## 2 Materials and Methods
### Theory
#### 2.1.3 Pre-Crash Variable Selection  
The simplified deformation energy formulation for pre-crash conditions provides the a priori form for most post-crash parameters. Crash parameters used for IR assessment can be broadly classified into two categories: velocity-based and system-based indicators. Velocity-based indicators define the kinematic features by characterizing the vehicle motion at the impact and have always been considered as a primary index for risk evaluation. Accordingly, the system-based indicators highlight the differences in impact configurations of crash systems with a dimensionless form, which focus on the effects of crash types, vehicle parameters, and impact eccentricity on occupant injuries. Both types of indicators have been proven to have a substantial influence on the injury outcome because they effectively characterize the main factors affecting the crash consequences: crash morphology and energy transformation [19; 36; 37]. Therefore, the combination of these indicators can explain the crash mechanism from different dimensions.  
The following crash parameters were selected as the PCIs based on previous findings:  
1. Closing velocity at impact (\(V_{r}\)) represents the effective velocity of the crash system along the direction of PDOF, reflecting the potential maximum damage of the collision [23]. \(V_{r}\), as an a priori variable, can be calculated by vector subtraction of the vehicle speed (Figure 2a).
2. Velocity change (\(\Delta V\)) refers to the change in a velocity vector experienced by a road user during a crash. Larger \(\Delta V\) implies stronger external force on the car-occupant compartment and has been widely used in in-depth investigation crash databases. Unlike the traditional posteriori method, this study summarizes the a priori form of \(\Delta V\), which can be derived from the deformation energy and the system mass as follows: \[\Delta V_{i}=\frac{1}{m_{i}}\sqrt{2E_{d}m_{c}\frac{(1+\epsilon)}{(1-\epsilon)}}\] (9)
3. EES refers to the equivalent velocity for a given deformation energy under the rigid crash conditions [38]. Under the assumption that kinetic energy is completely converted into deformation energy, EES measures the velocity level of the vehicle at the time of impact. This variable can be derived from the deformation energy and the vehicle mass: \[\text{EES}=\sqrt{2E_{d}/m_{i}}\] (10)
4. CMI and CSI are system-based variables used to measure vehicle configuration and crash characteristics of two-vehicle crash systems. CMI characterizes the momentum change of the subject vehicle and depends on the mass ratio and elastic modulus [39]; CSI measures the relationship between the dissipated deformation energy and the approaching velocity and depends on the mass ratio and stiffness ratio between vehicles [40]. In the same energy dissipation, impact eccentricity significantly affects the impact outcomes, which can be measured by the dimensionless parameters. These indicators are widely used in risk analysis and can be calculated by the following equations: \[\text{CMI}\,=\frac{1+\epsilon}{1+R_{m}}=\frac{\Delta V}{V_{r}}\] (11)\[\text{CSI}\,=\sqrt{\frac{1-\epsilon^{2}}{(1+R_{m})(1+R_{K})}}=\frac{\text{EES}}{V_{ r}} \tag{12}\]  
where \(R_{m}=m_{1}/m_{2}\) and \(R_{K}=K_{1}/K_{2}\) are the mass and stiffness ratios between vehicles, respectively, denoted by the vehicle configuration of the crash system.  
The majority of the proposed PCIs require the key variable of deformation energy, which is derived in combination with vehicle configurations and contact plane. The PCIs will specifically refer to \(E_{d}\), \(V_{r}\), \(\Delta V\), EES, CMI, CSI, and \(\text{M}_{\text{c}}\) in the following for illustration.