Kevin GINSBURGER^{1}, Fabrice POUPON^{2}, Felix MATUSCHKE^{3}, Jean-François MANGIN^{2}, Markus AXER^{3}, and Cyril POUPON^{1}

We propose to extend the functionalities of the Diffusion Microscopist Simulator to design more realistic white matter phantoms without any input mesh and with few parameters. The biomimetic phantoms can represent crossing configurations with an arbitrary number of fiber populations, include a myelin sheath and Ranvier nodes and account for beading, tortuosity and angular dispersion of fibers.

The phantom generation algorithm takes a maximal number of $$$14N+2$$$ parameters to generate complex axonal geometries, where $$$N$$$ is the number of fiber populations. At each step, the absence of intersection between fibers is ensured.

First,
a set of oversimplistic fiber populations is constructed. Each
population is defined by an orientation, a Gamma distribution of
fiber envelopes radii and an intracellular
fraction, amounting to 4 parameters (fig. **1**). A population
corresponds to rectilinear and parallel fibers outer envelopes,
represented by control points with corresponding outer radii values.
In the case of multiple populations, the degree of interweaving of
fibers from different populations is ruled by 2 scale parameters controlling the distance between axons within the same population, thus
enabling to create aggregated structures or sheet
organizations^{6} where fibers find their way amongst other
populations.

The
second and third steps consist in inducing global orientation
dispersion (fig. **1**) and local tortuosity^{7} in the geometry
(fig. **2.**a), amounting to 2 parameters.

The axonal membrane is then
created within the fiber envelope with a radius computed from a
predefined $$$g$$$-ratio (ratio between the axonal
membrane and the outer fiber diameters). For each population,
the $$$g$$$-ratio follows a Gamma distribution (2 parameters). The
myelin sheath corresponds to the space between the axonal membrane and
the fiber outer envelope (fig. **2.**b).

The algorithm also accounts for
the presence of Ranvier nodes in the myelin sheath (fig. **3.**a). The
internode distance $$$d$$$ is set by the maximal conduction
relationship^{8} $$$ d / D = k g \sqrt{( \log( 1/g ) )}$$$ where
$$$k$$$ is some constant and $$$D$$$ is the external diameter of
myelin. The size of each Ranvier node is computed as a ratio of the
internodal length^{9}, following a Gamma distribution (2
parameters).

Our algorithm also represents beading caused by
cytoskeletal damage of the axon membrane: the contour of both axonal
and myelin cylinders is swollen using adequate functions like
sinusoidal lobes (fig. **3.**b). The amplitude and spacing of those
lobes both follow a Gamma distribution (4 parameters).

Figure
**4** represents configurations with one, two and three populations with
or without global angular dispersion (AD).
The
target $$$10°$$$ AD value is reached only in the single population case,
showing that the use of tortuosity is essential to reach high values
of AD in multiple population configurations since the maximum
attainable global AD strongly depends on the number of fiber
populations.
Indeed, in figure **5.**2, global AD enables to reach $$$5.6°$$$ of AD (for a
target of $$$10°$$$). The tortuosity induction (**5.**3) brings this AD up to
the $$$10°$$$ target . Figure **5.**4-5 illustrate the creation of myelin
sheath and Ranvier nodes, accounting for the actual structure of
myelinated fibers^{8,9}.
Beadings -appearing in ischemia^{10}- are also handled in figure
**5.**6, which presents a biomimicking geometry combining all the
putative deformations of membranes observed in real tissues.

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**8.** Rushton, W. A. H. (1951). A theory of the effects of fibre size in medullated nerve. The Journal of physiology, 115(1), 101-122.

**9.** Salzer, J. L. (1997). Clustering sodium channels at the node of Ranvier: close encounters of the axon–glia kind. Neuron, 18(6), 843-846.

**10.** Budde, M. D. and Frank, J. A. (2010). Neurite beading is sufficient to decrease the apparent diffusioncoefficient after ischemic stroke. Proceedings of the National Academy of Sciences 107, 14472–14477.