Bidirectional hyperelastic characterization of brain white matter tissue

Abstract

Biomechanical study of brain injuries originated from mechanical damages to white matter tissue requires detailed information on mechanical characteristics of its main components, the axonal fibers and extracellular matrix, which is very limited due to practical difficulties of direct measurement. In this paper, a new theoretical framework was established based on microstructural modeling of brain white matter tissue as a soft composite for bidirectional hyperelastic characterization of its main components. First the tissue was modeled as an Ogden hyperelastic material, and its principal Cauchy stresses were formulated in the axonal and transverse directions under uniaxial and equibiaxial tension using the theory of homogenization. Upon fitting these formulae to the corresponding experimental test data, direction-dependent hyperelastic constants of the tissue were obtained. These directional properties then were used to estimate the strain energy stored in the homogenized model under each loading scenario. A new microstructural composite model of the tissue was also established using principles of composites micromechanics, in which the axonal fibers and surrounding matrix are modeled as different Ogden hyperelastic materials with unknown constants. Upon balancing the strain energies stored in the homogenized and composite models under different loading scenarios, fully coupled nonlinear equations as functions of unknown hyperelastic constants were derived, and their optimum solutions were found in a multi-parametric multi-objective optimization procedure using the response surface methodology. Finally, these solutions were implemented, in a bottom-up approach, into a micromechanical finite element model to reproduce the tissue responses under the same loadings and predict the tissue responses under unseen non-equibiaxial loadings. Results demonstrated a very good agreement between the model predictions and experimental results in both directions under different loadings. Moreover, the axonal fibers with hyperelastic characteristics stiffer than the extracellular matrix were shown to play the dominant role in directional reinforcement of the tissue.

Appendix

1.1 Hyperelastic models

The plots of normalized Cauchy stress (i.e., the Cauchy stress divided by its corresponding maximum value at the maximum stretch level) obtained using the Ogden hyperelastic model and its special case, the neo-Hookean model with α = 2, are compared in Fig. 

Fig. 7

Normalized Cauchy stress variations obtained by Ogden and neo-Hookean hyperelastic models in comparison with the experimental data reported in Labus and Puttlitz (2016a) under uniaxial and biaxial tension in the axonal and transverse directions

7 with the experimental data (Labus and Puttlitz 2016a) in both the axial and transverse directions under uniaxial and equibiaxial tensile loadings. It is clear that the simple neo-Hookean model cannot capture the stiffening behavior of the WM tissue at larger deformations.

1.2 Statistical analyses

The normal probability, histogram, and versus fits and orders diagrams for three objective functions ${\text{OBJ}}_{{\text{U||}}}$, ${\text{OBJ}}_{{{\text{U}} \bot }}$, and ${\text{OBJ}}_{{\text{B}}}$ at the first iteration are shown in Fig. 

Fig. 8

Statistical adequacy evaluation diagrams including a normal probability plot, b histogram, c versus fits diagram, and d versus order diagram for three objective functions ${\text{OBJ}}_{{\text{U||}}}$, ${\text{OBJ}}_{{{\text{U}} \bot }}$, and ${\text{OBJ}}_{{\text{B}}}$ at the first iteration

8.

The coefficient of determination (R²) is a measure of proportional variation of the estimated stress values that are predictable from the experimental stress values and is defined as:

$$R^{2} = 1 - \frac{{\sum\nolimits_{i} {\left( {S_{i}^{{{\text{Exp}}}} - S_{i}^{{{\text{Fit}}}} } \right)^{2} } }}{{\sum\nolimits_{i} {\left( {S_{i}^{{{\text{Exp}}}} - \overline{S}} \right)^{2} } }}\quad (i = 1, \ldots ,n + 1)$$

(27)

where n is the number of divisions of the stretch range, and

$$\overline{S} = \left( {\frac{1}{n + 1}} \right)\sum\limits_{i} {S_{i}^{{{\text{Exp}}}} }$$

(28)

Normalized root-mean-square error (NRMSE) is a measure of normalized differences between the estimated stress values predicted by the model and the experimental stress values and is defined as:

$${\text{NRMSE}} = \frac{1}{{\overline{S}}}\sqrt {\left( {\frac{1}{n + 1}} \right)\sum\limits_{i} {\left( {S_{i}^{{{\text{Exp}}}} - S_{i}^{{{\text{Fit}}}} } \right)^{2} } }$$

(29)

1.3 Finite element micromechanical model—FEMM

The present FEMM is a three-dimensional RVE consisting of two sections including cylindrical axonal fibers of identical caliber of 0.5 μm uniformly distributed within the hollowed ECM section forming a unidirectional composite tissue with the fiber volume fraction of $\varphi \approx 32\%$, as depicted in Fig. 

Fig. 9

The present FEMM with meshed components

9.

These sections are completely tied together with coupled displacements at the commonly shared faces forming an affine boundary condition. Both components are assigned incompressible Ogden hyperelastic material models with different constants set to the optimum solutions obtained by the RSM optimizer at the last iteration. In addition, the geometry is entirely meshed by C3D8RH elements, the 8-node linear brick cubic elements with reduced integration and hybrid formulation considering the incompressibility condition. The analysis type is general static, and nonlinear geometry is available. Comprehensive studies on the mesh refinement and RVE edge sizes and independency of results are available in Yousefsani et al. (2018b).

Considering that repetition of the RVE along the main axes of the coordinate system shown in Fig. 9 must be able to mimic the entire tissue structure at macroscale, different sets of linear algebraic equations, known as the periodic boundary conditions (PBC), should be imposed to the paired boundary nodes of the opposite faces, edges, and corners of the RVE in order to couple the corresponding degrees of freedom and satisfy the conditions of deformation and orientation compatibility. General form of these equations is written as:

$$\left\{ {\begin{array}{*{20}l} {{\mathbf{u}}\left( {L_{X} ,Y,Z} \right) - {\mathbf{u}}\left( {0,Y,Z} \right) = {\mathbf{u}}^{X} } \hfill \\ {{\mathbf{u}}\left( {X,L_{Y} ,Z} \right) - {\mathbf{u}}\left( {X,0,Z} \right) = {\mathbf{u}}^{Y} } \hfill \\ {{\mathbf{u}}\left( {X,Y,L_{Z} } \right) - {\mathbf{u}}\left( {X,Y,0} \right) = {\mathbf{u}}^{Z} } \hfill \\ \end{array} } \right.$$

(30)

where u or u_j is the displacement vector of nodes on paired boundaries, ${\mathbf{u}}^{i} = u_{j}^{i} = \overline{F}_{ij} L_{j}$ stands for the displacements of the dummy nodes defined on the RVE boundaries for the application of external loads, L_j is the RVE edge length in j direction, and $\overline{F}_{ij} = {{\partial x_{i} } \mathord{\left/ {\vphantom {{\partial x_{i} } {\partial X_{j} }}} \right. \kern-\nulldelimiterspace} {\partial X_{j} }}$ represents the deformation gradient tensor with x and X indicating the current (deformed) and reference (undeformed) configurations, respectively. For further details, the interested readers are referred to Karami et al. (2009), Jiménez (2014), and Yousefsani et al. (2018a). It is noteworthy that at each increment of simulation, the strain energy density and volume of each element (which is constant due to incompressibility) were found by post processing the simulation results reports. The total strain energy of the RVE model was calculated by multiplying the strain energy density of each element by its volume and summation over the entire meshed domain. The strain energy density of the RVE then was calculated by dividing the total strain energy by its total volume.