupling represented on coarser grids. The representation of the nonlocal coupling on the local solutions is based on Anderson's Method of Local Corrections and does not require iteration between different resolutions. A distributed-memory parallel implementation of this method is observed to have a computational cost per grid point less than three times that of a standard FFT-based method on a uniform grid of the same resolution, and scales well up to 1024 processors.

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