Sensitivity Analysis in Bed Capacity Studies including the Medical

Staff’s Decision Making

Cristina Azcárate

, Julio Barado

and Fermín Mallor

Department of Statistics and Operations Research, Public University of Navarre, Campus Arrosadia, Pamplona, Spain

Intensive Care Unit, Hospital of Navarre, Irunlarrea 3, Pamplona, Spain

Keywords: Simulation, Decision Making Modelling, Sensitivity Analysis, Intensive Care Unit, Bed Capacity.

Abstract: This paper deals with capacity planning studies in intensive care units (ICU). Our aim is to provide a

framework in which the discharge policy from an ICU can be modelled and included in a simulation model.

This is a very unique contribution of this research. We highlight the influence of the assumed policy in the

ICU quality of service. A high quality of service means a low percentage of rejected patients and a length of

stay in the ICU as long as necessary for the patient recovery. We introduce a parameterized set of rules to

mathematically model the discharging decisions made by the physicians of an ICU. Then we present a

sensitivity study carried out for the ICU of the Hospital of Navarra in Spain. The set of discharge policies is

represented in the space of the performance measures to distinguish efficient from no efficient policies.

Finally, the sensitivity analysis is extended, firstly, by considering variation in the number of beds and, then,

by varying the patient arrival ratio.

1 INTRODUCTION

The Intensive Care Unit (ICU) is a key area within

hospitals caring for critically ill patients. The beds

and the specialized staff of an ICU are costly

resources and then the ICU managers should balance

the contradictory goals of providing a high quality

health service and minimizing the operating costs.

Simulation has been widely used to tackle

health-care system management and operation

problems. Recent reviews dealing with the

application of simulation modeling in health care

can be found in (Brailsford et al., 2009), (Eldabi et

al., 2007), (Günal and Pidd, 2010), and (Katsaliaki

and Mustafee, 2010). Many studies use simulation to

analyze hospital capacity and bed allocation, but

only a few deal specifically with ICUs. Among these

Kim et al., in a series of papers (1999, 2000 and

2002), developed a simulation model of an ICU in

Hong Kong to study the unit’s capacity utilization

and the quality of care provided to its patients. They

also considered, (Kim et al., 2000), the conflict

between ICU physicians and the operating surgeons

when these last ones proposed to reserve some ICU

beds exclusively for elective surgery. Authors used

the simulation model to explore the implications of

these bed-reservation strategies. In (Kim and

Horowitz, 2002) the analysis is extended by using a

daily quota system for the elective surgery together

with the knowledge of a 1-week or 2-week

scheduling window. Similarly, (Kolker, 2009) also

developed a simulation model to determine the

maximum number of elective surgeries per day that

should be scheduled in order to reduce diversion of

an ICU to an acceptable low level.

(Litvack et al., 2008) analysed the bed capacity

problem in the ICUs of several hospitals in a region

on the Netherlands. They proposed a cooperative

solution that is found by a mathematical method

inspired by overflow models in telecommunication

systems. Simulation is only used here to assess the

quality of the provided solution.

(Ridge et al., 1998) developed a simulation

model for bed planning in an ICU. They studied the

relationship between the number of beds and the

percentage of patients that have to be transferred

because of lack of bed space. The authors performed

a sensitivity analysis by varying the number of ICU

beds but also by considering different admission

rules by varying the planned patients deferral

periods and by changing the number of beds

reserved for emergency admissions.

They also pointed out that the “early discharge”

of the more able patients to alternative wards is a

548

Azcárate C., Barado J. and Mallor F..

Sensitivity Analysis in Bed Capacity Studies including the Medical Staff’s Decision Making.

DOI: 10.5220/0004165305480555

In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (HA-2012), pages

548-555

ISBN: 978-989-8565-20-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

solution commonly adopted to cope with insufficient

number of ICU beds. Nevertheless, they did not

included an “early discharge” operating rule within

the model. Costa et al. (2003) extended these models

and discuss detailed mathematical models for the

number of beds required by an ICU to meet its own

individual workload.

In the medical specialized literature we also find

studies reporting that admissions and discharges are

triaged when enough beds are not available (e. g.,

(Snuff et al. 2004) and (Costa et al. 2003)). As a

consequence, the number of patients who are

rejected from admission increases and the length of

stay gets shortened.

The aim of the research presented in this paper is

to provide a framework in which the discharge

policy from an ICU can be modeled and used in a

capacity planning study by using simulation. We

will highlight the influence of the assumed policy in

the ICU quality of service. We identify two

dimensions for this quality of service: the first one

-of a social character- is the percentage of

population that can benefit from it when needed. The

second one -of an individual character- is the degree

of recovery reached by a patient when is discharged

from the ICU. A high quality of service means a low

percentage of rejected patients and a length of stay

(LoS) in the ICU as long as necessary.

The paper is organized as follows. Section 2

describes the mathematical modelling of the

discharging decisions made by the physicians of the

ICU. In section 3 we present a sensitivity study

carried out for the ICU of the Hospital of Navarra in

Spain. We introduce the performance measures to be

considered and represent the set of discharge

policies in the space of the performance measures to

distinguish efficient policies from no efficient ones.

Then the sensitivity analysis is extended, firstly, by

considering variation in the number of beds and,

secondly, by varying the patient arrival ratio.

Finally, we end the paper with a section of

conclusions and final remarks about the usefulness

of our approach.

2 MATHEMATICAL

MODELLING OF

DISCHARGING DECISIONS

Decisions made by the ICU doctors concerning the

discharge of patients have the ultimate purpose of

controlling the level of bed occupancy in the ICU,

by balancing the full recovery of the current patients

and the bed availability for future ones. Then any

ICU mathematical model, in general, and simulation

model, in particular, developed to be used in the

study of the bed capacity problem should include

this doctor’s ability to control the number of

occupied beds.

In (Mallor and Azcárate, 2012) it is showed that

it is crucial to incorporate the management decisions

made by the clinical staff to obtain valid ICU

simulation models. Discharge decisions are made in

order to keep the number of occupied beds in levels

neither too high (to compromise the incoming of

new patients) nor too low (to “waste” expensive

resources). Depending on the clinical situation of the

patient and on the bed occupation rate at that

moment, the discharge of a patient may be slightly

advanced or delayed if considered safe to do it.

Furthermore, there is no written protocol to

automatically determine the patient discharge time;

these decisions are subject to the judgment of the

intensive care consultant. From a mathematical point

of view, this means that the state of “enough

recovery” of a patient to safely leave the ICU is not

a discrete event that can be easily identified on the

time axis but an ambiguous time interval subject to

the intensive care physician assessment.

We took into account these considerations to

model the discharge decision of a patient depending

on the bed occupancy level. Specifically, we

consider two kinds of discharge rules. These rules

are based on the idea that the recovery of a patient is

a continuous process that leads to its ICU discharge

in a time that take values in an interval of admissible

values (see Figure 1). The main idea is to compare

the time already spent in the ICU, TS, with regard to

the length of stay, LoS, which was simulated for

each one of the patients occupying an ICU bed. We

define the LoSR (LoS ratio) as TS/LoS. Then,

 if bed occupancy level i is high and the time

already spent in the ICU is sufficiently high

(specifically, if (1- LoSR) is less than a value

PRi% and (LoS-TS) less than DRi days), a

patient leaves the ICU in advance (the one in

the best health condition, which means the one

with the greatest LoSR).

Figure 1: Patient recovery and ICU discharge.

Entry Recovery

Limit for early

discharge

PRLoS 100%

DRDaysN º

High occup.

Low

occup.

Extended

Stay

Sensitivity Analysis in Bed Capacity Studies including the Medical Staff's Decision Making

549

 if bed occupancy level i is low then the LoS of

a patient is increased in one day with certain

probability, PIi. There is a maximum of days

DEi that the LoS of a patient can be extended.

These rules can be identically defined for all type

of patients or they can be different for different

groups of patients. For example, we could

distinguish the group of programmed-surgery

patients -usually whit a short LoS- and consider that

it can only be shortened in one day with certain

probability PC

when bed occupancy level is i.

These set of rules defines infinite management

policies for the ICU: one for each set of parameter

values. To choose the rule that better fit the decision

making process in the ICU, in (Mallor and Azcárate

2012), we formulated an optimization problem with

the aim of matching as much as possible the output

of the simulation with the ICU historical data (see

Figure 2).

The decision variables are the parameters PR,

DR, PI, DE and PC above defined. Constraints

represent realistic monotonous relationships into

each set of parameters and upper bounds for their

values (uPR, uDR, uPI, uDE and uPC). The number

of ICU beds is denoted by n and by k1 and k2 the

boundaries for low and high occupancy levels,

respectively. We set as objective function to

minimize the squared differences of both occupancy

bed frequency distributions: the one observed in the

real ICU, real_freq(i), and the one obtained from the

simulation output, simul_freq(i). The proposed

optimization problem can be solved by combining

simulation and optimization techniques. The

optimizer produces a sequence of solutions whose

performance is tested in the simulator.

Figure 2: Optimization problem to determine the

parameter values of the medical management rules.

3 SENSITIVITY ANALYSIS

The discharge policy is influenced by the occupancy

level which also depends on the bed availability and

the input rate of patients. Then a bed capacity study

should take into account variability in management

policies. We show in this section that main

performance measures greatly depend on the

adopted policy.

3.1 Case Study. Parameterization of

The Rules

In a first step of our research we developed a

simulation model that included the representation of

the medical decisions made at the ICU of the

Hospital of Navarre, in Spain, following the

methodology presented in Section 2.

The Hospital of Navarra is a general public

hospital with reference specialties in the Community

of Navarra (Neurosurgery, cardiac surgery, vascular

surgery, oncology, infectious diseases, etc.). It has

483 beds, 2015 members of staff and 10 surgery

rooms. The ICU of this hospital has 20 beds and 86

physicians and nurses. It receives patients from 3

sources (emergency, operating theatre and ward).

A thorough data analysis was conducted to

obtain good statistical models for the arrival pattern

and LoS of each of the 8 groups of patients

considered. The necessary data for the statistical

estimation were recorded and provided by the

Hospital administration. We used two files: a patient

file and a bed occupancy file, containing 9 years of

data. The patient file includes all records of patients

attended in the ICU. For each patient the following

variables are known: age, arrival date, illness group

(8 groups were considered), output date, APACHE

(illness severity), infections in the ICU, and exitus

(recovered or died). The bed occupancy file records

the number of occupied beds at 4 p.m., each day. It

was used to validate the simulation model.

Cardiac surgery patients are special patients in

the performance of this ICU and represent 1/3 of the

total amount of patient arrivals.

We simulate the ICU model under different rules

to analyse their influence in a set of performance

measures. In order to make an easier comparison of

the results we simplify the structure of values that

can take the decision rule’s parameters. We

distinguish two levels for the state of high bed

occupancy:

 moderate high occupancy (75%-85%): when

there are a number of occupied beds from 15

to 17. It is denoted by level h1, and

 very high occupancy (+85%): when there are a

number of occupied beds from 18 to 20. It is

denoted by level h2.

Low occupation levels are also reduced: level l1

from 1 to 8 beds, and level l2, from 9 to 13 beds.

 



























21 ,11 2integer

0...

0.....

0...

)(_)(_

kk,...,kj,...,nkni ,DIDR

PCPCPCuPC

PIPI uPI

DEDEuDE

PRPRPR uPR

DRDRDR uDR

tosubject

ifreqsimulifreqrealMin

knnn

SIMULTECH 2012 - 2nd International Conference on Simulation and Modeling Methodologies, Technologies and

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550

In the presentation of the results we focus on the

decisions concerning the situation of high

occupancy, because they are more difficult to

manage and more important for the patient health.

We consider the following parameters related with

early discharges:

 PC

: the probability of reduction in 1 day the

LoS of cardiac surgery patients, with normal

post-surgery evolution, when bed occupancy

level is hj, for j=1,2.

 PR

: the percentage of reduction in LoS when

bed occupancy level is hj, for j=1,2.

 DR

: upper limit for the reduction in the

number of LoS days when bed occupancy

level is hj, for j=1,2.

Observe that the last four parameters do not

affect to cardiac surgery patients, with normal post-

surgery evolution.

Consequently, an early discharge policy is

described by a vector of 6 values: two for the cardiac

surgery patients and four for the other patient

groups.

We also consider the following four parameters

related with extended discharges:

 PI

: the probability of one-day increase in LoS

when bed occupancy level is lj, for j=1,2.

 ER

: upper limit number of LoS days

increased when bed occupancy level is lj, for

j=1,2.

3.2 Efficient Discharge Policies

Two are the main objectives of an ICU: it should

provide service to all patients that can benefit from it

and it should provide a full service to all admitted

patients. The first objective means that no patient

should be rejected because the ICU is full and the

second one means that no LoS should be reduced,

risking a full recovery, because the occupied bed is

needed. Based in these two objectives, two

performance measures can be defined:

 percentage of rejected patients. Emergency

patients arrive at random and they are

transferred to other hospitals -which is no

desirable- if they cannot be immediately

admitted. In the past, the lack of beds caused

to postpone surgeries. Nowadays, no surgery

is canceled due to the lack of operating rooms,

and the patients are also transferred to other

health facilities in the region, if necessary.

 percentage of shortened days. To calculate

this measure the truncated LoS and the “ideal”

LoS for each patient should be known. This

information is not included in ICU databases.

At least this is not reported in the literature

neither the ICU of the Hospital of Navarra

records it. Nevertheless, this information is

collected from the simulation model, because

a LoS -which is considered as the “ideal” LoS

- is simulated from the estimated statistical

model, when a patient is created. The

performance measure is obtained as the ratio

of the sum of the shortened days for all

patients to the sum of the LoSs of all patients.

To conduct the sensitivity analysis, we vary the

value of some parameters into different ranges while

other parameters have a fixed value. Varying and

fixed parameters are included in tables 1 and 2,

respectively.

Table 1: Set of variation for parameters PR

and PC

Parameters

Set of values

for j=1,2

0, 5, 10, 15, 20

for j=1,2

0, 10, 20, ... , 90, 100

Table 2: Parameters with fixed values.

Parameters

Fixed values

for j=1,2

Observe that monotonicity conditions imply that

>= PC

and

>= PR

h1.

This leads to

15x66=990 different combinations of values for

parameters PR

h1,

, PC

and PC

h2.

Each

combination denoted by (PR

, PR

, PC

)

defines a different discharge policy. For example, (5,

10, 30, 60) means that in case of occupancy level

from 15 to 17, with probability 0.3, the LoS of a

cardiac surgery patient is shortened in one day

while, for the rest of patients, in a maximum of 5%.

In case of bed occupancy level from 18 to 20, the

probability of one-day reduction increases to 0.6 and

the maximum of LoS reduction increases to 10%.

For each scenario, 100 replications of 50-year

simulation experiments were run, with a 3-year

warm-up period.

Figure 3 shows the representation of the 990

different discharge policies in the objective space.

Policies are grouped by the values of the PR

parameters defining the discharge for a general

patient (that is, a patient of any other group different

of cardiac surgery). We see that the set of all

Sensitivity Analysis in Bed Capacity Studies including the Medical Staff's Decision Making

551

Figure 3: Discharge policies for the present situation: 20 beds.

policies approximately fills a triangle and also the

policies within each group. We can analyse the

trade-off between both objectives. The policy (0, 20,

0, 100) located in one of the corners provides a

percentage of 1.5% of rejected patients and a

percentage of shortened stay of 1.8%. Any

movement from this policy to other policy to get a

reduction of the rejected patients implies a sharp

increase in the shortened stay: from the point (1.8,

1.5) (associated to policy (0, 20, 0, 100)) to the point

(4.9, 1.4) (associated to policy (20, 20, 100, 100) in

the right corner) there is a line of slope -0.03, over

which is located a piece of the Pareto frontier. Then

for each unit added to the percentage of shortened

stay the percentage of rejected patients is reduced

only by 0.03 units. Reasoning in the same way, by

using the policy (0, 0, 0, 0), located in the upper

corner and with associated point (3.3, 0), we get that

reducing by one unit the percentage of shortened

stay implies to increase, approximately (and in the

best of the cases) by one unit the percentage of

rejected patients. Figure 4 shows the same graphical

representation of Figure 3 but now distinguishing

among efficient and no efficient policies. There are

98 efficient policies, which are included in table 3

(see appendix).

Figure 4: Discharge policies: efficient vs. no efficient.

3.3 Bed Capacity Analysis: Increase in

the Number of Beds

Simulation models in health care are frequently used

to assist decision makers in capacity analysis, that is,

the optimal number of resource units to achieve a

determined objective. We have seen in the previous

subsection that the management policy is critical to

reach the desired levels for each objective when

there are conflicting objectives.

SIMULTECH 2012 - 2nd International Conference on Simulation and Modeling Methodologies, Technologies and

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We know that the answer of a what if question of

type, “what is the performance of the ICU when one

bed is added?”, will depend on the discharge policy

applied by the medical staff. To answer this question

we have adapted the decision rules to the case of 21

beds. The moderate high occupancy level h1 is now

defined when there are a number of occupied beds

from 16 to 18 and the very high occupancy level, h2,

when there are a number of occupied beds from 19

to 21. The set of parameters for the rules does not

change but we reduce the number of possible values

for the percentage of shortened stay to 3: {0, 10,

15}. The number of policies tested is 396.

We obtain a structure for the representation of

the policies in the objective space similar to the case

of 20 beds. Figure 5 compares both sets of efficient

solutions. We observe that the efficient frontier for

the 21- bed case is almost a translation of the

efficient frontier for the 20-bed case. Each policy

with 21 beds dominates the equivalent with 20 beds.

That is, when the resource is increased and the same

policy is applied then both objectives are improved

at the same time. In both cases, in the corner, is the

policy (0, 20, 0, 100). Its associated point in the 21-

bed case, is (1.24, 0.95) and the objective values

reached by the policies (0,0,0,0) and (20, 20, 100,

100) are (0, 2.23) and (3.73, 0.83), respectively.

Then when we moved out of (0, 20, 0, 100), per each

unit of improvement in the shortened stay, we need

to increase approximately by one unit the percentage

of rejected patients. And conversely, if we wish to

improve the 0.95% rejected patients we would

worsen the shortened stay in one unit to get a

decrease of 0.05 in the rejected percentage.

3.4 Bed Capacity Analysis: Increase in

the Arrivals Rate and the Number

of Beds

When the capacity analysis is done to determine the

resources needed to provide a service in the future it

is necessary to include in the model a prediction of

the future demand.

In this subsection we study the performance of

the ICU under the hypothesis that the patient’s

arrival ratio is increased due to the increase and the

aging process of the population. We consider three

different scenarios: increments of 5%, 10% and 15%

in the arrivals rate. These three scenarios are studied

with both the current capacity and the increased

capacity to 21 beds. Figure 6 shows the 6 Pareto

frontiers. We observe a shape for these frontiers

similar to the one we found in the analysis of the

present arrival rate case. Again in the corner is the

policy (0, 20, 0, 100) and the trade-offs between

objectives keep similar proportions to those found in

section 3.2 and 3.3. The efficient frontier

corresponding to a stress of 1.05 and 21 beds is

similar to the results found for 20 beds in the present

arrival rate. This is because both the service capacity

and the service demand are increased by a 5%. This

Figure 5: Efficient policies for the ICU with 20 and 21 beds.

Sensitivity Analysis in Bed Capacity Studies including the Medical Staff's Decision Making

553

Figure 6: Efficient policies for the ICU with different levels of stress in the arrival rates (20 and 21 beds).

argument also justifies the closeness of frontiers

corresponding to scenario 20 beds and stress 1.05

and scenario 21 beds and stress 1.10.

4 CONCLUSIONS AND FINAL

REMARKS

In this paper we have shown that a bed capacity

analysis in an ICU requires the consideration of the

discharge policy that is applied by the medical staff.

The main effect of this management policy is a

reduction in the LoS of some patients. Then the

percentage of total shortened stay is considered as a

performance measure to catch the degree of

intervention of the medical staff when there is a high

pressure due to lack of beds. This new measure is

studied together with the traditional performance

measure of percentage of rejected patients.

The doctors can find useful the representation of

the discharging policies in the space of goals to learn

about the trade-off between objectives that can be

achieved by modifying the parameters of the rules.

These rules can be interpreted and used by the

doctors as bench marks for their own decision

processes. This constitutes a normative approach to

the discharging policies in the sense that it indicates

how to proceed to get certain levels of quality of

service.

We have also simulated the ICU under the rules

estimated according to the methodology exposed in

section 2, that is, under the rules that better describe

the decision process in the real ICU. The

representation of the results in the space of goals

showed that the point is very close to the Pareto

frontier; specifically, it is located close to the corner

point, in the upper part. Thus, we can conclude that

the medical staff makes decisions almost efficiently

according to both objectives.

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APPENDIX

In table 3, we include the 98 efficient discharge

policies obtained when considering the 20-bed ICU

with the present patient arrival ratio. Each policy is

represented by a vector with four components:

(PR

, PR

, PC

Table 3: Efficient discharge policies. The policies are

ordered from the top left to the bottom right of the

objective space.

100

Table 3: Efficient discharge policies. The policies are

ordered from the top left to the bottom right of the

objective space (cont.).

100

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