Using probabilistic patient flow modelling helps generate individualised intensive care unit operational predictions and improved understanding of current organisational behaviours

Abstract

Purpose

We sought a bespoke, stochastic model for our specific, and complex ICU to understand its organisational behaviour and how best to focus our resources in order to optimise our intensive care unit’s function.

Methods

Using 12 months of ICU data from 2017, we simulated different referral rates to find the threshold between occupancy and failed admissions and unsafe days. We also modelled the outcomes of four change options.

Results

Ninety-two percent bed occupancy is our threshold between practical unit function and optimal resource use. All change options reduced occupancy, and less predictably unsafe days and failed admissions. They were ranked by magnitude and direction of change.

Conclusions

This approach goes one step further from past models by examining efficiency limits first, and then allowing change options to be quantitatively compared. The model can be adapted by any intensive care unit in order to predict optimal strategies for improving ICU efficiency.

The UK has one of the lowest numbers of intensive care beds per 100,000 population in Europe.¹ As such, intensive care units (ICUs) are under significant capacity pressures and maintaining a safe and efficient ICU is challenging. In the UK winter of 2016/17, acute and general bed occupancy was >99% (www.evidence.nhs.uk at the time of writing). This level of occupancy prevents ICU discharges when the ICU service is oversubscribed. Strained ICU capacity appears to have an indirect effect upon ICU mortality as a result of increased acuity of admissions and appears to result in reduced ICU length of stay suggesting an effect of strain upon admission and discharge behaviour.² There are also implications for elective surgical procedures, and requirement for a post-operative critical care bed increases the likelihood that a procedure will be cancelled on the day of surgery.³ Understanding ICU patient flow behaviour, limitations, and potential areas of optimisation under such constraints is crucial for improving care delivery under these stresses.

It has been reported that a safe ICU functions at 70–75% occupancy,⁴ meaning that 25–30% of ICU resources are unused. Safety in this context is defined as the point at which patients begin to be ‘rejected’ by an intensive care unit,⁵ or the point at which discharges begin to be delayed.⁶ A unit functioning at 70–75% of resource use is a significant loss under current financial and resource pressures. Simply creating more intensive care beds is often not possible due to funding constraints and may not be appropriate if efficiency can be improved. A different approach is required in order to understand and study how an individual intensive care unit’s practical functioning becomes compromised, and to develop strategies to minimise this impact. Important to this process is the identification of potential change options and assessment of how successful they are likely to be. We chose to better understand our unit functioning using patient flow models.

The traditional concept of a patient flow model consists of ‘items’ arriving, waiting, being served and leaving. These elements are typically described by probability distributions with means and variances. Waiting time depends on how quickly the queue builds from the rate of arrival, the time taken to serve an ‘item’, and how many servers there are. The service time depends on the nature of what is being done, and how quickly it takes for an ‘item’ to leave; though usually represented as a single process. By its own nature, the entire flow is a single linear path.

Originally, the elements were defined as closed probability distributions reflecting the collected data from the system of interest. The entire model is described as an A/S/C model and forms the classical ‘queueing theory’. A represents the arrival time distribution, S the service time, and C the number of servers. More recently the term ‘queueing theory’ may also be used to describe any patient flow model. A commonly employed model is M/M/C – Markov arrival time i.e. Poisson process, Markov service time (exponential process), and any number of servers. These models are dependent on their initial state and parameters. McManus et al.⁵ was able to use this M/M/C approach for their intensive care unit to demonstrate that bed occupancies exceeding 80–85% resulted in a rapid rise in ICU rejections. Williams et al.⁷ created an A/S/C type model to predict optimal critical care capacity for a new hospital. Green et al.⁸ used queueing theory to increase the effectiveness of emergency department staffing.

With more powerful computing, it is possible to move away from the A/S/C queueing model towards more descriptive, non-linear and complex models that are not ‘closed down’ by fixed equations. Barados et al.⁹ used dedicated ARENA simulation software to predict the daily bed occupancy for their unit, using patient factors such as APACHE score and infection type. The ARENA simulation package (www.arenasimulation.com) is a discrete event simulator and models individual events. Kolker¹⁰ used process modelling to understand the effect of daily elective surgical admissions on ICU diversion. Process modelling here permitted the inclusion of two patient populations to examine the impact of elective caseload on emergency admissions. Mathews and Long¹¹ investigated the impact of ICU bed allocation on patient flow by creating a non-linear flow model where patients passed through the different stages of the model depending on randomisation of their attributes such as priority class, arrival date and service time.

What these last three studies have in common is the use of simulating individual, stochastic, elements within a queue model. They limit the model to defining the elements travelling through a system, and in doing so free up the system to be manipulated and tested for non-linear, recursive, time or state dependent, multiple dependent and independent variables. Defining such models in their entirety in equation form would be exceedingly complex. Computers are powerful enough now that it is easier and faster to stochastically simulate the process of elements flowing through the system to solve the relationship between the inputs and the outputs, rather than trying to describe the entire model analytically.

The further advantage of the stochastic approach is that it lends itself to capturing multiple common and rare events under ‘normal and extreme’ conditions. The model results are often described as distributions making quantitative analysis and comparison accessible. This approach is employed in other industrial sectors for capacity planning, inventory management, process engineering manufacturing, resource allocation, scheduling and strategy.^12–14

We chose to use a stochastic Monte Carlo simulation of our ICU to find the thresholds at which practical unit function becomes compromised. Our intensive care unit has 13 level-3 staffed beds, 2 ‘reserve’ level-3 beds staffed in excess of funding, a potential ventilated bed in the adjacent theatre recovery area, and a complex pathway, with patient subtypes of different urgencies. We then used the model to explore potential operational changes, in order to quantify how successful each change would be, and to characterise the type of change produced.

Our outcome markers were the number of refused elective and emergency admissions, the number of unsafe days per year, and the proportion of the reserve capacity used per year. Refused admissions and unsafe days have been used in the past to identify the balance between bed use and practical ICU function.⁵ Unsafe days were classified when appropriate nurse:patient ratios were lost.

Methods

Our study did not require ethical approval as it used retrospective anonymised data which had already been collected for the hospital’s ICU information registry, and is an evaluation of our current service provision. The model itself was created using RStudio 1.1.383 and whose code is available in the supplementary material.

The model replicated our neurosciences intensive care unit with 13 beds, a reserve of 2 beds and 1 ventilated bed in theatre recovery. Patients were referred, admitted, inhabited the unit and then discharged. There were two patient subpopulations: elective neurosurgical admissions, and predominantly emergency neurosurgical admissions. The patient flow model is shown in Figure 1 and demonstrates how our unit currently handles patient flow. Emergency patients were prioritised over elective patients as is our current policy when bed capacity is limited. If the unit exceeded 13 occupied beds, elective patients would wait in a queue for a bed and were classified as a failed elective admission. Emergency patients were admitted until all 15 beds and the ventilated recovery bed were occupied. Following this, any further emergencies were not admissible and were classified as a failed emergency admission. Bed occupancy was determined by the number of occupied beds at the end of the day. Our unit demographics are shown in Table 1.

Figure 1.

Patient flow pathway for a 13-bed tertiary intensive care unit, with 2 flexible beds.

Table 1.

Population and ICU characteristics for the year 2017 used in the simulation.

		N	Length of stay					Survival				Age			Sex						Ventilated days
								Deceased		Alive					Female			Male
								%		%					%			%
Elective		158	µ(σ) mode	4.54 (9.34) 1 day				1.3		98.7		53.81 (16.01)			57.9			42.1			2.57 (8.62)
Emergency		375	µ(σ) mode	9.37 (9.83) 1 day				17.6		82.4		54.18 (16.22)			42.5			57.5			7.03 (9.86)
Neuroscience intensive care admission data and failed admission rates 2017
					Jan	Feb	Mar		Apr		May		Jun	Jul		Aug	Sep		Oct	Nov	Dec
Admissions 2017	Elective				10	12	11		9		16		27	14		14	11		3	18	14
	Emergency				30	23	37		36		31		33	36		29	36		34	28	33
Occupancy Patient bed days, % (13 + 2 bed unit) 2017					96	82	92		99		95		91	104		101	100		105	107	104
Denied elective admissions (Jan–Dec) 2017											48		Denied emergency admissions (Jan–Dec) 2017								15
Denied elective admissions (Jan–Dec) 2016											56		Denied elective admissions (Jan–Dec) 2016								18

We used current Faculty of Intensive Care Medicine Guidelines for the Provision of Intensive Care Services (GPICS)¹⁵ nurse:patient ratios for intensive care units in our model. Level-3 and level-2 patients in our model required 1:1 and 1:2 nursing respectively. For our 13-bed ICU, these guidelines recommend an additional supernumerary nurse as well as the clinical nurse coordinator. What this effectively means, is that when the ICU is full of ventilated patients and occupancy requires more than 13 nurses, nurse:patient ratios are compromised.

We used population data from our ICU to assign patient demographic data in the model. The data were analysed to produce probability distributions from which patients were assigned characteristics randomly. This included whether or not they were elective or emergency, and for what proportion of their stay they were likely to be ventilated. Table 1 gives the demographic statistics for our ICU which formed the basis for the population in the model. Patients admitted and leaving on the same day were still classified as occupying the bed for that day. Admission rates were predominantly new patients, and included the small proportion of re-admissions; however, it is not possible to separate these in the data.

Our ICU patient length of stay between January 2017 and December 2017 was exponentially distributed. For the purpose of accurate ICU modelling, these data were converted into a cumulative distributive function (CDF) from which the model would randomly sample with replacement. A stepwise linear regression analysis on the data (F statistic of inclusion p < 0.05, and exclusion > 0.1) found that the number of ventilated days was directly proportional to the length of stay, and was the most significant contributor (93% of explained variance). Age was not a contributing factor. Case urgency, gender and mortality contributed in total to < 1% of the explained variance of length of stay, and were still modelled, but were non-contributory to the model outcome variables because of their total minimal contribution.

Direct linear equations derived from the linear regression were used for assigning the number of ventilated days for each patient depending on the length of stay, and was important for classifying patients as level-2 or level-3 care. Since age was not significantly different between the elective and emergency groups (p = 0.805), age was drawn from a pooled normal distribution with mean = 54.07 and standard deviation of 16.145. Gender and mortality was assigned using a discrete distribution with the population characteristics shown in Table 1.

The model simulated an 18 month period with the first six months discarded as a ‘burn in’ period. This time length was chosen based on stabilisation of outcome variables, and is standard practice for randomised models to double the excluded length of time required to reach stability. The remaining 12 months of simulated data were used to produce probability functions and histograms to show how different model output parameters change under normal and differing operational conditions. The full probability spectrum achieved by running the model a total of 500 times to include rare events and impact of randomness; equating to 500 years of simulated ICU operation.

Referral rates to the ICU were modelled using a customised discrete distribution (Table 1) based on our 12 month admission data for 2017 and were separated into elective and emergency referrals. Basing the model on a combination of unit and patient data, means that unknown subtle hidden relationships are also incorporated.

To validate the model, we aimed to achieve a simulated bed occupancy and failed admission statistics that were statistically comparable to our 2017 data. In the context of a stochastic model, the outputs have a distribution with a mean, mode, median and variance. If our simulated data captured our real data within its 95% confidence intervals, then we were reassured that the real data are not significantly different to that of the model. For normally distributed data such as occupancy, standard T tests were applied. For asymmetrical non-normal distributions Mann Witney U tests were applied. Where this was not possible if the real data were within the bounds of the upper and lower quartiles of the simulated variable, giving a p value between 0.25 and 0.75, this meant it was regarded as not significantly different. We aimed to achieve these goals whilst retaining as many of the original data characteristics possible.

Determining the optimal occupancy under stress

We define the optimal occupancy as the balance between using as many beds possible, with minimal use of the two reserve beds; and so they are not accounted for in occupancy calculations. We expect from a practical perspective, to need to encroach into the reserve two beds < 5% of the time over a year using them under extreme circumstances only. As such the occupancy that achieved this was defined as the optimal occupancy.

We explored the optimal occupancy by varying the referral rates to the ICU by factors of 0.8 to 1.4, where 1 is normal rates based on the 2017 data and 1.4 is extreme stress, and observed how the outcome measures depended upon occupancy levels.

Redesigning the ICU for efficiency

We modified the model to incorporate one of the following changes to observe the impact upon occupancy, failed admissions and unsafe staffing days:

• Adding 2 beds so the new ICU had 15 beds, 2 ‘reserve’ plus 1 ventilated recovery bed; total capacity 17 + 1. Bed occupancy statistics are calculated for a 15-bed ICU.

• Permanently staffing the 2 ‘reserve’ beds, so total capacity is 15 level-3 beds and 1 recovery bed. Bed occupancy statistics are calculated for a 15-bed ICU.

• Reducing patient length of stay by one day arbitrarily.

• Transferring a maximum of three long-term ventilated (LTV) patients to a separate bed bay until their discharge; LTV defined as length of stay >22 days.¹⁶ This frees up to three beds for elective/emergency cases which would expand and contract according to demand. Bed occupancy statistics were produced for the ICU (13 beds) and for the separate bed bay.

The modification success was assessed by the change in bed occupancy, failed admission rates and the change in number of unsafe nurse:patient days. The change options were then ranked by the size of change. Statistical comparisons were undertaken in the same manner described above for validation. The better the improvement in occupancy, the lower the ranking unless changes in failed admissions and unsafe days needed to be taken into account.

Results

Validation

For the simulated 12-month period, the ICU had an average number of admissions of 535.1 (13.6), which was not significantly different to our year 2017 data (p = 0.438, Student’s T test). The model produced a mean number of 61.9 (18.3) failed elective admissions and an IQR of 14 to 65 for failed emergency admissions. These are not statistically different (Student’s T test) to our 2016, 2017 failed elective admissions (48;56), or to the 2016/2017 failed emergency admissions (18;15). The simulated ICU had a 92.22% occupancy, which is not significantly different (p = 0.88, Student’s T test) to the 98% average monthly occupancy for Oxford ICU in 2017.

Achieving this outcome was challenging. It required a balance between seeking as close an occupancy to 98% as possible whilst balancing data parameters such as elective:emergency admission ratios. In order to achieve this, we required an elective:emergency admission proportion of 0.75:0.25, where our sample data had 0.71:0.29. Our length of stay data for the model was not statistically different to our true sample data (Student’s T test). Striving for a closer simulated occupancy to 98% resulted in failure to achieve statistical similarity for both occupancy and failed admissions. Likewise, pushing for an elective:emergency ratio closer to the real data had a similar impact.

Considering the model parameters were as optimised as we could get them, and that the model occupancy and failed admissions were not statistically different to our data, we concluded that the model had reached its peak in optimisation, and was a valid representation of the ICU’s real-life behaviour for the year 2017–2018. To help support this conclusion, we plotted the ICU’s occupancy behaviour from 2013. Whilst the model differs by about 6% to the 2017 occupancy, all other years from 2013 were very close to the simulated occupancy and all resided well within the 95% confidence interval of the modelled normal state occupancy of the ICU (Figure 2).

Figure 2.

Displayed is the average daily bed occupancy for a 13-bed ICU with 2 beds in reserve capacity. The first six months are discarded in the data analysis due to the burn in period. The simulated period shows a static occupancy which is quite similar to the monthly occupancies for the years 2013 to 2017 inclusive (coloured lines). These lines approximate to the black simulated line closely, and are well within the 95% confidence interval limits (dotted) of the simulated data range.

Determining the optimal bed use under stress

Figure 3 shows the relationship between bed occupancy and failed elective admissions, diverted emergencies and unsafe staffing days produced by varying referral rates. As bed occupancy increases, all outcomes rise slowly and then beyond 90% occupancy, failed emergency admissions escalate rapidly. The number of unsafe days and failed elective admissions rise almost linearly with occupancy. The most rapidly changing factor in relation to occupancy is the number of failed emergency admissions.

Figure 3.

Graph showing the three outcome measures (black lines) against the cumulative probability distribution of occupying the reserve capacity (>13 level beds) in red. The y scale is a proportion scale, where elective fails and diverted emergencies are proportions relative to total admissions (successful + failed admissions), and unsafe days is a proportion of the 365 days per year. Being a proportion between 0 and 100, the y scale is also a probability scale for occupying the reserve capacity based on occupancy.

We originally defined the optimal occupancy cut off to the use of reserve capacity more than 5% of the time. The red line in Figure 3 is the probability of using more than 13 beds and thus reserve capacity, derived from probability density functions of the different occupancy states from the simulation. The shaded blue area represents occupying the reserve capacity less than 5% of the time, giving a cut off at 92–93% occupancy of the 13 beds.

Optimal strategy for implementing ICU changes

All change options were compared to our existing 13 + 2 + 1 configuration. Table 2 shows the outcomes measures under these differing conditions. All strategies were successful in reducing occupancy, and were statistically different to the normal unchanged model (p < 0.001, Student’s T test). The larger ICUs (15 + 1 and 15 + 2 + 1) were successful simply because occupancy was defined by 15 and not 13 beds; there was more capacity. There was no statistical difference in occupancy between the two larger ICUs (Student’s T test, p = 0.76). The LTV strategy reduced occupancy by moving a LTV patient out who would have prevented admission of other patients, and instead occupied on average 30% of the LTV bed capacity. The LTV strategy was not statistically different to the occupancy for the reduced length of stay strategy with the latter having marginally lower occupancy (p = 0.0588, Student’s T test). Adding physical beds or additional staffing reserve beds were both statistically better at reducing bed occupancy (p < 0.001, Student’s T test). Whilst statistical differences exist, all measures were approximately equally powerful in reducing the number of failed elective admissions. All modelled scenarios produced occupancies between 79% and 81%. Occupancy with two extra beds was 79.7%, additional staffing reserve was 79.8%, reduced length of stay was 81.0%, and LTV was 81.5%).

Table 2.

Simulated outcome measures for ICU change options and the change compared to the Normal ICU setup.

	Occupancy		Failed electives		Diverts		Unsafe days		Rank
	µ (σ)	Change	Median (IQR)	Change	Median (IQR)	Change	Median (IQR)	Change
Normal (13 + 2 + 1)	92.0 (4.7)	–	60 (73–49)	–	32 (63–14)	–	30 (42–20)	–
RLOS	80.7 (5.2)*	−11.3	30 (41–21)*	−30	7 (21–1)*	−25	13 (22–6)*	−17	2
LTV (13 + 2 + 1) +3 bed flexible	81.5 (3.3)* 31.3% (1.8%)	−10.5	27 (34–21)*	−33	2.5 (8–0)*	−30	9 (15–5)*	−21	3
Staffed reserve (15 + 1)	80.0 (4.3)*	−12.4	25 (35–17)*	−35	55 (102–30)*	+23	40 (57–28)*	+10	4
Two extra beds (15 + 2 + 1)	79.4 (4.5)*	−12.3	22 (31–15)*	−38	2 (8–0)*	−30	5 (10–1)*	−25	1

LTV: long-term ventilated; RLOS: reduced length of stay

^*p < 0.05 difference (T test/Mann–Whitney U).

Permanently staffing the reserve capacity (15 + 1) resulted in a marked increase in the number of diverted emergencies and unsafe days. For this reason it was ranked last as unlike the other simulated changes, it worsened ICU failed emergency admissions. Reducing patient length of stay was broadly similar and not as powerful at reducing the number of diverted emergencies or unsafe days compared to other change options.

Discussion

Using the Monte Carlo modelling approach for our ICU, we have shown that optimal ICU resource use under stress is a balance between bed occupancy and minimising failed elective admissions without compromising emergency admissions and levels of unsafe operating. For our ICU, this is achieved at approximately 92% occupancy. This is different to the current published 70–75% recommendation, which if we pursued, would result in underutilisation of our limited resources and high refusal of elective and emergency admissions. This 92% threshold is impacted by length of stay, number of ventilated days and the way an individual ICU runs with our patient population predisposed to long-term ventilation. Operating at 92% occupancy is a better resource use when under stress, and under current NHS pressures is more realistic, and from our results no less ‘safe’ in terms of nurse:patient ratios. Whilst the results are specific to our unit, the ability to model and understand an individual unit’s relationship between occupancy, failed admissions and safe nurse ratios is a powerful tool for any ICU. We encourage others to do the same with their own ICU setups.

Other than understanding current limitations and relationships, the additional utility of this modelling approach is the quantitative evaluation of change options prior to their implementation. Without the model’s stochastic nature this would have been difficult as the comparison depends on the generation of confidence intervals. We encourage others to model and understand the implications of their change options before implementing them, as the results are not always intuitive.

It was interesting that for our set of circumstances the model results contradicted the idea that permanently staffing the reserve capacity seemed like a suitable fix for an ICU unable to physically expand. The reason behind the failure of this idea is that elective admissions enter the reserve which emergency admissions depend upon, raising failed emergencies and unsafe days. To mitigate this effect, elective admission entry into the reserve would have to be prevented resulting in no patient flow change and negating the point behind this change option. The best way of improving efficiency for our unit is to add more permanent beds, which we are unable to do. However, this change option was equivalent in broad terms to reducing length of stay by 1 day. We note for our ICU, ventilated days is the single biggest contributor to length of stay. This makes strategies for reducing ventilated days an ideal target for clinical improvement projects on our unit.

The validation process of any model is important, and requires understanding of what the model does and does not capture. Our stochastic approach is designed to link dependent to independent variables through simulation, avoiding an equation-based analytical approach. This method is disadvantaged by how accurately the model reflects real life. With any simulation as model complexity increases, there is a trade-off between model accuracy and bias. For our model we were not quite able to reach exact matching of all the parameters. Despite this, the loss of accuracy and introduction of bias was acceptable for our purposes. Being a consistent error, we do not feel that it negatively impacts our conclusions. The fact that the model does not match the exact occupancy may be a consequence of several subtle factors. For example the model time frame is whole days and not hourly, and we did not simulate direct bed swaps, or doubling up low requirement level-3 patients. We have tried to minimise the impact of these factors by using techniques such as variable property selection from sampling real data, e.g. length of stay. Relationships which may not be immediately assessable are invariably included in the model e.g. early and late discharge effects on interpatient arrival times. As a result of sampling from real data, modelling such relationships becomes redundant. We also recognise that our flow model is the closest we could make it to real life, accepting that deviations occur on an assumed infrequent basis.

Modelling how a patient system works seems both sensible and incumbent firstly to understand it, and secondly prior to making system changes at the local hospital level. A model can better inform management decisions by closely simulating the impact those decisions have, both good and bad, and by capturing results which are not intuitive and may not have been recognised were it not for the model. What we would like to see is the growth of patient flow modelling as a standard practice for helping to inform healthcare decisions at the local hospital level. This model has been developed locally to meet our needs and we hope it will serve as an example that others can follow. We publish our code in the supplementary material with annotation, and advice on customising the code for local use can be obtained by contacting the first author.

In conclusion, our modelling approach requires minimal resources, and as long as the model accurately reflects the intended system, its results can be exceptionally informative. Modelling seems more important now with limitations on funding and physical space, and on already stretched systems. Such a model can be generalised to other ICU patient systems and beyond, or can produce personalised recommendations based on local demographic characteristics. This would allow each ICU to understand its system better, where the operational limitations are and to identify appropriate change options that will bring about the results they seek within their resource availability and local capabilities.

Authors’ contribution

GH: Concept, statistical modelling, design and analysis, write up; JT: Data collection; CH: Data collection; RS: Model design, write up; HM: Model design, write up.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.