Fiber ball imaging

Abstract

By modeling axons as thin cylinders, it is shown that the inverse Funk transform of the diffusion MRI (dMRI) signal intensity obtained on a spherical shell in q-space gives an estimate for a fiber orientation density function (fODF), where the accuracy improves with increasing b-value provided the signal-to-noise ratio is sufficient. The method is similar to q-ball imaging, except that the Funk transform of q-ball imaging is replaced by its inverse. We call this new approach fiber ball imaging. The fiber ball method is demonstrated for healthy human brain, and fODF estimates are compared to diffusion orientation distribution function (dODF) approximations obtained with q-ball imaging. The fODFs are seen to have sharper features than the dODFs, reflecting an enhancement of the higher degree angular frequencies. The inverse Funk transform of the dMRI signal intensity data provides a simple and direct method of estimating a fODF. In addition, fiber ball imaging leads to an estimate for the ratio of the fraction of MRI visible water confined to the intra-axonal space divided by the square root of the intra-axonal diffusivity. This technique may be useful for white matter fiber tractography, as well as other types of microstructural modeling of brain tissue.

Keywords: Brain; Diffusion MRI; Fiber orientation density function; Funk transform; High-angular-resolution diffusion imaging; Q-ball imaging.

Figure 1

The function g_2l(bD_a) defined by Eq. (18). The solid lines show the exact results for l = 0, 1, 2, 3, and 4, while the dotted lines show approximations derived from Eq. (19). For large bD_a, g_2l approaches unity, but the drop off as bD_a is reduced is more rapid as l is increased. The approximation of Eq. (19) is quite accurate for bD_a ≥ 1. For l = 0, this approximation is simply g₀ = 1 and so coincides with the top of the plotting frame. The broken vertical line shows bD_a =1.

Figure 2

Numerical simulations of exact fODFs, for four different examples, together with the corresponding fiber ball, corrected fiber ball, and q-ball approximations, all shown with solid curves. The dotted arrows indicate the directions of the peaks (local maxima) for each of the angular profiles. The fiber balls and q-balls all give smoothed approximations of the fODFs. The fiber balls were obtained from the inverse Funk transform of the dMRI signal, which was calculated from the exact fODF by using Eq. (17) with bD_a = 5. The q-ball was found by applying a Funk transform to the same signal and is necessarily smoother than the fiber ball because of the intrinsic properties of the Funk transform. The corrected fiber ball was derived by using Eq. (23) with bD₀ = 12 to partially compensate for the smoothing effects of a finite b-value. For the last example (bottom row), the q-ball only detects one of the three peak directions.

Figure 3

In vivo fiber balls and q-balls for a white matter region (centrum semiovale) with crossing fibers. A: selected region on anatomical MPRAGE image. B: fiber balls calculated with the inverse Funk transform of Eq. (9). C: corrected fiber balls calculated from Eqs. (15) and (23) with bD₀ = 12. Note that the corrected fiber balls have somewhat sharper profiles than the uncorrected fiber balls. D: q-balls calculated with the Funk transform of Eq. (11). Fiber balls and q-balls were overlaid on the corresponding FA image, and the overall scalings were adjusted for visual clarity.

Figure 4

Axial maps of $\zeta \equiv f_a/\sqrt{D_a}$ for a single human subject as estimated with Eq. (14). White matter regions, in which fiber ball theory is expected to apply, have elevated values in comparison to gray matter regions. Gray matter also contains a substantial fiber component in the form of axons and dendrites, but it is less clear whether the assumptions underlying fiber ball imaging are accurately fulfilled. The scale bar is in units of ms^1/2/µm.

Figure 5

Histograms for ζ and η in white matter. These distributions give mean values of ζ = 0.453 ± 0.081 ms^1/2/µm and η = 1.129 ± 0.185 ms^1/2/µm.

Figure 6

Plot of ζ as a function of FA for the whole brain, showing a monotonic increase. The error bars indicate standard deviations.

Figure 7

Relative variances of Eqs. (B.10) and (B.11) for the uncertainties in the fiber ball spherical harmonic coefficients due to signal noise as a function of the degree l. The plots are for parameter choices that match those of the experiment performed in this study (N = 256, SNR = 6.3, bD₀ = 12). Notice the rapid increase for large l in the variance for the corrected fiber ball coefficients. For this reason, the estimates of the higher degree coefficients may be substantially less precise than for the lower degree coefficients.