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. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.