## 2 Methodology  
### Model Development  
This study utilized nonlinear dynamic analysis explicit solver in LS-DYNA(r) FE software to simulate a rear-end vehicle collision with the Total Human Model for Safety (THUMS v.4) (Figure 1). This human model consists of a skeletal structure, muscular structure, organs, tendons, ligaments, and nerve system. The solid elements were used to model the bulky muscles, intervertebral discs (IVDs), and cancellous bones, whereas ligaments, spinal cord, and cortical bones were modeled with shell elements. The majority of the components were meshed with hexahedron elements, except for cortical shells and thin muscles composed of a tetrahedron and one-dimensional elements. Table 1 presents the material properties of the cervical spine components. Johnson and Cook elasto-plastic model was defined for the cortical and cancellous bony structures of the occipital condyle (OC), the entire cervical vertebra (C1-C7), and the first thoracic vertebra (T1) (Figure 1c) [11]. Their intervertebral nucleus pulposus and annulus fibrosus were defined as fluid and Hill hyper-elastic materials, respectively [12; 13]. Anterior longitudinal ligament (ALL), posterior longitudinal ligament (PLL), intertransverse ligament (ITL), ligamentum (LF), capsular ligament (CL), and interspinous ligament (ISL) were modeled using generalized Maxwell and Kelvin-Voigt viscoelastic equations [14; 15]. Fenghyper-elastic behavior was used to describe the thick bulky neck muscles, whereas thin muscles were modeled using nonlinear curves [16]. The seat's base and backrest cushioning material were made of SAF 6060 polyurethane foam [18], whereas a softer foam was assigned to the head restraint [19]. Ogden hyper \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Component** & **Constitutive Model** & \multicolumn{3}{c}{**Material Properties (\(\boldsymbol{\rho}\) 10\({}^{-6}\) kg/mm\({}^{3}\), \(\boldsymbol{\sigma}\) MPa)**} & **Ref.** \\ \hline \multirow{2}{*}{Cortical} & \multirow{2}{*}{Johnson and Cook Elasto-plastic} & \(\rho=1.82\), \(\sigma_{f}=155\), \(E=16800\), \(\sigma_{y}=110\), & \multirow{2}{*}{[11]} \\  & & \(\
u=0.3,\ b=100\), \(n=0.1\), \(c=1\) & & \\ \hline \multirow{2}{*}{Cancellous} & \multirow{2}{*}{Johnson and Cook Elasto-plastic} & \(\rho=0.17\), \(\sigma_{f}=2.23\), \(E=100\), \(\sigma_{y}=1.92\), & \multirow{2}{*}{[11]} \\  & & \(\
u=0.29\), \(b=20\), \(n=1\), \(c=1\) & & \\ \hline \multirow{2}{*}{Nucleus} & \multirow{2}{*}{Fluid} & \(\rho=1\), \(K=1720\) & \multirow{2}{*}{[12]} \\  & & & & & \\ \hline \multirow{3}{*}{Annulus} & \multirow{3}{*}{Hill hyper-elastic} & \(\rho=1.2\), \(\sigma_{f}=27.5\), \(\sigma_{f}=0.3\), \(n=2\), & \multirow{3}{*}{[13]} \\  & & \(C_{1}=0.115\), \(C_{2}=2.101\), \(C_{3}=-0.893\), & & \\  & & \(\beta_{1}=4\), \(\beta_{2}=-1\), \(\beta_{3}=-2\) & & \\ \hline \multirow{3}{*}{Spinal cord} & \multirow{3}{*}{Quasilinear viscoelastic} & \(E=1.37\), \(\varepsilon_{f}=10\%\), \(A=0.028\), \(B=25.9\), & \multirow{3}{*}{[17]} \\  & & \(\tau_{1}=4.38\), \(\tau_{2}=0.554\), \(\tau_{3}=234.4\), & & \\  & & \(G_{1}=0.104\), \(G_{2}=0.121\), \(G_{3}=0.377\) & & \\ \hline \multirow{3}{*}{Muscles} & \multirow{3}{*}{3D Feng hyper-elastic (bulky muscles), 1D nonlinear (thin muscles)} & \(\rho=1.05\), \(K=4.59\), \(\mu=0.1\) & \multirow{3}{*}{[16]} \\  & & \(E\) & \(\
u\) & \(E_{t}\) & \(\
u_{t}\) & \(\eta_{0}\) & \(\lambda\) & \(\sigma_{f}\) \\ \cline{1-1}  & & & \(50\) & \(0.4\) & \(10\) & \(0.42\) & \(28\) & \(1e6\) & \(31.9\) \\ \cline{1-1}  & & & \(50\) & \(0.4\) & \(10\) & \(0.42\) & \(28\) & \(1e6\) & \(29.3\) \\ \cline{1-1}  & & & \(50\) & \(0.4\) & \(11\) & \(0.42\) & \(28\) & \(1e6\) & - & [14,15] \\ \cline{1-1}  & & & \(50\) & \(0.39\) & \(0.4\) & \(0.42\) & \(28\) & \(1e6\) & \(4.5\) \\ \cline{1-1}  & & & \(50\) & \(0.4\) & \(28\) & \(1e6\) & \(5.6\) & \\ \cline{1-1}  & & & \(50\) & \(0.39\) & \(0.42\) & \(28\) & \(1e6\) & \(3.5\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the cervical spine components.