Here is a new random disks packing algorithm for numerical simulation of the white matter.
White matter tissue is divided in three compartments: axons, myelin sheath and extra-axonal space. Axons are assumed to be parallel cylinders, therefore the invariance along the fiber axis makes it possible to consider this problem in 2D. The dense packing of axons is thus equivalent to the generation of random 2-dimensional packing of N perfectly round and non-compressible disks. Axon diameter distributions follow a Gamma distribution (defined by its mean µ and variance σ2). Interestingly the g-ratio is fairly constant across species and white matter regions (31,32) and is dependent mostly on the diameter of the axon according to the relationship presented in (Ikeda M, Oka Y. Brain Behav. 2012): gratio= 0.220 * log(DIAMETER_unmyelinated) +0.508.
The different steps to process packing are the following: first, the diameters of the disks are randomly chosen using a gamma or lognormal distribution parameterized with the mean (d_mean), variance (d_var) and number of axons (N). Then, the positions of disks are initialized on a grid, and they migrate toward the center of the packing area until the maximum disk density is achieved.
The software packing provides microstructure features (Fiber Volume Fraction FVF, Myelin Volume Fraction MVF, Axon Volume Fraction AVF, fraction restricted FR).
1Institute of Biomedical Engineering, Polytechnique Montreal, Montreal, QC, Canada
2Functional Neuroimaging Unit, CRIUGM, Université de Montréal, Montreal, QC, Canada