The subject of this study is the controversial choice of directions in diffusion tensor MRI (DT-MRI); specifically, the numerical algebra related to this choice. In DT-MRI, apparent diffusivities are sampled in six or more directions and a least-squares equation is solved to reconstruct the diffusion tensor. Numerical characteristics of the system are considered, in particular the condition number and normal matrix, and are shown to be dependent on the relative orientation of the tensor with respect to the laboratory frame. As a consequence, noise propagation can be anisotropic. However, the class of icosahedral direction schemes is an exception, and icosahedral directions have the same condition number and normal matrix for direction encoding as the ideal scheme with an infinite number of directions. This normal matrix and its condition number are rotationally invariant. Numerical simulations show that for icosahedral schemes with 30 directions the standard deviation of the fractional anisotropy is both low and nearly independent of fiber orientation. The recommended choice of directions for a DT-MRI experiment is therefore the icosahedral set of directions with the highest number of directions achievable in the available time.
Copyright 2003 Wiley-Liss, Inc.