Traumatic Brain Injury in Pedestrian–Vehicle 1 Collision: Convexity and Suitability of some 2 functionals used as Injury Metrics 3 D. S´anchez-Molina, C. Arregui-Dalmases, J. Velazquez-Ameijide, M. Angelini, J. Kerrigan, J. Crandall 4 August 12, 2016 5 Abstract 6 Background and Objective: Abrupt accelerations or decelerations 7 can cause large strain in brain tissues and, consequently, different 8 forms of Traumatic Brain Injury (TBI). In order to predict the effect 9 of the accelerations upon the soft tissues of the brain, many different 10 injury metrics have been proposed (typically, an injury metric is a real 11 valued functional of the accelerations). The objective of this article 12 is to make a formal and empirical comparison, in order to identify 13 general criteria for reasonable injury metrics, and propose a general 14 guideline to avoid ill-proposed injury metrics. 15 16 Methods: A medium-size sample of vehicle-pedestrian collisions, 17 from Post Mortem Human Subject (PMHS) tests, is analyzed. A 18 statistical study has been conducted in order to determine the dis- 19 criminant power of the usual metrics. We use Principal Component 20 Analysis to reduce dimensionality and to check consistency among the 21 different metrics. In addition, this article compares the mathematical 22 properties of some of these functionals, trying to identify the desirable 23 properties that any of those functionals needs to fulfill in order to be 24 useful for optimization. 25 26 Results: We have found a pair-wise consistency of all the currently 27 used metrics (any two injury metrics are always positively related). 28 1