T- and T2-WeightedImaging

The simplest example is one where there is no flux present in the system, and only the Bloch terms contribute. In this case the equation reduces to the Bloch equation. In the vector form developed above, this is

The solution is a the matrix exponential in R. This can be written as which contains the familiar expressions describing T and T2 decay.

Diffusion Tensor Imaging

The next simplest case is Diffusion Tensor Imaging [7]. DTI is directly related to the canonical form of the Bloch-Torrey equation, which has a flux defined by the tensor version of Fick’s law for the local magnetisation i.e.

where D is the diffusion tensor.

The solution of the Bloch-Torrey equation with a tensorial Fickian flux is

It is common not to write the decay term explicitly and simply call the Bloch equation pre-factor S0 and the measured signal S. S is related to M via an integral over the local spin phase distribution, which is also implicit in the g-space approximation [33]. Determining the elements of D requires several measurements of this attenuation term. This is usually performed by formulating and inverting a linear system, the structure of which is derived from this solution (see, e.g. [30]).

From this we can see that the diffusion tensor follows simply from the flux. The form of the expression is the result of this assumption and the q-space approximation—an inverse Fourier Transform leads us back to a Gaussian spin displacement and time dependence. We also note that diffusion terms enter this equation entirely via the magnitude of the signal. Diffusion attenuates via a scalar term, rather than as an additional transformation on its orientation.

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