Harmonic viscoelastic response of 3D histology-informed white matter model

White matter (WM) consists of bundles of long axons embedded in a glial matrix, which lead to anisotropic mechanical properties of brain tissue, and this complicates direct numerical simulations of WM viscoelastic response. The detailed axonal geometry contains scales that range from axonal diameter (microscale) to many diameters (mesoscale) imposing an additional challenge to numerical simulations. Here we describe the development of a 3D homogenization model for the central nervous system (CNS) that accounts for the anisotropy introduced by the axon/neuroglia composite, the axonal trace curvature, and the tissue dynamic response in the frequency domain. Homogenized models that allow the incorporation of all the above factors are important for accurately simulating the tissue's mechanical behavior, and this in turn is essential in interpreting non-invasive elastography measurements. Geometric and material parameters affect the material properties and thus the response of the brain tissue. More complex, orthotropic, or anisotropic material properties are to be considered as necessitated by the 3D tissue structure. An assembly of micro-scale 3D representative elemental volumes (REVs) is constructed, leading to an integrated mesoscale WM finite element model. Assemblies of microscopic REVs, with orientations based on geometrical reconstructions driven by confocal microscopy data are employed to form the elements of the WM model. Each REV carries local material properties based on a finite element model of biphasic (axon-glial matrix) unidirectional composite. The viscoelastic response of the microscopic REVs is extracted based on geometric information and fiber volume fractions calculated from the relative distance between the local elements and global axonal trace. The response of the WM tissue model is homogenized by averaging the shear moduli over the total volume (thus deriving effective properties) under realistic external loading conditions. Under harmonic shear loading, it is proven that that the effective transverse shear moduli are higher than the axial moduli when the axon moduli are higher than the glial. Methodologically, the process of using micro-scale 3D REVs to describe more complex axon geometries avoids the partition process in traditional composite finite element methods (based on partition of finite element grids) and constitutes a robust algorithm to automatically build a WM model based on available axonal trace information. Analytically, the model provides unmatched simulation flexibility and computational power as the position, orientation, and the magnitude of each tissue building block is calculated using real tissue data, as are the training and testing processes at each level of the multiscale WM tissue.