Introduction
t
is of importance to improve the performance and throughput of patients in a
heart surgery department, especially when the resources are limited. A typical
heart surgery department is a ward where patients enter before going to the
operation theater. Following operation and before returning to the ward for
final discharge, they stay in the intensive care unit (ICU) for a period of
time. It is probable that some patients die within the system.
It is of interest to observe the behavior of the
system with varying numbers of beds in the ward and ICU in order to maximize the
throughput of patients in the system. The running of the department can be
viewed as a complex queuing system. To achieve the mentioned goals for the
system, an analytical solution would be very difficult to obtain and possibly
not feasible. So, using Monte-Carlo simulation1 is essential to
investigate such a system.
The application of
simulation processes to hospitals has received attention in recent years. O’Keefe investigated the
operation of an outpatient department using a qualitative approach.2 Cox et al used a
simulation study in order to optimize the running of an ear, nose, and throat
outpatient clinic.3 Brahimi and Worthington applied simulation
models for an outpatient clinic department.4 Hashimoto and Bell5
investigated the behavior of an outpatient clinic to improve the
performance of the clinic. Bagust et al6 used a stochastic
simulation model to dynamics of bed use in accommodating emergency admissions. Mui7 applied
a computer simulation model to simulate individuals’ coronary heart disease history
over time for a sampled Australian population, characterized by major coronary
risk factors. Goldman
et al8 used a computer simulation model to estimate the effects of
actual investments, made to change coronary risk.
The objective of this study was to calculate the bed
occupancy rate in the ward and ICU with different numbers of beds in each of
them and use the results to maximize the throughput of patients in the system. Since
the behavior of the ward is a complex queuing system, an analytical solution
would be very difficult to obtain and perhaps not possible. So, using
Monte-Carlo simulation1 is essential to investigate the system.
Monte-Carlo simulation is a simulation technique using
random numbers and probability distributions. For example, time taken for an
operation has a specific probability distribution, say exponential
distribution. So, simulating such a system is known as Monte-Carlo simulation.
Patients and Methods
A set of data was given to us by Mounsey et al 9
consisting of the throughput and associated data of 431 patients who had
coronary artery bypass surgery during a 9-month period. They underwent
coronary artery bypass grafting prospectively at the Freeman Hospital in Newcastle upon Tyne, England. Data were collected from a review of patients’
records and included baseline demographic characteristics, details of
cardiovascular risk factors, angina grade and treadmill exercise tolerance, and
the presence of serious coexistent pathology. Coronary angiographic data were also
obtained from the patients’ records. The patients who spent some days in the
department for other types of surgical operations were excluded from this
study. The coronary surgery patients normally occupy ICU beds for 24 hours or
less though some spend longer. Patients spending 24 hours or less in ICU are considered
as fast track patients while those spending less than 24 hours are deemed slow
track patients. If a patient stays in the intensive care unit for 24 hours or
less, then, for the next routine operating day, there will be a free bed in the
unit allowing for a new operation.
Assume that there are M beds in the ward, N
beds in the ICU, K operating theaters, and an infinite supply of
patients. At any time there are m free beds in the ward and n in
the intensive care unit.
Patients enter the ward from a waiting list, then go
to the theater for operation, to the ICU for special postoperative cares, and finally
back to the ward again for discharge. There is also the probability of death
occurring in the theater, the ICU or in the ward after operation. Each patient
spends one day in the ward. If there is a bed available in the ICU, she/he then
undergoes surgery after which she/he enters the ICU for at least a day. The
patient is then supposed to spend varying number of days in the ward, depending
on her/his clinical conditions. Beds in the ward are allocated to the ICU
patients and those on the waiting list with the ICU patients considered prior
to the latter group, however, if there are sufficient free beds in the ward,
some patients may wait in the ward for an operation.
The following are the possible events that may occur
in the system:
1.
patient enters the
ward from the waiting list;
2.
patient transfers
from the ward to the operating theater and then to the ICU;
3.
patient returns from
the ICU back to the ward;
4.
patient exits the
ward after completion of treatment; and
5.
patient dies in the
system.
The death of a patient during operation is allowed for
by having the postoperative ward stay time set to zero, nevertheless, the time
spent in the ICU has to be allocated for that patient before the operation. Without
a more complex stochastic formulation in the simulated system, death is not
predictable. Therefore, it would not be possible to perform another operation
in order to utilize the bed in the ICU, left by the deceased patient.
The sequence of the events that occurs, as system runs,
is controlled by a time index with a unit of time equal to one day. Each
patient in the system has to be allocated various lengths of time to spend in
the system. The simulation program works by discrete time; for each day, the
various events of patients’ movements are ascertained and then placed in order
of priority. Beds in the ward are allocated with priority to patients in the ICU
over those on the waiting list. The ward beds can not be totally occupied by
new patients waiting for surgery because, in that case, no beds would be left for
the patients returning from the ICU, and so the whole system would be blocked.
|
|
|
Figure 1. Time (days) spent in the ICU based on actual model and the
fitted model.
|
Some assumptions follow:
1.
the time spent in
the ward after the operation was presumed to start when the patient returned to
the ward from the ICU;
2.
at each time there
are some patients in the waiting list;
3.
if some patients are
waiting in the ward for an operation, the rule is first come-first served;
4.
operations, one per theater,
are only performed from Monday to Friday;
5.
patients may be
admitted from Sunday to Thursday;
6.
no patient can be
discharged on Sunday;
7.
admission and
discharge are at 9.00 am; and
8.
priority in the
system is for a discharged patient, then for a patient who is coming back from
the ICU into the ward, followed by a patient coming from the ward to the ICU,
and finally for the admission of a new patient.
|
|
|
Figure 2. Time (days) spent in the ward after operation based on
actual model and the fitted model.
|
The simulation work was carried out using a Fortran-77
program, supporting mark 16A NAG library subroutines, on a Sun-Solaris machine.
For simulating the system, the distribution functions
of time, taken in the ward and ICU, are necessary. It is rare for a patient to
spend more than one day in the ward before operation. Hence, for every patient
the preoperative ward time is given the value of one unit. Figures 1 and 2 show
periods of times for which patients stay in the ICU and the ward after
operation. As the Figure 1 shows, most patients usually spend a day in the ICU though
there are some who may spend from two to more than seven days. So, a geometric
distribution was fitted to the data. As Figure 2 shows, it is not possible to
fit a unique distribution for the time spent in the ward after operation. Therefore,
a mixture of geometric and uniform distributions was fitted to the data using
maximum likelihood for parameter estimation.
The parameters for the times spent in the ICU, in the
ward before operation, in the ward after operation and also the number of the beds
in the ICU and that of the ward were fed into the system, and the system was
simulated 40 times.6
Results
Table 1 shows the average bed occupancy rates in the
ward and ICU for the case of 2 beds in the ICU and varying numbers of beds in
the ward. When there were 5 beds in the ward, the bed occupancy rates in the
ward and ICU were 90% and 46%, respectively. The bed occupancy rate in the ward
was 81% and that of the ICU was 78% while having 11 beds in the ward.
|
Table 1. Bed occupancy rate in the ward and the ICU
when there are two beds in the ICU.
|
|
No.
of beds in the ICU = 2
|
|
No. of beds in the ward
|
Bed occupancy in the ward
|
Bed occupancy in the ICU
|
|
Mean
|
SD
|
Mean
|
SD
|
|
5
|
90%
|
5%
|
46%
|
8%
|
|
6
|
90%
|
5%
|
53%
|
8%
|
|
7
|
89%
|
5%
|
58%
|
8%
|
|
8
|
88%
|
5%
|
63%
|
8%
|
|
9
|
84%
|
6%
|
68%
|
7%
|
|
10
|
84%
|
6%
|
75%
|
7%
|
|
11
|
81%
|
6%
|
78%
|
6%
|
|
SD = standard deviation.
|
Table 2 shows the mean and standard deviation of bed
occupancy rate in the ward and ICU when there were 3 beds in the ICU, for
different numbers of beds in the ward. When there were only 5 beds in the ward,
the average occupancy rate was 96% and 37% in the ward and ICU, respectively.
The mean occupancy rate in the ward decreased as the number of the beds in the
ward increased. As the mean occupancy rate in the ICU increased, the number of
beds in the ward also increased so that the mean occupancy rate in the ICU was
79% when there were 16 beds in the ward.
|
Table 2. Bed occupancy rate in the ward and the
ICU when there are three beds in the ICU.
|
|
No. of beds in the ICU = 3
|
|
No. of beds in the ward
|
Bed occupancy
in the ward
|
Bed occupancy
in the ICU
|
|
Mean
|
SD
|
Mean
|
SD
|
|
5
|
96%
|
3%
|
37%
|
8%
|
|
6
|
96%
|
3%
|
41%
|
8%
|
|
7
|
94%
|
4%
|
45%
|
8%
|
|
8
|
93%
|
4%
|
49%
|
8%
|
|
9
|
93%
|
4%
|
53%
|
8%
|
|
10
|
91%
|
5%
|
57%
|
8%
|
|
11
|
90%
|
5%
|
63%
|
8%
|
|
12
|
89%
|
5%
|
63%
|
8%
|
|
13
|
88%
|
5%
|
69%
|
7%
|
|
14
|
87%
|
5%
|
73%
|
7%
|
|
15
|
85%
|
6%
|
76%
|
7%
|
|
16
|
84%
|
6%
|
79%
|
6%
|
Table 3 shows an instance of 9 beds in the ICU and 11
in the ward while the mean occupancy rate was 99% in the ward and 31% in the ICU.
Overall, the mean occupancy rate in the ward decreased as the number of the beds
in the ward increased. The maximum occupancy rate in the ICU was about 72% while
having 38 beds or more in the ward.
|
Table 3. Bed occupancy rate in the ward and
the ICU when there are nine beds in the ICU.
|
|
No. of beds in the ICU = 9
|
|
No. of beds in the ward
|
Bed occupancy
in the ward
|
Bed occupancy
in the ICU
|
|
Mean
|
SD
|
Mean
|
SD
|
|
11
|
99%
|
1%
|
31%
|
7%
|
|
12
|
99%
|
1%
|
31%
|
7%
|
|
13
|
99%
|
1%
|
34%
|
7%
|
|
14
|
99%
|
1%
|
36%
|
8%
|
|
15
|
99%
|
1%
|
39%
|
8%
|
|
16
|
99%
|
1%
|
39%
|
8%
|
|
17
|
98%
|
2%
|
39%
|
8%
|
|
18
|
98%
|
2%
|
40%
|
8%
|
|
19
|
98%
|
2%
|
43%
|
8%
|
|
20
|
97%
|
3%
|
44%
|
8%
|
|
21
|
97%
|
3%
|
45%
|
8%
|
|
22
|
97%
|
3%
|
47%
|
8%
|
|
23
|
96%
|
3%
|
48%
|
8%
|
|
24
|
96%
|
3%
|
50%
|
8%
|
|
25
|
96%
|
3%
|
53%
|
8%
|
|
26
|
96%
|
3%
|
55%
|
8%
|
|
27
|
95%
|
3%
|
55%
|
8%
|
|
28
|
95%
|
3%
|
57%
|
8%
|
|
29
|
94%
|
4%
|
59%
|
8%
|
|
30
|
94%
|
4%
|
60%
|
8%
|
|
31
|
94%
|
4%
|
62%
|
8%
|
|
32
|
93%
|
4%
|
63%
|
8%
|
|
33
|
92%
|
4%
|
65%
|
7%
|
|
34
|
92%
|
4%
|
67%
|
7%
|
|
35
|
91%
|
5%
|
67%
|
7%
|
|
36
|
91%
|
5%
|
69%
|
7%
|
|
37
|
91%
|
5%
|
71%
|
7%
|
|
38
|
91%
|
5%
|
72%
|
7%
|
|
39
|
90%
|
5%
|
72%
|
7%
|
|
40
|
90%
|
5%
|
72%
|
7%
|
Tables 4 and 5 show the mean required time for the throughput
of 500 patients in the department while having varying numbers of beds in the
ward and 2, 3, or 9 beds in the ICU; when there were only 2 beds in the ICU and
5 in the ward, the mean required time for the throughput of 500 patients was 911
days which declined to 853 days with 3 ICU beds and the same number of ward
beds. The mean decreased when the number of beds in the ward increased; for
instance, with 18 beds in the ward and 2 ICU beds, the required time for the throughput
of 500 patients in the department was 434 days. The mean decreased to 325 days
for 3 beds in the ICU and 18 in the ward. The mean declined when the number of
beds in the ward and ICU increased. For 2 or 3 beds in the ICU, increasing the
number of ward beds to above 18 did not lower the mean. The mean required time
for the throughput of 500 patients in the department was 219 days when there were
9 beds in the ICU and 20 in the ward. The mean was only 130 days with the same
number of beds in the ICU and 39 beds in the ward.
|
Table 4. The mean and standard deviation of the required time
(days) for throughput of 500 patients for different number of beds in the
ICU.
|
|
No. of beds in the ward
|
No. of beds in the ICU = 2
|
No. of beds in the ICU = 3
|
|
Mean
|
SD
|
Mean
|
SD
|
|
5
|
911
|
15
|
853
|
9
|
|
6
|
778
|
10
|
718
|
4
|
|
7
|
684
|
6
|
630
|
7
|
|
8
|
624
|
7
|
560
|
5
|
|
9
|
577
|
7
|
505
|
7
|
|
10
|
537
|
6
|
465
|
8
|
|
11
|
513
|
5
|
431
|
7
|
|
12
|
487
|
7
|
410
|
6
|
|
13
|
479
|
7
|
385
|
4
|
|
14
|
467
|
7
|
370
|
5
|
|
15
|
459
|
6
|
357
|
6
|
|
16
|
452
|
5
|
342
|
4
|
|
17
|
441
|
5
|
335
|
4
|
|
18
|
434
|
6
|
325
|
4
|
|
Table 5. The mean and standard deviation of the required time
(days) for throughput of 500 patients for nine beds in the ICU.
|
|
No.
of beds in the ward
|
No. of beds
in the ICU=9
|
No. of beds in
the ward
|
No. of beds in the ICU=3
|
|
Mean
|
SD
|
Mean
|
SD
|
|
20
|
219
|
4
|
30
|
157
|
2
|
|
21
|
207
|
4
|
31
|
151
|
3
|
|
22
|
200
|
5
|
32
|
149
|
2
|
|
23
|
191
|
2
|
33
|
147
|
3
|
|
24
|
185
|
3
|
34
|
142
|
2
|
|
25
|
177
|
3
|
35
|
141
|
4
|
|
26
|
174
|
2
|
36
|
138
|
4
|
|
27
|
170
|
4
|
37
|
135
|
2
|
|
28
|
164
|
2
|
38
|
133
|
3
|
|
29
|
159
|
2
|
39
|
130
|
3
|
Discussion
Overall, the study showed that the mean occupancy rate
in the ward decreases as the number of beds in the ward increases. As the mean
occupancy rate in the ICU increases, the number of the beds in the ward rises too.
Nevertheless, the throughput of patients in the system does not differ markedly
with a large number of beds in the ward compared to that in the ICU.
Cox et al3 studied percentage of busy time
for audiometricians in an ear, nose and throat outpatient clinic and the
results. In this study, bed occupancy rate in a heart surgery department was
studied. To project the incidence rates of coronary heart disease, the number
of hospitalized incident coronary heart disease cases and the hospital costs
associated with the first hospital admission in Australia, a research work was
carried out by Mui.7 A computer simulation model using a simulation
technique was developed to simulate individuals’ coronary heart disease history
over time for a sampled Australian population, characterized by major coronary
risk factors.7 Brahimi and Worthington4 used a simulation
model where a doctor played the role of the server, whereas in the present
study the servers were the beds in the ward and ICU. Hashimoto and Bell5
found that by consideration of appointment time and also increasing the number
of doctors in the clinic, the average patients’ time in the clinic decreases
significantly. In the present paper it was shown that the throughput of
patients in the system usually rises as the number of beds in the ward and ICU
increases. Ridderstolpe et al10 described the implementation of a
model for process analysis and activity-based costing management at a heart center
in Sweden as a tool for administrative cost information, strategic
decision-making, quality improvement, and cost reduction. A commercial software
package (QPR Company, USA) containing two interrelated parts, "Process Guide
and Cost Control", was used. All processes at the heart center were mapped
and graphically outlined. They concluded that a process-based costing system is
applicable and has the potential to be used in hospital management.
In this study, when
there are a reasonable number of beds in the ward, there will be no queuing
time in the ICU, waiting for a bed when a patient returns back to the ward. While
having few beds in the ward compared to those in the ICU, some queuing in the ICU
may ensue, especially when all of the beds in the ward are in use. When there
are few beds in the ward compared to the ICU, because of queuing time in the
ICU, the mean queuing time in the ICU is higher. The mean time, spent in the
hospital, does not depend on the number of the beds in the ICU or ward when
there are enough beds in the ward. For a fixed number of the beds in the ward,
the average bed occupancy rate is higher for a big ICU. It is worth mentioning
that to prevent a long stay in the ward and ICU, while the number of free beds
in the ward is less than or equal to that in the ICU, the system should not let
a new patient get into the ward if all of the beds in the ICU are in use. In
the present paper, since death is not predictable, it is assumed that it would
not be possible to perform another operation in the theater in order to utilize
the bed in the ICU, left by the deceased patient. However, using a more complex
stochastic model such as Markov Chain Monte-Carlo (MCMC) simulation and prior
distributions may be possible. An example from dentistry is presented by
Helfenstein et al11 using MCMC simulation in order to demonstrate
their application and use. MCMC methods may be of great value to dentists in
allowing analysis of the data sets exhibiting a wide range of different forms
of complexity.12 Some other simulation approaches in medical fields
were performed by different authors. A research work was carried out by Glance
and Osler13 for assessing the validity of using the standardized
mortality ratio, based on the New York State Cardiac Surgery Reporting System Prediction
Model, to compare coronary artery bypass grafting outcomes among hospitals,
using computer simulation. Also, Goldman et al8 used the Coronary
Hearth Disease (CHD) Policy Model, a validated computer-simulation model, to
estimate the effects of actual investments made to change coronary risk factors
between 1981 and 1990; they also studied the impact of these changes on the
incidence, prevalence, mortality, and costs of CHD during the same period and
projected to 2,015. O’Keefe2 designed an appointment system for
outpatient department using a simulation model. A similar approach can be done
for a heart surgery department. According to a report from the Management and Planning
Organization of Iranian government, the mean bed occupancy rate in Iran’s hospitals is 49%, which causes disinvestment of about 38 million dollars per year.14
Therefore, it can be
concluded that to prevent disinvestment, a simulation study can be performed to
observe the behavior of system prior to building a hospital or a new ward.
Also, using simulation methods in different aspects could help researchers and
managers in better understanding of the studied systems artificially. It is
necessary to mention that for simulating a heart surgery department in Iran, the related data from a hospital in the country is essential. However, the current
study has demonstrated the application of simulation studies in heart surgery
departments.
Acknowledgments
The work reported here is supported by the Iranian
Health and Medical Education Ministry for a full scholarship to get a PhD
degree for the first author. The authors appreciated for generous assistance
of Dr. D.R. Appleton from the Medical Statistics Department, who suggested the
simulation study. Also, many thanks to Dr. J.P. Mounsey, who offered the data
and to Dr. B.G. Watson from the Department of Heart Surgery at Freeman
Hospital, Newcastle upon Tyne, England.
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