College Station, TX, 77843, USA b Scott and White Clinic, 2401 S. 31st Street, Temple, TX, USA. Version of record first published: 02 Dec 2011.

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1 This article was downloaded by: [Texas A&M University Libraries] On: 10 September 2012, At: 07:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK IIE Transactions on Healthcare Systems Engineering Publication details, including instructions for authors and subscription information: Patient and resource scheduling of multi-step medical procedures in nuclear medicine Eduardo Pérez a, Lewis Ntaimo a, Wilbert E. Wilhelm a, Carla Bailey b & Peter McCormack b a Department of Industrial and Systems Engineering, Texas A&M University, 3131 TAMU, College Station, TX, 77843, USA b Scott and White Clinic, 2401 S. 31st Street, Temple, TX, USA Version of record first published: 02 Dec To cite this article: Eduardo Pérez, Lewis Ntaimo, Wilbert E. Wilhelm, Carla Bailey & Peter McCormack (2011): Patient and resource scheduling of multi-step medical procedures in nuclear medicine, IIE Transactions on Healthcare Systems Engineering, 1:3, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 IIE Transactions on Healthcare Systems Engineering (2011) 1, Copyright C IIE ISSN: print / online DOI: / Patient and resource scheduling of multi-step medical procedures in nuclear medicine EDUARDO PÉREZ 1,, LEWIS NTAIMO 1, WILBERT E. WILHELM 1, CARLA BAILEY 2 and PETER MCCORMACK 2 1 Department of Industrial and Systems Engineering, Texas A&M University, 3131 TAMU, College Station, TX 77843, USA eduardopr@neo.tamu.edu 2 Scott and White Clinic, 2401 S. 31st Street, Temple, TX, USA Received October 2010 and accepted August The rise in demand for specialized medical services in the U.S. has been recognized as one of the contributors to increased health care costs. Nuclear medicine is a specialized service that uses relatively new technologies and radiopharmaceuticals with a short half-life for diagnosis and treatment of diseases. Nuclear medicine procedures are multi-step and have to be performed under restrictive time constraints. Consequently, managing patients in nuclear medicine clinics is a challenging problem that has received little research attention. In this paper, we derive algorithms for scheduling nuclear medicine patients and resources. We validate our algorithms using simulation of an actual nuclear medicine clinic based on historical data and compare the performance of our algorithms with the methods currently used in the clinic. The results we obtain provide useful insights into managing patients and resources in nuclear medicine clinics. For example, results show that patient throughput can be increased when some clinic resources are reserved to exclusively serve specific procedures on those days when higher demand is expected. Keywords: Health care, nuclear medicine, patient service, scheduling, simulation 1. Introduction Nuclear medicine is a sub-specialty of radiology that provides highly specialized services by means of new technology for diagnosis and treatment of diseases. The increased demand for medical diagnostic procedures has been recognized as one of the contributors for the rise of health care costs in the U.S. Physicians are becoming more prone to asking patients to undergo specialized procedures in order to obtain more accurate diagnoses. Thus, the challenge of scheduling patients and resources in specialized clinics such as nuclear medicine is a problem of increasing concern. This paper holds two research objectives. The first is to derive algorithms to assist nuclear medicine managers in scheduling patients and resources more efficiently by considering the perspectives of both patients and managers. The second is to evaluate the scheduling algorithms computationally using data from an actual clinic to enhance insights concerning trade offs in patient and resource scheduling strategies. Corresponding author Nuclear medicine procedures (tests) require the administration of a radiopharmaceutical (radioactive isotope, e.g., iodine-131) to the patient and typically involve multiple steps and resources. Images of specific body organs are taken (i.e., scanned) using gamma cameras that sense radiation emitted by the radiopharmaceutical. Since radiopharmaceuticals have a short half-life (minutes), their decay imposes strict time constraints on scheduling patients and resources in order to get good quality scans. Thus, scheduling patients in nuclear medicine requires very strict procedure protocols, that, if not followed, can result in a waste of time, money and resources because the patient might have to be re-scheduled. Some nuclear medicine tests require only a single scan while others involve multiple scans in one day. Each scan takes several minutes to hours to complete and must occur within a narrow time window after radiopharmaceutical administration to assure scan quality. Multiple resources are required to perform each nuclear medicine procedure. Resources include technologists; gamma cameras; radiopharmaceuticals; and sometimes, a nurse or EKG (electrocardiography) technician. Gamma cameras can cost up to a million dollars and thus have to be used and managed effectively. Since many nuclear medicine clinics must have radiopharmaceuticals prepared C 2011 IIE

3 Multi-step procedure scheduling 169 by remote radio-pharmacies, scheduling their delivery, patient injection and image acquisition requires lead time and must be carefully managed. Each radiopharmaceutical may cost as much as several hundred dollars. Resources required to perform each procedure step must be available at the scheduled times. Therefore, scheduling patients, resources, and radiopharmaceutical preparation and delivery is a challenging problem for nuclear medicine departments. In fact, practical features that surround the management of patients and resources in nuclear medicine make it a unique problem with limited research reported in the literature. Consequently, providing a high quality of service to the patient through the use of mathematical techniques is of great interest to nuclear medicine managers. Furthermore, very few commercial packages are available for patient service management and those that are available do not incorporate algorithms for scheduling patients and resources efficiently. We implement our scheduling algorithms in JAVA and test them using discrete event simulation (Pérez et al., 2010). We obtain computational results that provide useful insights into managing patient service and resources utilization in nuclear medicine. While this paper focuses on nuclear medicine, we believe that results can be applied to similar health care settings that may not be as complex as nuclear medicine. For example, our algorithms can be applied to diagnostic imaging areas such as magnetic resonance imaging (MRI) and CT scan scheduling. The rest of the paper is organized as follows: we review closely related work in Section 2 and provide preliminaries on nuclear medicine resources and procedures in Section 3. We derive algorithms for scheduling nuclear medicine patients and resources in Section 4. We report on our computational study, which quantifies important trade-offs among patient and resource scheduling strategies, in Section 5. We also discuss the results and highlight new insights into the complexity of nuclear medicine patient service management. We end the paper with some concluding remarks and directions for further research in Section Literature review Facilities dedicated to the diagnosis and treatment of diseases are vital in comprehensive health care systems. Mettler et al. (2008) found a 5-to-6 fold increase in medical diagnostic procedures in the U.S. from 1972 to 2005, whereas the U.S. population increased by approximately 50%. This increase in diagnostic testing has been identified as one of the potential causes for increased health care costs in the U.S. (Zhang, 2007). Facilities that are hospital-based such as radiology clinics are highly specialized at providing different types of services to patients. Equipment utilized for diagnostic procedures is usually expensive and the unique characteristics of radiology procedures make finding an efficient way of scheduling patients and resources challenging (Vermeulen et al., 2009). A necessary condition for overall hospital efficiency is the effective utilization of medical diagnostic facilities, which are used by almost every patient that enters a hospital (O Kane, 1981). Operations research techniques such as simulation and mathematical optimization have been shown to be viable approaches for patient and resource scheduling in the literature. For example, Ho and Lau (1992) and Ho et al. (1995) used simulation to compare the performance of fifty appointment rules under various operating environments in outpatient clinics. They concluded that no rule will perform well under all environments and proposed a simple heuristic that uses the characteristics of the system to select one rule. Wang (1993) considered a single server system with exponential service times and used a recursive procedure to prescribe optimal appointment time. The author showed that it is optimal to schedule appointments that require shorter service times at the start and end of the day and longer ones in the middle of the day (dome-shaped). Wang (1997) extended these results to model service times as any phase-t type distribution. Klassen and Rohleder (1996) used simulation to study different strategies for sequencing patients that are classified according to their expected service time variability. They concluded that scheduling patients, whose service times have low variances, at the beginning of the day outperforms the Ho and Lau (1992) best performing rules. Extending this work, Rohleder and Klassen (2000) considered error in classifying patients. They concluded that even with classification errors, their rule still outperformed the rules presented by Ho and Lau (1992). Liu and Liu (1998a) and Liu and Liu (1998b) studied a clinic with multiple doctors where doctors arrival times are random. They identified properties shared by the best appointment schedules using simulation. Robinson and Chen (2003) determined optimal patient appointment times when the sequence of patients is given. They formulated the problem as a stochastic linear program and used simulation-based techniques to compare the performances of a variety of heuristic appointment rules. Denton and Gupta (2003) presented a two-stage stochastic linear programming model to determine the optimal appointment times for patients with arbitrarily distributed service times and showed that the optimal sequence of time intervals is dome-shaped. Cayirli et al. (2006) studied the scheduling of patients in ambulatory care clinics and investigated the interaction between patient characteristics and appointment system elements. They concluded that patient sequencing has a greater effect in the performance of the system than the choice of an appointment rule. Gupta and Wang (2008) used a Markov Decision Process (MDP) to model a primary care scheduling problem. They considered patient choices and showed that, if the clinic is served by one physician, the optimal booking policy can be characterized. We refer interested readers to recent surveys of outpatient appointment scheduling

4 170 Pérez et al. Table 1. Human resources responsibilities in nuclear medicine Technologists Nurses Physicians Managers Hydrate patient Hydrate patient Hydrate patient Hydrate patient Radiopharmaceutical Radiopharmaceutical Radiopharmaceutical Radiopharmaceutical preparation preparation administration administration Imaging Draw doses Draw doses Radiopharmaceutical administration research by Cayirli and Veral (2003) and Gupta and Denton (2008). Prior research on patient service management in nuclear medicine is very limited. However, some research has been reported in related health care environments such as radiology, radiotherapy, CT scan and MRI clinics. For example, Conforti et al. (2007) proposed an optimization model for outpatient scheduling within a radiotherapy department where patients must visit the treatment center several times during the week. Green et al. (2006) studied the problem of scheduling different types of patients that arrive randomly during the week to an MRI facility. They presented a finitehorizon dynamic program for a single server appointment schedule that allows at most one patient per time period. The problem of managing patients in a CT scan clinic was addressed by Patrick and Puterman (2007). They presented a mathematical formulation that prescribes appointment times to maximize the utilization of resources, subject to an overtime constraint. The authors assumed a pool of patients that can scheduled when time slots become available. They used simulation to show a decrease in patient waiting time for service. Vermeulen et al. (2009) devised an adaptive approach to optimize resource schedules in a CT scan facility. They used simulation for case analysis and demonstrated that their approach makes efficient use of resource capacity. This research differs from earlier studies in a number of ways. First, appointment scheduling in nuclear medicine deals with lead times required to obtain radiopharmaceuticals. Second, procedures in nuclear medicine involve multiple steps, each of which requires the use of two or more resources. Third, procedures in nuclear medicine follow sequential protocols that impose strict time constraints on the starting time of each step. Together, these characteristics make the problem of scheduling patients and resources in nuclear medicine challenging. In fact, Gupta and Denton (2008) identified the problem of scheduling patients in highly constrained health care environments as a current research open challenge. 3. Problem description We consider a nuclear medicine clinic that has a limited number of resources to serve patients. Let S be the set of stations and R be the set of human resources available at the clinic. Each station contains at least one type of equipment. Stations are classified according to the equipment they have. Nuclear medicine equipment include different types of gamma cameras and treadmills for cardiovascular tests. The staff includes four types of human resources: technologists, nurses, physicians, and managers. Each human resource has his/her own expertise and experience, determining the set of procedure steps or tasks that they can perform and the amount of time required to complete each one. Some of the procedure steps that can be performed by the human resources are listed in Table 1. Nuclear medicine procedures are requested by the patient s primary physician. A service request usually includes a procedure type from set P and also a preferred day of the week q. Patient appointments are scheduled when the request is received and decisions are made without taking future requests into account. Unlike patients in general outpatient clinics, those in nuclear medicine show up for their appointments most of the time. Nuclear medicine clinics perform P different procedures, each involving multiple steps or tasks. Table 2 presents a list of nuclear medicine procedures along with their current procedural terminology (CPT) codes. Procedure p P involves a total number of steps n p. Each step k of procedure p has an estimated duration β kp and the total duration of the procedure is given by n p k=1 β kp. At least one radiopharmaceutical from set A is administered to the patient at the beginning of each procedure. Radiopharmaceuticals are requested in advance and they need to be at the clinic by the time of the patient appointment. In addition, each step k of procedure p requires one station Table 2. Examples of nuclear medicine procedures CPT Code Name Cardiovascular Event (CVE) Myocardial Imaging (SP-M) Positron Emission Tomography (PET)/ Computed Tomography (CT) skull to thigh MSB-bone imaging (whole body) MSC-bone imaging (three phase) GIC-Hepatobiliary imaging CVJ-cardiac blood pool REB-Pulm perfusion/ventilation ENC-Thyroid imaging HEE-Lymphatic imaging CVD-Myocardial imaging (SP-R ORS)

5 Multi-step procedure scheduling 171 Table 3. Procedure 78315: MSC-bone imaging (three phase) Step (k) Task Average time (mins.) Station (S kp ) Human Resource (R kp ) 1 Radiopharmaceutical injection 20 Axis; P2000; Meridian; TRT Technologist; Nurse; Manager 2 Imaging 15 Axis; P2000; Meridian Technologist; Manager 3 Uptake delay 165 Waiting room 4 Imaging 45 Axis; P3000; Meridian Technologist; Manager s S and at least one human resource r R to be performed. Table 3 describes the MSC-bone imaging procedure, CPT This procedure has four steps and an average completion time of 245 minutes. Table 3 lists for each procedure step; the required task, the expected time duration, and the set of feasible stations and human resources. We define set S kp as the set of stations at which step k of procedure p can be performed. Similarly, let R kp be the set of human resources qualified to perform step k of procedure p. Each task must be performed by one of the qualified human resources at one station among the feasible stations S kp. We evaluate the performance of our scheduling algorithms using measures that take into account the perspectives of both the clinic patient and the manager. These two perspectives are relevant to nuclear medicine patient and resource scheduling. Table 4 describes the performance measures used to quantify the level of patient satisfaction in health care clinics, while Table 5 shows performance measures that consider the manager s perspective. To provide a unified perspective of all the relevant components, we now give a simple example to illustrate patient/resource scheduling in nuclear medicine. Figure 1(a) depicts two of the procedures performed in nuclear medicine. We list for each procedure the corresponding requirements of each step as follows: time duration (time), station (s), and human resource (r). For the purpose of this example, only one station and one human resource is associated with each procedure step. Figure 1(b) depicts a schedule for procedure (PET/CT skull to thigh) where the patient is assigned to arrive at the beginning of the day. The schedule shows that four resources are required at different times of the day to perform this procedure: nurse, technologist, treatment (TRT) station and axis station. Figure 1(c) shows the schedule for procedure and procedure (CVE, SP-M). Since some of the resources are unavailable at the beginning of the day, the second procedure, in dark color, has to be scheduled later in the day. The schedule shows that for procedure five resources are required namely; EKG technologist, technologist, TRT station, treadmill station, and axis station. Also, observe that no other procedure can be fitted into the schedule due to the unavailability of the resources at particular times. For instance, an additional procedure 78465, represented with dot lines, cannot be scheduled on this day because the axis station and the technologist are unavailable during the time slots that would be required for the last step of the procedure. 4. Patient and resource scheduling We now turn to patient and resource scheduling and derive two algorithms: fixed resource (FR) and procedure resource assignment (PRA). Both algorithms schedule patients following a general scheduling structure. The algorithms first search for the patient s preferred day for the appointment. If the search results in an appointment for which the patient has to wait more than a month, then an earlier appointment on an alternate day is considered. We first derive the FR algorithm, which involves fixing some of the members of the human resources to specific stations. This algorithm is based on the practical experience of the last two authors of this paper. This algorithm will provide a benchmark for our methodology using the PRA algorithm which is explained later in subsection 4.2. We assume that no more than one patient can be scheduled to use the same resource at the same time and that the scheduling horizon is long enough so that all patient requests are satisfied. For convenience, we list the notation we use to describe our scheduling algorithms in Table 3. In addition, we use the following symbols: denotes assignment; == denotes Table 4. Performance measures for patient s perspective Name Description Reference Waiting time type 1 Waiting time from the time of the procedure request until the time of Rohleder and Klassen (2002), the appointment Robinson and Chen (2003) Preference ratio Number of times patients are scheduled on the date requested above all Green et al. (2006) patient requests Cycle time Time patient spends in the system Podgorelec and Kokol (1997), Rohleder and Klassen (2000)

6 172 Pérez et al. Table 5. Performance measures for manager s perspective Name Description Reference Equipment utilization The amount of time an equipment is used during Rohleder and Klassen (2002), operating hours Denton et al. (2006) Human resource utilization The amount of time a human resource is used during Denton et al. (2006) operating hours Centeno et al. (2000) Patient throughput Number of patients served per day Ramakrishnan et al. (2004) (equality) comparison, and && denotes logic and. We define the set of day and time slot pairs (d, t)forresourcer as U r ={(d, t) 1 d h, 1 t τ}. Similarly, we define the set of day and time slot pairs (d, t) for station s as V s = {(d, t) 1 d h, 1 t τ}. ThesetsU r and V s include all the time slots that are already scheduled. The set of day and time slot pairs (d, t) for patient j schedule is defined as L j ={(d, t) 1 d h, 1 t τ}. Fig. 1. Example showing one and two scheduled procedures (color figure available online).

7 Multi-step procedure scheduling 173 Table 6. Scheduling problem sets and parameters Sets J : set of patients, indexed j P : set if procedures, indexed p J p : set of patients requesting procedure p, indexed j T : set of time periods t H : set of days, indexed h I : set of resources, indexed i S : set of stations, indexed s R : set of human resources, indexed r P : set of nuclear medicine procedures, indexed p A : set of radiopharmaceuticals, indexed a S kp : set of stations where step k of procedure p can be performed R kp : set of human resources qualified to perform step k of procedure p I kp : set of resources that can be used to perform step k of procedure p, I kp ={R kp S kp } L itj : set of appointment star times that require the use of resource i at time-slot t for patient j K itj : set of procedure steps that require the use of resource i at time-slot t for patient j T ij : set of time-slots where resource i could be used to serve patient j T aj : set of time-slots where radiopharmaceutical a could be used to serve patient j L j : Set of day and time slot pairs, (d, t), for patient j. U r : Set of day and time slot pairs, (d, t), for human resource r schedule. V s : Set of day and time slot pairs, (d, t), for station s schedule. Parameters i : subscript, for the i resource j : subscript, for the j patient a : subscript, for the a radiopharmaceutical p : subscript, for the p procedure k : subscript, for the k step of a procedure l : subscript, for the l starting time-slot for a patient appointment t : subscript, for the t time-slot, incremental time τ : total number of time-slots in a day, indexed t,...,τ β kp : number of time-slots required to complete step k of procedure p n p : total number of steps for procedure p, indexedk,...,n p ρ : variable representing resource r or station s δ p : total duration of procedure p ω : Number of days in a week µ : Number of days in a month m : Number of days before arrival of radiopharmaceutical after placing order q : day of the week requested by patient, indexed q = 1,...,5, where 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday For ease of exposition, we first describe a function, CheckSchedule(), which is implemented by both the FR and PRA algorithms (Fig. 2). This function checks the availability of a human resource (when ρ = r) or a station CheckSchedule (G ρ,d,t,β kp ) 1 G G ρ ; 2 for time = t to t + β kp do 3 if (d, time) G then 4 return false; 5 else 6 time time +1; 7 end 8 end 9 return true; Fig. 2. Pseudocode for CheckSchedule(). (when ρ = s) during a given time interval [t, t + β kp ], and returns a boolean indicating whether (true) or not (false) that time interval is available. If any of the time slots within the interval [t, t + β kp ] is occupied, meaning the time slot is in the set G, the function returns false;otherwise,itreturns true. The function simply checks whether or not any of the time slots from time t to t + β kp are included in the current schedule. The FR and PRA algorithms share the same overall structure described by the pseudocode in Fig. 3. The index setofpatientsj is initialized in line 1. Lines 2 and 3 define the time horizon (day and time) over which patient requests are to be received. Upon arrival (line 4) patient requests are incorporated in set J as they arrive (line 5). A function called ServeRequest-Algorithm(), where Algorithm denotes FR or PRA, uses patient information

8 174 Pérez et al. Scheduling-Algorithm 1 J { }, j =0; 2 while d h do 3 while t τ do 4 (p j,q) GetPatientRequest(j); 5 J {p j }, d j d, t j t, j j +1; 6 L j ServeRequest-Algorithm (j, p j,d j,t j,q); 7 end 8 end Fig. 3. Pseudocode for Scheduling-Algorithm. (p, q, d, andt) and finds an appointment L j (line 6). The two algorithms differ in the way they implement this function The fixed resource (FR) algorithm The FR algorithm allows the patient ( j) toprovideapreferred day of the week (q). Using this preferred day, a search for an appointment is performed based on the patient s requested procedure p. After the earliest feasible appointment is found, the algorithm determines how long the patient would have to wait. If waiting time is longer than a month, the algorithm searches for an alternative appointment (different day of the week) with a shorter waiting time. We use a boolean variable θ to store the output returned by the CheckSchedule() function. Recall that θ takes a value of true if a resource (human r or station s) is available, and false otherwise. In the FR algorithm a group of human resources (e.g., two technologists) are assigned to always serve patients at specific stations. For example, technologists 1 and 2 must serve all patients whose procedures require the use of stations s 1 and s 2, respectively. We use ŝ to denote the stations that may have human resources assigned. Figure 4 presents the pseudocode for FR, which is invoked by the function ServeRequest-FR(). Line 1 of FR initializes parameters. The first day on which FR attempts to schedule an appointment is determined in line 2 using the following information: day when the request was received (d), patient preferred day (q), and the number of days (m) needed to obtain the radiopharmaceutical required for the procedure. The scheduling horizon, which ends on day h, is defined in lines 3 and 4. In line 5 we begin a search for an available combination of station and human resource for each procedure step k. A breadth-first search is conducted to select a station from set S as well as a human resource from set R (line 6). We use breadth-first search to balance work assignments on all resources (load balancing). Line 9 identifies those stations to which human resources have been fixed. A fixed human resource always stays in the same station. Consequently, when the availability of a station with a fixed human resource is confirmed, the algorithm also confirms the availability of the human resource because they share the same schedule (lines 14 15). If the station found does not have a fixed human resource, a search is performed to find a qualified human resource from set R (lines 17 19). After finding the resources required for the first step of the procedure, we check that the waiting for the appointment does not exceed a month (line 10 and line 21). If no station is available for one of the procedure steps (line 31), the current start time for the procedure is increased (line 32) by one time slot. Using the new start time, the algorithm checks if the amount of time required to perform the procedure exceeds the remaining time on the day searched (line 33). If the time available in the day is not enough, the algorithm moves to the following week to perform a new search on the day requested by the patient (line 34). Otherwise, the algorithm begins a new search for an appointment using the new time as a starting time. This same process is repeated if no human resource is available for one of the procedure steps (lines 30 37). If the waiting time exceeds a month, we use the same routine to find an appointment but do not take the patient s preferred day into account (lines and lines 22 23). In other words, we will schedule the patient in the earliest time available in the scheduling horizon. The algorithm returns a patient schedule if a combination of human resource and station is found for each procedure step and if the waiting time for the appointment is less than a month The procedure resource assignment (PRA) algorithm We now turn to the derivation of the PRA algorithm. We will now assume that the nuclear medicine department observes a high demand pattern for at least one day of the week (e.g., Friday is usually the day of the week requested the most). The idea behind the algorithm is to assign procedures to resources on days with expected high patient demand based on representative historical demand. We derive an integer programming (IP) model to perform the procedure-resource assignment. Two binary variables are use in this problem, x ik jptl and wik jl. Variable xik jptl =1 if patient j requesting procedure p is scheduled to use resource i at time-slot t when procedure is started at time l for step k of the procedure, otherwise x ik jptl =0. Variable wik jl =1 if resource i is selected to serve patient j in step k when procedure is started at time l, otherwisew ik jl =0. We now state model IP: IP : Max : w i1 jl (1a) p P j J p i S 1p t T l L itj s.t. x ik jptl 1, i I, t T p P j J p k K itj l L itj (1b) w ik jl 1, p P, j J p, i R kp t T l L itj k = 1,...,n p (1c)

9 Multi-step procedure scheduling 175 Fig. 4. Pseudocode for FR. w ik jl 1, p P, j J p, i S kp t T l L itj k = 1,...,n p x ik jptl wik jl = 0, p P, j J p, i I kp, t T, l L itj, k = 1,...,n p w ik jl i R kp w i(k 1) jl = 0, p P, i R (k 1)p (1d) (1e) j J p, t T, l L itj, k = 1,...,n p w ik jl w i(k 1) jl = 0, p P, i S kp i S (k 1)p j J p, t T, l L itj, k = 1,...,n p jl jl = 0, p P, i R 1p w i1 i S 1p w i1 j J p, t T, l L itj (1f) (1g) (1h)

10 176 Pérez et al. x ik jptl,wik jl {0, 1} (1i) Model IP allocates a subset B J of the requests to the resource schedules so that their capacities are not exceeded. Each human resource can only serve one patient at a time. The objective function (1a) maximizes the number of patients scheduled during the day. Variables x ik jptl and wik jl are connected through constraint (1a) and together they control patient volume. Constraint (1a) assures that each resource is assigned to at most one patient each time period. Constraints (1a) and (1a) are used to select the staff and station for each procedure step, respectively, and also to decide the start-time of the appointment for each patient. Constraint (1a) assures that the same resource is scheduled for the duration of a particular procedure step. Constraints (1a) and (1a) are used to verify that the staff and stations, respectively, selected to serve a patient follow the procedure sequence. Constraint (1a) is used to match a station to a staff member for each step of the procedure requested by the patient. Finally, constraints (1a) require each variable to be binary. IP can be solved by a commercial solver but demand for the day must be known. Since patients make requests one-at-a-time, actual systems do not provide the decision situation supposed by model IP. Therefore, we implement IP using a forecast of demands and now derive a heuristic to provide good solutions. The key idea behind the heuristic is to choose a representative peak day (RPD) h H from a given demand trace and solve IP corresponding to this day. Let F H be a subset of days containing the day(s) of the week with the highest weekly demand over the entire scheduling horizon (i.e., all Fridays in the scheduling horizon). An RPD is determined by considering the procedure demand at each day h F. The heuristic selects the day with the highest demand in F and solve model IP for that day. The solution provides the procedure to resource assignments. These assignments are used to schedule patients online on those days of the week when high demand is expected. denote the patient procedure-resource assign- be an index set comprising all procedure-resource-time triplets (p,i,t) P I T such that (p, i, t) G p for ˆx ik jptl =1. We now define a function, CheckSchedulePRA() which uses set G p to search for an appointment. Figure 5 gives a pseudocode for the function. The CheckSchedulePRA() function gets Let ˆx ik jptl ment solution of model IP. Also, let G p Fig. 5. Pseudocode for CheckSchedulePRA(). the information of the procedure p requested by the patient and checks if there is a feasible appointment in set G p.ifan appointment is available, the function returns a schedule for the patient; otherwise, the function returns the empty set. Figure 6 gives a pseudocode for PRA, which employs CheckSchedulePRA(). PRA follows the major steps of FR with two major differences. First, PRA invokes an additional condition (line 4) that checks if the current day is one of the days where a high demand of requests is expected (day F). If the condition is satisfied, PRA calls the CheckSchedulePRA() function to find a feasible appointment for the patient. If the condition is not satisfied, PRA proceeds as FR does. The second major difference between these two algorithms is that PRA does not fix human resources to any of the stations. 5. Application To test and validate our methodology we applied it to the Scott & White Nuclear Medicine Clinic. The algorithms were implemented in JAVA and tested using the nuclear medicine department discrete event specification (DEVS) simulator by Pérez et al. (2010). A nuclear medicine clinic at an abstract level contains multiple entities which can be classified as human resources, stations, radiopharmeceuticals, and patients. Figure 7 shows the main components of the nuclear medicine DEVS simulator. The experimental frame (EF) allows the modeler to specify the experiments that will be performed using the simulation to answer the questions of interest. The EF contains the call generator (CGENR), scheduler (SCHED), radiopharmaceutical generator (RPGENR), patient generator (PGENR),and the transducer (TRANSD) models. The CGENR is in charge of generating random patient appointment requests. The SCHED model allows for selecting a scheduling algorithm to schedule patients into the system. Patient appointment information is passed from SCHED to RPGENR and PGENR models. RPGENR generates radiopharmaceutical arrivals to clinic at specified times. PGENR generates patient arrivals to the clinic at their appointment times. The TRANSD computes the performance measures of interest. The NMD model is an abstraction of the nuclear medicine department (NMD) and contains human resource models (TECH, NURSE, MANGR, PHYSN) and station models (STATION). Figure 7 only shows models for TECH, NURSE, MANGR, and one STATION due to space limitations. The EF provides input to the NMD model and after entities are served, the NMD provides input the EF model. The key assumptions of the DEVS simulation model include Poisson arrivals for patient appointment calls, which is based on historical data; generation of patient procedures is based on an empirical distribution of the number of different procedures performed in a given year; equipment located inside the stations are

11 Multi-step procedure scheduling 177 Fig. 6. Pseudocode for PRA. not subject to failures; and radiopharmaceuticals arrival on time. In the next subsection we describe the configuration of this actual nuclear medicine clinic and, in Section 5.2, we present our experimental setup. We report computational results and findings in Section Real nuclear medicine setting The Scott & White Nuclear Medicine Clinic is one of the largest, fully accredited nuclear medicine clinics in the country and is located in Temple, Texas. This nuclear medicine clinic operates nine hours a day, five days a week, and it is not open during weekends. The clinic has 12 stations, each named according to the type of equipment it comprises. Table 5.1 lists the names of the stations and the equipment at each station. The staff comprises technologists, nurses, and a manager. There are eight technologists and two EKG technologists at this clinic. The technologists have several responsibilities, including drawing doses and acquiring images. The nurse assists the technologists in drawing doses. EKG technologists perform stress exams for cardiac tests at treadmill stations. In the absence of one of the staff members, the

12 178 Pérez et al. Fig. 7. The nuclear medicine department model components. division manager can perform that staff member s tasks (see Table 1). Table 3 shows the procedures that were performed more frequently at the clinic during the year of our study Experimental setup The clinic configuration used for testing and validating our algorithms is based on the stations listed in Table 7 and historical patient demand data for one year, which was provided by the clinic. The staff comprises twelve members: Technologist(1), Technologist(2), Technologist(3), Technologist(4), Technologist(5), Technologist(6), Technologist(7), Technologist(8), Technologist(9), Technologist(10), Nurse(1), and Manager(1). From this group Technologist(9) and Technologist(10) are EKG Technologists. A Table 7. Stations of the Scott & White nuclear medicine clinic Station name Axis(1) Axis(2) Axis(3) P2000(1) P2000(2) P2000(3) Meridian(1) Treadmill(1) Treadmill(2) TRT(1) TRT(2) TRT(3) Equipment Philips Axis Camera Philips Axis Camera Philips Axis Camera Philips PRISM 2000 Camera Philips PRISM 2000 Camera Philips PRISM 3000 Camera Philips Meridian Camera Treadmill Treadmill Patient preparation Patient preparation Patient preparation Poisson process was assumed for procedure request arrivals based on historical data. The monthly call interarrival times in minutes followed an exponential distribution with the following means: January, 6.00; February, 6.25; March 6.58; April, 6.67; May, 6.75; June, 6.88; July, 6.96; August, 7.04; September, 7.10; October, 7.29; November, 7.34; and December, Empirical distributions based on historical data were used to generate a procedure request and an appointment preferred day for each patient. Procedure time durations were based on historical data and modeled using random variables. Monday and Friday were identified as the days of the week requested the most by patients. Since patient interarrival times were based on data estimates, our experiments considered the impact of having different demand patterns. A sensitivity analysis was performed using the base demand scenario described above as a benchmark for comparison. We considered an alternative scenario in which the monthly mean interarrival times are increased by one minute, and another scenario in which the monthly interarrival times are decreased by one minute. We call these scenarios low demand and high demand, respectively. We conducted experiments to evaluate the robustness of our scheduling algorithms. In our computational study, we considered only the nuclear medicine procedures listed in Table 2. The performance measures listed in Section 3 were used to quantify service levels based on both patient and management perspectives. In the FR algorithm Technologist(1) and Technologist(2) are fixed to stations Axis(1) and Axis(2). The experiments involved 100 replications. Each replication was based on one year of operations. To maintain independence among replications, different seeds were used in the pseudo random number generators for each simulation

13 Multi-step procedure scheduling 179 Pa ents served Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec FR Actual clinic Fig. 8. Patients served per month: FR versus actual (color figure available online). run. All experiments were executed on a DELL Optiplex GX620 with a Pentium D processor running at 3.2GHz with 3.0GB RAM Computational results We now report computational results to evaluate the robustness of both FR and PRA algorithms for three patient demand scenarios: (a) base patient demand, (b) low patient demand, and (c) high patient demand. Recall that FR is a scheduling algorithm representing current practice and has been validated (Pérez et al., 2010). A comparison of the average number of patients served per month using FR versus historical values is shown in Fig. 8 for a particular year. It can be seen that FR is within 10% of historical values Base patient demand scenario Table 8 shows the throughput and computational times for both FR and PRA for the base demand scenario. We report the mean, standard deviation (Std.Dev.), and a 95% confidence interval for each performance measure. Results show that FR and PRA obtained similar values for the average number of patients served during the year. The computation times (CPU seconds) were about the same for both algorithms. Table 8. Number of patients served, system throughput and simulation time Patients Patients CPU time Algorithm Statistic served served/day (secs) FR Mean Std. Dev CI Lower CI Upper PRA Mean Std. Dev CI Lower CI Upper Figure 9 shows two graphs that depict the average number of patients served (throughput) per month for both algorithms. The first graph on the left shows a decreasing trend of the average patient throughput as time progresses. This behavior was validated by comparison with the clinic s records, which showed that demand declined over the year studied. The graph on the right shows the average difference in the number of patients served per month by PRA and FR each month. This graph reveals that PRA is able to accommodate more patients early in the year (first three months) when patient demand is higher. After the third month, FR begins to schedule more patients per month, which is simply the demand that was pushed into the future when resource capacity was reached at the beginning of the year. This observation does not impact the overall number of patients served over the year. However, it results in patients having to wait longer for their appointments. This phenomenon is discussed later in this section. Figure 10 shows the percentage utilization of the equipment and human resources under both algorithms. There is an overall increase of 1% in the utilization of both equipment and human resources under PRA. This can be explained by the slight increase in the number of patients served under this algorithm. Both algorithms exhibit a low utilization for the TRT (treatment) stations. These stations are used mainly for patient preparation, which on average Fig. 9. Patients served per month under base demand scenario.

14 180 Pérez et al. Fig. 10. Resources utilization under base demand scenario. takes around 10 minutes for most of the procedures. FR provides a more balanced utilization for camera stations. We can see that PRA schedules more procedures at stations Axis(1) and Axis(2). However, PRA is able to provide a more balanced distribution of the tasks for the human resources. FR gives a higher utilization of Technologist(1) and Technologist(2), the human resources assigned to stations Axis(1) and Axis(2), respectively. Based on the utilization of the human resources, we argue that providing a more balanced distribution of the tasks is beneficial for the staff and that PRA performs better than FR, in that respect. Next, we report performance measures relative to patient perspective, waiting type 1 and preference ratio. Recall that patient waiting type 1 is the time a patient must wait from making a request to the appointment time, while preference ratio is the portion of patients scheduled on their requested day. Table 9 shows results obtained for the two performance measures. PRA showed better performance for both performance measures. About 9% improvement is obtained for patient waiting time, meaning that a patient would wait on average, about a day less under PRA than under FR. In terms of patient preference ratio, PRA shows a sight improvement over FR. Table 9. Patient waiting Type 1 and preference satisfaction ratio Patient waiting Patient preference Algorithm Statistic type 1 satisfaction ratio FR Mean % Std. Dev % CI Lower % CI Upper % PRA Mean % Std. Dev % CI Lower % CI Upper % Low patient demand scenario Table 10 shows the results for patient throughput and CPU time when there is low patient demand. As expected, the overall number of patients is lower compared to the results obtained for the base patient demand scenario (Table 8). Consequently, CPU times are also lower for both algorithms and the simulation time for FR is less than that for PRA. Both algorithms perform similarly relative to the average number of patients served over the entire year. Figure 11 shows the average number of patients served per month under each algorithm. Even though the average number of patients served per year is about the same for both algorithms, PRA accommodates more patients at the beginning of the year. The results for resource utilization are reported in Fig. 12. PRA gives a higher utilization compared to the FR algorithm. This can be attributed to the slight increase in the average number of patients served per year. Similar to the results obtained for the base case scenario, FR provides a more balanced utilization of equipment while PRA gives a more balanced utilization of human resources. Table 10. Number of patients served, system throughput and simulation time Patients Patients CPU time Algorithm Statistic served served/day (secs) FR Mean Std. Dev CI Lower CI Upper PRA Mean Std. Dev CI Lower CI Upper

15 Multi-step procedure scheduling 181 Fig. 11. Patients served per month under low demand scenario. Table 11. Patient waiting Type 1 and preference satisfaction ratio Patient waiting Patient preference Algorithm Statistic type 1 satisfaction ratio FR Mean % Std. Dev % CI Lower % CI Upper % PRA Mean % Std. Dev % CI Lower % CI Upper % Table 11 show the results relative to patient perspective performance measures. PRA results in a lower average patient waiting time. As for patient preference ratio, the results show that, when lower patient demand is observed at the clinic, both algorithms provide better ratios. This is due to the fact that, since fewer requests are managed by the Table 12. Number of patients served, system throughput and simulation time Patients Patients CPU time Algorithm Statistic served served/day (secs) FR Mean Std. Dev CI Lower CI Upper PRA Mean Std. Dev CI Lower CI Upper clinic, there is more flexibility in scheduling patients based on their preferences High patient demand scenario Table 12 shows patient throughput and CPU times for the high patient demand scenario. PRA shows a 1% increase Fig. 12. Resource utilization under low demand scenario.

16 182 Pérez et al. Fig. 13. Patients served per month under high demand scenario. in the average number of patients served over the year and has a slightly higher average CPU time. The higher CPU time can be attributed to the algorithm scheduling and more patients. Figure 13 reports the average number of patients served per month under each algorithm. Both graphs show that PRA schedules more patients each month. Observe that, as expected, the number of patients served each month under either algorithm is higher compared to the low demand and base case scenarios. Figure 14 shows the results for resource utilization. PRA obtained a 3% higher overall utilization of the resources compared to FR. Again, PRA is able to provide a more balanced utilization of the human resources, especially the technologists. In terms of patient perspective performance measures, PRA shows better performance. Its patient waiting time is about 16% lower, meaning that patients would wait about two days less on average compared to the FR algorithm. As for the patient preference ratio, PRA schedules patients on their requested preferred day about 52% of the time, which is an improvement of about 48% over the FR algorithm. Table 13. Patient waiting Type 1 and preference satisfaction ratio Patient waiting Patient preference Algorithm Statistic Type 1 satisfaction ratio FR Mean % Std. Dev % CI Lower % CI Upper % PRA Mean % Std. Dev % CI Lower % CI Upper % 6. Discussion and conclusions Managing patients in nuclear medicine departments is a challenging problem with limited research reported in the literature. The complexity involved in this health care setting makes this problem unique. In this paper, we derive and implement two algorithms for scheduling nuclear medicine patients and resources. The fixed resource (FR) Fig. 14. Resources utilization under high demand scenario.