Markov chain-based cost-optimal control charts

Control charts can be found in many areas in industry and engineering and even in more remote disciplines. When I started working with control charts, I already had a background in biostatistics, thus I immediately started thinking about healthcare applications. The most basic role of a control chart is to monitor and control a process. Such problems also occur in healthcare, you can think about, for example, the monitoring and controlling of blood lipid or blood sugar levels. This is very similar to the traditional, industrial setting: we have a characteristic we are interested in, we take regular samples and we try to keep the process in-control. The presence of control charts in healthcare is no novelty, there are several examples of quality and financial monitoring. However, we chose to focus on the cost-optimal monitoring of a single patient at a time.

This proved to be a challenging task, as the application of control charts in industry is facilitated by several assumptions which cannot be made in patient monitoring. For the sake of clarity, let us go through some notions used in control chart theory and pair them with their healthcare equivalent: An in-control process represents a healthy patient, thus a shift to out-of-control means some level of disease development. Repair is some kind of healthcare intervention (treatment) such as the administration of medication. Regular sampling can be interpreted as control visits to a medical professional. It quickly became apparent that almost every part of this process involves substantial randomity. In industrial settings, it is often assumed that the shift in the process has a fixed, known size, the repair is perfect and we have total control over the sampling intervals. We had to develop generalisation for all of these parts to be able to properly model the disease development, treatment and sampling.

Our cost-optimal control charts model the disease progression and treatment with distributions, instead of fixed and known values and permits the patients to skip control visits, modelling non-compliance [1]. We associate costs with all the relevant parts of the model: we take into account the healthcare burden generated by the disease and costs due to treatment and sampling. This enables us to find a cost-optimal treatment and monitoring setup, by finding the optimal time between control visits, intervention-requiring disease level and treatment type. The estimation of the total expected cost over a unit time is accomplished through the discretisation of the continuous distributions involved in the model. This way we can describe the model as a discrete time, discrete state space Markov chain, where the states of the chain are the disease level (e.g. blood lipid level of the patient) which are defined (i.e. measured) at the sampling times (i.e. control visits). The expected cost can be calculated using the stationary distribution of this Markov chain. The utility of the model is not exhausted by cost estimation or optimisation: it is also able to model the effect of changes in the setup such as different treatment types, disease stages, changes in patient compliance etc.

Now, I present an example of use for the above described control charts model. We were able to gather data of diabetic patients through the Healthware Consulting Ltd. (Budapest). The monitored characteristic will be the blood sugar level of the patient, which is measured by the glycated haemoglobin level (HbA1c). Insulins and oral antidiabetics (OAD) and these together are considered as treatment types. We were able to estimate all of the model parameters and reached an acceptable level of fit to the empirical data, measured by the HbA1c distribution and expected costs. We were able to estimate the relationship between the expected daily cost, the time between control visits and the critical HbA1c level (which requires intervention). This is summarised by the contour plot attached to the post, which contains the data of both type of therapies together. We can assess that less time between samplings and lower critical level results in lower expected costs. This is as expected, but we can gather further important information, such as the fact that cost dependence on the critical value is only moderate.

One can also compare different therapies using the model. Here, you can view the same contour plot, but separately for the insulin and OAD therapies. It can be seen that OAD therapies are much cheaper. Of course, we did not take into account the context where each type of therapy is used. Still it can be useful to see how costs compare and their relationship to other parameters.

We have several plans for the future of the model. We intend to publish an R package, so the method would become available for public use. We are also looking into ways to further generalise the model and to apply it to other healthcare processes.

[1] Dobi B, Zempléni A. Markov chain-based cost-optimal control charts for health care data. Qual Reliab Engng Int. 2019;1–17. https://doi.org/10.1002/qre.2518