Capacity planning for intensive care units

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 105 (1998) 346-355

Capacity planning for intensive care units J . C . R i d g e 4, S . K . J o n e s a, M . S . N i e l s e n b, A . K . S h a h a n i

a,*

~ Institute of Modelling jor Health Care, Faculty of Mathematical Studies, Universi O, of Southampton, SO17 1BJ. UK b General Intensive Care Unit, Southampton General Hospital, Tremona Road, Southampton SO, 6 6 YD. UK

Received 1 February 1996: revised 1 April 1997

Abstract

In this paper we describe a simulation model for bed capacity planning in Intensive Care. The model shows that there is a non-linear relationship between numbers of beds, mean occupancy level and the numbers of patients that have to be transferred through lack of bed space. Furthermore, there is a heavy trade off between bed occupancy and the number of transfers. The model employs a rudimentary deferral rule for the elective patients when their admission is blocked through lack of bed space; from this we conclude that there is scope for a more effective elective patient admission scheduling system. As a result of this work, a multi-disciplinary Intensive Care Modelling Task Group has been set up to guide the development of a more sophisticated model. © 1998 Elsevier Science B.V. Keywords: Simulation; Queuing; Health services; Capacity planning; Intensive care

1. Introduction

The provision of adequate supplies of hospital beds, and the question of waiting times and waiting lists are amongst some of the most fiercely debated issues within the health service. A recent report on Intensive Care by the Department of Health (Metcalfe and Mcpherson, 1994) highlighted the uneven spread of Intensive Care beds between hospitals, and showed that patient 'refusal rates' were strongly linked to local bed allocation. Deciding on just how m a n y ICU beds to provide is however not simple. Intuitively one might observe the mean

* Corresponding author.

monthly arrival pattern, and then calculate the required number of beds based on a confidence interval related to the mean. There are however a number of complicating factors which mean that this kind of simple calculation is inappropriate: • Emergency patients arrive at random, often in quick succession, and must be admitted with a minimum of delay. The build up of 'emergency queues' and the need to transfer patients to other hospitals is highly undesirable. • Planned (elective) patient admissions are subject to the constraints imposed by other hospital services, such as surgeons hours and theatre space, and the number of free beds in the ICU. • Planned patient admission profiles can be highly correlated with the time and day of the week.

S0377-2217198/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PHS03 7 7-22 1 7 ( 9 7 ) 0 0 2 4 0 - 3

J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346-355

• Patient lengths of stay (LOS) are frequently distributed with a bias towards shorter (than longer) LOS. Sometimes, however, a patient will stay a very long time, which can cause a disproportionate 'blocking' effect in the ICU with respect to subsequent referrals. • Different patient types have different LOS distribution profiles. These features point towards a need for sophisticated bed capacity planning models from the discipline of operational research. There are a number of different ways of approaching the problem and a considerable body of literature has already been published on the subject; the reader is referred to a very informative review article by Smith-Daniels et al. (1988). In this paper, we present in the main, a simulation model, other examples of which can be found in Goldman et al. (1968), Kao and Tung (1981), Williams (1983), Cohen et al. (1980), Parry and Pitroda (1992) and Dumas (1984, 1985). We also briefly demonstrate the potential of queuing theory models, see Cohen (1956), Saaty (1961), Bailey (1954, 1955), and Cooper and Corcoran (1974). Although generally rather limited in scope, they can be useful as a means of pre-calculating certain limiting output values for simulation models. We have used this as a method of cross-validating the essential logic of our simulation model code. Given the wealth of work that has already been done in this area, it is both surprising and disappointing that it has not found greater application. We aim to remedy this by bringing together past achievements under a single, flexible and sufficiently detailed model. In this paper we describe a self-contained part of this modelling work which has helped to inform capacity planning decisions at Southampton General Hospital.

2. Methods

The simulation model was encoded in PASCAL using a 3-phase simulation shell developed at Southampton University. The model encapsulates the essential features of a 'virtual ICU', having a certain number of 'virtual beds' and allowing 'virtual patients' to arrive, stay and depart at rates

347

which reflect the true historical (or any other ) patterns as required. Thus, arriving patients and events associated with them are able to be tracked through time which allows the model to collect data on longitudinal processes such as resource usage. The queuing model was also encoded in PASCAL as a separate program. The historical pattern of arrivals and LOS of 2000 patients admitted to the ICU at the Southampton General Hospital was determined using five years of past data which had been collected on a database by the ICU staff. The data was sorted into a number of different patient types (using criterion supplied by the ICU) using a spreadsheet; various statistics were derived using statistical modelling package. The length of stay ( L O S ) of the emergency patients and planned patients were described using either the negative exponential curve or a Weibull curve fitting routine (Shahani et al., 1994). Planned patients can be admitted subject to 'admission operating rules' which are detailed below. • Planned patients are deferred for a certain time period if, upon arrival at the ICU, the number of free beds has fallen below a minimum level. • Planned patients can only be deferred for a userdefined maximum number of times after which a planned patient is deemed to have emergency patient 'status'; emergency status planned patients and 'true emergency' patients are always admitted if there is a free bed. • Planned patient are deferred for either a fixed number of weeks or a random variable up to a maximum number of weeks. • Emergency patients were assumed to arrive at random at a rate independent of the day of the week. The planned patients were allowed a variable arrival rate dependent on the day of the week but on any one day the planned patients were then assumed to arrive on a random basis. The simulation logic flow diagram shown in Fig. 1 illustrates the flow of patients through the ICU. Other stages (not illustrated) generate the length of stay and arrival time of each patient. The logic of the nearest equivalent queuing model (formally type Mi/M/S/NPRP/oc/~) can be represented by Fig. 2. In this model the following hold:

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J. C Ridge et al. / European Journal of Operational Research 105 (1998) 346-355 outside world

patient type i 4 arrives

t--

YES

bed

NO

NO YES

admit patient

increase priority/status and defer admission

4

]

patient stays then leaves hospital

transfer to another ICU 'elsewhere'

Fig. 1. Simulation model flow diagram.

• a simple bed allocation priority rule favours emergency patients over planned; • no planned patient deferral rule; • no ability to check for a minimum number of free beds for planned patients; • no ability for emergency status patients to be "transferred elsewhere" if all beds are full; • patient queues can grow to infinite length if no beds are free;

• patient inter-arrival times follow a negative exponential distribution. Although quite drastic, these simplifications still provided a suitable 'test-bed' to check the basic logic (temporarily modified) of the more sophisticated simulation model. A n a l y t i c a l solution: L e t S be the number of beds; 2e the emergency arrival rate which has high priority, J.p the planned arrival rate which has low

J.C. Ridge et al. I European Journal of Operational Research 105 (1998) 346-355

349

outside woHd

planned patient arrives

emergency patient arrives

@ [

highest priority patient stays if bed free then leaves hospital

Fig. 2. Queuing model (MiIMISINPRPl~c/oc) flow diagram.

priority, h the average LOS for all arrivals; (x) the state in which x beds (0 ~
)-e -- ~p

(.)

f ( x ) = S - Ae -- ,'~p q- ('~e q- "~p) E(S, )~e or- "~p) where (.) = x! N(S, 2e + 2p) where N(a, b) = ~-,a

L.~i=0

bi/i ! "

ba and E(a, b) - a! N(a, b)

At this level of logical complexity, the analytical solution is relatively straightforward to derive and encode. Including a level of detail that approximates more closely to a real ICU is however a very tedious exercise, and in most instances completely intractable; the general approach of simulation is much easier to set up and tune to particular needs, and though it provides only a 'slow convergence' numerical approximation to the results produced by the queuing theory, computers are now sufficiently fast that even the speed of computation of a queuing model is not generally sufficient justification for encoding them in this sort of application. We have used this queuing model to find suitable 'warm up' and 'run times' for the simulation model as part of its validation process.

SE(S, ~e -]- Ap)

O z

S - -)~e - ~-p q- (2e q- A p ) E ( S , "~e q- ~p)"

2.1. Summary of data received from Southampton ICU

Average wait on emergency queue hD S - )~e Average wait on planned queue ShD (S -

)~e)(S

-- ~e -- ~;~p)

A large data set consisting of five years of computerised ICU patient record forms from 2000 patients was provided by Southampton General Hospital ICU for the project. The data were supplied in both dBase IV and Lotus 1-2-3 file format. Data for the years 1988-1993 were used.

J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346-355

350

Each patient record had up to 74 fields of information. Immediately useful data included date and time of arrival; number of days/length of stay (or in hours if <24 hours) and a coding system giving the patient type. Other data on the forms included Apache II, Injury Severity Score and details of interventions during ICU stay. The ICU paper based 'log-book' was also made available, since this gave the numbers of patients who were not able to be admitted to the ICU each year; this complemented the dBase and Lotus files which only gave information relating to those patients who actually stayed in the ICU. The cumulative and averaged data for the six patient types included in the model is summarised in Table 1. The calculated referrals column represents an average pattern of referrals which the ICU might expect to see each year; the exact figures were not easily retrieved from the data set.

In 1993 there were 598 referrals, 434 admissions, and therefore 164 refusals (27%). • M e a n length o f stay (LOS) was expressed in a three parameter format to facilitate the use of a Weibull distribution curve-fitting routine within the simulation model. • Mean admissions is equal to the mean number of patients of a particular type who actually stayed in the ICU. • Calculated referrals is calculated as follows: mean annual admissions x 1993 total referrals total mean annual admissions The relative daily arrival rates of the six patient types was also determined; this showed a random arrival pattern for the emergencies and a more structured pattern for the planned patients related to the working practices of the hospital consultants (Table 2).

Table 1 Southampton I C U data six patient type data summary Patient type

Mean LOS/days

LOS percentage point/days

LOS percentage point/%

Mean annual admissions

Calculated referrals for 1993

4.698 2.639 4.128

15.0 11.0 13.0

91.20 93.78 92.20

20.07 197.0 103.4

29.47 289.3 151.8

Type 1 Type 2 Type 3

1.637 2.802 2.260

4.00 10.0 6.00

94.12 95.20 92.30

Totals

4.171

-

-

Emergency Type 1 Type 2 Type 3

Planned 6.745 7.000 73.11

9.904 10.27 107.3

407.3

598

Table 2 Relative daily arrival rates Patient type

Mon.

Tue.

Wed.

Thur.

Fri.

Sat.

Sun.

10 33 23

7 27 8

4 31 20

10 32 11

5 34 7

13 20 22

8 31 12

3 0 25

2 3 38

4 7 81

5 2 57

4 7 49

0 1 18

0 1 5

Ernergen~T Type 1 Type 2 Type 3

Planned Type 1 Type 2 Type 3

J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346 355

3. Simulation model validation Firstly the simulation model was temporarily modified so that it worked according to the queuing model logic. This provided a method for finding a suitable 'warm up' time to remove initial transient states and subsequent 'run time' such that the simulation model outputs converged on the limiting values predicted by queuing theory (to 3 significant figures). After some experimentation it was found that a warm up time of 5 years and a run time of 100 simulated years period was sufficient. The core of the simulation logic was deemed to be correct and was then re-modified back to its more sophisticated form. The following simulation model parameters were then chosen to be as similar as possible to those observed in practice: • planned patients were accepted if there were two or more free beds; • planned patients were deferred a m a x i m u m of two times; • deferral times were selected by the model as a r a n d o m variable between 1 and 4 weeks; • the I C U had six beds; • emergency status patients are "transferred elsewhere" if there is no free bed. With this set up the model predicts an overall transfer rate of 23% + 1% at 95% confidence to be compared with 27% as seen in 1993. In general, the model would be expected to underestimate the percentage transfers because it takes no account of

351

the hourly variation in the arrival rate of the patient types each day, nor the occasional non-availability of nursing staff. In real life, we expect both these events to happen. In addition the referral rates were calculated from past data; overall it was decided that the model showed a sufficiently acceptable match to reality to justify looking at the effect of the different model parameters.

4. Results

(1) The effect of varying the number of ICU beds (Table 3). • Deferral times were selected by the model as a random variable between 1 and 4 weeks. • Planned patients were deferred a m a x i m u m of 2 times. • Planned patients were accepted if there were 2 or more free beds.

(2) The effect of varying the planned patients deferral period (Table 4). • • • •

The number of beds was fixed at 6. Deferral times were always of fixed length. Planned patients could be deferred only once. Planned patients were accepted if there were two or more free beds.

(3) The effect of changing the number of beds reserved for emergency admissions (Table 5). • Deferral times were selected by the model as a random variable in the range 1-4 weeks.

Table 3 Mean % emergency transfers vs. number of beds and overall % bed days occupied vs. number of beds Beds

6

7

8

9

10

11

Mean % emergency transfers Overall % bed days occupied

19.9 73.1

12.7 70.3

6.80 63.7

5.25 61.7

2.30 55.2

0.45 49.4

Table 4 Emergency and planned patient transfers vs. planned patient deferral time Planned deferral time weeks

Emergency transfers % Planned transfers %

1

2

3

4

5

6

7

8

23.2 17.4

24.3 14.7

24.7 15.3

25.1 17.5

26.1 17.7

24.7 16.1

24.5 16.0

24.8 16.5

Mean

SD

24.7 16.4

0.8 1.1

J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346-355

352

Table 5 Emergency, planned and all patient transfers vs. number of reserved beds Number of beds reserved for emergency admissions

0

1

2

3

Emergency transfers % Planned transfers % All referrals transfers %

25.4 8.2 21.7

24.0 14.8 22.0

23.7 19.7 22.9

23.8 22.6 23.5

Table 6 Number of free beds at midnight vs. day of the week Free beds

Mon.

Tue.

Wed.

Thur.

Fri.

Sat.

Sun.

Mean SD

2.0 1.5

1.8 1.6

1.5 1.2

1.4 1.3

1.5 1.2

1.6 1.3

2.2 1.5

• Planned patients could be deferred only once. • The number of beds was fixed at 6.

(4) Typical number of free beds at midnight vs day of the week (Table 6). • Deferral times were selected by the model as a random variable between 1 and 4 weeks. • Planned patients were deferred a maximum of two times. • Planned patients were accepted if there were two or more free beds.

o,] 0.3

0.2

(5) Typical free bed probability distributions. • Parameters as for (4) (Figs. 3 and 4).

0

1

2

3

4

5

6

Fig. 3.6 Bed ICU.

5. Discussion

5.1. Model results It is clear from Table 3 that percentage emergency patient transfers are very non-linear with respect to the number of beds. With increasing numbers of beds there is an initial steep drop in the number of expected transfers tending asymptotically to zero with ever larger numbers of beds. The trade off for having a low transfer rate is of course a lower mean occupancy level. With six beds, the mean occupancy is predicted to be 88%, with an emergency patient transfer rate of 24%; with 11 beds the mean occupancy is reduced to 62% with an emergency patient transfer rate of 2%. The random arrival of the emergency patients

0.15

,]

0.1

0.05

0, 0

I

2

3

4

5

6

7

Fig. 4. 11 Bed ICU.

8

9

10

11

J.C. Ridge et al. I European Journal of Operational Research 105 (1998) 346-355

means that it becomes ever more expensive to guarantee a free bed for all admissions at all times, since there is always a small probability of having a sudden rush of emergency patients for which there will never be enough beds. The model was also able to show the probability distribution of numbers of free beds at any particular time in the day. From Figs. 3 and 4 it is apparent that with only six beds the current ICU most often has either none, one or two free beds, but that there is also a finite but significantly reduced probability of having 3, 4, 5 or even 6 free beds. With 11 beds, the distribution has shifted such that the ICU could expect to have 4-8 free beds a fair proportion of the time. Though the expected free bed probability distribution is an interesting feature of the overall compromise between average bed occupancy levels and transfer rates, hospital planners should be aware that the distribution itself is probably not a useful tool for deciding how many beds to supply. Table 4 suggests that the planned patient deferral time did not influence the mean transfers. This is not surprising, since the simple deferral rule that we used was not a reactive scheduling policy related to current and anticipated needs. Further, Table 6 shows that the expected variance in the numbers of free beds each day is large relative to the mean level. This result implies that the process of scheduling elective patients is likely to be difficult, hinging on a knowledge of both probabilities of free beds and future emergency patient arrivals. We draw attention to the need and scope for effective scheduling of planned patient arrivals since this is clearly an important factor in determining ICU occupancy levels, nursing requirements, bed numbers and overall operating efficiency. Table 5 demonstrates that the common plan of keeping beds free at all times for emergency admissions, raises the percentage transfers of planned patients quite considerably whilst only making a small improvement in the percentage of emergency transfers. Overall transfers are minimised when no beds are reserved at all. In an ideal situation, the results produced by the model would be used as part of the capacity planning decision process. If this were done one might reasonably expect that any predicted results would hold in practice. One should be aware, how-

353

ever, of the effects of the change process, which may introduce an element of 'feedback' into the system, for example higher bed capacity might induce higher demand. In this sense the model is an invaluable guide to 'ideal' performance and it is a matter of local conditions which determines the final outcome.

5.2. Data analysis A major challenge in building models of this sort lies in the initial data collection and analysis. We were fortunate in having access to a well developed ICU database system and a well defined means of classifying the patients into different types useful to the ICU. Most ICUs have found it necessary to set up their own patient classification systems which makes it hard, if not impossible, to compare case mix profiles, medical outcome, and other data from different hospitals, where reported in the literature. A standardised data system such as that described in the INF O R M initiative (Leaning et al., 1991) might help to resolve this issue, at least at the data collection level. Health related groups (HRGs) as used in Britain, and Disease related groups (DRGs) commonly used in the USA, have been criticised for their inability to distinguish different levels of severity of illness between otherwise equivalent patients, which is clearly linked to their expected LOS and cost of treatment. It is arguable that these means of classification produce patient groupings that are not the best for a capacity planning model. What is required is a means of grouping patients that minimises the variability in LOS for any one patient group; this would then afford greater opportunity for more efficient and effective elective patient scheduling. A suitable technique might be CART (Breiman et al., 1993) and we are developing an application using this algorithm for just this purpose.

5.3. Lengths of stay A solution commonly adopted for coping with insufficient numbers of ICU beds, is the 'early

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J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346~55

discharge' or 'bumping' of the more able patients to other hospitals or alternative wards. From a purely mathematical point of view, this results in individually truncated LOS figures being mixed up with 'ideal' LOS figures. It is apparent from the literature however that few, if any ICU databases include records of which patients were early discharged and by approximately how much. This would be a very useful addition to an ICU database if one was being set up with capacity planning in mind. No attempt was made to correct for early discharge in the model. In this sense, the LOS data was assumed to be 'ideal'. We also did not include an 'early discharge' operating rule within the model, which could have been used to offset the number of transfers and deferrals. Obviously the ability to accurately predict LOS is a useful step towards developing ICU patient scheduling systems. There are a number of articles detailing work related to cardiac patients, notably Tu and Guerriere (1993), who developed a neural network method for predicting the risk of a cardiac patient staying longer than two days: patients were able to be stratified pre-operatively into low, intermediate, and high risk groups accordingly. More recently Tu et al. (1994) have used discriminant analysis for post-operative prediction of LOS and other outcomes. Rawlings et al. (1993) have also described a nomogram for predicting the LOS of neonatal patients.

5.4. High dependency units (HDU's) Within the NHS, there is some blurring of distinction between HDUs and ICUs, a fact that is brought out in the report produced for the Department of Health (Metcalfe and Mcpherson, 1994). It is well known that not all patients are 'appropriately' referred to the best level of care; sometimes this is because of pressure on bed space, other times it is more related to local referral practices, the uncertainty of medical prognosis, and the question again of effective patient grouping techniques. The situation is often complicated still further since some larger hospitals have separate areas for ICU patients from medical specialities such as neurosurgical, neonatal and cardiothoracic

surgical wards. This allows for greater specialisation of the separate services, but weighs in against some of the advantages of pooling resources. Noteworthy are the hospital models developed by Dumas (1984, 1985) which included several different levels of care and allowed exploration of alternative patient transfer policies. A number of useful results were presented which gave an indication of the scope for minimising inappropriate patient settings using simulation. Different levels of care are clearly an integral part of the problem of calculating ICU bed numbers and was not considered in our model.

6. Further w o r k

We propose that a next generation model should be developed to include the following features: • An elective patient scheduling system for maxiraising admission probability, given a fixed 'planning time horizon'. This could be used for optimising both day to day scheduling on the ward and for long term capacity planning. • A range of different levels of care, e.g. ICU and H D U beds. The interaction between different levels of care was not considered in our model and clearly complicates capacity planning quite considerably. • A system for describing other constraints including operating theatre hours, surgeons hours, nurse availability, bed borrowing, etc. • A well defined data input/output format exploiting a range of commonly used data systems, suitable for use by a range of health professionals. • Semi-automated historical data analysis. • A statistical method for defining patient groups appropriate for capacity planning models. • Effective dissemination to all potential users through influential co-ordinating bodies representative of the interests of both the health profession and the general population. We have already set up a multidisciplinary working group in association with the Intensive Care National Audit Research Centre (ICNARC) to steer the development of the next generation model.

J.C. Ridge et al. / European Journal of Operational Research 105 (1998) 346-355

7. Conclusion This paper presents a simulation model that demonstrates the power of operational research methods for capacity planning at a single level of care in a single hospital. It is clear that there is a heavy trade off between the number of transfers and mean occupancy; there is also a strong nonlinear relationship between the number of beds and the number of transfers. Recent reports have shown that there is both an uneven distribution of I C U beds between hospitals, and an inefficient overlap between H D U and I C U facilities. There is a clear need then to address both the issue of capacity planning at each different level of care within individual hospitals and also relative levels of care across different hospitals in a region. The use of simulation models is clearly an appropriate means for capturing the complexities of these issues.

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