Diffusion Imaging allows us to probe the microstructural organisation of the living human brain and it is extensively used today to map connectivity and biological changes occurring in healthy brain development, as well as psychiatric or neurological disorders. However, most diffusion indices available today provide only an average description of the underlying microstructural organisation at the voxel level. Recently, a new generation of tract specific metrics based on spherical deconvolution (SD) and fiber orientation distribution functions (fODF) have been proposed^1,2,3. From the fODF, multiple fibre orientations can be extracted and looking at the amplitude of the recovered fODF, metrics like hindrance modulated orientational anisotropy (HMOA) or apparent fibre density (AFD) have been shown to be sensitive to differences in microstructural organization, fibre density and diffusion parameters of each resolved fiber^1,2. More recently, fitting single Bingham distributions directly on individual fODF lobes was suggested as a way to map fiber dispersion^3,4. Despite the huge potential of mapping dispersion, reliably extending Bingham fitting to multiple peaks has proven to be more challenging than expected³. In this study, a new approach is presented to obtain improved fiber dispersion estimates by simultaneously fitting multi-Bingham distributions to all fODF lobes. To evaluate the proposed method performance, numerical simulations were performed and in-vivo human data was used to map dispersion along white-matter tracts.

Multi-Bingham:The sum of $$$N$$$ multi-Bingham distributions is fitted to all fODF samples $$$f_i$$$ using a nonlinear procedure that solves the following problem:

$$$\min\sum_i\left[f_{i}-\sum_{n=0}^Nf_0^n\exp\left(-k_1^n\left(\overrightarrow{\mu_1}.\overrightarrow{\eta_i}\right)^2-k_2^n\left(\overrightarrow{\mu_2}.\overrightarrow{\eta_i}\right)^2\right)\right]$$$

where, for each Bingham distribution $$$n$$$, $$$f_0$$$ is the peak distribution amplitude, $$$k_1$$$ and $$$k_2$$$ are the concentration parameters along the = perpendicular axis of the distribution, $$$\overrightarrow{\mu_1}$$$ and $$$\overrightarrow{\mu_2}$$$ and $$$\overrightarrow{\eta}$$$ are the sampling directions of the fODF. To increase this nonlinear procedure performance, Bingham parameters were initialized to the values obtained from fitting a single Bingham distribution to individual fODF peaks. After the nonlinear convergence, the concentration parameters $$$k_1$$$ and $$$k_2$$$ are converted to dispersion angles $$$a_1 = \sin^{-1}\sqrt{1/(2k_1)}$$$ and $$$a_2 = \sin^{-1}\sqrt{1/(2k_2)}$$$3.

Simulations: Two crossing fibers with varying intersection angles were first simulated based on the multi-compartmental simulations⁶. The simulations' multi-compartmental model was then modified to take into account different levels of dispersion along the axis $$$\overrightarrow{\mu_1}$$$ for a single fiber population.

MRI data: A healthy subject was recruited for this study. MRI experiments were performed in a 3T GE Signa HDx TwinSpeed system (General Electric, Milwaukee, WI). Sixty axial diffusion-weighted images covering the whole brain were acquired along 60 diffusion-weighted directions (b-value = 3000 s/mm-2) and 7 non diffusion-weighted volumes, using a spin-echo single-shot echo-planar imaging (EPI) sequence (ASSET factor of 2). Other parameters were as following: voxel size = 2.4x2.4x2.4mm, matrix=128x128, field of view = 307x307 mm, TE=93.4 ms.

Data processing: For both simulated and MRI data fODFs were generated using the damped Richardson-Lucy spherical deconvolution algorithm⁵. Tractography was then performed along the fODF peaks and the corresponding Bingham parameters were mapped along all tractogram streamlines.

Graphical visualisation of simulated fODF and respective fitted sum of multi-Binghams distributions are shown in Figure 1. In Figure 2, dispersion measures are plotted as function of the real crossing angle (top panels) and as a function of the real dispersion angle (lower panels). In figure 3 the two dispersion indices $$$a_1$$$ and $$$a_2$$$ are mapped along the Inferior fronto-occipital fasciculus and the uncinated fasciculus.The major geometrical features of fODF profiles can be be well characterised by a sum of multiple Bingham distributions (Figure 1). As expected, our simulations showed that even when no dispersion is simulated fODFs always have an intrinsic dispersion angle. However, for simulated dispersion angles larger than 10^o the estimated dispersion angle quickly converge to precisely match the expected value of the fibers. Our simulations also showed that fitting single Bingham to the lobes of crossing fibers can sometimes generate also implausible high values of dispersion. On the contrary, by applying a multi-Bingham fitting, this provided a more stable and reliable solution across all configurations. On in-vivo human data, dispersion angle estimates seem to be consistent with the known anatomy of the selected tracts. For instance, larger angle estimates of $$$a_1$$$ seem to correctly correlate with macroscopic region of fanning while small and uniform values of $$$a_2$$$ with a single axis of dispersion of the bundle.In this study we have shown that it is possible to reliably fit fODF profiles by applying multi-Bingham fitting. Moreover, derived metrics of dispersion can be already applied to real in-vivo data as new tract specific indices and preliminary results shows a good agreement with known anatomical organisation.