Theory of Q-ball imaging redux: Implications for fiber tracking

Q-ball imaging is widely used to determine fiber directions for fiber tracking. In q-ball imaging the directional dependence of water diffusion in tissue is described by Tuch's orientation distribution function (ODF); a different function, the q-ball orientation distribution function, is measured using high angular resolution magnetic resonance diffusion imaging (HARDI). Tuch's ODF is assumed to be well approximated by the q-ball ODF. In this study it is shown that: 1) the q-ball ODF is not a good approximation to Tuch's ODF; 2) the properties of the q-ball ODF depend strongly on q, the area of the diffusion sensitization gradients; and 3) the q-ball ODF for a composite system is the weighted average of the q-ball ODFs for each subsystem, but the weighting factor is the product of the percent composition and a renormalization factor. In addition, a derivation is presented of the q-ball ODF for a system described by a Gaussian distribution and expressions are derived for both the dependence of the angular resolution on q and for the relation between the angular resolution and the signal loss. These findings might be useful in the design and interpretation of fiber-tracking experiments.