Development of a multi-parametric model predictive control algorithm for insulin delivery in type 1 diabetes mellitus using clinical parameters

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Abstract

A multi-parametric model predictive control (mpMPC) algorithm for subcutaneous insulin delivery for individuals with type 1 diabetes mellitus (T1DM) that is computationally efficient, robust to variations in insulin sensitivity, and involves minimal burden for the user is proposed. System identification was achieved through impulse response tests feasible for ambulatory conditions on the UVa/Padova simulator adult subjects with T1DM. An alternative means of system identification using readily available clinical parameters was also investigated. A safety constraint was included explicitly in the algorithm formulation using clinical parameters typical of those available to an attending physician. Closed-loop simulations were carried out with daily consumption of 200 g carbohydrate. Controller robustness was assessed by subject/model mismatch scenarios addressing daily, simultaneous variation in insulin sensitivity and meal size with the addition of Gaussian white noise with a standard deviation of 10%. A second-order-plus-time-delay transfer function model fit the validation data with a mean (coefficient of variation) root-mean-square-error (RMSE) of 26 mg/dL (19%) for a 3 h prediction horizon. The resulting control law maintained a low risk Low Blood Glucose Index without any information about carbohydrate consumption for 90% of the subjects. Low-order linear models with clinically meaningful parameters thus provided sufficient information for a model predictive control algorithm to control glycemia. The use of clinical knowledge as a safety constraint can reduce hypoglycemic events, and this same knowledge can further improve glycemic control when used explicitly as the controller model. The resulting mpMPC algorithm was sufficiently compact to be implemented on a simple electronic device.

Introduction

People with type 1 diabetes mellitus (T1DM) may have a life expectancy ten years less than their normal glucose tolerant counterparts due to complications resulting from chronic hyperglycemia, such as cardiovascular disease and strokes [1]. Hyperglycemia is an elevated blood glucose (BG) concentration, with a threshold defined as BG greater than 180 mg/dL [2]. Aggressive treatment with intensive insulin therapy (IIT), involving up to a total of 12 manual capillary glucose measurements and insulin injections per day, reduces hyperglycemia and can lead to a reduction in the prevalence of these complications [2]. IIT also increases the risk of hypoglycemic events and increases the burden on the caregiver and/or patient administering the therapy [3]. Hypoglycemia is any lower than normal BG; symptoms of hypoglycemia, such as tachycardia and nausea, occur at around 50–70 mg/dL [4], [5].

The attraction of an efficient closed-loop device is thus threefold: increased life expectancy, decreased hypoglycemia, and reduction in the burden of administering effective therapy. Innovations in real-time continuous glucose monitoring (CGM) sensors and continuous subcutaneous insulin infusion (CSII) pumps mean that the components necessary for a closed-loop device suitable for use in ambulatory conditions are maturing [6], leaving the control algorithm as the limiting factor in development.

CGM sensors and CSII pumps use the subcutaneous (SC) route for glucose measurement and insulin delivery, respectively. Other routes, such as intravenous and intraperitoneal [7], [8], offer reductions in lag time [9], but are associated with an increase in the risk of infection at the site of insertion [10], [7]. The lag time associated with SC insulin infusion is an obstacle for a control algorithm: absorption of glucose into the blood from carbohydrate (CHO) raises BG faster than simultaneously injected SC insulin can lower it. Insulin delivery rates may also be limited by the physical limitations of the CSII pump and safety constraints driven by clinical parameters [11]. A controller framework known to be suitable for systems with large lag times and constraints is model predictive control (MPC); this control algorithm has been proposed as a candidate controller architecture for insulin delivery [12], [13], [14] and has been implemented in manual closed-loop trials overnight [15]. Central to each of these MPC implementations has been a dynamic model of the effects of subcutaneous insulin on glycemia.

The metabolic processes underlying insulin action involve complex interactions of hormones [16], which lead to significant variation in insulin sensitivity [17], [18], [19], [20]. Insulin absorption variability is less than insulin action variability [21], but can be affected by biofilms and inflammation. Gut glucose absorption is highly dependent upon the composition of a meal [22]. Tracer studies involving radioactive isotopes have been reported in order to characterize subcutaneous insulin and gut glucose absorption [22], [23], [24]; such models may be representative of the variation inherent in a population, but are not practical for use in an MPC algorithm because adapting these models to an individual subject would require the repetition of an expensive experiment. The use of data driven, empirical models based only on data collected from ambulatory subjects is a more practical method for development of a personalized model; auto-regressive exogenous input models have been presented in literature [25], [26]. The main caveat of developing models from data obtained from ambulatory subjects is that typically both model inputs—SC insulin and oral CHO—occur simultaneously, which gives an identifiability problem; the “best” empirical models can therefore have physically counterintuitive characteristics, such as an incorrect sign for a process gain [27]. Classic process control techniques, such as impulse response tests, have been executed in clinical trials in which insulin boluses and meals were separated by 3 h, and support the notion that a simple model—with a gain, a time constant, and a time delay—can capture the critical bandwidth behavior of glucose–insulin interactions [28].

Physicians are calculating “gains” through interactions with their patients with T1DM, and these clinical parameters are the standard of care in endocrinology practices. The “correction factor” (CF) is the lowering effect of BG from administering one unit of rapid-acting insulin; the “insulin-to-carbohydrate ratio” (ICR) is the amount of carbohydrate offset by one unit of rapid-acting insulin. These parameters are used to guide insulin dosing decisions and are often refined throughout the life of the person with T1DM [29]. Due to this refinement process, the parameters obtained are considered reliable and should be included as a safety constraint in a closed-loop control algorithm [11].

Constrained MPC can necessitate the on-line solution of a quadratic program. This on-line optimization can be replaced with a single set of a priori optimizations via multi-parametric programming; the on-line problem is reduced to the evaluation of an affine function obtained from a lookup table [30]. This reformulation is valuable in any application where on-line computation should be minimized, due to low computational power, or in order to extend battery life by minimizing computation, or to minimize the footprint on a chip. Multi-parametric MPC (mpMPC) has been evaluated in several biomedical applications [31], and has been investigated for use with intravenous insulin delivery in an intensive care unit in silico [32].

This study was designed to show the feasibility of using simple models, clinical parameters, and mpMPC for control of glycemia via the SC route. Elements of modeling for control were, for the first time, unified with clinical parameters derived by physicians. The resulting models and safety constraints were used to develop a novel mpMPC control algorithm for glucose control via subcutaneous insulin delivery. Closed-loop simulations were performed on two clinically verified virtual subject cohorts.

Section snippets

Virtual subject cohorts

Two clinically validated simulation models were used to represent a virtual subject with T1DM. Both models are semi-physiological, semi-empirical, and represent the effects of SC insulin and oral CHO on BG. The models are divided into three compartments as shown in Fig. 1: gut absorption of glucose from oral CHO, absorption of SC insulin into the plasma, and their effects via a glucose-insulin kinetic model on BG.

The first cohort of virtual subjects was based on the model of Hovorka et al. [23]

Model development

Comparison of each of the models A–F with the subject data was performed. For the Hovorka cohort, the critical bandwidth frequencies were captured in each of the personalized models, i.e., models A, B, D, and E. The calibration data from the Hovorka cohort and each model’s response to a 1 U insulin bolus is shown in Fig. 2; models C and F, which have gains based upon clinical parameters, do not capture the asymptotic behavior as well as models A, B, D, and E. The model response to a meal is

Conclusions

Low order transfer function models can have physically meaningful parameters and the necessary degrees of freedom to characterize the principle characteristics of clinically validated models of SC insulin effects on BG. The identification procedure for parameters for such models used only ambulatory data and would be relatively inexpensive to implement. Clinical parameters served a similar function in characterizing the most important information and could be used as the basis of models useful

Acknowledgments

Financial support from the Juvenile Diabetes Research Foundation (JDRF) grants 22-2009-796, and 22-2009-797, Institute for Collaborative Biotechnologies (ICB) grant DAAD 19-03-D-004 from the US Army Research Office, and National Institutes of Health (NIH) grant R01-DK085628 are gratefully acknowledged. MWP would also like to thank Urban Mäder for his helpful comments. The UVa patent foundation is gratefully acknowledged for the use of the metabolic simulator.

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