THIS 22.1

DAMAGE Criterion (DMG)

The DAMAGE value is determined by creating a second-order system that mimics the behavior of a physical mass-spring-damper setup. This model is used to capture the maximum strain exprienced by the brain when the head rotates around each axis. 

The equation of motion for this damped, three-degree-of-freedom mechanical system, which is influenced by external forces acting on the masses, can be represented in matrix form:

\[ \begin{bmatrix} m_x & 0 & 0 \\ 0 & m_y & 0 \\ 0 & 0 & m_z \end{bmatrix} \begin{bmatrix} \ddot{\delta}_x \\ \ddot{\delta}_y \\ \ddot{\delta}_z \end{bmatrix} + \begin{bmatrix} c_{xx} + c_{xy} + c_{xz} & -c_{xy} & -c_{xz} \\ -c_{xy} & c_{xy} + c_{yy} + c_{yz} & -c_{yz} \\ -c_{xz} & -c_{yz} & c_{xz} + c_{yz} + c_{zz} \end{bmatrix} \begin{bmatrix} \dot{\delta}_x \\ \dot{\delta}_y \\ \dot{\delta}_z \end{bmatrix} + \begin{bmatrix} k_{xx} + k_{xy} + k_{xz} & -k_{xy} & -k_{xz} \\ -k_{xy} & k_{xy} + k_{yy} + k_{yz} & -k_{yz} \\ -k_{xz} & -k_{yz} & k_{xz} + k_{yz} + k_{zz} \end{bmatrix} \begin{bmatrix} \delta_x \\ \delta_y \\ \delta_z \end{bmatrix} = \begin{bmatrix} m_x & 0 & 0 \\ 0 & m_y & 0 \\ 0 & 0 & m_z \end{bmatrix} \begin{bmatrix} \ddot{u}_x \\ \ddot{u}_y \\ \ddot{u}_z \end{bmatrix} \]

Here, 

\( m = \text{mass}, \quad c_{ij} = \text{damping coefficient}, \quad k_{ij} = \text{stiffness coefficient} \)
\( \delta, \dot{\delta}, \ddot{\delta} = \text{displacement, velocity, acceleration} \)
\( \ddot{u} = \text{applied angular acceleration} \)
\( m_x = 1 \, \text{kg}, \quad m_y = 1 \, \text{kg}, \quad m_z = 1 \, \text{kg} \)
\( k_{xx} = 32142 \, \text{N/m}, \quad k_{yy} = 23493 \, \text{N/m}, \quad k_{zz} = 16935 \, \text{N/m} \)
\( k_{xy} = 0 \, \text{N/m}, \quad k_{yz} = 0 \, \text{N/m}, \quad k_{xz} = 1636.3 \, \text{N/m} \)
\( a_1 = 5.9148 \, \text{ms}\) 

\( [c] = a_1 \times [k] \)

The solution to this equation , \( \vec{\delta}(t) = \begin{bmatrix} \delta_x(t) & \delta_y(t) & \delta_z(t) \end{bmatrix}^T \) , is a vector containing the displacement time (t) histories of the three coupled masses. These displacements are assumed to be analogous to some measure of brain deformation under rotational motion about each axis of the head. The maximum magnitude of the system displacement is then fitted to the maximum brain strain (MPS), and is referred to as the DAMAGE metric.[E1]

\( \text{DAMAGE} = \beta \max_t \left| \vec{\delta}(t) \right| \quad \text{where } \beta = 2.9903 \, m^{-1} \, (\text{scale factor}) \)

The scale factor relates the maximum resultant displacement of the system to the MPS value from the FE brain model. 

The equations of motion can be solved using various numerical methods, each with its advantages and specific applications. The calculation methods used by T/HIS are:
  1. Runge-Kutta 4th Order (RK4)
  2. Runge-Kutta Fehlberg (RKF45)
  3. Newmark Beta method

References

[E1] Lee F. Gabler, Jeff R. Crandall, and Matthew B. Panzer. Development of a Second-Order System for Rapid Estimationof Maximum Brain Strain