Towards an improved resolution of radiotherapy scheduling Yoan Jacquemin, Eric Marcon

Pascal Pommier

LASPI IUT de Roanne 20, avenue de Paris 42334 Roanne Cedex - France [email protected] (Corresponding author), [email protected]

GCS ETOILE 60, avenue Rockfeller 69008 Lyon – France [email protected]

Abstract— In any cancer treatment, waiting times are critical, the faster you treat, the more effective the treatment is. Minimizing waiting times is especially difficult in the context of radiotherapy where material and human resources are usually scarce. Although healthcare scheduling is extensively documented, few studies have addressed the specific radiotherapy problem. In this paper we develop an integer linear optimization of a non-block scheduling strategy to improve existing models accuracy. After analyzing the weakness of the previous works, we suggest a new mathematical model, which better fits the reality. First we increase the planning horizon to ensure efficiency over several weeks then we take into account several new constraints such as critical human resources and protocol-dependant constraints while using near real-life data from the studied hospital. Results achieved show an improvement of radiotherapy treatment room utilization, a higher number of patients treated along with a decrease in patient waiting times. Keywords-component; Scheduling , radiotherapy, integer linear programming, optimization, healthcare

I.

INTRODUCTION

For several decades, the number of radiotherapy patients have been steadily increasing, this growth in demand can be explained for one part by the global ageing of the population and for another part by the easier access to more accurate radiotherapy treatment. More people can be treated and more pathology can now be cured. Radiotherapy treatment is a complex process involving several steps. Most of these steps have to be performed in a specific order and place at a precise time and with highly skilled workers. Almost all of them have to be done with the proper delay between them. Furthermore, linear accelerators (linacs) are expensive machines, and oncologists, nurses and physicians working in synergy are critical resources to treat patients. These constraints result in long waiting-times in a process where short waiting-times are directly proportional to the effectiveness of the treatment. In other words, the faster you treat, the better you cure. Minimizing waiting-times is a well studied problem in industry. Numerous advanced methods were, and still are, developed to solve this kind of complex problems. This provides us a great amount of tools and methods, from reengineering the facilities, to modifying the process through defining the best scheduling to use.

A large amount of these methods were adapted to healthcare problems, especially in costliest areas such as emergency department and operating theatre. Operating rooms being one of the most costly facilities in hospitals, numerous studies were published on the better way to schedule surgeries and staff shifts while choosing the appropriate staff size [1]. However, for the specific subject of radiotherapy planning, few studies have been published, and in our knowledge nothing relevant has been published before 2006. Two main approaches were designed; the first one from Coventry and Nottingham Universities resulted in a constructive algorithm providing optimized scheduling [2]. In [3], after a mathematical formulation of the radiotherapy scheduling problem, the authors classified it as a Stochastic Dynamic Job-Shop Problem (JSP). Due to the highly combinatorial nature of this kind of scheduling problem, proved NP-Hard (non-deterministic polynomial-time hard) by [4], the authors first advised towards the use of metaheuristic approach (i.e., Tabu search). Then in [7] they designed two constructive algorithms based on due date. ASAP (As Soon As Possible) booked patients forward from the earliest feasible start date, while the other, named JIT (Just in Time), booked patients backward from the latest feasible start date. Both of them work with the same prioritized patients’ list ranked by emergency level, softer constraints were applied to top-priority patients. Furthermore, their algorithms try to avoid useless starts at the end of the week and harmful long breaks within the course of treatments. Their objective function maximized: i) amount of tardy patients, ii) total length of waiting time (rapid decrease in satisfaction after 28 days of wait) and iii) amount of treatment breaks. They found that JIT algorithm provided better solutions according to their quality criteria. In [8], they merged this two approaches into one called the Target approach which tried to book patients near a target date by using forward and backward method. They also developed three improvement they applied incrementally to Target approach: i) a Threshold approach which assigned a maximum number of patients per day for each priority-level; ii) a Creation Day approach which delayed the planning construction for few days, excepted for urgent patients and iii) MNDA (Maximum Number of Days in Advance) approach which allowed to wait for several days before booking a new patient. And finally they optimized obtained schedules with a

GRASP (Greedy Randomized Adaptive Search Procedure) approach from [6]. Their best scheduling results on simplified real-world data over 18 months (from Nottingham University Hospitals, NHS Trust) were obtained with 10% of the time assigned for emergency patients, a schedule created 3 times a week (Mondays, Wednesdays and Fridays) and a one day wait before booking routine patients as advised by the MNDA approach. Finally their GRASP optimization improved schedules for 38% of the experiments and worsed for 23% of them. The second approach from [7,8], described an integer linear optimization program specifically designed to solve radiotherapy scheduling. In their first publication, they introduced their model and in the second one they extended and improved it. In the last version their algorithm works with two patients’ list, one of Booked Patients and one of Waiting Patients weighted accordingly to their emergency status and their arrival order. Their objective function is to maximize the number of new patients starting their treatment. They included several constraints ensuring an accurate treatment in terms of numbers of sessions per week, session duration, maximum capacity of linac and availability of already booked patients. Results are obtained for one-week from a set of artificial data. In our approach we combined the best of both approaches by scheduling simplified real-world data during 15 successive weeks using a linear integer optimization approach. After establishing the efficiency bounds of the Conforti’s approach [8], we show our approach allows a better linacs utilization, a better quality of service and shorter delays. This paper is structured as follow. In the second part, we describe data used and differences between Conforti et al algorithm and our own. In the next part, we present results obtained with each approach, which are discussed in the fourth part. . Finally the fifth part gives concluding remarks and points direction for future work. II.

MATERIALS AND METHODS

Through our experiment, we tested and compared the performance of 4 algorithms: the first one named Conforti is the original version of [8]. ii) Overtime, a slightly modified version which authorize adding time to each shift in order to treat already booked patients, iii) Waitav, which improve multiple weeks planning by taking in consideration patients’ availability from the beginning and iv) Rav, which aim to ensure patient’s radiotherapist availability on the first day of treatment. A. Patients Data Our virtual center performs treatments on 2 identical linacs although when one patient started on one of them, his/her treatment will always take place on this one. Each linac is open 300 minutes in the morning and 300 minutes in the afternoon, 6 days on 7. Two lists of patients are used, one for Booked

Patients (BP) i.e. patients who already started their treatments and one for Waiting Patients (WP) i.e. patients who are waiting to start their treatment. The dataset is based on a French oncology centre (Centre Léon Bérard, CLB) data. These data give us an order of magnitude for the number of patients per week, and the distribution of parameters such as the number of treatment weeks and the prevalence of emergency cases. We gathered patient files from 2008 (i.e., from January 1st to December 31th) which were filtered against incomplete files and outliers. During the first 4 weeks, number of WP slowly grows to fill the center from 20 new patients the first week to 23 the third, while BP starts with 45 patients. After these 4 weeks of warm-up, random amounts of patients were assigned between 25 and 30 patients per week. We decided to use 1 to 6 weeks treatment protocols with 20% of them with a high priority. We also kept Conforti choices in terms of number of treatment sessions per week: either 4 or 5 for every patient. Instead using Conforti duration data, we conducted a short survey on two linacs of the CLB to assess more realistic session times. According to these data, we chose session time between 7 and 15 minutes for regular session and twice or thrice this time for the first session. And finally, we randomly generated some of the values like patient and radiotherapist availability and last day of treatment of previous week for patients already booked in the first week. B. Algorithms Our formulation of the radiotherapy scheduling problem is mainly based on Conforti’s so we only provide a brief reminder of their notation. Furthermore, we only describe changes made to the original Conforti’s algorithm [3]. The simulated center operates M machines (i.e., 2 by default), for K work days per week (i.e., 6 days), divided into F shifts (i.e., 2 shifts) of Tmkf minutes (i.e., 300 minutes). Two lists of patients are used: BP, for each already booked patients p and WP for each waiting patients j ranked first by priority then by arrival order. Each list has some attributes which describe each patients treatment s , ii) of first protocol with i) duration time of regular session ~ session ~ s , iii) last possible day of the week for starting treatment ld, iv) availability of BP avp, v) t the number of sessions per week, vi) wj the weight of each j from WP, based on emergency status prj and arrival order, vii) weekly allowed breaks which we individualized for each protocol (2 and 3 days respectively allowed for 5 and 4 sessions/week protocol). Several decision variables are also used: i) zmkf for the decision of starting the first session on machine m on shift f the day k, either for j or p patient, ii) ykf for regular session either for j or p patient, iii) dmkf for overtime on machine m on shift f of day k and iv) xmj for the decision of starting treatment for patient j on machine m. In order to avoid weight scale issues, we had to redefine Conforti’s weight function: for the patient j from WP, wj=w(j-1)+(|WP|-j) + 100prj + 1. 1

TABLE I. WP

ATTRIBUTES OF THE 15TH FIRST PATIENTS OF FIRST WEEK ORDERED BY DECREASING WEIGHT

D1_7

D1_9

D1_14

D1_1

D1_2

D1_3

D1_4

D1_5

D1_6

D1_8

D1_10

D1_11

D1_12

D1_13

D1_15

20

18

33

18

22

45

26

22

45

14

16

27

30

45

30

10

9

11

9

11

15

13

11

15

7

8

9

10

15

10

Priority (pr)

10

10

10

1

1

1

1

1

1

1

1

1

1

1

1

Weight (w)

4910

3890

2871

1853

1736

1620

1505

1391

1278

1166

1055

945

836

728

621

Session/week (t)

5

5

5

5

4

5

5

5

5

5

5

5

5

5

4

~ s1

~ s

Total sessions

15

20

25

25

24

20

25

25

25

25

20

25

15

20

20

Last day (ld)

2

2

2

2

3

2

2

2

2

2

2

2

2

2

3

Radiotherapist (rt) 1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

1) Experiment #0: From Conforti to Overtime algorithm: In this algorithm overtime is allowed in order to treat already booked patients. Indeed, with Conforti’s algorithm, patient’s availability is taken in consideration only for BP, which can results in infeasible linear programming when planning over several weeks. We decided to limit the use of this overtime to BP, hence closing these extended shifts to WP. The decision variable used is dmkf which add a constant time OT per shift. Three constraints were added to the 17 defined in [8] to ensure that: (1) the time used by BP is lesser to the available time, with or without overtime. (2) no time is available for WP if there is overtime needed and (3) combined time of WP and BP do not exceed regular capacity when overtime is not needed. We also add a penalty in the objective function based on the sum of wj. WP M

max

∑∑ j =1 m =1

M

w j x mj −

K

F

∑∑∑

WP

d mkf

m =1 k =1 f =1

∑w

j

(Obj) OT

j =1

subject to BP

∑ ~s

p y pmkf

av pkf ≤ Tmkf + d mkf OT

∀m, ∀k , ∀f (1)OT

p =1

WP

∑ ~s

1

j

z jmkf + ~ s j y jmkf ≤ (1 − d mkf )Tmkf

∀m, ∀k , ∀f

(2) OT

j =1 WP

∑ j =1

BP

~ s 1 j z jmkf + ~ s j y jmkf ≤ Tmkf −

∑ ~s

p y pmkf

av pkf + d mkf M

p =1

∀m, ∀k , ∀f , M > Tmkf (3) OT

2) Experiment #1: From Overtime to Waitav algorithm: As the need of extended shift duration arise from the patient’s availability not taken into account during the first week, two choices exists to solve this problem. The first and the easiest one is to never consider patients’ availability. The second one is to book accordingly to patients’ availability from the beginning. Since our first goal is to improve quality of patient’s care we chose the second approach although corresponds more to the reality of care. To achieve this goal, we introduce a new parameter waitav for waiting patients. This parameter describes patient’s availability from the first to the

last week and factorizes each occurrence of zjmkf or yjmkf in overtime formulation. For example: WP

∑ ~s

1

j z jmkf

+~ s j y jmkf ≤ (1 − d mkf )Tmkf

∀m, ∀k , ∀f

(2) OT

j =1

Became WP

∑ ~s

1

j z jmkf

waitav jkf + ~ s j y jmkf waitav jkf ≤ (1 − d mkf )Tmkf

j =1

∀m, ∀k , ∀f (2) Waitav 3) Experiment #2: From Waitav to Rav algorithm: In the French center studied, radiotherapists have to check treatment’s parameters and patient’s position before allowing the first session begin. Most of the time, patient’s radiotherapist, (i.e. which already encountered the patient and is likely to have done the dosimetry for this patient), perform better and faster than any other radiotherapist. Furthermore, the continuity of care contributes to improve quality of patient’s care. To reach this objective we decided to take into account the combination of patient’s availability with radiotherapist’s availability on the first treatment session. This new constraint is a soft one, meaning a patient is allowed to start his/her treatment with another radiotherapist. To manage this softness, we introduce a new penalty in the objective function dependent on the sum of a hundredth of patient’s weights multiplied by the decision to change the starting radiotherapist i.e. it cost one hundred times less to start a patient despite a radiotherapist change than to delay him/her to next week. Three parameters were created, rav for describing radiotherapist availability, rtj to keep track of each patient’s radiotherapist and chrtmj for the decision of changing the radiotherapist. The new objective function is:

WP M

max

∑∑ j =1 m =1

M

x mj w j −

K

F

∑∑∑ m =1 k =1 f =1

WP

d mkf

∑ j =1

WP

wj − j

∑ chrt

mj ( w j

j =1

(Obj) Waitav

/ 100)

We also had to split the first constraint of [8] in two new constraints to handle the possibility of change: ld j

x mj =

∑z

jmkf

waitav jkf

jmkf

rav rt j kf waitav jkf + chrt mj * 100 ∀j , ∀m (4-a)Rav

jmkf

rav rt j kf + (1 − chrt mj ) * 100

∀j , ∀m (4)Waitav

k =1

Became ld j

x mj ≤

∑z k =1 ld j

x mj ≤

∑z

∀j , ∀m (4-b)Rav

k =1

III.

RESULTS

Our results were obtained by Integer Linear Programming under LINGO©. Input data and results were managed by Microsoft Excel 2007© combined with LINGO VBA module and several Excel macros.. Results were obtained on a 1.73Ghz CPU with 1Gb of DDRAM. Even if we are using an exact resolution method, resolution times over 15 weeks result in intolerable total resolution time therefore each ach algorithm was allowed to run a maximum of 240 seconds for each week resolution, although our experiments showed that solutions obtained with 120 seconds had almost the same quality and that on the contrary solutions after 500 seconds were not better (data not shown). Moreover until the center is full, several solutions are equivalent from our objectives functions’ point of view and randomly chosen by the branch anch and bound algorithm. The combination of these facts results in variability in global performances. To overcome this issue we ran 30 replications of each algorithm to estimate itss magnitude hence each performance indicators shows average and extreme (i.e., ( min/max) results. Presented results are focused on three performance indicators, the first compares quality of care obtained with each algorithm, the second focuses on delayed treatments t and the third deals with linacs utilization. These results are organized by following experiment designs previously explained. Results obtained with Conforti’s algorithm are not shown due to its inability to perform on successivee weeks with several datasets without leading to infeasible models (cf. Fig. 1) because of patients’ availability, weights scale issues or because no solutions was found within 240.. Overtime model can be used as a replacement as it is specifically designed to overcome this issue without affecting performances. A. Exp #1: Overtime and Waitav Fig. 2 shows the percentage of patients which were treated per week among total patients in the center. Indeed, not every WP can always lways be treated and starting patients are chosen on their respective weights. It’s also important to stress out that algorithms both respect every patient’s availability. Overtime algorithm needs an average of 3 overtime shifts to achieve this planning although variability ariability of results increases with Overtime algorithm.. Overtime performance drops since week 4 and gradually falls under 40% by week 15 while Waitav performances stay above 95% over the 15 weeks period.

Figure 1. Amount of new patients treated each week over 15 weeks using Conforti’s original algorithm. Infeasible models are observed in weeks 5 and 8 because of availability incompatibilities. The algorithm is infeasible from 10 to 15 because of a scale issue with patients’ weights and/or an impossibility lity to find a solution within 240 seconds.

Figure 2. Orange: Overtime, Blue: Waitav. Percentage of patients treated (BP + WP starting treatment) among patients in the system (BP ( + WP WP).

Second performance indicator relies on delay between week of patient’s arrival in the center and their first week of treatment (Fig. 3). Waitav achieves the least patients’ delay, as only 1 patient is delayed for 1 week while Overtime results in more delayed patients with 54, 19, 5 and 1 patient delayed respectively for 1, 2, 3 and 4 weeks. However both algorithms ensure the start of high-priority priority patients as soon as they arrive in the waiting list. The total amount of patients treated is also higher with Waitav than with Overtime (i.e., ( 382 and 329 respectively). Third and fourth performance indicators describe Linacs utilization. The first one (Fig. 4) shows a higher and steadier use of opened hours over the 15 weeks when using Waitav algorithm while the second one (Fig. 5) shows the mean utilization of opening hours urs during a week of work. Saturdays are under-used (i.e.,