A performance bound for optimal insulin infusion in individuals with Type 1 diabetes ingesting a meal with slow postprandial response

Abstract Recent work, Goodwin et al. (2015, 2018), has established that, subject to an assumption regarding the relative blood glucose response times of meal and insulin, it is optimal for the minimisation of the peak blood glucose excursion to apply a bolus of insulin at the time of (or slightly prior to) meal ingestion. A key assumption underlying this result is that the impact of an impulse of insulin “lasts longer” than the impact of an impulse of food. The result is consistent with clinical trials reported in Lopez et al. (2014) . In the current paper we consider the converse situation where the response of the meal lasts longer than the response of an insulin impulse. The latter situation is known to arise in certain circumstances e.g., when low glycaemic index and/or hi-fat/hi-protein meals are consumed (Bell et al., 2015; Lodefalk et al., 2008; van der Hoogt et al., 2017). We consider the same set-up as in Goodwin et al. (2015). However, here we show that, when the response of the meal lasts longer than the response of an insulin impulse, it is optimal to use an open-loop policy combining an insulin bolus (applied with the meal) and a specific form of decaying insulin flow thereafter. This gives support to results for clinical trials, reported in Lopez et al. (2017). However, as far as we are aware, no earlier work has given a methodology to quantify the best insulin flow after the initial bolus nor the best split between bolus and post-bolus insulin flow. In support of the theory, we apply the results to a model obtained from clinical trials on a real subject with Type 1 diabetes and include a discussion of open-loop versus closed-loop policies.