In radiation therapy, the objective is to deliver a dose to a target volume while protecting the surrounding organs. To ensure accuracy and proper dose delivery, quality assurance (QA) of a linear accelerator (linac) is performed daily, weekly, monthly, and annually, to a clinically acceptable commissioned data tolerance. The status of the linac is determined by contrasting the measured and analyzed data to those in the relevant guidelines [1,2,3].

Traditionally, linac output QA tests are conducted on the Daily QA3 (QA3) system (Sun Nuclear Corporation; Melbourne, USA) before any patient treatment. Recently, a novel system, machine performance check (MPC) derived from the TrueBeam Edge 2.0 platform (Varian Medical Systems, Palo Alto, CA, USA) has been suggested. It relies on a fully integrated and automated imaging system that includes an electronic portal imaging device (EPID), kilovoltage (kV), megavoltage (MV), and an on-board imager (OBI). In addition, the system is combined with a vendor issued IsoCal phantom aiding dosimetric and geometric tests.

MPC-EPID based can perform daily output check. It is a two-dimensional (2D) detector array attached to the linac ushered in by Baily et al [4]. At first, intended for patient position verification, it has emerged as a dosimetric and now as a linac output verification [5,6]. In this study, the variation in daily output in terms of photon and electron energies was monitored using the MPC-EPID system over a month to follow trends and patterns in the measurement. Hosain et al [7] suggested that the linac output variation may be due to an environmental and seasonal deviation thereby, causing cyclical changes affecting electronic response. Others [8] have stated that the uncorrected output can increase or decrease in some cases and attribute the disparity to the monitor chamber design differences [9]. Equally important, the representation of the level (α), trend (β), and seasonality (γ), in the output data is triggered by infrequent output modifications and tunings made to the linac, thereby creating gaps in the measured data caused by service maintenance event. Therefore, forecasting is vital and warranted for decision-making and strategic planning.

Commonly, forecasting involves three main stages: short, medium, and long-term strategy dependent on the planning quality assurance (QA) schedule. In this report, we sought to use the Holt-Winters (HW) method [10], a triple exponential smoothing, to reduce irregularities in the time series data. Historical data are used as input to make informed estimates that are predictive in terms of determining the future trends direction. This linac asset enhanced the overview of the true fundamental performance. As a result, preventive maintenance could be initiated, improved consistency and efficiency observed, and a reduction in the linac downtime. To the best of our knowledge, this is a first comparative report based on MPC output variation modelled by HW. Hence, the purpose of this work is to propose a forecasting solution.

Output measurements from an Edge TrueBeam (Varian Medical Systems, Palo Alto) linac were carried out based on clinically available photons (6 and 10 MV, 6 and 10 FFF) and electrons (6, 9, 12, 15 MeV) energies. First, the linac was calibrated annually using the TG51 protocol [11]. Then, MPC baseline data was acquired and set. Finally, daily output constancy checks were performed over 30 days by MPC.

MPC Variation Forecast

MPC output variations forecast was carried out using an HW time series. This is a combination of triple exponential smoothing that includes level (α), trend (β), and seasonality (γ), and is characterized by the following equations as:

Overall Smoothing

The total forecast is given by:

where 0 < α < 1; 0 < β < 1; 0 < γ < 1, and

α is the smoothing factor for the level. 

β is the smoothing factor for the trend. 

γ is the smoothing factor for the seasonality.

y and S are the actual and smoothed observations.

b is the trend factor.

I is the seasonal index.

F is the forecast m steps ahead.

L is the cycle length.

t is the period.

This method is based on five equations that calculate the value of the sequence in the past (level), the future tendency (trend), and a seasonal tern (seasonality), which allows to develop the presence of repetitive patterns Finally, equations (4) and (5) calculate the weighted sum of the previous terms as a forecast. The method comprises of forecast and smoothing equations for level, trend, and seasonal components with corresponding smoothing parameters α, β, and γ, respectively. The general architecture of this study is shown in Figure 1. Table 1 summarizes the HW additive (HWA) and multiplicative (HWM) approaches.

Figure 1: Flow chart of the MPC output variation forecast simulation.

Table 1: Comparative equations for the multiplicative and additive Holt-Winters models.

Optimization

Goodness-of-fit is the measure of the accuracy of the predicted model compared to actual values. A classic approach is based on the closeness between the predicted and actual values. Three measurement criteria were employed in this study: the mean squared error (MSE), mean absolute percentage error (MAPE), and mean absolute deviation (MAE). For all three metrics, the smaller the value, the better the predicted accuracy.

Mean squared error.

The MSE is a measure of the dispersion of forecast errors. The smaller the value of the MSE, the more stable the model.

Mean absolute percentage error.

The MAPE is an error measurement that does not emphasize large errors. The MAPE is given by:

Mean absolute error.

The MAE can be described as the average of the absolute error value without regard to whether the error was overestimated or underestimated.

An Excel-based nonlinear optimizer solver namely a generalized reduced gradient (GRG) optimization was used to identify the values of the smoothing constants by minimizing MSE and deduce optimized (α, β, ?) factors. MSE, MAPE, and MAE were determined to assess the model fitting.

The HW forecasting method involves the selection of many parameters for optimum outcome prediction. HW was applied to the MPC dataset from different energy with initial smoothing parameters (α₀, β₀, γ₀). Optimization of the HW parameters was achieved by minimizing the MSE (prediction error) using GRG. A flow chart for this procedure is shown in Figure 1. The three smoothing parameters (α, β, γ) were optimized to attain a better convergence of the observed and estimated MPC output data. The limits on the smoothing parameters were between zero and one (0 < α < 1; 0 < β < 1; 0 < γ < 1).

Tables 2 and 3 show the values of the smoothing parameters and the performance metrics MSE, MAPE, and MAE for seasonal HWA and HWM. Values close to zero for the smoothing parameters suggest that little weight is placed on the most recent observations when predicting future values. In contrast, values closer to one mean that much more weight is associated with observations in the far distant past when acquiring forecast values. However, the estimated value of zero for β indicates that the slope b of the trend component of the seasonal HWA is not revised over the time series, but instead is set equal to its initial value. In this study, both models produced relatively small MSE errors, meaning that this could be used as a better metric than MAPE or MAE. Yet, HWM had a lower MAPE than HWA. Highly accurate models, in which the MAPE is less than 10 %, can be used as a benchmark [12]. Both models had a MAPE of less than 10, indicating high performance. The results for the accuracy metrics of MSE, MAPE, and MAE are illustrated in Figure 2 (a-c). They revealed an average MSE values of 0.0257 ± 0.02655 and 0.0255 ± 0.0184 for HWA and HWM seasonality, respectively. The MAPE values were estimated as 2.4927 ± 3.4354 and 1.9460 ± 2.86669 for HWA and HWM seasonality, respectively, and the MAE values as 0.1181 ± 0.0710 and 0.1210 ± 0.04739, respectively. The majority of the MPC output variation predictions had an MSE of less than or equal to 0.05. A lower MSE corresponds to a predictive model that better correlates with the actual variation in the MPC output. As a primary objective of the simulation, the best combination of the three-performance metrics was selected against the lower MSE values of the forecast. Optimized values of the smoothing parameters (α, β, γ) were found by converging the MSE to its lowest possible value for each energy. The convergence behavior of the MSE is illustrated in Figure 2d. The same patterns were observed for the MAPE (Figure 2e), and MAE (Figure 2f) for each energy level. The final convergence behavior for all performance metrics for each model is displayed in Figures 2g and 2h for HWA and HWM, respectively.

Figure 2 (a-h): MSE, MAPE, and MAE values for MPC datasets output variation using the additive and multiplicative Holt- Winters methods. Symbol a and m denote additive and multiplicative, respectively.

Table 2: performance quantities of the HW model additive.

An HW forecasting model was applied to the MPC dataset over 30 days. The process is displayed in the flow chart (Figure 1), and the initial values are specified for the level, trend, and seasonal components, which are instrumental in the forecasting procedure. The analysis showed that both HW models were suitable for MPC output variation forecasting, since the value of MSE was smaller for all energies compared to the other performance metrics, such as the MAPE and MAE. The results were obtained using a GRG solver to minimize the MSE and optimize the smoothing parameters (α, β, γ), which are energy and HW-specific. The need to select the initial values for the model is one of its main disadvantages. These parameters affect the level, trend, and seasonality.

In addition to the optimization, the HW method of MPC output variation forecasting for different energies is subject to inherent long-term pixel stability of the EPID panels. They exhibit a variation of between 0.29% and 0.6% per pixel, and 99% of all pixels show a deviation of less than 1%, as noted by several investigators [13,14]. Maintenance and recalibration procedures are the most likely sources of unexpected systematic changes in MPC output values, as reported by Barnes et al [15]. Other events may include the EPID ghosting effect that usually depicts the modification of detector response due to previous irradiation. The magnitude of the ghosting will depend on the number of monitor units delivered, as well as the time interval used between the two radiation fields. This will change the sensitivity of the detector and will affect the image gain correction to the pixel sensitivity distribution. Several authors [16-19] have also reported on the flux of the dose-response reproducibility of the EPID system characterized by 1.0% variability from the nominal output and have suggested that the beam flatness is a major contributor to the MPC output. Others attribute this drift in the MPC response to the continuing fading in panel sensitivity. These events will affect MPC output and hence the input data for the forecast model.

The aim of the study was to forecast MPC output and identify patterns via Holt-Winters smoothing. In summary, the study showed that HW both additive and multiplicative is adequate for MPC output modelling. Performance metrics such as RMSE, MAPE, and MAE could assess HW goodness fit. The model robustness could be evaluated using the residuals through ACF and cumulative periodogram. Here, 6 MeV was examined, and the result showed that the standardized residuals are normally distributed and white noises. Hence, the model is considered dependable.

Future works will involve the use of neural network (NN) model that will house the correlation between two nonlinear variables. Also, various moving average models such as autoregressive–moving-average (ARMA), autoregressive integrated moving average (ARIMA), autoregressive integrated moving average- neural network (ARIMA-NN) are of greatest interest and could be used for comparison.

Finally, Theil's U statistics will be used to verify the accuracy test of the model and will allow us to compare the predicted results to the actual model results with minimal historical data.

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