An Analytical Model of Traumatic Diffuse Brain Injury

Diffuse axonal injury (DAI) with prolonged coma has been produced in the primate using an impulsive, rotational acceleration of the head without impact. This pathophysiological entity has been studied subsequently from a biomechanics perspective using physical models of the skull-brain structure. Subjected to identical loading conditions as the primate, these physical models permit one to measure the deformation within the surrogate brain tissue as a function of the forces applied to the head. An analytical model designed to approximate these experiments has been developed in order to facilitate an analysis of the parameters influencing brain deformation. These three models together are directed toward the development of injury tolerance criteria based upon the shear strain magnitude experienced by the deep white matter of the brain. The analytical model geometry consists of a rigid, right-circular cylindrical shell filled with a Kelvin-Voigt viscoelastic material. Allowing no slip on the boundary, the shell is subjected to a sudden, distributed, axisymmetric, rotational load. A Fourier series representation of the load allows unrestricted load-time histories. The exact solution for the relative angular displacement (V) and the infinitesimal shear strain (ε) at any radial location in the viscoelastic material with respect to the shell was determined. The strain response for brain tissue (μ = 345 Poise, G = 1.38 × 10⁴ Pa) was examined at the non-dimensional radial location R = 0.3. The size (1.8<R₀<6.8 cm) and constitutive properties of the brain, and the magnitude, duration (2<T_D<100 msec) and waveform (sine, square, triangle) of the applied load were varied independently; each contributed to the response of the tissue. Strain increased linearly with increasing magnitude of the applied load and exponentially with increasing brain size (where the exponent n was frequency-dependent). Peak angular acceleration (Θ¨_p) and peak angular velocity (ΔΘ˙_p) were defined as appropriate load descriptors. As ΔΘ˙_p increased, strain rose rapidly and was independent of Θ¨_p, then levelled off and was highly sensitive to Θ¨_p.