Modelling the non-linear viscoelastic behaviour of brain tissue in torsion

TL;DR

This work tackles the challenge of characterizing the non-linear viscoelastic response of brain tissue by combining a novel torsion-based testing protocol with the Modified Quasi-Linear Viscoelasticity (MQLV) model. A full set of relaxation parameters is estimated via simultaneous fitting to torque and normal force data across three twist rates, and results are validated through Finite Element simulations in FEniCS. The MQLV framework provides physically plausible parameter values consistent with prior torsion studies, and the dual-dataset torsion protocol offers a robust, broadly applicable approach for soft tissues beyond the brain. The findings have potential impact on traumatic brain injury modelling and protective equipment design, while enabling extension to other soft tissues in biomechanical research.

Abstract

Brain tissue accommodates non-linear deformations and exhibits time-dependent mechanical behaviour. The latter is one of the most pronounced features of brain tissue, manifesting itself primarily through viscoelastic effects such as stress relaxation. To investigate its viscoelastic behaviour, we performed ramp-and-hold relaxation tests in torsion on freshly slaughtered cylindrical ovine brain samples ($25\,\,\text{mm}$ diameter and $\sim 10\,\,\text{mm}$ height). The tests were conducted using a commercial rheometer at varying twist rates of $\{40,240,400\}\,\,\text{rad}\,\,\text{m}^{-1}\,\,\text{s}^{-1}$, with the twist remaining fixed at $\sim 88\,\,\text{rad}\,\,\text{m}^{-1}$, which generated two independent datasets for torque and normal force. The complete set of viscoelastic material parameters was estimated via a simultaneous fit to the analytical expressions for the torque and normal force predicted by the modified quasi-linear viscoelastic model. The model's predictions were further validated through finite element simulations in FEniCS. Our results show that the modified quasi-linear viscoelastic model - recently reappraised and largely unexploited - accurately fits the experimental data. Moreover, the estimated material parameters are in line with those obtained in previous studies on brain samples under torsion. When coupled with bespoke finite element models, these material parameters could enhance our understanding of the forces and deformations involved in traumatic brain injury and contribute to the design of improved headgear for sports such as boxing and motorsports. On the other hand, our novel testing protocol offers new insights into the mechanical behaviour of soft tissues other than the brain.

Figures (11)

• Figure 1: Procedure for preparing cylindrical brain samples of radius $12.5\,\,\text{mm}$ and height $10\,\,\text{mm}$ for testing: (a) long cylindrical sample excised from the cerebral hemisphere using a steel punch; (b) top face cut flat using a cutting guide of height $13\,\,\text{mm}$; (c) opposite face cut flat using a cutting guide of height $10\,\,\text{mm}$ and (d) flat cylindrical sample ready for testing.
• Figure 2: (a) Anton Paar MCR 302e rotational rheometer with parallel plate geometry used to perform the torsion tests and (b) side view of a twisted sample during testing.
• Figure 3: (a) Twist and (b) twist rate profiles for our proposed torsion testing protocol. The twist for the red, blue and orange data was increased linearly to a final value of $\phi_0=88\,\,\text{rad}\,\,\text{m}^{-1}$ over durations of $t^{\star} \in \{2.2,0.367,0.22\}\,\,\text{s}$, corresponding to twist rates of $\dot{\phi_0} \in \{40,240,400\}\,\,\text{rad}\,\,\text{m}^{-1}\,\,\text{s}^{-1}$ respectively. After reaching the final twist value, the twist was held constant for $200\,\,\text{s}$.
• Figure 4: Raw output data for sample $S_{16}$ from a torsion test performed at a twist rate of $240\,\,\text{rad}\,\,\text{m}^{-1}\,\,\text{s}^{-1}$: (a,b) twist and twist rate profiles; (c,e) measured torque and normal force for the first second of the test (including experimental artefacts) and (d,f) measured torque and normal force for the entire duration of the test (excluding experimental artefacts). Both the torque data generated when the upper plate was accelerating (black) and decelerating (red) were excluded from the proper torque data in (c), whereas only the normal force data generated when the upper plate was accelerating were excluded from the proper normal force data in (f). A dashed line indicates the end of the ramp phase.
• Figure 5: Representative torque, normal force and filtered data for samples (a) $S_2$, (b) $S_{16}$ and (c) $S_{24}$ from torsion tests performed at twist rates of $\{40, 240, 400\}\,\,\text{rad}\,\,\text{m}^{-1}\,\,\text{s}^{-1}$. The insets show the ramp phase and the initial part of the hold phase in more detail.