Several MR methods have been presented that aim to characterize the tissue microstructure using diffusion-weighted MRI. Most of these methods are based on a conventional diffusion weighting applied in various directions with different strengths. During the past few years, interest in so-called double-diffusion-encoding (DDE) or double-wave-vector (DWV) experiments [1, 2] has increased, in particular because these experiments show a signal behavior that is specific for restricted or locally anisotropic diffusion. Thus, these experiments could provide more direct insight into tissue microstructure than conventional experiments, especially when targeting axon diameters [3]. A previous DDE-based approach to estimate axon diameters was limited to a known fiber orientation, assumed that only cylindrical cells are present within the tissue, and used a coarse approximation of the MR signal to obtain axon diameters [3]. In this study, this approach is extended (i) to be applicable without prior knowledge of the fiber orientation, (ii) by considering a more complex tissue composition including spherical cells to model glial cells and a freely diffusing compartment for extracellular space, and (iii) using the multiple correlation function (MCF) framework [4, 5] that provides a more accurate approximation of the MR signal.

A 3D gradient combination scheme based on 18 directions obtained from three orthogonal circles (each sampled with 8 directions) was applied with parallel and antiparallel orientations yielding 36 combinations in total.

The MCF algorithm was implemented in Matlab (The Mathworks Inc.). The tissue model involved cylinders, spheres, and an unrestricted compartment with isotropic diffusion using the cylinder and sphere radii and volume fractions, the cylinder orientation (azimuthal and polar angle), and the diffusion coefficient of the unrestricted compartment as free parameters. For the diffusion coefficient within the restricted compartments a fixed value of 2.4 µm² ms^-1 was used in accordance with literature values [6] that were adjusted to human body temperature. To test the feasibility of the approach, numerical simulations of the diffusion signal for a variety of model parameters were performed using a random-walk algorithm.

Healthy volunteers were investigated with a 3 T whole-body MR system (Magnetom TIM Trio) using a 32-channel head coil after their informed consent was obtained. DDE experiments were performed with b values of 125 s mm^-2 and 250 s mm^-2 per diffusion weighting and an echo-planar imaging readout (voxel size 3.0×3.0×3.0 mm³) covering 25 slices (gap 1.5 mm) yielding an echo time (TE) of 215 ms and using a repetition time (TR) of 6 s. The total acquisition time (TA) was 29.6 min for 8 averages.

The signal curves obtained with the numerical simulations could be very well approximated with the MCF model (Fig. 1). The fit parameters derived were in very good agreement with the simulation parameters (well below 1° and 5%) indicating a correct MCF algorithm and fit routine.

Figure 2 shows the signal curves for the 36 direction combinations in two regions-of-interest. In the pyramidal tracts, the data clearly show higher signal amplitudes for the antiparallel combinations which reflects the presence of restricted diffusion, e.g. within the cells. The agreement of the MCF fit with the data is reasonable and corresponds to a cylinder and spherical radii of 4.9 µm and 12.1 µm, respectively. For the splenium of the corpus callosum the difference between parallel and antiparallel is less pronounced which is typical for smaller cell sizes (fit parameters 2.6 µm and 10.4 µm, respectively). Taking into account a shrinkage of the cells due to the preparation process for the histology and that the acquired data represent a volume-weighted average of the size distribution present, these results could be considered to be in a good agreement with the values that could be derived from the literature (effective axon radius around 3.7 µm and 1.0µm in the pyramidal tracts and the corpus callosum, respectively) [7, 8]. Maps and histograms of the diameters and volume fractions for the cylinders (modelling axons) and spheres (modelling glial cells) are presented in Fig. 3 and 4, respectively.

In conclusion, a previous DDE-based approach to estimate axon diameters has been (i) generalized to be applicable to arbitrary fiber orientations, (ii) refined to provide a more accurate signal modelling with the MCF formalism, and (iii) extended to consider the influence of spherical cells like glial cells.