Calculation of Tensor elements

Following data acquisition, you calculate an ADC map for each direction. However, these maps are not immediately the elements of the diffusion tensor. In order to keep track of the different ADC maps, let's introduce the notation ADC_ijk to denote a diffusion map calculated from the data acquired along (G_x, G_y, G_z) = (i,j,k). For example, ADC₁₁₀ is the ADC map measured with both G_x and G_y applied simultaneously. The elements of the diffusion tensor, on the other hand, will be denoted by ADC_xx, ADC_yy, ADC_xy, etc. If only G_x is applied, then

and so on. If, however, more than one gradient is applied simultaneously, then the expression will contain the off-diagonal elements of the diffusion tensor. The relationship between for example ADC₁₁₀ and the elements of the tensor is given by

Consider the gradient scheme is the dual-echo scheme commonly used: (G_x, G_y, G_z) = {(1,1,0), (1,0,1), (0,1,1), (-1,1,0), (-1,0,1), (0,-1,1)}. A listing of all six measured ADC maps expressed in terms of the tensor elements is:

This can also be expressed in matrix form:

or, more compressed,

where ADC_m is a vector of the measured elements and ADC_e is a vector of the diffusion tensor elements, and M is the transformation matrix relating ADC_m and ADC_e. The transformation matrix depends only on the diffusion directions that are applied. The elements of the tensor is then calculated via the inverse relationship:

For a square matrix, taking the inverse is straight-forward. For a non-square matrix (i.e., when more than 6 directions are applied), the inverted matrix can be estimated by a least squares formalism:

Go to the next step: Diagonalization.