Optimization of data acquisition and analysis for fiber ball imaging

Abstract

The inverse Funk transform of high angular resolution diffusion imaging (HARDI) data provides an estimate for the fiber orientation density function (fODF) in white matter (WM). Since the inverse Funk transform is a straightforward linear transformation, this technique, referred to as fiber ball imaging (FBI), offers a practical means of calculating the fODF that avoids the need for a response function or nonlinear numerical fitting. Nevertheless, the accuracy of FBI depends on both the choice of b-value and the number of diffusion-encoding directions used to acquire the HARDI data. To inform the design of optimal scan protocols for its implementation, FBI predictions are investigated here with in vivo data from healthy adult volunteers acquired at 3 T for b-values spanning 1000 to 10,000 s/mm², for diffusion-encoding directions varying in number from 30 to 256 and for TE ranging from 90 to 120 ms. Our results suggest b-values above 4000 s/mm² with at least 64 diffusion-encoding directions are adequate to achieve reasonable accuracy with FBI for calculating axon-specific diffusion measures and for performing WM fiber tractography (WMFT).

Keywords: Axon; Diffusion MRI; Fiber ball imaging; Fiber orientation density function; Funk transform; White matter.

Figure 1:

Logarithm of the mean ratio between the direction-averaged dMRI signal ($\overline{S}$) and the signal without diffusion weighting (S₀) over all WM voxels from the three subjects as a function of the logarithm of the ratio between a chosen reference b-value (b₁₀ ≡ 10,000 s/mm²) and the b-value (b) used to acquire the HARDI data. The leftmost data points correspond to b = b₁₀ while the rightmost data points correspond to b = 1000 s/mm². The theory underlying FBI predicts that the direction-averaged signal obeys a simple power law scaling of the form $\overline{S}$ ∝ b^−1/2, for large b-values. Thus, the data points should approach a line having a slope of one-half as the b-value is increased as is apparent for b ≥ 4000 s/mm² (four leftmost b-values). Some of the subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars indicate standard deviations.

Figure 2:

Measured harmonic power (p_l) for l = 2, 4, and 6 as a function of b-value, normalized by the power for l = 2 at b = 1000 s/mm² (p₂*), together with theoretical fits based on Equation (16). Each data point represents a mean for the WM voxels of an individual subject. The solid lines are global fits using the full dataset, while the dashed lines are restricted fits for just the data points with b ≥ 4000 s/mm². The global fits treat the intra-axonal diffusivity, D_a, as an adjustable parameter for each subject, whereas the restricted fits assume a literature value of D_a = 2.25 μm²/ms (Dhital et al., 2019). Thus, the full set of data for each subject was fit with only four adjustable parameters for the global fits and three adjustable parameters for the restricted fits. The theoretical predications are in reasonable accord with the experimental measurements, except for the restricted fits to some of the data points with b < 4000 s/mm² particularly for l = 2. These discrepancies may be due to contributions from extra-axonal water to the dMRI signal, which are not included in Equation (16). Error bars are omitted since the standard errors of the mean are small in comparison to the size of data symbols.

Figure 3:

Parametric maps of FBI-derived measures from one subject for a single anatomical slice. The columns, from left to right, show ζ , FAA, and color FAA maps for b-values ranging from 1000 to 10,000 s/mm². The calibration bar for ζ has units of ms^1/2/μm while FAA is dimensionless. The color FAA depicts directionality where the convention is left-right (red), anterior-posterior (green), and superior-inferior (blue). FBI predictions are only meaningful in WM regions and are expected to become more accurate with increasing b-value as the contributions from extra-axonal water are further reduced.

Figure 4:

Mean values for ζ and FAA over all WM voxels from the three subjects as functions of b-value. Both parameters are fairly constant for b ≥ 4000 s/mm² suggesting this to be approximately the minimum diffusion weighting needed for reasonable accuracy with FBI. The calculations utilized 128 diffusion-encoding directions. Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars indicate standard deviations.

Figure 5:

Voxel-wise standard deviations for ζ and FAA (σ_ζ and σ_FAA) averaged over all WM voxels as functions of b-value. These were calculated from numerical simulations based on adding Rician noise to the previously processed HARDI data from three subjects (Subjects 1-3). The voxel-wise standard deviations for ζ are relatively insensitive to the b-value, but they grow noticeably with increasing b-value for FAA. Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars reflect the spread over all WM of the voxel-wise standard deviations.

Figure 6:

Mean values for ζ and FAA over all WM voxels from a single subject (Subject 4) as functions of the number of diffusion-encoding directions with b = 6000 s/mm². The number of diffusion-encoding directions used were 30, 64, 128, and 256. Similar values are found with 64 directions and above indicating that 64 directions are sufficient for estimation of these two microstructural parameters. However, discrepant values are obtained when only 30 directions are used showing this to be inadequate sampling. Error bars signify standard deviations.

Figure 7:

Mean values for ζ and FAA over all WM voxels from a single subject (Subject 5) as function of the TE with b = 4000 s/mm² and 128 diffusion-encoding directions. For TE varying between 90 and 120 ms, little change in the parameter values is observed. Error bars signify standard deviations.

Figure 8:

Voxel-wise standard deviations for ζ and FAA (σ_ζ and σ_FAA) averaged over all WM voxels as functions of TE. These were calculated from numerical simulations based on adding Rician noise to the previously processed HARDI data from Subject 5. For both parameters, the voxel-wise standard deviations tend in grow with increasing TE. The error bars reflect the spread over all WM of the voxel-wise standard deviations.

Figure 9:

A diffusion ellipsoid calculated from DTI (b = 1000s/mm²), a dODF calculated from QBI (b = 6000s/mm²), and fODFs calculated from CSD and FBI (b = 6000s/mm²) for voxels with single and crossing fibers. The two FBI fODFs are calculated with D₀ = ∞ (uncorrected fODF) and with D₀ = 3.0 μm²/ms (corrected fODF). The QBI-derived dODF improves upon the DTI ellipsoid by resolving the fiber crossings while the fODFs of CSD and FBI are substantially sharper than the dODF. Note also that the corrected fODF is slightly sharper than the uncorrected fODF.

Figure 10:

The ODF estimate across b-value for the QBI dODF, the CSD fODF, and the uncorrected (D₀ = ∞) and corrected (D₀ = 3.0 μm²/ms) FBI fODFs. The ODFs depicted for each method comes from the same three voxels of Figure 9 containing single and crossing fibers. As b is increased, notice that the two FBI fODF variants converge to similar ODF representations by approximately b = 4000s/mm². There are minor observable differences as b is increased to 10,000 s/mm². The CSD fODF has sharp features, even in the low b-value regime around 2000 s/mm², and is as similar to the corrected FBI fODF. The QBI dODF is blurred until roughly b = 4000s/mm² where definite peak directions begin to show through.

Figure 11:

Difference between FAA values for the corrected (D₀ = 3.0 μm²/ms) and uncorrected (D₀ =∞) FBI fODFs as a function of b-value for three subjects. Due to sharper features, the FAA for the corrected fODF is always larger than for the uncorrected fODF. The difference between the fODF variants decreases as the b-value is increased. Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars indicate standard deviations.

Figure 12:

Fraction of WM voxels from three subjects with one, two, and three or more fODF peak directions as a function of b-value. The number of directions in each voxel was determined by using peak threshold factors of 10% and 40% of the maximum peak amplitude. At the 40% threshold, the voxel fractions are nearly constant for b ≥ 4000s/mm². For lower b-values, the voxel fractions deviate substantially, implying that the fODFs are not reconstructed accurately. For the 10% threshold, most of the WM voxels have three or more peak directions, but only a small number have three or more peak directions at the 40% threshold and b ≥ 4000s/mm².

Figure 13:

Mean angular difference (θ) between the principal fODF direction and a reference direction (principal direction at b = b₁₀ ≡ 10,000 s/mm²) over all WM voxels from each of three subjects as a function of b-value for maximum spherical harmonic degrees of l_max = 4, 6, and 8. The angular differences are all similar for 4000 s/mm² ≤ b ≤ 8000 s/mm² but increase for smaller b-values, depending on l_max. The l_max = 8 values tend to be larger, which may reflect the greater noise sensitivity of the higher degree harmonics. The data points for b = 10,000 s/mm² are zero by definition. Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars indicate standard deviations.

Figure 14:

Mean angular difference (θ) between the principal fODF directions as calculated with D₀ = 3.0μm²/ms (corrected fODF) and D₀ =∞ (uncorrected fODF) over all WM voxels from each of three subjects as a function of b-value for maximum spherical harmonic degrees of l_max= 4, 6, and 8. The angular differences are approximately two degrees or less for b ≥ 4000s/mm². This shows that the principal fODF directions are insensitive to the choice of D₀ for b ≥ 4000s/mm². Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars indicate standard deviations.

Figure 15:

Voxel-wise standard deviations for angular difference (σ_θ) between the principal fODF direction and a reference direction (principal direction at b = b₁₀ ≡ 10,000 s/mm²) averaged over all WM voxels from each of three subjects (Subjects 1-3) as a function of b-value. These were calculated from numerical simulations based on adding Rician noise to the previously processed HARDI data of the three subjects. The voxel-wise standard deviations are all similar for 4000 s/mm² ≤ b ≤ 8000 s/mm². The data points for b = 10,000 s/mm² are zero by definition. Some subject data points have been slightly displaced in the horizontal direction to improve readability, and the error bars reflect the spread over all WM of the voxel-wise standard deviations.

Figure 16:

Angular differences (θ) between the principal fODF direction and a reference direction (principal direction for 256 diffusion-encoding directions) averaged over all WM voxels from a single subject (Subject 4) as a function of the number of diffusion-encoding directions for l_max = 4, 6 and 8. Similar values are found with 64 and 128 directions, with little dependence on l_max, but substantially higher values are obtained when only 30 directions are used with l_max = 6 or 8. The data points for 256 directions are zero by definition. Error bars signify standard deviations.

Figure 17:

Angular differences (θ) between the principal fODF direction and a reference direction (principal direction for TE = 90 ms) averaged over all WM voxels from a single subject (Subject 5) as a function of TE for l_max = 4, 6 and 8. Similar values are found for all echo times other than for TE = 90 ms, which is zero by definition. There is also with little dependence on l_max. Error bars signify standard deviations.

Figure 18:

Voxel-wise standard deviations for angular differences (σ_θ) between the principal fODF direction and a reference direction (principal direction for TE = 90 ms) averaged over all WM voxels from a single subject (Subject 5) as a function of TE. These were calculated from numerical simulations based on adding Rician noise to the subjects’ previously processed HARDI data. Similar values are found for all echo times other than for TE = 90 ms, which is zero by definition. The error bars reflect the spread over all WM of the voxel-wise standard deviations.

Figure 19:

Average NI versus b-value for three subjects with maximum spherical harmonic degrees of l_max = 4, 6, and 8. The square data points are for the corrected fODF, while the circular data points are for the uncorrected fODF. A substantially lower NI is found for the uncorrected fODF. In all cases, the NI is small in comparison to one indicating that most fODF values are positive. That the NI is higher for the corrected fODF is likely a consequence of signal noise, the effects of which are amplified as D₀ is decreased. The error bars represent inter-subject standard deviations.

Figure 20:

Whole-brain, deterministic WMFT generated from DTI, QBI, CSD and FBI. Identical seed points, fiber tracking algorithms, and track termination criteria were used in all cases, so that any discrepancies are due solely to differences in the number and directions of the peaks identified by the four methods. FBI-based WMFT produces the most extensive set of fiber tracks, due to its greater sensitivity to fiber crossings, and appears similar whether the uncorrected (D₀ = ∞) or corrected (D₀ = 3.0 μm²/ms) fODF is used. The extent of the CSD-based WMFT is similar in breadth to the FBI-based WMFT. Noticeable differences between DTI-based, QBI-based and FBI-based WMFT are apparent, for example, in the corpus callosum (shown in red) where it crosses the superior longitudinal fasciculus (shown in green).