Relationship between structural modeling and hyperelastic material behavior: application to CNS white matter

Recent measurements of the material properties of brain tissue allow an examination of the underlying microstructural basis in both physiological and pathophysiological conditions. The purpose of this study is to develop a mathematical relationship between microstructurally based models of the central nervous system (CNS) white matter and equivalent hyperelastic material models. For simplicity, time dependent material behavior is not included in this formulation. The microstructural representation is used to formulate structural property relationships for highly oriented white matter, and is mathematically compared to one isotropic and two anisotropic hyperelastic formulations. For the anisotropic characterizations, the population of axons in the white matter is assumed to align along one preferred direction of the material, yielding a transversely isotropic formulation. Relatively simple strain-energy functions incorporating material anisotropy provide sufficient flexibility to model the nonlinear behavior predicted from structurally based models, although the tangential stiffness of the hyperelastic approaches does not follow completely the behavior of the structurally based formulations. This analysis is an initial step towards linking microstructural aspects of the tissue to material models commonly used for large deformations, and may be an important step in relating predicted tissue deformation to the deformation and stress of cellular and subcellular structures.