The Continuity Equation for Vector-Valued Quantities

Conservation laws and conserved quantities are familiar concepts to every physicist. The idea is that the total amount of a particular quantity is constant. Whilst conservation is very general and useful, in many situations it is helpful to require that the distribution of the conserved quantity is smooth. The presence of jumps means that derivatives are not defined, and hence approaches to studying dynamics that rely on differential equations, such as the Bloch-Torrey equation, will fail. This extension of conservation to include local smoothness is called Continuity. Continuity is a widespread approach in physics. The first usage was probably by Euler who used the idea as early as 1757 [3]. This section describes how continuity is formalised for the vector-valued quantities of interest in MR physics.

We can think about this by considering the amount of a quantity in a particular control volume V. Let M(r) be the density of some vector quantity at a point r in space. The total amount in an arbitrary volume V at time t is then

Unlike a global conservation law, this quantity is not guaranteed to be constant. In fact, its rate of change can be readily defined as

This change can also be expressed in terms of the processes that cause it. The change in the total amount of a quantity in the volume V is given by the net amount of the quantity entering and leaving the volume, and also by the intrinsic change in the amount of quantity present due to its creation and destruction. Figure 1 illustrates these two processes—a process transferring a quantity into (or out of) a region is

Fig. 1 The two mechanisms by which the amount of a quantity of interest in a region V can change. (a) Transport in or out via a flux process J, or (b) intrinsic changes in the quantity itself via a source/sink process R. Black particles decay away (dotted circles), and new particles emerge (ringed circles)

known as a flux (a). A quantity may also change in the absence of movement (b) for example chemical reactions can occur or radioactive decay may transform one quantity into another. Let J(r, t) be the net flux of M in to/out of a point r and time t, and define the net intrinsic change in the quantity in the volume at time t as Ъ (t). This means we can write the net change in M_{net}(t) as

where S is the surface that bounds V. The source and sink process is described by a net additive vector, but the flux term is more subtle. In the scalar case the flux is a vector, but here we need to describe the transport of a vector quantity in a vector direction, necessitating a rank-two object. We will write this as J(M), giving

Equating Eqs. (4) and (6) we have

With a small amount of manipulation we can put this in a more convenient form. A generalisation of the divergence theorem using tensor contraction [4] gives

and the source term can be written in terms of a local density

Substituting these into Eq. (7) gives

Notice now that all three terms now contain volume integrals over V. Since V is arbitrary, the only way in which Eq. (10) can hold is if the integrands are equal, i.e.

This is the continuity equation for a vector quantity. It captures the changes in a conserved quantity in the presence of a flux, J and a source and sink process a. This is a very general equation, describing local conservation of a continuum tagged with a vector quantity which is not strongly coupled to the flux motion. As yet, we have made no assumptions about the nature of this quantity other than the fact that it is a vector. In MR physics we are interested in particles tagged with a vector quantity: Magnetisation. We shall see that looking at diffusion MRI through the lens of continuity is a powerful unifying principle which reveals different assumptions made in different models.