Geometrical determinations of IMRT photon pencil-beam path in radiotherapy wedges and limit divergence angle with the Anisotropic Analytic Algorithm (AAA)

Purpose: Static wedge filters (WF) are commonly used in radiation therapy, forward and/or inverse planning. We calculated the exact 2D/3D geometrical pathway of the photon-beam through the usual alloy WF, in order to get a better dose related to the beam intensity attenuation factor(s), after the beam has passed through the WF. The objective was to provide general formulation into the Anisotropic Analytical Algorithm (AAA) model coordinates system (depending on collimator/wedge angles) that also can be applied to other models. Additionally, second purpose of this study was to develop integral formulation for 3D wedge exponential factor with statistical approximations, with introduction for the limit angle/conformal wedge. Methods: The radiotherapy model used to develop this mathematical task is the classical superposition-convolution algorithm, AAA (developed by Ulmer and Harder). We worked with optimal geometrical approximations to make the computational IMRT calculations quicker/reduce the planning-system time. Analytic geometry/computational-techniques to carry out simulations (for standard wedges) are detailed/developed sharply. Integral developments/integral-statistical approximations are explained. Beam-divergence limit Angle for optimal wedge filtration formulas is calculated/sketched, with geometrical approximations. Fundamental trigonometry is used for this purpose. Results: Extent simulation tables for WF of 15o, 30o, 45o, and 60o are shown with errors. As a result, it is possible to determine the best individual treatment dose distribution for each patient. We presented these basic simulations/numerical examples for standard manufacturing WF of straight sloping surface, to check the accuracy/errors of the calculations. Simulations results give low RMS/Relative Error values (formulated) for WF of 15o, 30o, 45o, and 60o. Conclusion: We obtained a series of formulas of analytic geometry for WF that can be applied for any particular dose delivery model. Simulations results gave acceptable trigonometrical approximations/data that can be used for LINAC applications/planning-system software. The integral formulas presented are practical for dose delivery calculations/3D-approximations when using WF/other similar types of beam modification devices. Limit angle formulation and conformal wedge concept was also presented.