^{1,a)}, Matthew Whitaker

^{2}and Arthur L. Boyer

^{3}

### Abstract

Every quality assurance process uncovers random and systematic errors. These errors typically consist of many small random errors and a very few number of large errors that dominate the result. Quality assurance practices in radiotherapy do not adequately differentiate between these two sources of error. The ability to separate these types of errors would allow the dominant source(s) of error to be efficiently detected and addressed. In this work, statistical process control is applied to quality assurance in radiotherapy for the purpose of setting action thresholds that differentiate between random and systematic errors. The theoretical development and implementation of process behavior charts are described. We report on a pilot project is which these techniques are applied to daily output and flatness/symmetry quality assurance for a photon beam in our department. This clinical case was followed over . As part of our investigation, we found that action thresholds set using process behavior charts were able to identify systematic changes in our daily quality assurance process. This is in contrast to action thresholds set using the standard deviation, which did not identify the same systematic changes in the process. The process behavior thresholds calculated from a subset of the data detected a 2% change in the process whereas with a standard deviation calculation, no change was detected. Medical physicists must make decisions on quality assurance data as it is acquired. Process behavior charts help decide when to take action and when to acquire more data before making a change in the process.

The authors would like to thank Karen Mellenberg and Scott Tanaka for careful acquisition of daily quality assurance data and many discussions on the sources of errors as determined by the process behavior charts. They would also like to thank Ally Xu for stylistic input on the presentation of this work.

I. INTRODUCTION

II. METHODS

II.A. Definition of systematic error in SPC

II.B. Process behavior charts

II.C. Hypothetical cases

II.C.1. A process with only random errors

II.C.2. A process with both random and systematic errors

III. CLINICAL CASE

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Ionization chambers
- 17.0
- Quality assurance
- 17.0
- Cancer
- 10.0
- Dosimetry
- 8.0
- Radiation therapy
- 8.0

## Figures

(a) The average chart for theoretical data from a normal distribution. The data are sampled from a normal distribution with mean and standard deviation and . Sixty subgroup averages are plotted based on a total of 240 values sampled from the normal distribution. The subgroup size was . The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart for theoretical data from a normal distribution. Each point represents the range of each subgroup that was plotted in Fig. 1(a). The open circles indicate the subgroup ranges that were used to calculate the process behavior limits indicated in the figure by a dashed line.

(a) The average chart for theoretical data from a normal distribution. The data are sampled from a normal distribution with mean and standard deviation and . Sixty subgroup averages are plotted based on a total of 240 values sampled from the normal distribution. The subgroup size was . The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart for theoretical data from a normal distribution. Each point represents the range of each subgroup that was plotted in Fig. 1(a). The open circles indicate the subgroup ranges that were used to calculate the process behavior limits indicated in the figure by a dashed line.

(a) The average chart for theoretical data with known systematic variations. The data are sampled from a normal distribution with varying mean and standard deviation as shown in Table I. Sixty plotted values and a total of 240 values subgroup size . The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart for theoretical data with known systematic errors. The range of each subgroup in Fig. 2(a) is plotted. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for theoretical data with known systematic variations. The data are sampled from a normal distribution with varying mean and standard deviation as shown in Table I. Sixty plotted values and a total of 240 values subgroup size . The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart for theoretical data with known systematic errors. The range of each subgroup in Fig. 2(a) is plotted. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for the clinical case. Subgroup size is used to verify the constancy of the flatness and symmetry. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart of the four periphery ion chambers in the RBA-5 device. The range of each subgroup in Fig. 3(a) is plotted. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for the clinical case. Subgroup size is used to verify the constancy of the flatness and symmetry. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The range chart of the four periphery ion chambers in the RBA-5 device. The range of each subgroup in Fig. 3(a) is plotted. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for the clinical case. Subgroup of size 1 is used to verify the constancy of central axis output. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The moving range chart of the central axis ion chambers in the RBA-5 device. Consecutive values in Fig. 4(a) are plotted as the range. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for the clinical case. Subgroup of size 1 is used to verify the constancy of central axis output. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by one standard deviation. Data with open circles indicate those points that were used to calculate the limits. (b) The moving range chart of the central axis ion chambers in the RBA-5 device. Consecutive values in Fig. 4(a) are plotted as the range. Data with open circles indicate those points that were used to calculate the limits.

(a) The average chart for the clinical case. Subgroup size is used to verify the constancy of the flatness and symmetry. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by the standard deviation. The process behavior limits have been recalculated after a change in the process was detected as shown in Fig. 3(a). (b) The range chart of the four periphery ion chambers in the RBA-5 device. The range of each subgroup in Fig. 5(a) is plotted with new process behavior limits.

(a) The average chart for the clinical case. Subgroup size is used to verify the constancy of the flatness and symmetry. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by the standard deviation. The process behavior limits have been recalculated after a change in the process was detected as shown in Fig. 3(a). (b) The range chart of the four periphery ion chambers in the RBA-5 device. The range of each subgroup in Fig. 5(a) is plotted with new process behavior limits.

(a) The average chart for the clinical case. Subgroup of size 1 is used to verify the constancy of central axis output. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by the standard deviation. The process behavior limits have been recalculated after a change in the process was detected as shown in Fig. 3(a). (b) The moving range chart of the central axis ion chambers in the RBA-5 device. The range of each subgroup in Fig. 6(a) is plotted with new process behavior limits.

(a) The average chart for the clinical case. Subgroup of size 1 is used to verify the constancy of central axis output. The thick dashed lines are the process behavior limits and the thin dashed lines are the limits calculated by the standard deviation. The process behavior limits have been recalculated after a change in the process was detected as shown in Fig. 3(a). (b) The moving range chart of the central axis ion chambers in the RBA-5 device. The range of each subgroup in Fig. 6(a) is plotted with new process behavior limits.

This flow chart describes the interactive nature of SPC and the use of process behavior charts. It outlines the activity of identifying and removing systematic errors to reduce variation in the process of daily output and flatness/symmetry quality assurance.

This flow chart describes the interactive nature of SPC and the use of process behavior charts. It outlines the activity of identifying and removing systematic errors to reduce variation in the process of daily output and flatness/symmetry quality assurance.

## Tables

The mean and standard deviation of the normal distribution from which the data are sampled for our hypothetical cases. Each data point consists of four values (subgroup size ) sampled from the normal distribution for a total of 240 individual values sampled from the normal distribution.

The mean and standard deviation of the normal distribution from which the data are sampled for our hypothetical cases. Each data point consists of four values (subgroup size ) sampled from the normal distribution for a total of 240 individual values sampled from the normal distribution.

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