Home Computer Science Computational Diffusion MRI: MICCAI Workshop, Athens, Greece, October 2016

# Models with Boundary Conditions

The idea of multiple, non-exchanging continua can be extended to include assumptions about geometry. By imposing boundary conditions on the fluxes in each compartment it is possible to impose hard boundaries with certain shapes and orientations. Solutions for Fickian fluxes can be constructed over a unit interval, from which we can construct expressions for geometries such as cylinders, spherical shells, and parallel planes. These can be combined together with models such as the tensor in multi-compartment models which can then be fitted to the signal [36].

These models are solutions of multiple Bloch-Torrey equations with uniform relaxation and without exchange. Each compartment has a chosen boundary condition which is expressed parametrically in the expression for the signal. Superpositions of expressions for diffusion in multiple compartments with different geometries provides expressions for the signal which can be fitted to diffusion- weighted measurements. Of course, there is no guarantee that the combination of geometries chosen is appropriate for the tissue being measured, so care is required in both construction and interpretation of results.

Distributions as Compartments If we fix the relaxation terms across all the system, it is mathematically possible to add more and more compartments without limit—provided there is enough data to support all the parameters. In this case the signal can be written as

where fk is the volume fraction of the kth compartment, with signal Sk and subject to the constraint that J]f=1 fk = 1. This represents K non-exchanging compartments with any chosen geometry, all with the same T2. This approach can be extended to give access to a new class of compartment: one with a continuous distribution of compartments. The approach is to let K and change the sum to an integral,

In this case the signal has a contribution from a continuous set of compartments with a weight function describing the contribution from each member of the set. Here R(y) is the response function of an individual component compartment and f (x, y) is an unknown function describing their distribution. Different approaches make different assumptions about the form of this integral, although in some cases this is not immediately obvious.

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