Descriptive Time Series Analysis for Downtime Prediction Using the Maintenance Data of a Medical Linear Accelerator

Abstract

A medical linear accelerator (LINAC) delivers high-energy X-rays or electrons to the patient’s tumor. In this study, we categorized failures and predicted downtime leading to discontinuous radiation treatment using a descriptive time series analysis of a 20-year maintenance dataset of a medical LINAC. A LINAC dataset of failure records for 359 instances was collected from 2001 to 2021. Next, we performed institution-specific seasonal autoregressive integrated moving average (ARIMA) modeling to analyze the causes of the failure categories and predict the downtime. Furthermore, we evaluated the performance of the predictive model using standard error metrics and statistical methods. Our results show that the downtime will increase by 95 h/year after 2022 and 100 h/year after 2023. The accumulated downtime in 2029 is predicted to be a maximum of 2820 h. The modeled seasonal ARIMA showed statistical significance (p < 0.001) with a residual error of ${\sigma }^2$ (328.33 ± 9.4). In addition, the forecasting performance of the model was assessed using the mean absolute percentage error (MAPE). The failure parts where the major downtime occurred were the multileaf collimator (25.2%), gantry and couch motion part (15.4%), dosimetric part (11.7%), and computer console (10.0%). Using the development of the ARIMA model specific to our institution, the downtime is predicted to reach up to 2820 h.

1. Introduction

Linear accelerator (LINAC)-based cancer treatment was first implemented at Hammersmith Hospital in London, England, in the 1950s. The treatment started with a 6MV LINAC in 1956 [1]. A medical LINAC accelerates electrons using a waveguide, in which the RF power generates a standing wave. However, modern LINAC machines have a shorter vertically mounted waveguide and a longer horizontal waveguide for a higher energy output, which has a bending magnet that directs the beam vertically toward the patient [2]. A LINAC can generate a stable and accurate radiation beam using various electromechanical parts. A LINAC with a single-energy electron beam between 4 and 25 MeV can provide an X-ray output including electron energy when electrons strike a high-density (e.g., tungsten) target [2]. Brain, spine, lung, breast, prostate, liver, and skin tumors, among others, can be treated by electron or photon beams generated using a LINAC with appropriate treatment plans [3].

A LINAC is mainly composed of a gantry head, body, and treatment couch. Some differences exist in some parts depending on the manufacturer, for example, the bore and robot type. In detail, it consists of jaws, multileaf collimators (MLCs), a flattening filter, a primary collimator, an X-ray target, an accelerating waveguide, an electron gun, a feed wavelength, a magnetron, a vacuum pumping line, a vacuum pump, and another motion system [4]. The onboard imager and electronic portal imaging device are attached to LINAC’s body for image-guided radiation therapy. Additionally, the body-tracking device of the patient is mounted on the ceiling and floor of the treatment room to perform additional accurate imaging, while the patient treatment in fractionation is performed using a surface-image-guided radiotherapy system [5,6,7].

Meanwhile, abrupt discontinuation of a medical LINAC when treating patients using high-energy radiation has serious consequences in that the planned dose cannot be delivered to the patient. Thus, performing quality assurance procedures for consecutive patient care without sudden LINAC downtime composed of complex electromechanical components is a top priority for medical institutions. Nevertheless, LINAC machine downtime is caused by harsh operating environments, such as large accumulated amounts of monitor unit (MU) use, a prolonged operation period, and higher cumulative dose rate operations. Therefore, the task group (TG)-100 of the American Association of Physicists in Medicine (AAPM) provided guidelines to each medical institution on establishing a quality management program that determines the risk of LINAC failures according to its quality control procedures [8]. Each medical institution can perform TG-142, which provides detailed items for periodic quality assurance of medical accelerators [9]. Thus, the best way to avoid downtime is for those responsible for quality control, such as radiation therapy technologists and medical physicists, to follow the LINAC’s guidelines for quality control processes, risk analysis, and management practices [8,9,10]. Currently, all medical institutions comply with the strict radiation management regulations of their respective countries for the safety of patients and healthcare professionals, and stringent quality control is stipulated. Nevertheless, LINAC failures remain difficult to predict, and it is impossible to determine which component will cause a failure at a certain time.

Several studies have published the results of operating status analysis to maintain the optimal condition of LINACs. Ma et al. developed a stacked long short-term memory prediction model to predict the machine operating status and records of daily quality control lists (beam output, symmetry, etc.) for two radiotherapy machines [11]. The goal was to boost the confidence of medical staff or physicists in predicting the future malfunctioning behavior of machines before they reach their tolerance levels and taking planned preventive action. However, a time lag occurred when adjusting the parameters of the predictive model, thereby causing the inaccuracy of the status prediction. Thus, they reported a limitation in which the model prediction was delayed upon adjusting the length of the time lag.

Yun et al. dealt with unformatted maintenance logs to identify the LINAC failure modes and trends over time using natural language processing techniques [12]. As a result, a periodic pattern over years for the machine parts, such as the MLC, couch, and pendant, was discovered. However, the results could be affected by inaccurate trend analysis due to an insufficient data volume.

Unlike previous studies that have reported the analysis of LINACs’ status information, predicting downtime is required to prepare for predictive maintenance. In our neurosurgery clinic, a medical LINAC, Novalis^® (BrainLAB AG, Feldkirchen, Germany; Varian Medical Systems, Palo Alto, CA, USA), has been used to treat brain and spine tumors. Additionally, a rigid quality assurance protocol has been maintained from 2001 to the present [13,14,15,16]. Using the long-term maintenance records acquired during 20 years of operating the LINAC, we aimed to analyze downtime trends and predict downtimes that cause sudden patient treatment discontinuation to enable a preventive measure. Therefore, it is necessary to predict the downtime concerning the life span of a LINAC operating in a harsh environment.

This work investigated the fault error classification according to the cause of failures and downtime prediction over the years using the 20-year maintenance records of LINAC operation. In this procedure, performance checks were evaluated using error metrics.

2. Materials and Methods

2.1. Overall Process

Figure 1 shows a research flowchart for predicting downtime using LINAC maintenance records to classify failures using a descriptive time series analysis. The time series dataset (n = 359) was collected from 599 cases, excluding the monthly periodic maintenance cases (n = 240) from 1 January 2001 to 31 December 2021. The autoregressive integrated moving average (ARIMA) model was adopted for the predictive modeling process.

As part of the preprocessing process, the data for causes of failure were arranged by date, and the stationarity was checked in advance for ARIMA modeling. Additionally, the data were detrended and differentiated. Finally, ARIMA modeling was performed using the error metrics, evaluated using standard error measures and statistical methods (Figure 1).

2.2. Modeling for the Descriptive Time Series Analysis

The ARIMA model, which is an integrated model of the autoregressive (AR) and moving average (MA) models, was applied to the modeling process to develop the time series analysis [17]. The model was fitted to the time series data to predict future points (i.e., downtime) in the series. Recall that the ARIMA model is applied when the data show substantiation of non-stationarity in the sense of the mean, where an initial differencing stage can be used one or more times to eliminate the non-stationarity (i.e., fault trend) of the mean function. Therefore, it predicts the future value using a linear combination of past observations of a specific variable to be predicted by composing autocorrelation using time series modeling. That is, early observations affect later observations. Additionally, the MA error will affect the predicted value during the prediction phase. Thus, the AR model uses lags of the dependent variable as independent variables. However, the MA model uses past errors that follow a white noise distribution as explanatory variables. If y is denoted as the d-th difference, $y_t^{\prime }$ is the differenced series. Thus, the general forecasting model can be expressed as Equation (1). This model assumes stationarity and is analyzed after preprocessing, such as log transformation and differencing for forecasting or prediction (Figure 1 and Equation (1)) [17,18].

(1)$y_t^{\prime }={\phi }_0+{\phi }_1{y}_{t-1}^{\prime }+{\phi }_2{y}_{t-2}^{\prime }+\cdots +{\phi }_p{y}_{t-p}^{\prime }+{\theta }_1{ϵ}_{t-1}^{}+{\theta }_2{ϵ}_{t-2}^{}+\cdots +{\theta }_q{ϵ}_{t-q}^{}+{ϵ}_t^{}$

where y regresses on itself lagged by the n-th period; ${\phi }_i\left(i=1,\dots ,p\right)\mathrm{and}{\theta }_j\left(\mathrm{j}=1,\dots ,q\right)$ are defined by the weights for the AR and MA parameters; ${\phi }_1$ and ${\theta }_1$ are therefore the coefficients of the first AR and MA terms, respectively; and ${ϵ}_t$ is a residual term with mean zero and variance ${\sigma }_{ϵ}^2$. In Equation (1), the error term ($ϵ$) reflects the previous state at present. This implies that the MA model estimates the rate of change using autocorrelated errors.

Thus, the resulting integrated model, the ARIMA model, is expressed as the ARIMA (p, d, q) for the AR, MA, and autoregressive moving average (ARMA). p is the order of AR terms, d is the number of nonseasonal differences needed for stationarity, and q is the order of MA terms. To select the best ARIMA model for the fitting process through estimation, we used information criteria to determine the potential AR/MA orders [19]. Here, parameter combinations for the ARIMA (p, d, q) were chosen for the seasonal ARIMA as follows (0 < p = d = q < 2).

For example, an ARIMA (0, 0, 0) model is white noise, which means that the errors are uncorrelated across time. An ARIMA (1, 0, 0) model is a first-order AR model, which is a stationary and autocorrelated series. It can be predicted as a multiple of its previous value with a constant ($y_t^{\prime }={\phi }_0+{\phi }_1{y}_{t-1}^{\prime }$). For the ARIMA (0, 0, 1), the MA model processes means identically as the infinite sum of exponentially weighted past observations of the process.

For model estimation, the Akaike information criterion (AIC) was used to seek a simple model with lower orders [20,21]. During this evaluation process, the ARIMA model parameters were found in various combinations using p, d, and q.

For evaluating and comparing ARIMA models with different p, d, and q parameters, the models were ranked according to how well the data fit or the precise future data point prediction. That is, a model that fits the data very well is one that uses fewer orders to achieve the same good fit. A grid search function was used to iteratively search different combinations of the p, d, and q parameters. The seasonal ARIMA model was then fitted using the SARIMAX() function from the Python statsmodels module for each combination of parameters.

2.3. Model Evaluation Using Error Metrics

To evaluate the regression model, the MSE, RMSE, and MAPE were used [22]:

(2)$\mathrm{MSE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(y_i^{}-{\widehat{y}}_i\right)}^2$

(3)$\mathrm{RMSE}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_i^{}-{\widehat{y}}_i\right)}^2}{n}}$

(4)$\mathrm{MAPE}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|\frac{y_i^{}-{\widehat{y}}_i}{y_i^{}}\right|\times 100$

where n is the sample data points, $y_i^{}$ denotes the actual values, and ${\widehat{y}}_i$ denotes the prediction values.

Python 3.8.3, statistical computation model 0.11.1, and Python-Arima version 1.8.4 were used to develop the predictive model, statistical analysis, and model evaluation in a programming environment.

3. Results

3.1. Classification by Cause of Failure

Figure 2 illustrates the causes of failure for the total instances (n = 359). The main downtime occurrence sites were the MLC (25.2%), gantry and couch motion parts (15.4%), dosimetric part (11.7%), and PC (computer console either in the treatment control room or treatment room) (10.0%). Causes of failure for the PC included computer system errors, such as boot failure and unexpected shutdown. Another 7.8% of the error was a hand pendant problem with the accessory. This can be attributed to internal board failure or a problem with the pendant’s body and the jack connected to the couch. A simple release of warnings accounted for 6.9% of the error resolution. That is, the problems were solved by software processing, lubricating mechanically operated parts, or cleaning contaminants without replacing parts through an engineer’s visit. Additionally, failures of dosimetric or beamline parts were considered serious faults, such as the replacement of internal parts, and required additional internal quality assurance procedures.

Figure 3 shows a cumulative downtime of 1500 h and an average downtime of 72 h ± 33.44 for the entire dataset. To analyze the distribution of causes of failure, we analyzed the trend of the monthly average downtime for each event (instance). In particular, in 2009 and 2016, the downtime rapidly increased to 170 h and 136 h, respectively, due to the failure of the dosimetric part (electron gun, ion chamber) and beamline components. The downtime increased in those years because the part to be replaced took a long time to supply, and it entailed a complex dosimetric quality assurance (QA) procedure. However, other years saw an average downtime of 63 h ± 20.98.

3.2. Downtime Prediction Modeling Using Descriptive Time Series Analysis

We decomposed the time series for the LINAC downtime dataset into several components representing trends, seasonality, and residuals, as shown in Figure 4. First, Figure 4A shows that downtime increased over the years. The time series has seasonality, which is unaffected by the specific seasonal factor in Figure 4B. For the residuals, a downtime surge occurred due to major LINAC part replacement from 2006 to 2007, as shown in Figure 4C.

Figure 5 shows the prediction of downtime from 2022 to 2030 from the downtime dataset between 2001 and 2021. The prediction boundary indicated by the dark gray band (mean square error = 0.25) is shown along with the predicted data from 2022. As time moves past the first forecast point, 2022, the uncertainty increases, thereby increasing the amplitude of the forecast boundary. The ARIMA model predicts that the downtime will increase by 95 h/year after 2022 and 100 h/year after 2023. However, the forecast error is also expected to accumulate at an annual rate of 25%. Moreover, the cumulative downtime in 2029 is expected to be about 2820 h.

Replacing the previously failed parts makes the LINAC operation more stable. In other words, the lower limit of the gray error region in Figure 4 can be interpreted as the period to maintain the stable performance of the LINAC due to the replacement of the previously failed parts. Conversely, an upper bound in the gray area indicates that critical part replacement or subsequent QA procedures can cause more downtime than predicted.

3.3. Diagnosis for the ARIMA Model

Figure 6 shows the residual plot analyzed using the ARIMA model. As shown in Figure 6A, the residual errors fluctuate around a mean of zero and have a uniform variance. In Figure 6B, the kernel density estimation plot suggests a normal distribution (n) with a mean of zero. Figure 6C on the bottom left shows the model’s quantile–quantile (Q–Q) plot. All the dots should fall perfectly in line with the solid red line. Any significant deviations denote that the distribution (quantile axis) is skewed in dark blue dots. In the interval where the theoretical quantity is between −2 and 2, the population tends to follow normality with the red trend line. Figure 6D shows the correlogram, and the autocorrelation function (ACF) plot shows that the residual errors are not autocorrelated. Any autocorrelation implies some pattern of an unexplained residual error in the model. Thus, this shows that the stationary status was satisfied by gradually converging to zero.

3.4. ARIMA Model and Error Metric Analysis

Table 1 presents the seasonal ARIMA model. It was built as the best model with the lowest AIC, as shown in Table 1. There were four main lagged AR and MA variables. The first set of AR and MA variables lagged by one time step (AR.L1 and MA.L1, respectively). The second set lagged by 12 and 24 time steps (AR.S.L12 and AR.S.L24). The coefficient column represents the weight or importance of each set of AR and MA variables and how the variables impact the time series to be stationary. The coefficients of the AR and MA terms are less than 1, and the sum of the coefficients of the two seasonal AR terms is less than 1. The p-values are statistically significant as <0.001 in the AR, MA, and seasonal ARIMA models (p < 0.05) of the lagged AR and MA terms, which significantly affect the model forecast. All standard deviation errors show values close to zero. The value for sigma squared (σ²) was calculated, indicating the variance of the residual. This means that the normality of the residual against the alternative of non-normality was tested for the ARIMA model.

Table 2 presents the error metric analysis using the MSE, RMSE, and MAPE. The MAPE is less than 10%.

4. Discussion

4.1. The Use of LINAC in Radiosurgery

Novalis is dedicated to radiosurgery. First, a conical collimator is one of the most important components used for 4−20 mm-size tumors. Few cases of Novalis electrical contact failure have been reported; thus, we conclude that the electrical contact with the external cone insert is relatively robust. Second, the m3-MLC is 3 mm thin in the center field at a field size of 10 × 10 cm. As shown in Figure 2, the MLC occupied about 25.2% of the defect rate of motor replacement and leaf movement error for 20 years. When errors occur in the MLC leaf’s movement over the planned position, a serious incident occurs and causes downtime by delaying patient treatment. Thus, verifying the leaf position accuracy and MLC motion is the top priority in periodic maintenance and QA. Third, the gantry and couch had a failure rate of 15.4%. For brain tumor treatment, multiple noncoplanar beams are used according to the patient’s plan, and the couch moves in the longitudinal, lateral, vertical, and rotational directions while performing elaborate motions to treat a few-millimeter- to centimeter-size tumors, causing fatigue in the couch operation. Therefore, our institution established and periodically verified QA procedures for couch movement. Fourth, the failure rate of the ExacTrac system, which is an image-guided radiation therapy system, was reported to be 3.3% (Figure 3). The system is a major component for patients being treated for spine tumors. Specific failures were related to IR reflector errors. Treatments for brain and spinal tumors are the main reasons for radiosurgery. Here, preventive measures are being taken using periodic maintenance (PM) procedures and QA. Therefore, the importance of PM is increasingly recognized.

4.2. Developing Predictive Models to Maintain Optimal LINAC Performance

Various studies have been conducted to maintain LINACs’ optimal condition. Overall, we conducted the downtime prediction using 20-year data records through time series modeling. Table 3 summarizes the characteristics of each study. Puyati et al. demonstrated QA feasibility using daily MPC test results on 490 data. It follows that machine performance was checked using previous records for the MLC, gantry, couch, beam, and imager to maintain the optimal LINAC performance. Furthermore, the performance, prediction, and distribution of the beam power variance were calculated using predictive analysis. Compared with other studies, our study used various performance metrics, including the MSE, RMSE, MAPE, CC, and ACF. Additionally, we developed the model based on excellent autocorrelation using ARIMA descriptive time series analysis. The MAPE was calculated as 3%. The metric shows a highly stable prediction with an error rate of less than 5%. Thus, the superiority of the performance test in our study is shown in Table 3. The results of our study show that a predictive maintenance approach using LINAC downtime data is possible. Moreover, by analytically classifying faulty parts, the system can be used clinically for patients at all times when important parts, such as the MLC, have a high percentage of failures. In summary, our study shows that a predictive maintenance approach using LINAC downtime data is feasible. In addition, the presented prediction results indicate that the predicted downtime can be expected for important parts such as MLCs by analytically classifying faulty parts with high failure rates. However, it shows the characteristics of a machine used for a single fraction or hypofractionation (<5 fractions) in brain and spinal tumor treatment in a specific area in a single neurosurgery clinic. Thus, the results do not reflect the characteristics of a general LINAC condition.

4.3. The Use of AR Models and the Limitations of Time Series Analysis

The ARIMA model for time series analysis, which is a generalization of the ARMA model, is widely used in statistics and econometrics. Both models are well suited for time series data to better identify the data or to forecast future points in the series (prediction). In an ARIMA model, the initial difference step (the term “d”) can be fitted one or more times to eliminate non-stationary mean functions.

The AR model used in this study is used in other fields. Han et al. used a spatiotemporal change series with ARIMA modeling for drought forecasting. The AR model predicted the time series of vegetation temperature condition index images (time change of pixel values and spatial information) compared with general predictions for series that change with time [27]. Jan et al. used the AR model and an estimator for the weakly dependent process corresponding to the deterministic component together with the functional AR model. Relatively better results were obtained by reducing the prediction error compared with the prediction methods using only nonfunctional prediction techniques, such as the AR model [28]. Shah et al. applied modeling to the parametric AR, nonparametric AR, ARMA, and vector AR models to predict the spot price of electricity. Filtering techniques were important processes for reducing errors in the time series data. The results of their study showed that filtering techniques, such as the recursive filter on prices and the moving window filter on prices (MFP), combined with attenuation schemes and threshold substitution produced lower error values [29]. In this study, the seasonal derivative was applied to remove the seasonal component from the time series. Moreover, preprocessing including differencing was performed to create a stationary dataset. A fundamental part of this study was how well the MA was reflected using autocorrelation: that is, whether it predicts a sharp increase in downtime from the regression error after 2022 or whether it converges gently to a specific time. Therefore, we focused on the model that obtained the performance shown in Table 1 by appropriately finding p, d, and q using the ARIMA (p, d, and q) model.

Nevertheless, time series analysis has the following limitations considered in this study [18,30,31]. First, time series analysis does not support missing values like other models. Therefore, missing values are treated with preprocessing. Second, the relationship between the data points must be linear. Here, the dataset had 2 years of abnormal peaks in 2009 and 2016. It was not judged to have a specific periodic correlation. Thus, we smoothened the observation curve to treat the preprocessing procedure. Third, time series analysis requires data transformation to continuous time values. However, our model works mainly on univariate data.

4.4. Characteristics of LINAC Used for Single-Fraction or Hypofractionated Treatment

Although this study used a 20-year dataset, it focused on the LINAC downtime dataset used for patient care at a single institution. This has a limitation in that, as the data are predictive, it applies only to the characteristics of a specific institution. This means that more than 80% of the cancer patients are being treated with our LINAC in a single-fraction or 3–5 fractionations with high doses of 12–20 Gy. However, the operating environment differs from most radiation oncology LINACs, which treat conventional fractionation patients with relatively low doses of 1.8–2.2 Gy at one time. Therefore, it is necessary to use LINAC downtime data for neurosurgery or radiation oncology departments of various institutions to make a time series prediction that satisfies several operating environments. More diverse information, such as the number of patients treated, the MU used, and the operation time, must be collected for precise prediction under complex conditions. Multicenter data should be aggregated, and clinical application characteristics of the device should be investigated. Such research is proposed as a future project.

4.5. Failure Mode and Effects Analysis (FMEA)

This study applied analysis of more detailed datasets. Thus, there is the possibility of further studies applying FMEA. FMEA is the process of inspecting as many components, assemblies, and subsystems as possible to reveal the potential failure modes of a system and their causes and effects. For each part, the failure mode and the resulting impact on the entire system are recorded in a risk priority number (RPN) scoring metric [32]. However, although RPN scoring is a quantitative method, subjective factors that vary depending on the situation of individuals and institutions performing RPN scoring are included. Although FMEA is a qualitative analysis, it might rely on a quantitative basis using a mathematical RPN scoring for each institution. FMEA is highly recommended in medical physics using the AAPM TG-100 guideline [8]. Thus, it provides a QA procedure for the successful operation and management of LINACs with a high safety potential. Potential failure mechanisms, which impact the system operation, can reveal the severity and occurrence. Therefore, it is possible to provide a stable treatment environment without interrupting the patient’s treatment, and to minimize downtime.

5. Conclusions

The aim of this study was to predict downtime by utilizing long-term maintenance records to maintain a medical LINAC’s optimal condition. The parts of failure that caused major downtime were the MLC (25.2%), gantry and couch motion part (15.4%), dosimetric part (11.7%), and computer console (10.0%). The results of this study show that a predictive maintenance approach using a LINAC downtime dataset is feasible. Through the analytical classification of faulty parts, to enable the system to be used clinically for patients at all times, it is essential to prepare for important parts, such as the MLC, which have a high percentage of failures. However, our result shows the characteristics of a machine used for a single fraction or hypofractionation (<5 fractions) in brain and spinal tumor treatment in a specific area in a single neurosurgery clinic. Although data from a single institution were used, the ARIMA model predicted that downtime might increase by 95 h/year in 2022 and 100 h/year in 2023. Additionally, forecast errors were estimated over 25 years, but the total downtime in 2029 is projected to reach up to 2820 h.

Author Contributions

K.H.K. contributed to the study’s conception and design. Material preparation, data collection, and analysis were performed by K.H.K. and M.-J.S. The first draft of the manuscript was written by K.H.K., M.-J.S., S.L., A.K.M., H.-W.K. and S.-W.Y. corrected and evaluated the results of the manuscript. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. A research flowchart for descriptive time series analysis for LINAC maintenance records. Failure instances (n = 359) were collected over time, and an ARIMA model was adopted for predictive modeling. AR, autoregression; MA, moving average; p, term for the number of AR; d, term for the integration of AR and MA; q, the number of lagged forecast residual errors for MA; RMSE, root mean square error; MAPE, mean absolute percentage error.

Figure 2. LINAC causes of failure for Novalis in our institution.

Figure 3. Annual downtime analysis from 2001 to 2021. In this period, excluding 2009 and 2016, the average downtime was 63 h ± 20.98. The downtime increased for years when replaced parts took longer to supply and needed complex QA procedures, e.g., 2009 IQR [(median:14, Q1:6.5, Q3:19.88) (5–25.5)].

Figure 4. Time series decomposition for the LINAC downtime dataset. (A) The downtime increased over the years. (B) The time series has seasons uninfluenced by the specific seasonal factor. (C) The downtime surge occurred due to major LINAC part replacement from 2006 to 2007.

Figure 5. Downtime prediction with the ARIMA model (mean square error = 0.25, represented by dark gray bands).

Figure 6. Cross-validation using an autocorrelation function, estimated density function, and quantile–quantile analysis. (A) Standardized residual was analyzed over 20 years, and downtime increased due to major component replacements between 2006 and 2007. (B) KDE (kernel density estimation) with a histogram in a normal distribution. (C) Verification for the normal distribution using a Q-Q analysis. (D) ACF shows that the stationary status was satisfied by gradually converging to zero in a correlogram.

Table 1

Results for the seasonal ARIMA model in this study.

Model	Coefficient	STD Error	z	p > ∣z∣	95% CI (0.025–0.975)
AR.L1	0.9497	0.038	25.24	<0.001	0.876	1.023
MA.L1	−0.7114	0.041	−17.432	<0.001	−0.791	−0.631
AR.S.L12	−0.6484	0.013	−51.846	<0.001	−0.673	−0.624
AR.S.L24	−0.3435	0.013	−26.597	<0.001	−0.369	−0.318
${\mathit{\sigma }}^{\mathbf{2}}$	328.3387	9.421	34.854	<0.001	309.875	346.803

Note: The model has four main lagged AR and MA variables. The first set of AR and MA variables lagged by 1 time step for AR.L1 and MA.L1, respectively. Meanwhile, the second set lagged by 12 and 24 time steps for AR.S.L12 and AR.S.L24, respectively.

Table 2

Error metric analysis using the MSE, RMSE, and MAPE.

Metrics	Values
Mean square error (MSE)	204.23
Root mean square error (RMSE)	14.29
Mean absolute percentage error (MAPE)	3%

Table 3

Comparison with various prediction studies for maintaining LINACs.

Authors	Purpose	KeyFeature	Data, Methods	Results	Downtime Prediction	Year	References
Puyati et al.	To demonstrate the feasibility of predictive quality assurance using MPC tests that allow preventive maintenance procedures	QA	490 data, ARIMA modeling	The predictive model accuracy was 85% and RMSE < 0.05.	NA	2020	[23]
Wojtasik et al.	A multivariate approach to the analysis of trajectory log data and used to predict upcoming MLC replacement needs	MLC	4280 log files, PCA using PI	Identified 8 out of 13 logged MLC replacements (weight = 0.05).	NA	2020	[24]
Donon et al.	To increase the LINAC’s reliability by early detection and prediction of anomalies in its operations, down to the component level	RF power sources	9 million entries from different RF sources, ARI, and signal processing	Anomaly detection was used to record jitter periods in 2 MHz RF generators.	NA	2020	[25]
Able et al.	To predict component failure or system dysfunction of the LINAC	QA	Machine status dataset, mean analysis	A customized graphical user interface provided a means to review the machine parts.	NA	2016	[26]
This study	Descriptive time series analysis for LINAC maintenance history	Cause of failures (MLC, gantry, couch, dosimetric parts, etc.) with downtime	20 years (359 instances), seasonal ARIMA	The accumulated downtime in 2029 is predicted to be a maximum of 2820 h. ARIMA model (p < 0.001, SD ≈ 0.0, ${\sigma }^2$ = 328.33 ± 9.4, CC = 1.00, and ACF = 0.01). Error metric (MAPE = 0.03).	Downtime prediction using a time series analysis	Present	-

Abbreviations: MPC, machine performance check; QA, quality assurance; RMSE, root mean square error; PCA, principal component analysis; PI, performance index; ARI, adjusted Rand index; NA, not applicable.

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