Performance of orientation distribution function-fingerprinting with a biophysical multicompartment diffusion model

Abstract

Purpose: Orientation Distribution Function (ODF) peak finding methods typically fail to reconstruct fibers crossing at shallow angles below 40°, leading to errors in tractography. ODF-Fingerprinting (ODF-FP) with the biophysical multicompartment diffusion model allows for breaking this barrier.

Methods: A randomized mechanism to generate a multidimensional ODF-dictionary that covers biologically plausible ranges of intra- and extra-axonal diffusivities and fraction volumes is introduced. This enables ODF-FP to address the high variability of brain tissue. The performance of the proposed approach is evaluated on both numerical simulations and a reconstruction of major fascicles from high- and low-resolution in vivo diffusion images.

Results: ODF-FP with the suggested modifications correctly identifies fibers crossing at angles as shallow as 10 degrees in the simulated data. In vivo, our approach reaches 56% of true positives in determining fiber directions, resulting in visibly more accurate reconstruction of pyramidal tracts, arcuate fasciculus, and optic radiations than the state-of-the-art techniques. Moreover, the estimated diffusivity values and fraction volumes in corpus callosum conform with the values reported in the literature.

Conclusion: The modified ODF-FP outperforms commonly used fiber reconstruction methods at shallow angles, which improves deterministic tractography outcomes of major fascicles. In addition, the proposed approach allows for linearization of the microstructure parameters fitting problem.

Keywords: crossing fibers; diffusion MRI; fingerprinting; microstructure model fitting; multicompartment diffusion model; orientation distribution function; shallow angles; tractography.

FIGURE 1

Schematic illustration of a simplified hypothetical two-dimensional variant of ODF-FP applied for signal ODFs sampled at 36 gradient angles (pictured as gray radial lines). The surface plots show ODFs representing pairs of fibers crossing at arbitrary angles (10°, …, 80°), generated synthetically with SNR=10 (signal-to-noise ratio). The curves represent discretized input ODFs (blue) and their corresponding dictionary ODFs (red) found using pattern matching. This approach allows ODF-FP to identify fibers crossing at shallow angles below 40°, even when ODF peaks merge into one blob resembling a single fiber configuration.

FIGURE 2

Enumerated ODF-dictionary sizes grow exponentially with the number of samples taken per parameter of the diffusion model (illustrated on x-axes). The plots show memory requirements (in bytes) for storing enumerated ODF-dictionaries of the full model and its three simplifications (defined in Subsection 3.4) for N ≤ 2 fibers per voxel (left plot) or N ≤ 3 (right plot), assuming 321-point tessellation. The respective memory sizes of the randomized ODF-dictionaries with n = 10³, 10⁴, 10⁵, 10⁶ elements are shown for reference as green horizontal lines.

FIGURE 3

ODF-FP reached highest accuracy among all tested methods when identifying crossing angles below 40° in the synthetic data experiment. The plots show percent rates of correctly identified crossing angles under the error tolerance ε = 10°, 15°, 20°. The ground truth data are grouped into 9 intervals: 1 – 10°, 11 – 20°,…, 81 – 90°. The three variants of additive Rician noise with SNR=50, 20, 10 are presented in rows. In all cases, the ODF-dictionary contained 10⁵ elements generated with the full model. The number of fibers per voxel was N ≤ 2.

FIGURE 4

The ODF-dictionary sizes (left column) and most of the ODF-dictionary simplifications (right column) had little or no impact on the angular precision of ODF-FP in the synthetic data experiment. The box plots show the crossing angle reconstruction errors. The three variants of additive Rician noise with SNR=50, 20, 10 are presented in rows. All ODF-dictionaries were generated with the number of fibers per voxel N ≤ 2.

FIGURE 5

ODF-FP and the probabilistic approach were visibly more robust than other tested algorithms when reconstructing the number of fibers per voxel in the synthetic data experiment. The plots show percent rates of correctly identified voxels with N = 1, 2, 3 crossing fibers (respectively).

FIGURE 6

Results of fitting of the diffusivity parameters D_a,∥, D_e,∥, D_e,⊥ and the intra-axonal volume fraction f_in with ODF-FP in the synthetic data experiment. The plots present normalized root mean squared errors with standard deviations (RMSE±STD) when using the full model (a) and its three simplifications (b-d). The ODF-dictionaries contained 10³, 10⁴, 10⁵, or 10⁶ elements (respectively). The results are compared to the state-of-the-art fitting algorithms: MIX optimizer and Brute2Fine.

FIGURE 7

Cortical terminations of the fascicles were most abundantly reconstructed with ODF-FP, and slightly less with the probabilistic approach or CSD. The images show deterministic tractography reconstructions of the pyramidal tracts, arcuate fasciculus, and optic radiations virtually dissected in DSI Studio from an HCP subject in the original (1.25 mm isotropic) and reduced (2.5 mm isotropic) resolution, using 5 alternative techniques for determining fiber numbers and directions (presented in columns).

FIGURE 8

The low-resolution reconstructions of fibers of an HCP subject using CSD, the probabilistic approach, and ODF-FP reached highest agreement with the high-resolution reference data of the same subject (top-left slice) processed in DSI Studio. The first row of (a) presents the number of fibers per voxel as found by the tested approaches. The fibers reconstructed in the low-resolution data whose orientation matched the fibers in the respective high-resolution voxels are classified as true positives (TP; second row). The last two rows of (a) illustrate, respectively, the false positives (FP) and false negatives (FN) of this comparison. The relative TP and FP rates are plotted in (b) and (c)

FIGURE 9

Diffusivities and fraction volumes found with our method in vivo conform with the values reported in the literature. The microstructure parameters representing the dominating fiber of the full model $D_{a,\parallel }^{(1)}$, $D_{e,\parallel }^{(1)}$, $D_{e,\perp }^{(1)}$, $f_{in}^{(1)}$,p⁽¹⁾ and the free water fraction p_iso are given in rows. The second column from the left presents color maps plotted at a sample axial slice with color intensities determined by the respective intervals. The remaining three columns on the right show histograms calculated in the regions of genu and splenium of corpus callosum (CC) and the ensemble of WM. The diffusion-weighted images were provided by the Human Connectome Project.