A Bayesian control chart based on the beta distribution for monitoring the two-dimensional gamma index pass rate in the context of patient-specific quality assurance.

Abstract

In the context of quality assurance in intensity modulated radiation therapy (IMRT), the aim of this work was two-fold: (a) to show that the beta distribution characterizes the two-dimensional gamma index pass rate (GIPR), and that the quantiles of the distribution should be used in order to compute the control limit (CL) for the detection of abnormally low GIPR, and (b) to introduce a Bayesian control chart that allows calculation of CLs from the first measurement. In order to enable monitoring of the GIPR from the first measurement, we developed a Bayesian control chart based on the beta distribution, elaborated according to the following two steps: (a) an iterative bayesian inference approach without any prior information on the GIPR distribution was used at the start of monitoring and the CL was progressively updated; and (b) when sufficient in-control arcs had been recorded and the estimators of the parameters of the beta distribution were sufficiently accurate, the CL of the chart was fixed to a constant value corresponding to the quantile of the beta distribution. The clinical utility of this approach is illustrated through a real data case study: monitoring the GIPR of patients treated with a moving gantry IMRT technique RapidArcTM on a Novalis TrueBeam STx (Varian Medical Systems) linear accelerator equipped with an aS1200 electronic portal imager device. We showed that some commonly used distributions for monitoring GIPR in the literature, such as normal or logarithm transformation, are not appropriate. We compared the CLs of those solutions with the CL of our chart based on the BD (CL=95.14%). The comparison revealed that the CL for the normal law (CL=97.62%) generated too many false positives, and that the CL of the Logarithm transformation (CL=83.74%) could fail to efficiently detect (i.e., sufficiently early on or faster) changes in the process. Successful GIPR monitoring requires careful and rigorous application of well-established statistical concepts in the field of statistical process control. In this paper, we stress the importance of carefully analyzing the distribution of the monitored characteristic that is plotted on the control chart. We propose a Bayesian control chart that can be viewed as a practical solution for early implementation of GIPR monitoring, starting from the first arc. We demonstrate that beta distribution is a better method for characterizing the GIPR, and thus, the use of this approach is expected to improve patient-specific quality assurance plans in radiotherapy.