Background and Objective: The brain, a mixture of neural and glia cells, vasculature, and cerebrospinal fluid (CSF), is one of the most complex organs in the human body. In order to understand brain responses to traumatic injuries and diseases of the central nervous system, it is necessary to develop accurate mathematical models and corresponding computer simulations which can predict brain biomechanics and help designing better diagnostic and therapeutic protocols. Brain tissue has been modeled as either a poroelastic mixture saturated by CSF or as a viscoelastic solid. However, it is still not obvious which model is more appropriate when investigating brain mechanics under certain conditions. In this study we propose to model the brain as a Kelvin-Voight (KV) model of a one-phase viscoelastic solid as well as a Kelvin-Voight-Maxwell-Biot (KVMB) model of a two-phase (solid and fluid) mixture and explore the limit between these two models. In order to account for the evolving microstructure in brain, we replace the classic integer order time derivatives by Caputo fractional order derivatives and thus generalize the above mentioned models to corresponding fractional KV and KVMB models. In these fractional models, the fractional order is a measure of the amount of microstructures in brain.
Modeling Approach: We start with the classic KVMB model where the brain is a poroelastic mixture made of two phases: 1). CSF, an incompressible Newtonian fluid, and 2). brain tissue which is assumed to be an incompressible, linear elastic solid. The model links the dynamic viscosity of CSF, the permeability and tortuosity of brain. We solve the corresponding system of first order linear differential equations using the eigenvalue method. These eigenvalues are then used to build an equivalent classic KV model where the brain is seen as a one-phase viscoelastic solid. Similarly, we propose the fractional KVMB model whose analytic solutions, found using a generalization of the eigenvalue method, are used to determine the equivalent fractional KV model. Inspired by Michaels' work in soil mechanics (Michaels, 2006), we use the displacement of the solid phase in the classic (fractional) KVMB model and the displacement in the classic (fractional) KV model to define a poroelastic-viscoelastic limit.
Results: When the CSF and brain tissue in the KVMB (classical/fractional) model have similar velocities, then the KV (classical/fractional) and KVMB (classical/fractional) models are indistinguishable. As the fraction order approaches 1, the analytic solutions of the fractional KVMB model converge to the analytic solutions of the classic KVMB model. Our numerical simulations provide threshold values for poroelastic-viscoelastic limits for the classic as well as fractional models.
Michaels, P. (2006). Relating Damping to Soil Permeability. International Journal of Geomechanics, 6(3), 158-165.