Maths, baths and computer models: ways to make health services more efficient

How can mathematics and computer models improve referral to treatment times? Perhaps imagine running a bath. If you know its size and shape, you can work out a formula that enables you to predict the water level at any time, depending on how fast the water is coming out of the tap. If you don’t put the plug in, then you can adjust your calculations to take account of the size of the plughole.

Now, suppose you want to be really wasteful and only use the hot tap, and want the water in the bath to be as hot as possible. The smaller the plughole, the fuller the bath will get, the longer the water will spend in the bath, the cooler the water will become and hence the less likely you are to ensure it maintains an acceptably hot temperature. Understanding a bit about thermodynamics, you can now add an extra formula that predicts average water temperature in the bath.

Although the real details are a bit more complex, this can be used as an analogy for understanding referral to treatment times. Much as you can calculate the impact of turning the tap one way or another, or widening/narrowing the plug hole, you can predict the consequences of changes in referral volumes or specialist capacity and therefore work out efficient ways of managing list sizes or for solving bottlenecks.

This was a paper recently presented by Dr Richard Wood from NHS Bristol, North Somerset and South Gloucestershire CCG at the 60^th annual Operational Research Society conference hosted by Lancaster University. It was one of many papers in the health care applications stream describing how mathematical and computer models are used to improve the efficiency and effectiveness of health services.

Other examples include:

• Developing a predictive model using information on the referral forms for breast cancer screening to establish the risk of abnormal screening results. Then hospitals can investigate the consequences of choosing different thresholds for identifying high-risk women, using a computer simulation that models the processes of screening, diagnosis and treatment, and their influences on clinical outcomes.
• Using mathematical formulae of queueing behaviour to evaluate different approaches for reducing overcrowding in emergency departments and achieving the four-hour target. This has been used to explore what would happen if, for example, low acuity patients were managed differently, or if beds were freed up quicker.
• Developing a system for helping surgeons plan their schedules based on what is known about their case mix.
• Using computer modelling to help design the layout of a hospital.

Faster knowledge

A major value of these approaches is that they enable you to estimate the consequences of doing something before actually doing it. And in health care, direct observational studies are not always practical. For example, there may be too many possible ways for organising a service that can be tested in one observational study; the sample sizes may be too small; the effects of a change may not be visible for a long time; or it may be too disruptive to the service to test options in practice.  

Furthermore, modelling different options needs much less resource than setting up and testing multiple ways of delivering services. Often, however, the idea is not to replace direct observation but to help choose which of several options is likely to give the better outcomes, and therefore which to focus on in an observational study.

Becoming more commonplace

Although operational research and modelling applications are often being developed in health care, a common problem is getting them widely used. This can be due to expensive software for simulation applications, and a shortage of technical expertise within hospitals and the wider health care system. The Health Foundation-funded PLETHORA project has aimed to address this issue of wider use through setting up national working groups, targeting key areas such as building analytical capacity or how researchers engage with the NHS.

And progress is being made. CCGs are now employing modellers, as are some trust business intelligence units. Attempts are also being made to make simulation tools more accessible by providing open source software or reprogramming key elements in Excel. Some tools map out care pathways very clearly, making it easy for users to see how particular changes influence the wider system. And creating useful analogies like the bathtub scenario can help demystify what can appear to decision-makers as a complicated black box.

Further information and resources (click on each for more information)

• Change by design: Systems modelling and simulation in healthcare
• A selection of case studies 
• A YouTube clip explaining the work of the UCL Clinical Operational Research Unit
• Projects selected by the Health Foundation’s Advancing Applied Analytics programme
• PLETHORA project
• ACT Academy: Online library of quality, service and redesign tools – modelling and simulation