The manipulation of spin magnetic moments is central to NMR and MRI. Via a wide variety of mechanisms, it is possible to excite signals of various kinds from samples placed in the scanner’s magnetic field by applying controlled RF pulses and magnetic field gradients. Since the behaviour of spins is affected by the chemical and physical environment they experience, NMR and MRI measurements provide a vector for analysing the chemical make-up and physical structure of objects or living organisms.

The Bloch equation describes the change in magnetisation of a continuum of spins in a field B, assuming that spins are stationary. It is an effective theory which captures smaller-scale effects like spin-spin and spin-lattice interaction via decay constants. In many situations, however, it is useful to consider the effects of diffusive motion on spin magnetisation, which can have an important effect on NMR

M.G. Hall, Ph.D (H)

Developmental Imaging and Biophysics, UCL GOS Institute of Child Health, University College London, London, UK e-mail:
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A. Fuster et al. (eds.), Computational Diffusion MRI, Mathematics

and Visualization, DOI 10.1007/978-3-319-54130-3_1

measurements. Spin diffusion was first incorporated by Torrey [40], who included an additional additive diffusion term.

Torrey’s derivation assumes that spins have only a very small drift velocity, but in more recent literature it is common to also include a linear flow term. The full form is

where M = (M_{x}, M_{y}, M_{z})^{T} is the local magnetisation, M_{0} is the equilibrium magnetisation, i, j, k are unit vectors defining the lab frame, B is the static scanner field, T and T_{2} are relaxation constants related to spin-lattice and spin- spin interactions respectively, and у is the gyromagnetic constant for the medium, D describes the local diffusivity, and u is a vector describing coherent flow.

The Bloch-Torrey equation led to the development of various pulse sequences which allow the diffusive term to be quantified (see, e.g. [11, 39]), which in turn has lead to the development of diffusion-weighted imaging (DWI).

Diffusive motion encodes information about the environment experienced by diffusing particles. This encoding, however, is non-trivial and extracting environmental information from measurements of diffusion, particularly when it happens in the presence of microstructure, is hugely challenging. Nevertheless, the fact that the length scales of diffusive motion over the timescale of a typical MR pulse sequence are orders of magnitude smaller than a typical scan voxel has led to considerable research effort into how best to analyse diffusion-weighted measurements. This in turn has lead to a large number of models and approaches to diffusion imaging which can be confusing to someone new to the field.

All of diffusion imaging is ultimately grounded in the Bloch-Torrey equation— different solutions provide the models used to analyse diffusion-weighted data. This chapter reviews how the Bloch-Torrey equation is derived, and shows how a minor and straightforward generalisation of the transport term provides a useful unifying principle which we can use to reveal the relationships between different models.

The Bloch-Torrey equation treats magnetisation as a continuum. The presence of the spatial derivative requires that M be smoothly varying in space. Similarly, the time derivative assumes that the continuum is changing smoothly with time. This is compatible with the concept of magnetisation as a vector field rather than being defined discretely on separate, point-like particles. We can think of this as a “local magnetisation”—the magnetisation per unit volume—given by

where are individual magnetisation vectors due to point charges in the applied field B, V is a small control volume over which the derivatives are smooth, and M is a smooth continuum approximating the underlying discrete magnetisation distribution. This can be derived by considering the conservation of mass as applied to a vector quantity via the continuity equation. Note that V is not typically associated with a scan voxel—it is a theoretical volume over which the continuum approximation holds. For experimental purposes it may be regarded as vanishingly small.

We will review diffusion MRI models starting with a derivation of the continuity equation for a vector quantity, showing how this leads to the Bloch-Torrey equation. We then show how different models can be seen as choices of transport mechanism, and then explore generalisations to multiple continua and models with boundary conditions. The aim is to give a comprehensive overview of the diffusion MRI modelling literature whilst also providing a physical basis for the assumptions made in each one.