Reliability analysis of computed tomography equipment using the q‐Weibull distribution

Inspired by the successful application of the q‐Weibull distribution in other research fields, we took the lead to use it in the field of medical devices in this work. The parameter estimation of the q‐Weibull distribution was performed using the probability plot method. The CT failure data from Nanfang Hospital in Guangzhou, China, were used to study the reliability of CT equipment at two levels: the CT system and its seven main components. In terms of evaluation accuracy, the mean squared error, Akaike's information criterion, and the determination coefficient were used to compare the accuracy of fitting of different distribution models. The results show that the accuracy of fitting the q‐Weibull distribution is higher than that of the Weibull distribution in terms of determination coefficient and mean squared error. When considering the complexity of the model, the fit accuracy of the Weibull distribution is better. The results were analyzed using reliability and failure rate plots. The q‐Weibull distribution gives a good fit for the failure data of the CT system and components. Though the Weibull distribution fits better in a few cases, the q‐Weibull distribution can describe the entire “bathtub curve” with only a set of parameters. The findings of this study can be extended to other medical devices.


Weibull distribution
Long-term reliability research has revealed that the relationship between the failure rate and running time of many devices, especially electronic devices, obeys the law of the "bathtub curve" (Figure 1), 19 which is divided into three stages: (1) early failure, (2) random failure, (3) wear-out failure. Early failure is mostly caused by design and production defects and is characterized by a high initial failure rate but a rapid decrease in the failure rate with an increase in the running time. The random failure period is generally longer, and the failure rate tends to be stable and can be described by an exponential distribution. The final stage is the wear-and-tear period, where the failure rate increases rapidly with time, mainly due to the wear and aging of the equipment. The Weibull distribution has become the theoretical basis for reliability analysis because it can better fit the three stages of the bathtub curve. 20 Weibull distribution has been employed to characterize the yield strength of steels, 21 the fracture strength of glass, 22 adhesive wear in metals, 23 and failure of composite materials, 24 wind sector, 25,26 fracture statistics of ceramics 27  The probability density function (pdf) of the Weibull distribution at time t can be expressed as follows 28 : where > 0, − t 0 > 0, and t − t 0 ≥ 0. is a scale parameter representing the characteristic life at which 36.78% of the subjects can be expected to have failed and is also positive, 29 is the shape parameter, and t 0 is the location parameter, which indicates that no failure has occurred before this time. The reliability function R(t) can be defined as follows: The Weibull hazard function can be expressed as follows: When < 1, the failure rate decreases monotonically, corresponding to the "early failure" period of the bathtub curve. When = 1, the Weibull model is transformed into an exponential distribution, where the hazard function is constant over time. When > 1, the hazard function curve increases monotonically, indicating that the equipment's operating condition is deteriorating. Although the Weibull distribution produces good adjustment results, it can only exhibit monotonic and constant shapes for its hazard function. In this way, it cannot adequately describe a complex behavior system.

q-Weibull distribution
In the literature, 30 a generalization of the concept of entropy is introduced using a real parameter q: where k is a positive constant that gives dimensional consistency to the expression (Boltzmann's constant), p i is the probability of occurrence of the ith microstate, and W is the total number of microstates in the system. This expression retrieves the Boltzmann-Gibbs-Shannon entropy, S 1 = −k ∑ W i p i ln p i when q → 1. Recently, in non-generalized statistical mechanics, q-type distributions have been used to describe complex systems and have found applications in physics, mathematics, and other fields. [31][32][33] The q distributions are functions based TA B L E 1 q-Weibull failure rate for different values of q and q < 1 q = 1 1 < q < 2 on a non-generalized formalism introduced in non-generalized statistical mechanics. One of these functions is the q-exponential function 34,35 : Since the Weibull model is most commonly used to describe lifetime data, the corresponding q-Weibull distribution has received more attention than other q-type distributions (q-exponential, q-gamma, etc.). Jose found that the q-Weibull distribution gives a better fit for the ordered remission times of 137 bladder cancer patients 36 ; Isabel used the q-Weibull distribution to describe the compression device failure rate, and results showed that the q-Weibull was more suitable for the time to failure data and this distribution is very promising 37 ; Xu suggested that reliability issues such as optimal preventive maintenance policies and system design can incorporate the q-Weibull model. 38 The pdf of the q-Weibull distribution is 39 where > 0 and q < 2 are shape parameters, and > t 0 is a scale parameter. The pdf of the q-Weibull distribution as q converges to 1 is a Weibull pdf: The reliability function of the q-Weibull distribution can be expressed as The hazard function of q-Weibull is

Estimation of the q-Weibull parameter
In engineering applications, the most fundamental problem is the estimation of the q-Weibull parameters. The existing parameter estimation methods of the q-Weibull distribution include maximum likelihood estimates (MLEs), 34 least-squares estimates (LSEs), 35 probability plotting method, 39 adaptive hybrid artificial bee colony algorithm, 38,40 and the method of moments. 36 The disadvantage of the q-Weibull model is that it is difficult to estimate the parameters. Considering the complexity of the calculation process and the magnitude of the calculation amount for each method, we applied the probability plotting method to obtain the optimal parameters. Let ln q (x) denote the inverse function of exp q (x): The q-Weibull accumulated function F q (t) is the complement of the reliability function and can be defined as From Equation (11), we have By taking the logarithm again, we obtain the following.
Next, we have the linear form. with In this article, the failure times organized in ascending order constitute the sample data, and the non-reliability values are estimated using Bernard's approximation of the median ranking provided in the literature 41 : where n is the sample size and i is the fault order number ranging from 1 to n. For each time sample t i , we have and The parameters , , q, and t 0 are estimated by maximizing the coefficient of determination R 2 : where > 0, q < 2, − t 0 > 0, t 0 < t min , t min is the shortest sampling time, and The presence of the parameter q makes the computational procedure to solve the parameters of the q-Weibull model more complicated than that of the Weibull model, therefore, this study uses the Nelder-Mead algorithm to solve for the maximum value of R 2 . The advantage of this algorithm is that no derivation is required so it is suitable for cases where the derivation of the objective function is difficult or where the specific expression of the objective function is not known. but it should be noted that this is a local optimization method and usually requires several experiments to set different initial values to obtain the global optimal solution. Solving the optimal solution of equation over a wide range of initial parameter values confirms that the algorithm always converges and achieves the required accuracy.

Evaluation indicators
In this study, we applied the coefficient of determination R 2 to obtain the optimal parameters. We also used the mean squared error (MSE) 42 and Akaike information criterion (AIC) 43 as additional criteria to compare the fitting accuracy of the Weibull and q-Weibull models In the case of a small sample size, AIC is likely to overfit. 44 To address such potential overfitting, AICc was developed 45 : where k is the number of parameters in the statistical model, and n is the sample size. AICc is essentially AIC with an extra penalty term for the number of parameters, and as n → ∞, the extra penalty term converges to 0. 46 For small samples, AICc is more accurate than AIC.  is the maximized value of the likelihood function of the q-Weibull distribution expressed as follows: where > 0, > t 0 , t min > t 0 , and q < 2.

RESULTS AND DISCUSSION
While they are in use, the failure data of medical devices are available in the medical device information management system of a hospital, and the clinical engineers involved in this study conducted careful selection to ensure the authenticity and accuracy of the failure data. In reliability engineering, failures can be classified into two broad categories: soft and hard. Soft failures are the gradual deterioration of the performance of a device over time, whereas hard failures cause it to stop working. Currently, clinical engineers follow the manufacturer's recommended intervals for regular preventive maintenance to reduce the occurrence of soft failures. In contrast, hard failures are unpredictable and more appealing in reliability engineering. 4 In this study, we collected historical hard failure data of 6 CTs of the same brand in Nanfang Hospital and analyzed them at different levels: system level, and components level. The components we used in this study and their functions are shown in Table 2. We then pool the times between the n − 1th and nth failure events (Appendix C, Tables C1-C8) and conduct the q-Weibull and Weibull analysis for each category separately. This study uses both distributions to analyze the above failure data to further investigate the properties of Weibull and q-Weibull distributions and compare the pros and cons. The estimated parameters of the Weibull and q-Weibull distributions are listed in Tables 3 and 4. The coefficient of determination R 2 , AICc, and MSE values of both distributions are shown in Table 5. It can be found that both the fittings have R 2 values greater than 0.92, indicating that the two distributions fit the failure data well.
In Table 3, WCS has the lowest R 2 (0.9243). CT system and VSM have R 2 higher than 0.99 and exhibit same failure rate shapes. The hazard functions of the the gantry and PDS show an increasing trend, whereas those of the patient table and console decrease monotonically. VMS has the highest value (2.3514) and the smallest t 0 (−658.4669 days).
The parameter is the characteristic lifetime of the Weibull distribution rather than the one of the q-Weibull distribution. The literature 31 gives the formula for the characteristic lifetime q of the q-Weibull distribution: According to Equation (5), q < 1 is the cutoff. The time value corresponding to this cutoff point is t max and is obtained using the maximum likelihood function of the q-Weibull distribution 31 :

Gantry
The scanning frame is an important part of the CT machine, equipped with an x-ray tube, filter, collimator, reference detector, detector, and various electronic circuits. The gantry can be rotated and tilted backward.

Patient table
The scanning bed is a tool to complete the scanning task of transporting the subject, with a vertical motion control system and horizontal longitudinal motion control system, which can automatically enter and exit the scanning frame aperture according to the requirements of the program to complete the automatic positioning of the scanning position of the test object.

Console
It is the control center for generating scanning motion, processing data, and reconstructing images.
Power distribution system (PDS) Provides AC power to the system's electronic components to meet their operating requirements.
Image reconstruction system (IRS) Reconstruction of CT images from projection data acquired by the detector.
Vital signal module (VSM) Monitor vital physiological signals in the human body.
Water cooling system (WCS) Control the temperature in the gantry.  Table 4 lists all the fitting parameters of the q-Weibull model and the values of q and t max . The VSM has the longest characteristic life of 1226.7314 days, and the failure rate shows an increasing trend. Only the Console has a monotonically decreasing failure rate, and the CT system has the smallest characteristic life (42.6243 days). Compared to Table 2, it can be found that the R 2 of the q-Weibull distribution is closer to 1 than that of the Weibull distribution. From this point of view, the fitting of q-Weibull is better than that of Weibull. Table 5 shows the AICc and MSE values for the q-Weibull and Weibull distributions. From the perspective of AICc, the q-Weibull distribution is better for CT system, patient table, console, PDS, IRS, and WCS. For the other two components, the AICc value of Weibull is smaller. This situation can be explained because the q-Weibull distribution has more parameters and a more complex computational process than the Weibull distribution, resulting in a higher cost of parameter estimation. Moreover, AICc has an extra penalty term compared with AIC.

TA B L E 3 Parameters of the Weibull distribution
The MSE values of the q-Weibull distribution range from 1.4587 × 10 −4 to 5.5064 × 10 −3 . The component with the lowest MSE is VMS (1.3377 × 10 −4 ), and its R 2 (0.9987) is the highest among all components. The MSE values obtained from the q-Weibull distribution were lower in the CT system and in all components except for the console and IRS, whose MSE value calculated by the q-Weibull model is slightly larger than that from the Weibull distribution. Table 6 compares the shapes of the hazard function of the CT system and each component calculated from the Weibull distribution and the q-Weibull distribution, respectively. For console and VMS, both distributions show the same failure TA B L E 6 Failure rate shape of Weibull and q-Weibull distributions

Components
Weibull failure rate shape q-Weibull failure rate shape Abbreviations: ↓, decreasing; ↑, increasing; −, constant; ∪, bathtub-shaped; ∩, unimodal. For console and VSM, the failure rate shapes are the same for both distributions, and this fact can be visualized in Figure 2.

F I G U R E 2
The console showing a decreasing failure rate indicates that this component is in the infant mortality phase, and it is difficult to distinguish the two distributions from Figure 2 as they both seem to fit the data well, and the failure rate and reliability curves are virtually superimposed. But the AICc that resulted from the fitting are 380.6459 and 387.9521 for the q-Weibull and Weibull distributions, respectively, from this point of view, the fit of q-Weibull is better. The characteristic lifetime q obtained by the q-Weibull model is very close to that obtained by the Weibull model, confirming that regardless of which model is used, the clinical engineers should develop the same maintenance tactics for the console.
For VSM, both models show monotonically increasing shapes of the hazard function, and the reliability curves are close to each other, as both models show the coefficients of determination R 2 > 0.99. One reason for the good q-Weibull fittings Weibull fittings

F I G U R E 3
Parameters and plots of the CT system and power distribution system. First and second columns: fitted curves (red line) of q-Weibull and Weibull model; third column: reliability curves of both models; fourth column: the hazard function curves of both models fit is that the number of failure data, so far, is still small, and the time intervals between failures are close.
The results of both models show that the VSM has the largest characteristic life q (1226.7314 days) and minimum location parameter t 0 (−184.0805 days). The q-Weibull model is better in terms of R 2 and MSE, while the AICc model is more suitable for a small sample size, and the Weibull model is more appropriate from this perspective. Precise results require more failure data. A solution could be collaborating with other hospitals to aggregate more failure data from the same devices under similar operating and environmental conditions for our future work. The reliability curves and the hazard function curves of the CT system and PDS are presented in Figure 3. The Weibull failure rates show an increasing trend; for q-Weibull, the failure rates are unimodal. For the CT system, the reliability curves of both models are very close, as both models show the coefficients of determination R 2 higher than 0.99 and present very close characteristic lifetimes. According to the Weibull fittings, the shape parameter is close to 1, indicating that the CT system is in the random failure or aging phase of the bathtub curve. The failure rate shape is roughly the same for both models, only in lower time to failure, and the hazard function of the q-Weibull distribution reaches its peak at 35.0832 days and then shows a decreasing trend. The behavior of the hazard function curves moves away from each other after 44.2369 days. Note that the hazard function of the q-Weibull model can get closer to the sample data, especially at higher times. This is the result of the light tail of the q-Weibull distribution.
For the PDS, the failure rates derived by both models are the same at 825.5994 days, and the q-Weibull model's failure rate reaches its highest value at 338.2504 days. The q-Weibull model, whose coefficient of determination R 2 is higher and MSE lower, allows for a more accurate forecast of maintenance strategy. Figure 4 compares the fitting curves of both models for the patient table, image reconstruction system, and water cooling system. Their hazard function curves are decreasing for the Weibull distribution and unimodal shapes for the q-Weibull distribution.
For the patient table, the failure rate of the q-Weibull distribution is higher between 12.4061 and 568.3889 days, and the failure rates of both models drop below 0.002 after 954.1923 days. Note that the excessive failure rate of the Weibull distribution at 0d is inconsistent with the real sample data, which confirms that the q-Weibull distribution is a more appropriate model.
Interestingly, the Weibull distribution fits better to the IRS data according to MSE, and the reliability curves of both models are very close. The failure rate shape of the q-Weibull model was interpreted as increasing until 57.0714 days, then decreasing failure rates considering both distributions and intersects at 526.5957 days.
As in the WCS, the adjustment with q-Weibull gives lower MSE and AICc values, so the q-Weibull distribution leads to a better adjustment than the Weibull, and the q-Weibull model is the optimal choice for clinical engineers to employ q-Weibull fittings Weibull fittings Table   IRS WCS ln(t-t 0 ) ln{−ln

F I G U R E 4
Parameters and plots of the patient table, image reconstruction system, and water cooling system. First and second columns: fitted curves (red line) of the q-Weibull and Weibull model; the third column: the reliability curves of both models; the fourth column: the hazard function curves of both models more efficient maintenance tactics. In contrast, the MSE obtained by the Weibull model is the largest among all components. The WCS has the largest location parameter t 0 considering both distributions, it is less likely to fail at lower times. After the hazard function of the q-Weibull distribution peaks at 232.5543 days, it exhibits a decreasing trend like Weibull.
In Figure 5, we can see that the gantry has an increasing failure rate shape for the Weibull and Bathtub-shaped (q < 1 and 0 < < 1) failure rate for the q-Weibull model. values of MSE and R 2 indicate that for this component, the q-Weibull distribution is a more appropriate model. The hazard function of q-Weibull has the smallest value at 30. Four thousand three hundred sixty days and intersects with the hazard function of Weibull at 285.9868 days. The t max is 458.0545 days, which is compatible with the true maximum lifetime of 453 days.

q-Weibull fittings
Weibull fittings It is worth noting that although the "bathtub curve" theory plays a significant role in reliability practice, in this study only one component hazard function shows a bathtub curve. In contrast, five components have unimodal failure rates, and the remaining two components have monotonically increasing or decreasing failure rates. This study exposes the limitations of the "bathtub curve" theory to some extent, and Klutke et al. 11 point out that the bathtub curve can only describe 10% to 15% of applications, which reminds us that we cannot blindly apply the "bathtub curve" in engineering practice, especially for medical equipment such as CT. To avoid an excessive early failure rate, medical devices are subjected to strict inspection tests before being used. The reason why the Weibull distribution is the theoretical basis for reliability analysis and lift testing is that, for the instantaneous failure rate that is in the three stages of the bathtub curve, its life distribution can be uniformly given by the Weibull distribution. But when the bathtub curve cannot completely describe the characteristics of the failure data, the importance of Weibull decreases, while the hazard function of q-Weibull can flexibly describe some complex situations.

CONCLUSIONS
In this study, the reliability of the CT system and its seven components was analyzed using the Weibull and the q-Weibull distributions, respectively. The console has a decreasing failure rate for both distributions, indicating that it is in the infant mortality phase and preventive maintenance is unnecessary. VMS also presents the same increasing failure rate shapes for both models. The clinical engineers should employ the same maintenance strategies regardless of the adopted model; the CT system and PDS presents the greatest difference in the behavior of failure rates, and the q-Weibull model with lower MSE indicates that the failure rates are unimodal; the patient table, IRS, and WCS, according to the Weibull distribution, show decreasing failure rates, but the excessive failure rate in lower time to failure is inconsistent with the reality. For the gantry, the q-Weibull model is bathtub-shaped, but this trend is weak. The goodness of fitting performed by the probability plotting method is measured by the coefficient of determination R 2 , AICc, and MSE. The AICc not only considers the number of parameters but also has an extra penalty term. According to this criterion, the q-Weibull distribution fits better for the CT system, patient table, console, PDS, IRS, and WCS. For all the subjects studied, the R 2 obtained by the q-Weibull distribution is higher than that of Weibull. The q-Weibull model is better according to this criterion, and the lower MSE values obtained from the q-Weibull model confirm this view again.
In general, the q-Weibull distribution gives a good fit for the failure data of the CT system and components. Even though the Weibull distribution fits better in a few cases, the q-Weibull distribution can describe the entire "bathtub curve" with only a set of parameters. Reliability analysis among different brands can provide a scientific basis for acquiring more cost-effective equipment. However, the results of the q-Weibull model in this study show that only one component exhibited the bathtub-shaped failure rate while most of the others had unimodal failure rates. This may be because medical equipment will undergo rigorous tests to avoid excessive failure rates in the early stages before being put into use. The suitability of the "bathtub curve" theory for the reliability analysis of medical devices in the clinical application phase deserves further investigation.
In application, based on the q-Weibull model, clinical engineers can make different maintenance strategies according to different hazard functions presented by different CT components. For example, if the hazard function curve is unimodal, it means that clinical engineers should perform maintenance more frequently at the peak point time. When the hazard function is monotonically decreasing, more maintenance should be done in the early use stage. In contrast, more maintenance should be done in the wear-out stage when the hazard function is monotonically increasing. In the bathtub curve, more maintenance work should be done in the early use and wear out stages.
This work had several limitations. First, the data used for reliability analysis in this work were limited, which may affect the accuracy of fitting for the Weibull and q-Weibull functions. Second, solving the q-Weibull model parameters was more complicated and time-consuming than Weibull. Finally, in fact, some other models were also put forward in recent years. The comparison of various models should be done in further analysis. Our future study will focus on collecting more data, optimizing the solution of the parameters of the q-Weibull model, and introducing more models to compare with.
In conclusion, the q-Weibull model had a powerful ability to predict reliability for computed tomography equipment compared to the traditional Weibull model. The q-Weibull model that we reported above might assist clinical engineers in optimizing maintenance strategies and decreasing costs.

APPENDIX B. THE PDF, RELIABILITY FUNCTION, AND HAZARD FUNCTION OF THE WEIBULL AND q-WEIBULL DISTRIBUTIONS
In Figure B1, for the Weibull and q-Weibull, their probability density functions can be both represented as "monotonically decreasing" and "unimodal" with the change of parameters. With different combinations of parameters, their reliability function curves can both become steeper or smoother. For the hazard function of the Weibull and q-Weibull distributions, Weibull can only describe monotonically increasing, monotonically decreasing or constant situations, while q-Weibull can describe not only various situations of Weibull but also "unimodal" and "bathtub".