A model for managing patient booking in a radiotherapy department with differentiated waiting times

Abstract

Background. In Denmark, the waiting time from the ready-to-treat date to the first radiotherapy fraction is by national legislation guaranteed not to exceed 4 weeks. This guarantee has now been tightened for some specific diagnoses as it is required that e.g. intestinal and head and neck cancer patients have to be treated without unnecessary delays. Thus, patients with these tumour sites have to start radiotherapy treatment immediately after diagnosis, if it is their primary treatment modality. Previously, patients have been booked at the first empty time slot after their ready-to-treat date. Now, booking has to take the differentiated waiting times into account. To facilitate this, a model has been developed. It is used to manage the booking of patients, reserve accelerator capacity for patients with no waiting time and establish the waiting times for other patients. Methods. The patients are divided into categories according to their waiting time guarantee and for each category a maximum waiting time is defined. The required daily accelerator capacity and average new starts rate for each waiting time category has been determined from the actual patient case-mix in the department. To account for variations in treatment capacity, a prospective daily accelerator capacity is set. Based on the prospective capacity, preparation times, maximum waiting times, and new starts rates, a maximum booking curve (MBC) and a lower limit curve (LLC) are derived. They show the daily maximum and minimum limits, respectively, for booking at future dates. Results. The method is evaluated by a retrospective analysis of actual number of appointments booked compared to the MBC and LLC in situations of both excessive workload and ineffective use of capacity. Conclusion. The model represents a tool for effectively managing the capacity in a radiotherapy department with differentiated waiting times. It improves the transparency of the booking process and prospective waiting times can easily be derived on a daily basis.

It is a well known fact that if the capacity is less than the demand, a queue will appear and a waiting list will develop. This is the situation for radiotherapy (RT) in many countries [1–3]. Also in Denmark the waiting time for RT has increased [4] and it has been shown that this increase has resulted in tumour progression for head and neck cancer patients [5]. In a recent systematic review of the literature about the relationship between waiting time and clinical outcome [6], it is concluded that the risk of local recurrence increases with increasing waiting time for RT and therefore waiting times for RT should be as short as reasonably achievable. In Denmark there has for some time been a waiting time guarantee of 4 weeks by national legislation. This means that a patient is guaranteed to start RT treatment within 4 weeks from the ready-to-treat date. If the limit is exceeded due to shortage of accelerator capacity in a department, the National Health Authorities have decided that breast and prostate cancer patients are the most suitable patient groups to be offered to be re-referred to another RT department with a waiting time shorter than 4 weeks.

The waiting time guarantee has by the first of April 2008 been changed for some specific diagnoses (e.g. intestinal and head and neck cancer) as it is now required that these patients have to be treated without delays, that are not medically indicated. Thus, patients with these tumour sites have to start their RT treatment immediately after diagnosis, if it is their primary treatment modality. Previously, patients have been booked at the first empty time slot after their ready-to-treat date. Now, booking has to take the differentiated waiting times into account. In a department with a limited accelerator capacity, this is a challenge as it further has been shown by the aid of queuing theory, that the maximum acceptable percentage utilisation of an accelerator has to be in the range 70–90% [7], [8] to be able to limit the waiting time due to fluctuations in the patient referral rate.

To facilitate the booking of patients with various waiting time limits and to attempt to use as close as possible all available time slots on the accelerators, a model has been developed which is used to manage the booking of patients, reserve accelerator capacity for patients with no waiting time and prospectively establish the waiting time for other patients.

Methods

All patients in our department are booked for their RT course in the booking application of our Oncology Information System (ARIA®, Varian Medical Systems) when the referral is received, i.e. a 100% pre-booking is performed. One appointment is booked per treatment fraction. The length of the appointments for RT is individual and depends on the technique to be used and the site to be treated. The length of the appointment for the first fraction is longer than for the following fractions. In the Oncology Information System the appointments can be displayed in different views but it is not possible in a simple way to obtain information about the total number of accelerator appointments booked on a given day in the future. To obtain this, a SQL query has been created which directly import the actual number of appointments booked on the accelerators per day from the Oncology Information System database to a spread sheet (Microsoft Excel®). In the spread sheet the actual number of appointments booked is compared with curves describing the two limiting scenarios when it is required that the daily utilization of the accelerator capacity is going to be 100%. The maximum limit (maximum booking curve, MBC) on the number of appointments which can be booked on day t₀ to a given day t in the future is the accelerator capacity on day t minus the capacity to be reserved for patients which have waiting time limits (tw_max) which are less than the time interval t-t₀. The minimum limit (lower limit curve, LLC) is the number of appointments on day t which has to be booked on day t₀ to compensate for the number of patients which will finish treatment. If the number of new patients which is ready to start treatment (i.e. has preparation times less than t-t₀) is less than the number of patients which end treatment the utilization of the capacity will be less than 100%. On contrary, if the number of new patients ready for treatment is larger than the number of finished patients waiting time has to be introduced.

The patients are divided into categories according to their guaranteed waiting time limit and preparation time. The patient categories are shown in Table I. The table also includes information about the total number of patients of the different categories treated in 2007, the average number of appointments per day for each category i (A_i), and the average number of new starts per day in category i (Sr_i). Furthermore, the table shows minimum and maximum time limits from referral to start of RT. These limits are given in number of working days i.e. in number of days where the RT department is open for patient treatment. The minimum limit tp_i (preparation time) for category i includes a possible time interval from referral to the ready-to-treat date for the patient. Thus, the preparation time is defined as the sum of the time interval until the patient is ready to start treatment and the time interval needed to plan the patient for treatment. Therefore, the preparation time is long for breast patients receiving chemotherapy prior to RT and for prostate patients, reflecting the actual referral practice in our department that referral is performed during the chemotherapy course (breast patients) or prior to start of hormone treatment (prostate patients). The maximum number of working days from booking to the first RT treatment is given by the maximum waiting time limits for each category. For breast and prostate patients, the maximum limit in the model is set to 100 days although in principle no limit exists as these patients have to choose if they want to wait until there is capacity at their primary RT centre or to be re-referred to another RT centre to start treatment within 4 weeks. It also means that if Σ A_i is larger than the daily capacity, the number of appointments available for breast and prostate patients is decreased in order to fulfil the condition that the maximum number of appointments per day is equal to the accelerator capacity. The daily accelerator capacity in the department expressed in number of appointments has been determined from the case-mix of patients treated in 2007.

Table I.  The patient case-mix in 2007 is shown. The number of patients is the actual number of patients treated in 2007. The average number of appointments and the new starts rates are based on the actual number of patients treated. The minimum number of working days from booking to start of RT is the time from referral to ready-to-start for the various patient categories whereas the maximum time limit is determined by the maximum allowed waiting time for each patient category.

			Average no. of appointments per day	Average no. of new starts per day	No. of working days from booking to start of RT
Category	i	No. of pts	Ai	Sri	Min-tpi	Max–twmax,i
Head and neck and rectal cancer	1	216	31.0	0.9	1	3
Palliative patients	2	1350	15.6	5.4	1	10
Cervical cancer	3	38	4.1	0.2	5	15
All other curative patients except breast and prostate patients	4	473	42.8	1.9	5	20
Prostate patients	5	111	17.3	0.4	40	100
Breast patients RT+ chemotherapy	6	153	14.7	0.6	35	100
Breast patients RT only or RT + homone treatment	7	136	13.1	0.5	20	100
Total		2477	138.6	9.9

For patients in category i, the minimum number of appointments on day t in the future which needs to be booked on day t₀ to compensate for patients ending treatment can be described by:where t₀ is today and t –t₀ is the number of working days from today to day t in the future, and A_i /Sr_i is the average number of fractions for patients in category i.

The lower limit curve for all patient categories is then given by the sum:The maximum number of appointments on day t in the future which can be booked on day t₀ for patients in category i in order to reserve capacity for patient categories with shorter waiting times can be expressed as:and thenThe expressions for LLC and MBC have been set up in the spread sheet with the case mix data shown in Table I and is presented in a graph with the actual number of appointments booked obtained by the SQL query.

Results

Examples of the output obtained from the spreadsheet are shown in Figure 1 for two days ultimo April and medio May, respectively. The triangles show the actual number of appointments booked on the accelerators. The LLC is shown with a dashed line, the MBC with a solid line, and the daily RT capacity with a dashed dotted line. For dates close to the day where the graph was generated (t₀) the lines coincide, corresponding to the fact that the number of appointments which have to be booked is equal to the capacity to obtain a 100% utilisation of the accelerators. When going to the right in the figure the distance between the MBC and the capacity is larger reflecting the fact that capacity is reserved for patients in categories with shorter preparation times and waiting time limits. For dates more than 8 weeks in the future, the MBC shows the maximum number of appointments which at day t₀ can be booked for breast and prostate patients. It can be seen from the figure that there is a fixed distance between the dashed dotted line representing the accelerator capacity and the solid line representing the MBC for dates 2 months or more in the future. This is due to one of the assumptions in the model that if the capacity is reduced or increased it is the capacity for treatment of prostate and breast patients which is changed. In this way the capacity is unchanged for other patient categories and the breast and prostate patients are handled as a sort of buffer capacity.

Figure 1.  The number of appointments booked per day versus date is shown as triangles for a) April 25, 2008 and b) May 19, 2008. The dashed line shows the LLC, the solid line the MBC and the dashed dotted line the RT capacity versus date.

The actual waiting times for patients with a maximum waiting time limit of 4 weeks can be found from the figure as the date where the number of booked appointments is below the MBC. From the figure it can also be seen that the available window between the LLC and MBC is rather narrow especially when there are large variations in the capacity, which is the case at the start of the summer period.

Figure 2 shows booking data at four different dates from mid December 2007 to mid January 2008. On December 18 (Figure 2a) the actual number of appointments booked for the first half of January was lower than the LLC, indicating that an utilisation of the RT capacity of less than 100% might be expected in the first half of January. In Figure 2a it can further be noticed that the number of appointments booked on December 18 for the period after January 14 is between the LLC and MBC curves. This pattern continues on December 28 (Figure 2b) and January 7 (Figure 2c). Showing that in fact, in the first half of January the utilisation of the RT capacity was below 100% whereas from January 14 and on the utilisation increased to 100% (Figure 2d). This is in accordance with the pattern indicated in the data from December 18 that the number of appointments booked for January 14 and later was higher than the lower limit curve.

Figure 2.  The number of appointments booked per day versus date is shown as triangles for a) December 18, 2007, b) December 28, 2007, c) January 7, 2008, and d) January 14, 2008. The LLC is shown as a dashed line, the MBC is shown as a solid line and the capacity is shown as a dashed dotted line.

The opposite situation where the number of pre-booked appointments per day for a period was higher than the number given by the MBC is shown in Figure 3. The booking status on January 21 (Figure 3a) shows that in February the number of appointments exceeds the MBC. This is still the case on February 12 as shown in Figure 3b.

Figure 3.  The number of appointments booked per day versus date is shown as triangles for a) January 21, 2008 and b) February 12, 2008. The LLC is shown as a dashed line, the MBC is shown as a solid line and the capacity is shown as a dashed dotted line.

Discussion

The two situations with ineffective use of capacity (Figure 2) and excessive workload (Figure 3) are events which occurred before the model was established. They represent examples of booking practice based only on the agendas for the accelerators in the department. As it can be seen from the figures, the LLC is a useful predictor for the lower limit of number of appointments to be pre-booked in order to obtain an utilisation of the accelerator capacity of 100% on a given date in the future. The MBC, on the contrary, is a good measure of the maximum number of appointments which can be pre-booked for a day in the future on a given date in order not to exceed 100% utilisation when that future date arrives. Figure 1 shows examples of the number of appointments pre-booked per day in the future after the model has been utilized in the booking of patients for a period of time. From the figure, it can be seen that the actual number of appointments pre-booked per day is between the LLC and MBC curves until the beginning of July where the number of appointments exceed the MBC indicating that a situation with excess workload may occur. The sharp drop in accelerator capacity at the first of July is partly due to an old accelerator is closed down in order to be replaced by a new one and a replacement accelerator will first be ready for clinical operation after the summer period and partly due to a general reduction in accelerator opening hours in the department as the number of radiotherapist is reduced because of summer holidays. From the figure it is suggested that the chosen procedure of reducing the accelerator capacity sharply from one week to the next is a situation which may give rise to excess workload. The data indicates that a better approach for future summer periods may be a more stepwise reduction in capacity. Finally, it may be noted from Figure 1 that the LLC is increasing in August and this increase is more pronounced in Figure 1b than in Figure 1a. Thus, a narrow window between the LLC and MBC at the end of the summer period will develop during the next months. The increase of the LLC in August which can be observed more than 3 months in advance is because the capacity for treating prostate patients increases after the end of the summer period and due to the long period from referral to start of treatment for this patient category. Thus in order to obtain a 100% utilisation of the accelerators from mid August it is necessary to pre-book several prostate patients in the middle of May.

Originally, the model was based on the patient case-mix treated in 2007. However to take variations in case-mix over time into consideration, the case-mix is adjusted regularly. We have chosen to correct the case-mix quarterly so it always represents the distribution of diagnoses and proportion of palliative to curative treatments treated in the preceding one year period. However, a comparison of the case-mix in 2007 and for the first quarter of 2008 shows only minor changes in the distribution of patients (data not shown). It may be noted that the case-mix does not include a special category for emergency patients. They are not included as a special category as the number of patients referred to start RT within 24 hours is very low in our department and they more or less counterbalance the number of last minute cancellations of other patients.

The average treatment time per fraction is assumed to be the same for all patients. This is based on the fact that the length of the appointment for RT does not show large variations between most techniques applied in our department. One exception is the treatment time for stereotactic treatments. However, the patients receiving stereotactic (cranial and extracranial) treatments are included in the case-mix and therefore these time consuming treatment types contribute to the average number of daily appointments for the patient category where these patients are included. Another approach is the use of a basic treatment equivalent (BTE) which was presented by Delaney and co-workers [9]. In their BTE model of resources utilisation, the complexity of the various treatment types was incorporated. This model was developed when the use of shielding blocks was still common thus prolonging the delivery of the treatment when shielding blocks were going to be mounted for each field. Today, when MLCs are common on most accelerators and the fields are delivered in an auto sequence, the number of fields is not an important parameter. If an IMRT plan with a high number of MU is delivered, the beam on time will be longer due to the higher number of MUs. However, in our department the time used on positioning the patient in the treatment room and preparing the room for the next patient exceeds the beam on time for treating the patient.

A basic condition for using the model is that the patients are pre-booked for treatment when the referral is received. However, it seems to be a good practice as Martin et al. [10] found that patients who were pre-booked in their department for RT had shorter waiting times compared with patients who were not pre-booked. They also introduced a priority points system for the various patient diagnoses to aid booking and found that this also had a positive effect on the waiting times. The maximum waiting time limits for the different patient categories may be compared with a simple version of the points system described in ref [10].

The model is relatively simple as it is based on average values and some deviations from the curves may be expected. The referral of patients is subjected to random fluctuations and in order to simulate these variations it would be necessary to extend the model with Monte Carlo simulations. The use of Monte Carlo simulations for waiting time modelling has previously been demonstrated [11].

The practical use of the model in the daily booking of patients in our radiotherapy department has shown that the model is easy to use as the booking staff by pressing a button in the spread sheet can obtain an updated display of the future work load at the accelerators. From the booking curve the prospective waiting time is easily found as the date where the number of appointments pre-booked is below the MBC. Therefore it becomes easy to obtain updated information about the actual waiting times in the department.

Conclusion

The model represents a tool for effectively managing the capacity in a RT department with differentiated waiting times. It improves the transparency of the booking process and prospective waiting times can easily be derived on a daily basis.

Acknowledgements

Declaration of interest: The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.