The direct tensor solution and higher-order acquisition schemes for generalized diffusion tensor imaging

Abstract

Both in diffusion tensor imaging (DTI) and in generalized diffusion tensor imaging (GDTI) the relation between the diffusion tensor and the measured apparent diffusion coefficients is given by a tensorial equation, which needs to be inverted in order to solve the diffusion tensor. The traditional way to do this does not preserve the tensorial structure of the equation, which we consider a weakness in the method. For a physically correct measurement procedure, the condition number of the acquisition scheme, which is a determinant of the noise behavior, needs to be rotationally invariant. The method which traditionally is used to find such schemes, however, is cumbersome and mathematically unsatisfactory. This is considered a second weakness, closely connected to the first. In this paper we present an alternative inversion of the diffusion tensor equation, which does preserve the tensor form, for arbitrary order, and which is named the direct tensor solution ( DTS). The DTS is derived under the assumption that the apparent diffusion coefficient in any direction is known, i.e. in the infinite acquisition scheme. Whenever the DTS is valid for a given finite acquisition scheme and for a given order, the condition number is rotationally invariant. The DTS provides a compact, algebraic procedure to check this rotational invariance. We also present a method to construct acquisition schemes, for which the DTS is valid for the measurement of higher-order diffusion tensors. Furthermore, the DTS leads to other mathematical insights, such as tensorial relationships between diffusion tensors of different orders, and a more general understanding of the Platonic Variance Method, which we published before.