Solution of Lagrange's equation of motion form the first principle for volumetric modulated arc therapy delivery

Abstract

To calculate the Lagrange’s equation of motion from first principle for volumetric modulated arc therapy delivery. The delivery of volumetric modulated arc therapy (VMAT) is based on a rotational co-ordinate system. This delivery technique differs from any other delivery technique with static beams. VMAT delivery is a complex many-body problem, to represent the VMAT uniquely, it is necessary to find the equation of motion in a non-inertial (rotating) fame. In the non-inertial frame of reference, Newton's equations of motion do not hold, hence Lagrange's equation of motion (Lagrange's equation of motion of second kind) needs to be solved. If A is the aperture created by Multileaf collimator (MLC), MU is delivered Monitor Unit of radiation dose and θ is the gantry angle, then equation of motion for volumetric arc therapy delivery is represented by following three equations 1. $$\frac{d\theta }{dt}=\dot{\theta }=$$ constant between two control points, 2. $$m\ddot{x}-m x \dot{\theta }2= Fx$$ ; where Fx is the external force along $$\overrightarrow{x}$$ direction, causing the MLC movement along the x-direction only and m is the mass of the MLC, and $${MU}_{delivered}\left(t\right)={\int }_{t=0}^{t=t}MU{(t)}_{planned}dt$$ . This piece of derivation presents the calculation of the equation of motion for VMAT delivery in terms of MLC movement, gantry angle, and delivered dose by deriving Lagrangian formulation from the first principle.