Continuity of Magnetisation: The Bloch-Torrey Equation

We will now show that the Bloch and Bloch-Torrey equations, which are fundamental to NMR and MRI experiments may be derived from the continuity of a vector quantity, and are in fact both particular cases of the continuity of magnetisation in the presence of a strong applied field.

The Bloch Terms as Sources and Sinks

The source and sink terms of the continuity equation describe the evolution of the quantity of interest that occurs independently of transport in the medium. A vector quantity may change not only magnitude but also orientation. In the present context, they capture the interaction between a spin’s magnetisation and the local magnetic field—exactly the role of the terms of the Bloch equation. We may therefore consider the Bloch equation as a set of source and sink terms for our vector continuity equation and write Eq. (11) as

We can immediately see the similarity between Eq. (12) and the traditional form of the Bloch-Torrey equation. Equation (12) describes the change in magnetisation of a continuous quantity in an applied magnetic field with an arbitrary transport process given by J(M). This is, of course, the same approach used by Torrey in deriving his equation in the first place. The difference being that Torrey does not start from the general form with an unspecified J(M) but instead proceeds directly to the drift- diffusion case [40].

We note that Eq. (12) employs unit vector notation. This is perfectly valid, but obscures the essentially simple nature of the equation. We can re-write the T and T_{2} terms in matrix form as

The Bloch terms also contain a precessional term, which takes the form of a cross product with the applied field. A cross product can also be written in matrix form. Adding this to Eq. (13) gives

Thus the Bloch equation can be expressed as a single matrix multiplication. This notation also allows us a layer of abstraction with regards coordinate system. The matrix R can be written in the lab frame, as here, or can be transformed into the rotating frame. By writing our systems (and solutions) in terms of R we are free to specify what coordinate frame we use separately. Equation (12) then becomes

Table 1 Flux term choices and corresponding diffusion models