Renaud Hedouin^{1}, Kwok-Shing Chan^{1}, Riccardo Metere^{1}, and Jose P Marques^{1}

This study presents the creation of 2D white matter models, based on real histologically derived axon shapes, with

**Introduction**

White Matter Model creation: The segmentation of an electron microscopy image of the spinal cord of a dog [3] was performed using AxonSeg opensource software [4], to obtain a collection of ~600K myelinated axons. To create each model, 400 axons were randomly picked and located in a large 1000x1000 grid. Subsequently, an in-house developed axon packing algorithm derived from [5] was applied to achieve the desired FVF (see Fig 1). In addition, a method to change the g-ratio of the model was used (see Fig 1) resulting in artificial but realistic 2D representation of WM including 3 compartments (axon, myelin, and extra-axonal) with expected axon diameter distribution [6] and different microstructure related parameter (FVF, g-ratio).

The radial arrangement of phospholipids in the myelin sheaths results in an anisotropic component (Xa) in addition to the isotropic component (Xi) of susceptibility (see Fig 2). Its tensorial form was calculated using a method, similar to [2], better able to cope with the non-cylindrical axon geometries. The field perturbation was computed taking into account the B0 orientation with respect to the generated model [7] and then, the complex signal evolution along TE was derived (see Fig 3) assuming a different relaxation time (T2*) and proton density (ρ) specific to each compartment [8].

The complex GRE signal evolution in WM can be used to estimate fiber orientation when the sample is rotated with respect to B0 with more than 7 directions [1]. Yet, DWI is the state-of-the-art method to compute fiber orientation and here, we take advantage of the signal variations due to orientation to map microstructure properties of WM.

The signal was simulated with 9 different orientations that were concatenated into one single vector Stotal as follow (see Fig 4):

Stotal= [θ1, abs(S1), phase(S1), … , θ9, abs(S9), phase(S9)]

where θi is the angle of the fibers with B0, abs(Si) and angle(Si) are the normalized magnitude and phase at 12 different TE’s acquired/simulated with the sample at the i-th orientation with respect to B0.

This signal vector is defined by 6 parameters: FVF, g-ratio, T2_myelin, T2_IntraExtraAxonal, Myelin Water Concentration (ρ) and Xi. The remaining parameters were fixed (Xa = -0.1 ppm, Water concentration of intra and extra water = 1). The dictionary is composed of ~20M vectors, including 8 repetitions with different geometric models created with similar FVF and g-ratio and between 5 and 10 entries for each parameter. Deep learning was performed on this dictionary using Keras [7]. The neural network (2 dense layers, loss function : mse) was trained for multi-regression of the 6 parameters on 7 models and tested on the last model.

[1] Wharton, Samuel, and Richard Bowtell. "Gradient echo based fiber orientation mapping using R2* and frequency difference measurements." Neuroimage 83 (2013): 1011-1023.

[2] Xu, Tianyou, et al. "The effect of realistic geometries on the susceptibility‐weighted MR signal in white matter." Magnetic resonance in medicine 79.1 (2018): 489-500.

[3] Cohen-Adad, et al. (2018, October 16). White Matter Microscopy Database. https://doi.org/10.17605/OSF.IO/YP4QG

[4] Zaimi, Aldo, et al. "AxonSeg: open source software for axon and myelin segmentation and morphometric analysis." Frontiers in neuroinformatics 10 (2016): 37.

[5] Mingasson, Tom, et al. "AxonPacking: an open-source software to simulate arrangements of axons in white matter." Frontiers in neuroinformatics 11 (2017): 5.

[6] Pajevic, Sinisa, and Peter J. Basser. "An optimum principle predicts the distribution of axon diameters in normal white matter." PLoS One 8.1 (2013): e54095.

[7] Liu, Chunlei. "Susceptibility tensor imaging." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 63.6 (2010): 1471-1477.

[8] Wharton, Samuel, and Richard Bowtell. "Fiber orientation-dependent white matter contrast in gradient echo MRI." Proceedings of the National Academy of Sciences (2012): 201211075.

[9] Chollet, François, et al. “Keras”, https://keras.io/

** Fig 1. Left:** Axon packing illustration. 400 axons randomly picked are regularly placed on a 1000x1000 grid, the extra-axonal space is represented in blue and the green axons are surrounded by their yellow myelin sheaths. The axons are attracted to the center of the image and repulse each other to avoid overlap. The packing process occurs to achieve a high FVF value (0,85).

**Right: WM Models.** Axons are randomly removed from the packed area to reach an expected FVF. Then, keeping the same myelinated axon shapes,the mean g-ratio is modified by dilatation/erosion of the myelin to obtain a model with expected FVF and g-ratio

**Fig 2. Left panel**: Top left figure represent myelin phospholipid orientation of one axon. The 3 other images represent the simulated field with various B0 orientation (polar angle θ and the azimuthal angle Ф) and fixed susceptibility values (Xi = - 0,1 ppm, Xa = -0,1 ppm).

**Right pane**l: Corresponding magnitude and phase of the signal with different θ and Ф values. As expected, there is an important signal variation as a function of θ, while the Ф has a smaller but non-negligible effect. This observation leads us to take the entire orientation of the B0 magnetic field (θ and Ф) into our model.