The human brain is one of the most complex structures in nature. After centuries of research, our understanding of its structure and function is still inadequate. Therefore, having an accurate constitutive model (force-deformation relation) of the brain would definitely help. Take the case of a traffic accident, where the head/brain is injured. It would definitely help if we had an accurate model of the brain in such cases.
The brain tissue is nonlinear and viscoelastic. Viscoelasticity refers to the property of a material to be both elastic (like a rubber band) and viscous (like honey). The elastic response is time-independent, and the viscous response is time-dependent.
One of the main features of the brain is its tension-compression asymmetry and greater shear stiffness (Tension refers to pulling/stretching, and compression refers to pushing/squeezing. Shear refers to the deformation of a material that causes a change in its shape). In the case of the brain, shear stiffness is more in compression than in tension.
Tension-compression asymmetry refers to the distinct responses of a material under tension and compression. This asymmetry is further manifested in the shear response of the brain tissue under different pre-stretch states.
Although existing models can capture some degree of asymmetry, accurately representing the pronounced tension-compression asymmetry of brain tissue remains a challenge.
The analysis of several models showed that simpler models failed to reproduce the experimentally observed behaviour. So, more elaborate four-parameter models were developed, in order to capture the asymmetry more effectively.
A model known as the Ogden model, which is a two-parameter model, is currently the best model that captures the asymmetry of the brain. Budday et al. (2017) confirmed this.
However, at higher compression levels, the Mooney asymmetry (Mooney asymmetry refers to a metric that quantifies the tension-compression asymmetry) of the Ogden model increases monotonically (changing in a consistent, one-directional manner), whereas experimental brain data indicate a bounded value. The proposed model, in contrast, correctly predicts a bounded asymmetry, in line with what experiments actually show.
This discrepancy highlights the need for improved models to capture the behaviour of the brain more accurately and with fewer parameters.
Therefore, in this study, the authors Mr. Mani Reddipaga and Prof. Krishna Kannan from the Department of Mechanical Engineering, Indian Institute of Technology (IIT) Madras, Chennai, India (Prof. Krishna Kannan is also affiliated with Center for Soft & Biological Matter, IIT Madras, Chennai, India), have proposed a new constitutive model for the brain and compared it with other models.
A central feature of this work was the construction of the constitutive model with explicit incorporation of tension-compression asymmetry and maximising it, while ensuring compliance with Baker-Ericksen inequalities (Baker-Ericksen inequalities are important conditions to be satisfied in the mechanics of elastic materials to ensure a physically reasonable response).
Brain tissue is also able to undergo deformations well beyond the linear limit in the physiological regime. Apart from nonlinear elasticity, brain tissue also exhibits time-dependent behaviour (the response depends on how fast it is deformed). Moreover, due to the viscoelastic nature of the brain tissue, some of the mechanical energy gets dissipated.
In this study, a finite deformation viscoelastic model has been developed within the thermodynamic framework proposed by Prof. K. R. Rajagopal, which derives the governing equations systematically from two scalar functions: a stored energy (which captures how the material deforms elastically) and a rate of dissipation (which accounts for how energy is lost as heat during deformation). This framework ensures that the model is consistent with the second law of thermodynamics, specifically that energy dissipation is always non-negative.
Model predictions showed improved agreement with the experimental data of Budday et al. (2017), outperforming the Budday-Ogden model while requiring fewer parameters. This demonstrates both the efficiency and descriptive power of the proposed model.
Prof. Namrata Gundiah, from the Department of Mechanical Engineering, Indian Institute of Science (IISc), Bengaluru, India, gave her analysis and pointed out the importance of the work done by the researchers with the following comments: “In a recent study, Reddipaga and Kannan developed a new hyperelastic stored-energy function based on multiaxial experimental data that addresses several limitations with existing models that require many fitting parameters. Their model is constructed using mathematical principles to ensure that the material behaves in a physically realistic manner under complex deformations. Importantly, their formulation captures the experimentally observed tension–compression asymmetry and introduces a nonlinear shear modulus under combined shear and axial loading, which is particularly relevant when studying traumatic brain injury and in designing surgical procedures. They also extend this formulation to incorporate finite-deformation viscoelasticity using the thermodynamic framework developed by (late) Prof. K. R. Rajagopal, where constitutive equations are derived systematically using a stored-energy function and a dissipation function while enforcing thermodynamic consistency. This framework incorporates evolving natural configurations to represent viscous effects and successfully captures the experimentally reported stress relaxation, hysteresis, and rate dependence in brain tissues. Overall, the work combines mathematical rigor, thermodynamic consistency, and improved predictive capability, to provide a more realistic representation of the brain tissue mechanics under complex loading conditions which should be useful to researchers in biomechanics, neuroscience, and computational modelling.”
Article by Akshay Anantharaman
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