Skull motion during MRE at 50 Hz is well approximated as rigid-body motion [28]; standard 3D kinematic equations for rigid-body acceleration (Eqs. (1a)–(1c)) [29] are used. Here, $a$ is linear acceleration, $\alpha $ is angular acceleration, $r$ defines the distance between some point and the skull origin (approximated as the posterior clinoid process), and $\omega $ is angular velocity. Each of these vectors consists of three components, denoted as $x$, $y$, and $z$, to represent the right–left (RL), anterior–posterior (AP), and superior–inferior (SI) directions. Thus, ($axO$, $ayO,azO$) are, respectively, the RL, AP, and SI linear accelerations of the skull origin; ($\omega x$, $\omega y$, $\omega z$) are rates of rotation (angular velocity) about the RL, AP, and SI directions, respectively; and ($\alpha x$, $\alpha y$, $\alpha z$) are the components of angular acceleration. ($ax$, $ay,az$) are the components of linear acceleration of a point whose position, measured from the skull origin, is ($rx,ry,rz$). The right-hand side of these equations will contain nine unknowns (three components of motion for each of $a$, $\alpha $, and $\omega $ at the skull origin), which can be solved using the known values of linear acceleration at the positions of the tri-axial accelerometers
Display Formula

(1a)$ax=axO+(\alpha yrz\u2212\alpha zry)+(\omega x(\omega yry+\omega zrz)\u2212rx(\omega y2+\omega z2))$

Display Formula(1b)$ay=ayO+(\alpha zrx\u2212\alpha xrz)+(\omega y(\omega zrz+\omega xrx)\u2212ry(\omega x2+\omega z2))$

Display Formula(1c)$az=azO+(\alpha xry\u2212\alpha yrx)+(\omega z(\omega xrx+\omega yry)\u2212rz(\omega x2+\omega y2))$