Determination of an inflection point for a dosimetric analysis of unflattened beam using the first principle of derivatives by python code programming

Abstract

Background

Practice of Unflattened or Flattening filter free (FFF) beam has become the high dose standard in radiotherapy (RT), such as stereotactic radio-surgery (SRS) and stereotactic radiotherapy (SRT). The removal of a flattening filter (FF) from the path of a photon beam alters the characteristics of FFF beam. Since the conventional route for dosimetric analysis of FF beam cannot be applied to FFF beam, the procedure of analyzing beam characteristics for FFF beam based on inflection points (IPs) is used. IP is a point where the concavity change observed corresponds to its change in sign (±) of the second derivative.

Aim

The objective of the study is to determine IPs for dosimetric analysis of the FFF beam profile.

Methods and materials

In this study, IPs are determined through the python code programming based on the mathematical first principle of the derivative. They are compared with IPs estimated by the conventional graphical manual method using Microsoft Excel (MS). IPs and their dependent dosimetric parameters determined by both mathematical and graphical manual methods are compared.

Result

Percentage differences between the IPs determined by both methods, for 6MVFFF inline and crossline beam profile are found to be 2.7% and 0.8% respectively. Similarly, the average penumbra differences for 6MVFFF inline and crossline beam profile are found to be 0.15mm and 0.9mm, respectively. However, differences in the field width between both methods are found insignificant.

Conclusion

Graphical manual method is very time-consuming, tedious and user dependent. However, the mathematical method through python code programming is more precise, faster and independent of individual users.