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 ADCijk to denote a diffusion map calculated from the data acquired along (Gx, Gy, Gz) = (i,j,k). For example, ADC110 is the ADC map measured with both Gx and Gy applied simultaneously. The elements of the diffusion tensor, on the other hand, will be denoted by ADCxx, ADCyy, ADCxy, etc. If only Gx 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: (Gx, Gy, Gz) = {(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 ADCm is a vector of the measured elements and ADCe is a vector of the diffusion tensor elements, and M is the transformation matrix relating ADCm and ADCe. 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.