Viscoelastic Modeling of Brain Tissue: A Fractional Calculus-Based Approach

Abstract

In recent years, the mechanical study of the brain has become a major topic in the field of biomechanics. A global biomechanical model of the brain could find applications in neurosurgery and haptic device design. It would also be useful for car makers, who could then evaluate the possible trauma due to impact. Such a model requires the design of suitable constitutive laws for the different tissues that compose the brain (i.e. for white and for gray matters, among others).

Numerous constitutive equations have already been proposed, based on linear elasticity, hyperelasticity, viscoelasticity and poroelasticity. Regarding the strong strain-rate dependence of the brain’s mechanical behaviour, we decided to describe the brain as a viscoelastic medium. The design of the constitutive law was based on the Caputo fractional derivation operator. By definition, it is a suitable tool for modeling hereditary materials. Indeed, unlike integer order derivatives, fractional (or real order) operators are non-local, which means they take the whole history of the function into account when computing the derivative at current time t.

The model was calibrated using experimental data on simple compression tests performed by Miller and Chinzei. A simulated annealing algorithm was used to ensure that the global optimum was found. The fractional calculus-based model shows a significant improvement compared to existing models. This model fits the experimental curves almost perfectly for natural strains up to − 0.3 and for strain-rates from 0.64s ^− 1 to 0.64 10^− 2 s ^− 1.