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Review Article

Constitutive Modeling of Brain Tissue: Current Perspectives

[+] Author and Article Information
Rijk de Rooij

Department of Mechanical Engineering,
Stanford University,
Stanford, CA 94305
e-mail: rderooij@stanford.edu

Ellen Kuhl

Department of Mechanical Engineering,
Stanford University,
Stanford, CA 94305
e-mail: ekuhl@stanford.edu

1Corresponding author.

Manuscript received September 1, 2015; final manuscript received December 29, 2015; published online January 18, 2016. Editor: Harry Dankowicz.

Appl. Mech. Rev 68(1), 010801 (Jan 18, 2016) (16 pages) Paper No: AMR-15-1100; doi: 10.1115/1.4032436 History: Received September 01, 2015; Revised December 29, 2015

Modeling the mechanical response of the brain has become increasingly important over the past decades. Although mechanical stimuli to the brain are small under physiological conditions, mechanics plays a significant role under pathological conditions including brain development, brain injury, and brain surgery. Well calibrated and validated constitutive models for brain tissue are essential to accurately simulate these phenomena. A variety of constitutive models have been proposed over the past three decades, but no general consensus on these models exists. Here, we provide a comprehensive and structured overview of state-of-the-art modeling of the brain tissue. We categorize the different features of existing models into time-independent, time-dependent, and history-dependent contributions. To model the time-independent, elastic behavior of the brain tissue, most existing models adopt a hyperelastic approach. To model the time-dependent response, most models either use a convolution integral approach or a multiplicative decomposition of the deformation gradient. We evaluate existing constitutive models by their physical motivation and their practical relevance. Our comparison suggests that the classical Ogden model is a well-suited phenomenological model to characterize the time-independent behavior of the brain tissue. However, no consensus exists for mechanistic, physics-based models, neither for the time-independent nor for the time-dependent response. We anticipate that this review will provide useful guidelines for selecting the appropriate constitutive model for a specific application and for refining, calibrating, and validating future models that will help us to better understand the mechanical behavior of the human brain.

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Figures

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Fig. 1

Special cases of incompressible, uniaxial tension/compression (left) and simple shear (right) for different hyperelastic elastic brain tissue models. The top and bottom figures illustrate the responses for small and large deformations. The material parameters are chosen such that ∂σ11/∂λ|λ=1 is the same for all the models. The remaining material parameters are chosen to be consistent with the experimental data.

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Fig. 2

Axial stress versus stretch curves for the Ogden model in incompressible, uniaxial tension/compression loading for varying model parameters N and αi. In tension, the Ogden model displays a strain softening if |αi|<1 for all i=1…N, and a strain stiffening otherwise. In compression, the model predicts a strain stiffening for all the values of αi.

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Fig. 3

Time-dependent behavior of the brain tissue under uniaxial loading. The brain tissue displays viscous effects and stress relaxation under constant deformation.

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Fig. 4

Multiplicative decomposition model and standard linear solid model. Every component of the deformation gradient is associated with a rheological element: F, Fe, and Fv are related to the main elastic network, the elastic spring, and the viscous damper. These elements require individual constitutive relations to define the elastic stress σe, the viscous stress σv, and the viscous stretch rate dv.

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Fig. 5

Hrapko model [15] and Bilston model [14] as examples of multiplicative decomposition models. Both models are popular viscoelastic models for the brain tissue based on multiple parallel configurations.

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Fig. 6

Prevost model [16] as example of multiplicative decomposition model. The Provost model is a popular viscoelastic model for the brain tissue based on four configurations.

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Fig. 7

History-dependent response of the brain tissue under uniaxial loading–unloading. The brain tissue displays a preconditioning effect that converges after approximately seven preconditioning cycles.

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Fig. 8

History-dependent response under uniaxial loading–unloading displaying the Mullins effect and residual strains similar to rubberlike materials

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