Formulation of objective indices to quantify machine failure risk analysis for interruptions in radiotherapy

Objectives: To evaluate the effect of interruption in radiotherapy due to machine failure in patients and medical institutions using machine failure risk analysis (MFRA). Material and methods: The risk of machine failure during treatment is assigned to three scores (biological effect, B ; occurrence, O ; and cost of labor and repair parts, C ) for each type of machine failure. The biological patient risk (BPR) and the economic institution risk (EIR) are calculated as the product of B and O ( B (cid:1) O ) and C and O ( C (cid:1) O ), respectively. The MFRA is performed in two linear accelerators (li-nacs). Result: The multileaf collimator (MLC) fault has the highest BPR and second highest EIR. In particular, TrueBeam has a higher BPR and EIR for MLC failures. The total EIR in TrueBeam was signi ﬁ cantly higher than that in Clinac iX. The minor interlock had the second highest BPR, whereas a smaller EIR. Meanwhile, the EIR for the LaserGuard fault was the highest, and that for the monitor chamber fault was the second highest. These machine failures occurred in TrueBeam. The BPR and EIR should be evaluated for each linac. Further, the sensitivity of the BPR, it decreased with higher T 1 = 2 and α / β values. No relative difference is observed in the BPR for each machine failure when T 1 = 2 and α / β were varied. Conclusion: The risk faced by patients and institutions in machine failure may be reduced using MFRA. Advances in knowledge: For clinical radiotherapy, interruption can occur from unscheduled downtime with machine failures. Interruption causes sublethal damage repair. The current study evaluated the effect of interruption in radiotherapy owing to machine failure on patients and medical institutions using a new method, that is, machine failure risk analysis.

with multiple interruptions, such as intensity-modulated radiation therapy, may be less than those of the same dose without interruptions. Moreover, the effect of cell survival with SLDR appears to almost plateau after several hours of interruption. 6 Brenner et al.
suggested a linear-quadratic (LQ) model with the Lea-Catcheside time factor to analyze cell survival considering SLDR during irradiation at the cell population level. 7 In this study, the biological effect of the duration of interruption caused by machine failure was determined using the LQ model with the Lea-Catcheside time factor of a single interruption in one fraction, and a risk analysis with SLDR by machine failures was emphasized. For an institution, the cost associated with machine DT is a significant factor to consider as the costs associated with health systems must be economically sustainable. 8,9 Hence, machine failure presents a high risk for medical institutions.  Table 1 lists the total number of unplanned intra-fraction machine failures on a treatment session from April 2015 to April 2018. The total number of unplanned intra-fraction machine failures was 60.

2.A | Classification of machine failure
Based on the 60 failure modes identified, the machine failures were broadly classified into the following categories: multileaf collimator (MLC), potentiometer, radio frequency (RF) driver, minor interlock, water temperature and quantity, monitor chamber and LaserGuard.
The failure associated with the MLC includes failures associated with the motor, communication, leaf and carriage positioning, and power supply. If the water temperature increases, low water and gas pressures or other problems cause an interruption during irradiation to decrease the temperature. Failures associated with the potentiometer and RF driver involve an active interlock, which turns the beam off during irradiation. These machine failures must be addressed by replacing the parts. The TrueBeam linac has a collision detection system called LaserGuard, which comprises an infrared laser. Laser-Guard is used to replace the parts when the interlock associated with the collision cannot be released. The minor interlocks alert the operator to the existence of conditions that affect machine operation, such as filament time delay, calibration cycle timeout, and excess dose rate. This interlock does not require a significant amount of time to release the interlock, part replacement or a system restart.
It is released by re-mode up, username and password input.

2.B | Machine failure risk analysis
MFRA is performed to calculate the risk faced by patients and institutions by evaluating the cost of repair and biological effects when DT occurs. The risk of machine failure during treatment can be assigned to the following three scores: biological effect, B; occurrence, O; and labor and repair part, C. The current study focused on machine failure, without swapping clinical treatment plans among beam-matched linacs, and no rescheduling time is available after treatment of all patients.

2.C | Biological effects in treatment
In this study, we focused on the biological effects of an unplanned intra-fraction break in a treatment session caused by a machine malfunction, as shown in Table 1. We assume that, except for this delay, the total dose for the session was delivered as planned without swapping with a clinically beam-matched linac and that no rescheduling time was available for a patient to continue treatment after the daily treatment of all patients. Moreover, in the current study, it was assumed that biological effects follow the LQ model, which provides a simple relationship between cell survival and delivered dose. 10,11 More importantly, the standard LQ formalism, as applied to time-dose relationships, is not merely a truncated power series in dose. The key feature is a specific mechanistically based functional form for the protraction factor (G), which considers dose protraction or fractionation. This factor was derived by Lea and Catcheside. 12,13 Brenner et al. applied G to the biological dose calculation and calculated the survival fraction in the case of two acute dose fractions,D 1 and D 2 , separated by the DT using the LQ formalism that was incorporated as well as the protraction factor. 7 The biological effect with interruption depends on the DT and dose per fraction (DPF). In the current study, it was assumed that the interruption occurred during one-half of the irradiation. The survival fraction with interruption (SF with ) is calculated as follows: Here, λ is the repair rate for double-strand breaks, equal to 2/ T 1=2 , where T 1=2 is the repair half-time. Typical values of α/β are used for early responding tissues, as listed in Table 2.7 These were also used in the simulation by Brenner et al. 14 The survival fraction without interruption equivalent to the survival fraction with interruption is denoted as SF w=o . . The equivalent DPF is defined as D eq DT when SF with ¼ SF w=o , as shown in Fig. 1. SF w=o is calculated as follows: The D eq DT value can be calculated from Eqs. (1) and (2) as follows: Additionally, the biological effect of DT for each DPF is denoted as B DT,DPF . To calculate B DT,DPF , the maximum DT was used when the machine failures were classified every 10 min in the range of 0-100 min, as shown in Table 1.

2.D | Occurrence
Occurrence in MFRA is defined as the probability of machine failure for the DPF of each patient when DT occurs (o DT,DPF,MF ). Information regarding machine failures and DT is shown in Table 1. In practice, the occurrence is calculated as follows: where N DT,DPF,MF is the number of machine failures for the DPF of each patient when a machine failure with a DT occurs. The evaluation period used was 36 months, which was equivalent to the analysis period.

2.E | Cost of labor and repair parts
The cost of labor is directly related to the payment of the treatment staff. In the current study, it was assumed that a backup machine T A B L E 1 Total number of unplanned intra-fraction machine failures and downtime (DT; min).
would not be used. The cost was analyzed based on machine failures that occurred during an unplanned intra-fraction break in a treatment session, as shown in Table 1. Thus, this study focused on the labor cost and repair parts for an unplanned intra-fraction break in a treatment session. The labor cost incurred by two therapists and nurses in treating a patient when machine failures did not occur was considered. Meanwhile, various cases pertaining to the labor cost of an engineer exist. Some hospitals hire engineers or establish a maintenance contracts with vendors. Therefore, in the current study, the labor costs of two therapists and nurses were considered. Moreover, cost of a physicist was not included in the economic institution risk (EIR) analysis because flexible working hours are applied to most physicists and their salary is not paid hourly. The cost of radiotherapy has been categorized based on high and low-income countries by Van Dyk et al. 15 Using the monthly salary and working time specified by Van Dyk et al., the total salary of two therapists and nurses per minute S was calculated as follows: S was approximately $1.06/min. In addition, the cost of labor for the event i of machine failure for each DT and DPF c labor i was calculated using.
where t i is the maximum DT when machine failures are divided into intervals of 10 min in the range of 0-100 min, as shown in Table 1.
where BPR DT,DPF,MF is the BPR DT,DPF for each machine failure. BPR MF is calculated using the total DT at each DPF for each machine failure.
For economic analysis, the EIR for each machine failure (EIR MF ) is defined as.
An example of BPR calculation for water temperature and quantity faults is presented here. This fault occurred three times with DT of 0-10 min and B ¼ 3:3%, and once with DT of 30-40 min and B ¼ 5:4% in 2 Gy/fr patients. The calculation is given by:

2.H | Sensitivity of EIR
The sensitivity of EIR MF to the variation in labor cost was investigated. The labor cost was eliminated by assuming that a backup machine can be used, and that all patients with machine failures can be transferred to the backup machine without overtime. The EIR for each machine failure without a backup machine is defined by Eq. 12.
The EIR for each machine failure with a backup machine (EIR BA MF ) is defined as the ratio of the cost of repair parts to the total cost:   Figure 7 shows the variations in BPR MF for the machine failures caused by the MLC, potentiometer, RF driver, minor interlock, water temperature and quantity, cable, monitor chamber and LaserGuard.

3.D | Sensitivity analysis of BPR MF
The BPR MF decreased with higher T 1=2 and α/β values.

| DISCUSSION
The B DT,DPF increased with higher DPF and DT, as shown in Fig. 2.
In clinical practice, the treatment technique and DPF differ for each patient. The difference between the DT and DPF is considered in the BPR for machine failures. In this study, the effects of multiple treatments on BPR were not considered. We would expect the BPR to be reduced for multi-fraction treatments as opposed to single- In addition to BPR, an EIR analysis was conducted to estimate the risk of economic cost in a medical institution. As shown by the results in Figs. 3 and 4, the patient and economic cost risks differ. analysis of the EIR shows that using a backup machine can reduce the EIR. However, treatment with a backup machine poses another risk. Thus, a swift patient quality assurance for each treatment plan and a plan verification for the remaining MU and segment after a machine breaks down is necessary. Determination of the action level or threshold for the biological and economic risks by the machine faults is outside the scope of the current study. Biological and economic risks were independent. Therefore, the action level depends on the institution because the risk priority differs for each institution. The MFRA can be used to control and evaluate patient risk or economic risk. In future studies, the MFRA system will be expanded simulate risk reduction by performing MFRA after applying the riskreducing methods presented herein.

| CONCLUSIONS
In machine failure, the risks faced by patients and institutions differ.
The proposed MFRA contributes to the reduction in economic cost for institutions and biological effects on patients. Furthermore, the risk effects on patients and institutions differed between TrueBeam and Clinac iX. Identifying the machine failure risk faced by patients and institutions during treatment is critical for each institution and can offer prevention through model creation for preemptive maintenance to mitigate the risk, or through feedback to service engineering.

CONFLI CT OF INTEREST
The author have no other relevant conflict of interest to disclose.