# Modeling and Simulation of Head Trauma Utilizing White Matter Properties from Magnetic Resonance Elastography

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. MRE Acquisition and Inversion

#### 2.2. Finite Element Mesh Generation

#### 2.3. Material Properties

#### 2.4. Interface and Boundary Conditions

#### 2.5. Experimental Verification

## 3. Results and Discussion

#### 3.1. Simulation of Impact

#### 3.2. Stochastic Wave Propagation

#### 3.3. Limitations

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution of Loss (

**top**) and Storage (

**bottom**) modulus in the Finite Element (FE) model. Darker regions indicate higher magnitude of shear modulus.

**Figure 2.**(

**a**) Voxel-based finite element mesh segmented into skull (red), cerebrospinal fluid (CSF) (yellow), gray matter (gray) and white matter (white). (

**b**) Surface of cerebral cortex with sulci and gyri clearly resolved.

**Figure 3.**Input force time history (adapted from [17]).

**Figure 5.**Comparison of displacements in the $x\u2013$direction for two positions in the anterior (

**A**) and posterior (

**P**) to the Hardy C383-T1 experiment.

**Figure 6.**Comparison of displacements in the $z\u2013$direction for two positions in the anterior (

**A**) and posterior (

**P**) to the Hardy C383-T1 experiment.

**Figure 7.**Shear wave propagation due to frontal impact. Notice the attenuation of the wavefront as time progresses.

**Figure 8.**Comparison of pressure at three points on the sagittal plane. The difference in the heterogeneous and homogeneous models is most evident in regions of high relative stiffness.

**Figure 9.**Comparison of attenuation of pressure wave along the sagittal plane between homogeneous and heterogeneous model. (

**a**) At 3 ms and (

**b**) at 7 ms.

**Table 1.**Average values and standard deviations for different tissues within the model: Gray Matter (GM), White Matter (WM), Corpus Callosum (CC), Corona Radiata (CR).

GM | WM | CC | CR | |
---|---|---|---|---|

G${}^{\prime}$ (kPa) | $2.02\pm 0.09$ | $2.66\pm 0.30$ | $3.09\pm 0.39$ | $2.78\pm 0.37$ |

G${}^{\u2033}$ (kPa) | $1.04\pm 0.12$ | $1.54\pm 0.15$ | $1.23\pm 0.26$ | $1.97\pm 0.12$ |

$\mathbf{\xi}$ | $0.32\pm 0.03$ | $0.31\pm 0.03$ | $0.23\pm 0.07$ | $0.37\pm 0.05$ |

Tissue | Mass Density (kg/m${}^{3}$) | Bulk Modulus K (Pa) | Shear Modulus G (Pa) |
---|---|---|---|

Skull [25] | 2070 | 3.61 × ${10}^{9}$ | 2.7 × ${10}^{9}$ |

Grey Matter | 1040 | Hyperviscoelastic | |

White Matter | 1040 | Hyperviscoelastic | |

Mass Density (kg/m${}^{3}$) | Young’s Modulus E (Pa) | Poisson Ratio | |

CSF [26] | 1000 | 160 | 0.49 |

**Table 3.**Hyper-viscoelastic material properties [29].

‘Compliant’ | ‘Average’ | ‘Stiff’ | |
---|---|---|---|

${\mu}_{1}$ (Pa) | 26.9 | 53.8 | 107.6 |

${\mu}_{2}$ (Pa) | −60.2 | −120.4 | −240.8 |

${\alpha}_{1}$ | 10.1 | 10.1 | 10.1 |

${\alpha}_{2}$ | −12.9 | −12.9 | −12.9 |

${G}_{1}\left(kPa\right)$ | 160 | 320 | 640 |

${G}_{2}\left(kPa\right)$ | 39 | 78 | 156 |

${G}_{3}\left(kPa\right)$ | 3.1 | 6.2 | 12.4 |

${G}_{4}\left(kPa\right)$ | 4.0 | 8.0 | 16.0 |

${G}_{5}\left(kPa\right)$ | 0.05 | 0.10 | 0.20 |

${G}_{6}\left(kPa\right)$ | 1.5 | 3.0 | 6.0 |

${\beta}_{1}(1/s)$ | ${10}^{6}$ | ${10}^{6}$ | ${10}^{6}$ |

${\beta}_{2}(1/s)$ | ${10}^{5}$ | ${10}^{5}$ | ${10}^{5}$ |

⋮ | ⋮ | ⋮ | ⋮ |

${\beta}_{6}(1/s)$ | ${10}^{1}$ | ${10}^{1}$ | ${10}^{1}$ |

**Table 4.**Total excursions (in mm) for Hardy’s C383-T1 experiment compared to the predicted values for two models.

Location | Experiment | Homogeneous | Heterogeneous |
---|---|---|---|

A1 | 9.24 | 7.47 | 9.02 |

A2 | 8.04 | 4.22 | 6.88 |

P1 | 12.42 | 7.66 | 8.76 |

P2 | 9.80 | 4.01 | 10.14 |

**Table 5.**Difference of material properties and peak pressure, displacement response for three distinct points (indicated in Figure 8) along the sagittal plane within the white matter.

Location | % Difference in Shear Modulus (${\mathit{G}}_{\mathit{\infty}}$) | % Difference in Peak Pressure | % Difference in Peak Displacement |
---|---|---|---|

1 | 12.11 | −9.12 | −3.16 |

2 | 24.75 | −13.91 | −6.62 |

3 | 18.67 | −29.05 | −12.96 |

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**MDPI and ACS Style**

Madhukar, A.; Ostoja-Starzewski, M.
Modeling and Simulation of Head Trauma Utilizing White Matter Properties from Magnetic Resonance Elastography. *Modelling* **2020**, *1*, 225-241.
https://doi.org/10.3390/modelling1020014

**AMA Style**

Madhukar A, Ostoja-Starzewski M.
Modeling and Simulation of Head Trauma Utilizing White Matter Properties from Magnetic Resonance Elastography. *Modelling*. 2020; 1(2):225-241.
https://doi.org/10.3390/modelling1020014

**Chicago/Turabian Style**

Madhukar, Amit, and Martin Ostoja-Starzewski.
2020. "Modeling and Simulation of Head Trauma Utilizing White Matter Properties from Magnetic Resonance Elastography" *Modelling* 1, no. 2: 225-241.
https://doi.org/10.3390/modelling1020014