Impact of intra‐axonal kurtosis on fiber orientation density functions estimated with fiber ball imaging

Abstract

To determine the impact of an intra-axonal kurtosis on estimates of the fiber orientation density function (fODF) obtained with fiber ball imaging (FBI).Theory and Methods: Standard FBI assumes Gaussian diffusion within individual axons and estimates the fODF by applying an inverse generalized Funk transform to diffusion MRI data for b-values of 4000 s/mm 2 or higher. However, recent work based on numeric simulations shows that diffusion inside axons is non-Gaussian with an intra-axonal kurtosis of ∼ 0.4. Here, the theory underlying FBI is extended to incorporate an intra-axonal kurtosis. This is done to first order in the intra-axonal kurtosis without making assumptions about the details of the diffusion dynamics and to all orders for a specific model based on a gamma distribution of diffusivities. The first order approximation is used to assess the effect of an intra-axonal kurtosis on FBI estimates for the fODF and axonal water fraction. The gamma distribution model is used to test the validity of the approximation. Results:The first order approximation indicates the estimated fODF is altered by a few percent for an intra-axonal kurtosis of 0.4 in comparison to predictions of standard FBI. If one neglects the intra-axonal kurtosis, the angular resolution of the point spread function for the fODF is changed by <1 • , whereas the axonal water fraction is overestimated by ∼ 5%. The gamma distribution model shows that the first order approximation is accurate to within a few percent. Conclusion:The intra-axonal kurtosis has a small impact on fODFs estimated with FBI.

The effects of axonal beading and undulation on axonal diameter estimation from diffusion MRI: Insights from simulations in human axons segmented from three‐dimensional electron microscopy

“…This Gamma‐distributed axial diffusivity leads to an additional contribution of axial kurtosis, 63 with an estimate of 1.16 for the fit in Figure 8b. This axial kurtosis in human axons of this study is substantially higher than the axial kurtosis 0.44 in mouse axons of Lee et al 20 This is due to the stronger caliber variations and undulations in the human EM sample compared with the mouse EM sample.…”

“…The distribution of local axial diffusivity is described by a Gamma distribution . Similarly, the spherical mean signal summed over all axonal segments is Substituting Equations (7) and (22) into the above equation, we obtain 63 where is a Gauss hypergeometric function. We fit the above functional form to the simulated spherical mean signals in real axons (Figure 8b) and obtain 2.59, 0.58 m 2 /ms, and an estimated radius < 0.001 m.…”

“…Substituting Equations ( 7) and ( 22) into the above equation, we obtain 63 F I G U R E 8 To explain the underestimated axon radius due to caliber variations in Figure 5, we suggest accounting for this observation by using a Gamma distribution for variations of local diffusivity (a) transverse to or (b) parallel to axons in the intra-axonal signals (7).…”