  Traumatic axonal injury occurs when loads experienced on the tissue-scale are
transferred to the individual axons. Mechanical characterization of axon
deformation especially under dynamic loads however is extremely difficult owing
to their viscoelastic properties. The viscoelastic characterization of axon
properties that are based on interpretation of results from in-vivo brain
Magnetic Resonance Elastography (MRE) are dependent on the specific frequencies
used to generate shear waves with which measurements are made. In this study,
we aim to develop a fractional viscoelastic model to characterize the time
dependent behavior of the properties of the axons in a composite white matter
(WM) model. The viscoelastic powerlaw behavior observed at the tissue level is
assumed to exist across scales, from the continuum macroscopic level to that of
the microstructural realm of the axons. The material parameters of the axons
and glia are fitted to a springpot model. The 3D fractional viscoelastic
springpot model is implemented within a finite element framework. The
constitutive equations defining the fractional model are coded using a
vectorized user defined material (VUMAT) subroutine in ABAQUS finite element
software. Using this material characterization, representative volume elements
(RVE) of axons embedded in glia with periodic boundary conditions are developed
and subjected to a relaxation displacement boundary condition. The homogenized
orthotropic fractional material properties of the axon-matrix system as a
function of the volume fraction of axons in the ECM are extracted by solving
the inverse problem.
