Simulation of diffusion MRI signal in biological tissue
which uses a combination of applied magnetic fields to measure,
statistically, the diffusion of water molecules due to Brownian motion.
Its spatial resolution is on the order of millimeters.
The cellular structure inside the human brain varies on the scale of micrometers,
which is much smaller than the size of a voxel.
There may be thousands of irregularly-shaped cells within a voxel,
and they all contribute to the environment seen by water
molecules whose displacement is measured by the MRI scanner. In the typical DMRI experiment,
the time interval over which water diffusion
is measured is in the range of 50-100 milliseconds.
Using the diffusion coefficient of 'free' water at 37 degrees Celsius, D = 3e-9 m2/s,
we get an estimated diffusion distance of 15-25 micrometers. Clearly, in a DMRI experiment,
water molecules encounter numerous times
inhomogeneities in the cellular environment, such as cell membranes, fibers, and macromolecules.
We want to simulate the DMRI signal at the scale of a single voxel,
while taking into account cellular structure
and the shape and duration of the diffusion gradients.
I will describe the Bloch-Torrey partial differential equation
which is a phenomenological equation that describes the time evolution
of the nuclear magnetization in a sample. I will talk about
the numerical solution of the Bloch-Torrey PDE using Green's functions
and alternatively by finite elements discretization. Finally,
the measured diffusion MRI signal is the sum of the
magnetization in the sample and I give some analytical results on
the aggregate signal under some simplying assumptions.