Among many of our achievements as a race, we humans have set foot on other planets, invented machines for our comfort, developed medicines and treatment to cure our ailments, and currently ruling the planet Earth; all these are possible because of the ingenuity of the human brain that is the most amusing and complex structure of the human body. The brain is the control centre of our thoughts, memory, intelligence, the regulator of various voluntary functions, and one of the reasons for the distinct personality of an individual. Given the utmost importance of this organ, the brain is guarded by a tough skull that protects it from injuries. Nevertheless, accidents can lead to traumatic injuries in the brain, causing death or rendering a person to a vegetative state.
Upon delving deep into the brain research, Prof. Krishna Kannan and Durga Prasad from IIT Madras have proposed an analysis-driven hyperelastic model that can help in calculating stress and strain experienced by the brain due to blunt force injury and lesion (tumour) growth. The new model developed by Prof. Krishna Kannan’s team at IIT Madras will help neurosurgeons and researchers to understand how the damage due to injuries in the brain progresses with time and devise strategies to treat the damaged brain.
“Tissues are live matter and to develop mathematical models for them, one ought to recognize that their response characteristics in vivo is very different from their response in vitro, and all the modelling, including the one that is carried out in this paper concerns the in vitro response of tissues. Tissues probably present the most challenging issues with regard to mathematical modeling” commented Prof. K.R. Rajagopal who is a Distinguished Professor at the Texas A&M University and expert in the same domain.
In finite hyperelasticity, the stress tensor associated with a deformable body is derived from a potential. Once the potential is known, one can calculate the stress and strain field of an elastic body, and that how the body deforms due to the application of forces. “One of the main challenges in hyperelasticity is deriving the mathematical form of the potential with the fewest number of material parameters, from either a molecular-statistical basis or a priori analysis using proven constitutive inequalities, which describe plausible deformation of a material,” says Krishna Kannan, Professor at the Department of Mechanical Engineering at IIT Madras who leads the research.
He further adds, “Brain tissue consists of many types of interacting macromolecules infused in a fluid. Therefore, statistical techniques are seldom applied to the brain and the other living tissues resulting in one using the constitutive inequalities. Inequalities due to Baker and Ericksen ensure that an elastic body deforms in a physically acceptable manner, in that the principal stresses and stretches are ordered in the same way. It is interesting to note that there are a few materials that exhibit a negative Poisson ratio and do not obey these inequalities. Brain tissue shows pronounced asymmetry of shear modulus during compression and tension, which poses a unique difficulty in deriving the hyperelastic potential.”
Overcoming these challenges, the team used a combination of experimental data and a priori analysis using Baker-Ericksen inequalities to condense the mathematical form of the potential. The results of this study are published in the prestigious Journal of the Mechanics and Physics of Solids. As the team suggests, the potential derived with four material parameters in this research exhibited better quantitative and qualitative behaviour when compared with a famous four-parameter Mihai-Ogden model.
“Prof. Krishna Kannan and his student Mr. Durga Prasad have simplified the constitution of the tissues to be a single constituent nonlinear elastic solid and have developed a model for describing the response of brain tissue. However, within tissues one has numerous blood vessels that are essential for the delivery of oxygen, that is, as mentioned earlier, the tissue is multi-constituent. Also, in such thin blood vessels, blood cannot be described by a well-studied constitutive relation such as the Navier-Stokes constitutive relation. Thus, to model the tissue as a single constituent nonlinear elastic body is a drastic simplification. However, this is a reasonable first step in a long walk that needs to be taken to describe the full complexity of the tissue. This study is a significant and crucial first step and as it stands has a much firmer basis than previous attempts at describing the brain tissue as a nonlinear elastic body. The paper is very useful addition to the literature and will be helpful in the future development of models for brain tissue” adds Prof. Rajagopal.
“The growth of a brain lesion (tumour) due to uptake of nutrients from the blood or in the case of epidural hematoma characterized by lenticular shaped pooled blood between the skull and the outer layer of dura mater, pushes the surrounding brain tissue and causes excessive extension and shearing of axons. By using the stresses in brain tissue computed using the mathematical model, one can post-process using diffusion tensor imaging data and obtain the shearing forces and extension of axons. An extension/shear of more than a critical value causes irreparable damage to the brain tissue causing impairment of brain functions. Such quantifiable information can be used to record the progression of the disease and provide the clinicians with input for a better prognosis. The timing of surgical intervention for the excision of a brain lesion can also be planned based on such data,” adds Prof. Kannan while explaining the compelling applications of his research.
The mathematical form of the elastic potential derived in this paper can provide useful inputs for a more general viscoelastic model of the brain tissue, wherein both the mechanical energy storage and viscous dissipation are considered. Short durational blunt blow to the head such as during an automobile accident or a soldier injured due to a bomb going off can cause Diffuse Axonal Injury (DAI), which usually is not detected by an MRI. Simulations using the in-silico model of the human head with the brain described as a viscoelastic material as suggested by Prof. Kannan and his team can help in understanding the degree of axonal damage and providing quantifiable information for neurologists and neurosurgeons.
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