Encoding of anisotropic diffusion with tetrahedral gradients: A general mathematical diffusion formalism and experimental results

A diffusion imaging method with a tetrahedral sampling pattern has been developed for high-sensitivity diffusion analysis. The tetrahedral gradient pattern consists of four different combinations of x, y, and z gradients applied simultaneously at full strength to uniformly measure diffusion in four different directions. Signal-to-noise can be increased by up to a factor of about three using this approach, compared with diffusion measurements made using separately applied x, y, and z gradients. A mathematical formalism is presented describing six fundamental parameters: the directionally averaged diffusion coefficient D̄ and diffusion element anisotropies η and ε which are rotationally invariant, and diffusion ellipsoid orientation angles θ, φ, and Ψ which are rotationally variant. These six parameters contain all the information in the symmetric diffusion tensor D. Principal diffusion coefficients, reduced anisotropies, and other rotational invariants are further defined. It is shown that measurement of off-diagonal tensor elements is essential to assess anisotropy and orientation, and that the only parameter which can be measured with the orthogonal method is D̄. In cases of axial diffusion symmetry (e.g., fibers), the four tetrahedral diffusion measurements efficiently enable determination of D̄, η, θ, and φ which contain all the diffusion information. From these four parameters, the diffusion parallel and perpendicular to the symmetry axis (D(is parallel with) and D(is perpendicular to)) and the axial anisotropy A can be determined. In more general cases, the six fundamental parameters can be determined with two additional diffusion measurements. Tetrahedral diffusion sequences were implemented on a clinical MR system. A muscle phantom demonstrates orientation independence of D, D(is parallel with), D(is perpendicular to), and A for large changes in orientation angles. Sample background gradients and diffusion gradient imbalances were directly measured and found to be insignificant in most cases.