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Abstract
HIGHLIGHTS AxonPacking: Open-source software for simulating white matter microstructure.Validation on a theoretical disk packing problem.Reproducible and stable for various densities and diameter distributions.Can be used to study interplay between myelin/fiber density and restricted fraction.Quantitative Magnetic Resonance Imaging (MRI) can provide parameters that describe white matter microstructure, such as the fiber volume fraction (FVF), the myelin volume fraction (MVF) or the axon volume fraction (AVF) via the fraction of restricted water (fr). While already being used for clinical application, the complex interplay between these parameters requires thorough validation via simulations. These simulations required a realistic, controlled and adaptable model of the white matter axons with the surrounding myelin sheath. While there already exist useful algorithms to perform this task, none of them combine optimisation of axon packing, presence of myelin sheath and availability as free and open source software. Here, we introduce a novel disk packing algorithm that addresses these issues. The performance of the algorithm is tested in term of reproducibility over 50 runs, resulting density, and stability over iterations. This tool was then used to derive multiple values of FVF and to study the impact of this parameter on fr and MVF in light of the known microstructure based on histology sample. The standard deviation of the axon density over runs was lower than 10−3 and the expected hexagonal packing for monodisperse disks was obtained with a density close to the optimal density (obtained: 0.892, theoretical: 0.907). Using an FVF ranging within [0.58, 0.82] and a mean inter-axon gap ranging within [0.1, 1.1] μm, MVF ranged within [0.32, 0.44] and fr ranged within [0.39, 0.71], which is consistent with the histology. The proposed algorithm is implemented in the open-source software AxonPacking (https://github.com/neuropoly/axonpacking) and can be useful for validating diffusion models as well as for enabling researchers to study the interplay between microstructure parameters when evaluating qMRI methods.
Highlights
The white matter contains bundles of myelinated nerve cell projections
While some relationships can be derived using analytical equations (e.g., myelin content can be related to fiber volume fraction (FVF), assuming g-ratio is fixed), other mathematically complex relationships, such as the axon density as a function of axon diameter distribution, requires simulations
The disk density calculated in the red rectangle for this packing after 7000 iterations is FVF = 0.892, which is close to the theoretical values FVFHCP = 0.907 (2% error)
Summary
The white matter contains bundles of myelinated nerve cell projections (axons). Over the past years, Magnetic Resonance Imaging (MRI) has seen the development of quantitative metrics that can provide microstructural information about these axons, such as the myelin volume fraction (MVF), the intra-axonal volume fraction via the fraction of restricted water (fr), and the ratio of the inner to the outer diameter of the neuronal fibers (g-ratio) in white matter (Stanisz et al, 1997; Laule et al, 2007; Assaf et al, 2008; Fieremans et al, 2008, 2016; Alexander et al, 2010; Stikov et al, 2011). While some relationships can be derived using analytical equations (e.g., myelin content can be related to fiber volume fraction (FVF), assuming g-ratio is fixed), other mathematically complex relationships, such as the axon density as a function of axon diameter distribution, requires simulations. Assuming parallel fibers, which is typical for model-based quantitative diffusion metrics (Assaf et al, 2008), the geometrical simulation of a fiber bundle can be reduced to a two-dimensional polydisperse disks packing problem. Examples of applications are the modeling of granular media (Zhang and Makse, 2005; Isola, 2008) or powder and fluid (Bernal and Mason, 1960; Yu et al, 1997; Williams and Philipse, 2003), optimal arrangement of cylindrical products in a container (Dowsland, 1991) or electrical wires in a bundle (Sugihara et al, 2004) or lastly conformal mapping on a surface (Collins and Stephenson, 2003)
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