Constitutive Modelling of Brain Tissue for Prediction of Traumatic Brain Injury

Abstract

To develop protective measures for crash situations, an accurate assessment of injury risk is required. By using a Finite Element model of the head, the mechanical behaviour of the brain can be predicted for any acceleration and improved injury criteria can be developed and implemented into safety standards. Many head models are based on a detailed geometrical description of the anatomical components. However, for reliable predictions of injury, also an accurate constitutive model for brain tissue is required that is applicable for large deformations and complex loading conditions that occur during an impact to the head. This chapter deals with constitutive modelling of brain tissue. Different approaches towards modelling of the mechanical response of biological tissues are discussed. A short overview of the large strain behaviour of brain tissue and constitutive models that have been developed for this material is given. A non-linear viscoelastic model for brain tissue is then discussed in more detail. The model is based on a multi-mode Maxwell model and consists of a non-linear elastic mode in combination with a number of viscoelastic modes. For this model, also a numerical implementation scheme is given. The influences of constitutive non-linearities of brain tissue in numerical head model simulations are shown by comparing the performance of the model of Hrapko et al. with a simplified version, based on neo-Hookean elastic behaviour, and a third non-linear constitutive model from literature.

Appendix

In Sect. 4, a constitutive model for brain tissue was presented. In this appendix, a numerical integration scheme for this model based on explicit integration of Eq. 36 is given. For each increment, the state of the material, resulting from a given deformation and the solution at the end of the previous increment, is obtained from the following procedure:

• 1. Compute the deformation $\user2{F}(t)$ from the nodal displacements.

• 2. Compute

  $$\varvec{\sigma}^{\rm h}_{\rm e}(t) = K(J(t)-1)\user2{I}$$

• 3. Compute

  $$ \begin{aligned} \varvec{\sigma}_{\rm e}^{\rm d}(t) &= \frac{G_{\infty}}{\sqrt{I_{3}(t)}} \left[ (1 - A)\exp \left(-C\sqrt{b\tilde{I}_{1}(t) + (1-b)\tilde{I}_{2}(t) - 3}\right) + A \right]\\ & \quad \left[b\tilde{\user2{B}}^{\rm d}(t) - (1 - b) (\tilde{\user2{B}}^{-1}(t))^{\rm d} \right] \end{aligned}$$
• 4.

  • a. Retrieve $\user2{C}_{{\rm v}_i}(t-\Updelta t)$ and $\dot{\user2{C}}_{{\rm v}_i}(t-\Updelta t)$ from the previous time increment for each mode i = 1 to N.

  • b. For mode i = 1 to N, predict $\user2{C}_{{\rm v}_i}(t)$, $\user2{B}_{{\rm e}_i}(t)$, and $\varvec{\sigma}_{{\rm ve}_i}^{\rm d}(t)$

    $$\user2{C}_{{\rm v}_i}^{*}(t) = \user2{C}_{{\rm v}_i}(t-\Updelta t) + \dot{\user2{C}}_{{\rm v}_i}(t-\Updelta t)\Updelta t $$
    $$\user2{B}_{{\rm e}_i}^{*}(t) = \user2{F}(t)\cdot \user2{C}_{{\rm v}_i}^{*-1}(t) \cdot \user2{F}^{\rm T}(t) $$
    $$\varvec{\sigma}_{{\rm ve}_i}^{*{\rm d}}(t) = \frac{G_i}{\sqrt{I_{3}(t)}} \left[a\tilde{\user2{B}}_{{\rm e}_i}^{*{\rm d}}(t) - (1-a) (\tilde{\user2{B}}^{*-1}_{{\rm e}_i}(t))^{\rm d} \right] $$

  • c. Predict $\varvec{\sigma}^{\rm d}(t)$ and τ(t)

    $$\varvec{\sigma}^{*{\rm d}}(t) = \varvec{\sigma}_{{\rm e}}^{\rm d}(t) + \sum_{i=1}^{N}\varvec{\sigma}_{{\rm ve}_i}^{*{\rm d}}(t) $$
    $$\tau^{*}(t) = \sqrt{\frac{1}{2}\varvec{\sigma}^{*{\rm d}}(t):\varvec{\sigma}^{*{\rm d}}(t)} $$

  • d. For mode i = 1 to N, predict η _i (t), $\user2{D}_{{\rm v}_i}(t)$, and $\dot{\user2{C}}_{{\rm v}_i}(t)$

    $$\eta^{*}_i(t) = \eta_{\infty_i} + \frac{\eta_{0_i} - \eta_{\infty_i}} {1 + \left(\frac {\tau^{*}(t)} {\tau_0}\right)^{(n_i-1)}} $$
    $$\user2{D}_{{\rm v}_i}^{*}(t) = \frac{\varvec{\sigma}_{{\rm ve}_i}^{*{\rm d}}(t)}{2\eta_i^{*}(t)} $$
    $$\dot{\user2{C}}_{{\rm v}_i}^{*}(t) = 2 \user2{F}^{\rm T}(t) \cdot \user2{B}_{{\rm e}_i}^{*-1}(t) \cdot \user2{D}_{{\rm v}_i}^{*}(t) \cdot \user2{F}(t) $$

    and determine

    $$\user2{C}_{{\rm v}_i}(t) = \user2{C}_{{\rm v}_i}(t-\Updelta t) + \frac{1}{2} \left(\dot{\user2{C}}_{{\rm v}_i}(t-\Updelta t) + \dot{\user2{C}}_{{\rm v}_i}^{*}(t)\right)\Updelta t $$
    $$\user2{B}_{{\rm e}_i}(t) = \user2{F}(t)\cdot \user2{C}_{{\rm v}_i}^{-1}(t) \cdot \user2{F}^{\rm T}(t) $$
    $$\varvec{\sigma}_{{\rm ve}_i}^{\rm d}(t) = \frac{G_i}{\sqrt{I_{3}(t)}} \left[a\tilde{\user2{B}}_{{\rm e}_i}^{\rm d}(t) - (1-a) (\tilde{\user2{B}}_{{\rm e}_i}^{-1}(t))^{\rm d} \right] $$

    and store $\user2{C}_{{\rm v}_i}(t)$ for the next time increment.

  • e. Determine

    $$\varvec{\sigma}^{\rm d}(t) = \varvec{\sigma}_{\rm e}^{\rm d}(t) + \sum_{i=1}^N\varvec{\sigma}_{{\rm ve}_i}^{\rm d}(t)$$
    $$\tau(t) = \sqrt{\frac{1}{2}\varvec{\sigma}^{\rm d}(t):\varvec{\sigma}^{\rm d}(t)} $$

  • f. For mode i = 1 to N, determine

    $$\eta_i(t) = \eta_{\infty_i} + \frac{\eta_{0_i} - \eta_{\infty_i}} {1 + \left(\frac {\tau(t)} {\tau_0}\right)^{(n_i-1)}} $$
    $$\user2{D}_{{\rm v}_i}(t) = \frac{\varvec{\sigma}_{{\rm ve}_i}^{\rm d}(t)}{2\eta_i(t)} $$
    $$\dot{\user2{C}}_{{\rm v}_i}(t) = 2 \user2{F}^{\rm T}(t) \cdot \user2{B}_{{\rm e}_i}^{-1}(t) \cdot \user2{D}_{{\rm v}_i}(t) \cdot \user2{F}(t) $$

    and store $\dot{\user2{C}}_{{\rm v}_i}(t)$ for the next time increment.

• 5. Compute

  $$\varvec{\sigma}(t) = \varvec{\sigma}_{\rm e}^{\rm h}(t) + \varvec{\sigma}^{\rm d}(t) $$